Jan 17, 2017 - Winnipeg, MB ... Ron Mueller and Dr. David Swatek from Manitoba Hydro, as well as ...... Note, Emerson Process Management, Houston, pp.
An Infrared Spectrometer Based on a MEMS Fresnel Zone Plate for Measuring Dissolved Gases in High Voltage Equipment by Pawel Krzysztof Glowacki
A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirements of the degree of
master of science
Department of Electrical and Computer Engineering University of Manitoba Winnipeg, MB
Copyright © 2017 by Pawel Krzysztof Glowacki
An Infrared Spectrometer Based on a MEMS Fresnel Zone Plate for Measuring Dissolved Gases in High Voltage Equipment Pawel Krzysztof Glowacki 2017
Abstract This thesis presents a unique design for an infrared spectrometer based on a MEMS Binary Fresnel Zone Plate for the purpose of assessing the health of oil-impregnated high voltage (HV) equipment. It does so by measuring dissolved gases within it. These gases include carbon monoxide, carbon dioxide, methane, ethane, ethylene, and acetylene. These gases are currently measured using numerous technologies such as gas combustion, gas chromatography, photoacoustic spectroscopy, and FTIR spectroscopy. Each of these technologies have their advantages and disadvantages. The design presented in this thesis consists of an analysis of how the various Binary Zone Plate parameters affect its spectral resolution and transmission efficiency. Simulations show that increasing the number of zones and the focal length, as well as decreasing the aperture diameter, increases the spectral resolution of the spectrometer. Simulations also show that transmission efficiency is proportional to the number of zones and the aperture diameter. This thesis presents a theoretical argument for how one zone plate lens can be used to measure all dissolved gases present in HV equipment. Lenses for the visible and infrared ranges were fabricated in the University of Manitoba NSFL Cleanroom. The lenses were then tested in an optical setup. The results show that the visible light experiments were successful in achieving appropriate spectral discrimination by changing the distance between the aperture and the lens. The results from the infrared experiment show that a detector was able to discriminate between full and no incident radiation.
ii
Contributions This work has shown the following: • That a Binary Fresnel Zone Plate can be used to measure wavelengths ranging from the near to mid IR spectral range, as well as for visible light. The IR sweep covers the absorption bands of all possible dissolved gases present in oil impregnated high voltage equipment. • How the number of zones and focal length of Binary Zone Plates, as well as the diameter of an aperture prior to the detector, affect the spectral resolution and transmission efficiency of Binary Zone Plate based IR spectrometers. • A novel method for determining the widths of all Zone Plate slits to obtain optimal performance. • A theoretical analysis providing evidence for how one Binary Zone Plate Lens can be used to measure the concentrations of all key gases in Dissolved Gas Analysis. • That it is possible to fabricate a Binary Zone Plate lens designed for the near to mid IR range using 5µm technology. Such a lens is designed with sufficient spectral resolution in order to measure dissolved gases in high voltage equipment. • That the radial radiation profile from narrow annular slits in Circular Binary Zone Plates can be approximately modeled as the linear radiation profile from linear slits in Linear Binary Zone Plates. The error between such simulations is less than 2% for an f = 2” IR binary zone plate lens.
iii
Acknowledgements Firstly, I would like to thank Dr. Cyrus Shafai and Eric (Ireneusz) Witkowski for being my supervisors in academia and industry respectively during my years as a graduate student. I am grateful to them for sharing with me their intelligent approaches to problem solving, as well as large amounts of knowledge and wisdom during our meetings and pep talks. I consider myself lucky to have had the privilege of being their student. Thank you.
I would like to thank Dwayne Chrusch, Cory Smit, Dave Tataryn, Daryl Hamelin and everyone in the tech shop in E3-550 for teaching me a lot of very practical skills in the engineering trade. I appreciate their helpfulness in some of the more hands-on aspects of my thesis. Examples included cleanroom training, laser cutting services, teaching me some tricks in plumbing, and machining services. I am also grateful to Dr. Arkady Major and Dr. Sherif Sherif for sharing some valuable optical engineering knowledge with me, as well as some optical equipment. Special thanks to Dr. Mirosław Pawlak for his guidance on chapter 4.
From industry, I would also like to thank Dr. David Prystupa from Spectrum Scientific, Ron Mueller and Dr. David Swatek from Manitoba Hydro, as well as North Forge. I have learned a lot about optical systems, how spectroscopy is applied in the field, as well as practical prototyping and machining skills from them. I would like to thank all former colleagues at Manitoba Hydro whom I have worked with including Robert Armstrong, Nazra Gladu, Ian Naften, Amelia Au, Jana Brunel, and many more for whom there is not enough room to list.
iv
I appreciate the friendship of some friends I made in the MEMS Research Group including Yu Zhou, Byoungyoul Park, Ramin Soltanzadeh, Elnaz Afsharipour, Janaranjana (Sampath) Liyanage, Meiting Li, and more.
I markedly thank Yu for
providing me with significant amounts of cleanroom training.
Some of the best engineering knowledge and lessons I have learned during my graduate years were through being a member of the UMSATS team. Beyond the engineering aspect, I have also learned a lot about team dynamics while on the team. This experience has truly benefited me in many ways. I would like to thank the team advisors Dr. Witold Kinsner and Dr. David Levin for their help, knowledge and wisdom. I would also like to thank all current and former members of the team, namely Greg Linton, Michael Lambeta, Matthew Driedger, Dr. Jose Juan Mijares Chan, Dr. Ahmad Byagowi, and many more.
Finally, I would like to thank all of my friends as well as my father Jerzy Głowacki, mother Bożena Głowacka, and brother Tomasz Głowacki for all of their support, encouragement, and friendship along the way.
v
January 17, 2017
This Thesis is dedicated to the Country of Canada on the year of Her One Hundred and Fiftieth Birthday.
vi
Table of Contents
List of Tables
xi
List of Figures
xii
List of Acronyms
xvii
List of Commonly Used Symbols
xix
List of Chemical Formulas
xxi
1 Introduction
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Existing DGA Technologies . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
Gas Monitoring using Combustion
. . . . . . . . . . . . . . . . .
3
1.2.2
Gas Chromatography . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.3
Optical absorption . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Proposed Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Background
10
2.1
DGA Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Spectral Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.1
HITRAN Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.2
Spectral Line Profile . . . . . . . . . . . . . . . . . . . . . . . . .
16
Radiation Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
18
2.3
Available Technology . . . . . . . . . . . . . . . . . . . . . . . . .
vii
2.3.2 2.4
2.5
2.6
Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . .
20
Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4.1
Measuring molecular concentrations . . . . . . . . . . . . . . . . .
22
2.4.2
Window Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5.1
Zone Plate Geometry . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5.2
Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.5.3
Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.6.1
Thermal detectors . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.6.2
Photoelectric detectors . . . . . . . . . . . . . . . . . . . . . . . .
34
2.6.3
Photoacoustic detectors . . . . . . . . . . . . . . . . . . . . . . .
36
2.6.4
Sources of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.6.5
Detector specification . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.6.6
Signal conditioning . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3 Spectrometer Design
41
3.1
Systems Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.2
Fault Gas Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2.1
C2 H2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.2.2
Band Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.3
Calculation of Absorption Coefficients . . . . . . . . . . . . . . .
46
3.3
Radiation Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.4
Sample Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.5
Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.5.1
Lens Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.5.2
Lens Design Overview . . . . . . . . . . . . . . . . . . . . . . . .
54
3.5.3
Calculation of slit widths . . . . . . . . . . . . . . . . . . . . . . .
58
3.5.4
Calculation of aperture diameter . . . . . . . . . . . . . . . . . .
61
3.5.5
Calculation of focal length . . . . . . . . . . . . . . . . . . . . . .
63
3.5.6
Calculation of N . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.6
Radiation Detection
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.6.1
Visible Light Experiment . . . . . . . . . . . . . . . . . . . . . . .
70
3.6.2
Infrared Light Experiment . . . . . . . . . . . . . . . . . . . . . .
70
viii
3.7
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Multigas Measurement
74 75
4.1
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.2
Spectral Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.3
Gas Concentration Determination . . . . . . . . . . . . . . . . . . . . . .
82
5 MEMS Fabrication
83
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.2
Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.3
Fabrication Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6 Experimental Results and Analysis
89
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.2
Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.3
Reasons for Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6.4
Visible Light Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.4.1
Results for two inch lens . . . . . . . . . . . . . . . . . . . . . . .
94
6.4.2
Results for one inch lens . . . . . . . . . . . . . . . . . . . . . . .
97
Infrared Light Experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.5
6.5.1
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
6.5.2
Detector Circuit Design . . . . . . . . . . . . . . . . . . . . . . .
100
6.5.3
Thermopile Modeling . . . . . . . . . . . . . . . . . . . . . . . . .
103
6.5.4
Results with no Aperture
. . . . . . . . . . . . . . . . . . . . . .
104
6.5.5
Results with Aperture . . . . . . . . . . . . . . . . . . . . . . . .
106
6.5.6
Infrared Experiment Summary and Future Work . . . . . . . . . .
108
6.6
Sawtooth Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
6.7
Hologram Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
6.8
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
7 Future Work and Conclusion
112
7.1
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
7.2
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
Appendices
116 ix
A Derivation of surface reflection for an S polarized wave
117
B Annular Slit Modeling
121
C Matlab Code
126
C.1 Calculating blackbody curves . . . . . . . . . . . . . . . . . . . . . . . .
126
C.2 Calculating zone plate parameters . . . . . . . . . . . . . . . . . . . . . .
129
C.3 Calculating and plotting all gas absorption lines derived from HITRAN .
134
C.4 Cross-multiplication of gas absorption profiles with ARESR . . . . . . . .
138
D L-Edit C Code for generating concentric rings
142
E Lens Geometric Dimensions
145
E.1 Lens 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145
E.2 Lens 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
E.3 Lens 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
E.4 Lens 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
F Error measurement for calculating Voigt absorption profiles
157
G Solidworks Model
161
H Detector Circuit PCB
163
I
165
Cleanroom Photos
Bibliography
171
Index
176
x
List of Tables 2.1
Gas Ratio Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Rogers Ratio Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Classification based on Rogers Ratio Codes . . . . . . . . . . . . . . . . .
12
2.4
Concentration L1 for Doernenburg Ratio method . . . . . . . . . . . . .
13
2.5
Key Gas Ratio - Doernenburg . . . . . . . . . . . . . . . . . . . . . . . .
13
3.1
Sum absorption coefficients for C2 H2 spectral bands at 297K, 101.3kPa .
45
3.2
Lens Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3
Hammamatsu T11262-01 Specifications . . . . . . . . . . . . . . . . . . .
72
6.1
2” VIS Lens Results Overview between 500nm - 1110nm . . . . . . . . .
96
6.2
1” VIS Lens Results Overview between 500nm - 1110nm . . . . . . . . .
99
6.3
Detector Circuit component values for no aperture . . . . . . . . . . . . .
104
6.4
RMS voltages at key nodes for the case of a 100µm aperture . . . . . . .
106
E.1 Lens 1 - λ = 3045nm, f = 50.8mm . . . . . . . . . . . . . . . . . . . . .
145
E.2 Lens 2 - λ = 3045nm, f = 25.4mm . . . . . . . . . . . . . . . . . . . . .
149
E.3 Lens 3 - λ = 540nm, f = 50.8mm . . . . . . . . . . . . . . . . . . . . . .
154
E.4 Lens 4 - λ = 540nm, f = 25.4mm . . . . . . . . . . . . . . . . . . . . . .
155
xi
List of Figures 1.1
A Gas Chromatograph system. Image copyright of [14] . . . . . . . . . .
5
1.2
Systems Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
A Photoacoustic Analyzer. Image copyright of [16]
. . . . . . . . . . . .
7
2.1
Duval’s Triangle for mineral insulating oil. Image copyright of [20] . . . .
13
2.2
Infrared Absorption Spectrum for Key Gases [21] . . . . . . . . . . . . .
17
2.3
Absorption spectrum of CO2 absorption line at 667.9983cm−1 at the pressures of 0.1, 0.3 and 0.5MPa. Image copyright of [26] . . . . . . . . .
19
2.4
Radiation from a real surface. Image copyright of [27] . . . . . . . . . . .
21
2.5
Blackbody radiation curves . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.6
Gas Chamber System Setup . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.7
2D projection of S polarized wave. Image copyright of [30] . . . . . . . .
24
2.8
2D projection of P polarized wave. Image copyright of [30] . . . . . . . .
24
2.9
S and P polarization. Image copyright of [32] . . . . . . . . . . . . . . .
24
2.10 Ray tracing from aperture i to point P in far field . . . . . . . . . . . . .
26
2.11 Illustration of geometry showing aperture and observation planes . . . . .
27
2.12 Block diagram of thermal detector . . . . . . . . . . . . . . . . . . . . . .
32
2.13 Overview of thermopile . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.14 Radiation incident on pn junction in photovoltaic detector . . . . . . . .
35
2.15 Block diagram of a lock-in amplifier . . . . . . . . . . . . . . . . . . . . .
39
3.1
Systems Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.2
Spectrometer Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
xii
3.3
C2 H2 absorption spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.4
Sample C2 H2 spectral lines . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.5
Hawkeye IR-Si217 radiator with parabolic reflector . . . . . . . . . . . .
47
3.6
Hawkeye IR-Si217 blackbody radiation at 1385◦C excluding emissivity and losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.7
Gas Absorbance as a function of chamber length . . . . . . . . . . . . . .
50
3.8
Gas chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.9
Gas chamber windows . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.10 Sample Fresnel zone plate . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.11 Full Width at Half Maximum . . . . . . . . . . . . . . . . . . . . . . . .
52
3.12 ARESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.13 Ray tracing from slit to point in far field . . . . . . . . . . . . . . . . . .
56
3.14 Possible angles for maximum constructive interference . . . . . . . . . . .
57
3.15 ARESR and transmission efficiency as a function of a1 . . . . . . . . . .
60
3.16 ARESR as a function of φap . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.17 Spectral plot of ARESR as a function of φap . . . . . . . . . . . . . . . .
62
3.18 ARESR as a function of f . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.19 Spatial FWHM angle as a function of f . . . . . . . . . . . . . . . . . . .
64
3.20 Spatial FWHM as a function of f . . . . . . . . . . . . . . . . . . . . . .
65
3.21 Intensity profile in aperture plane as a function of angle for different f
65
.
3.22 Intensity profile in aperture plane as a function of radius from the center of the aperture for different f . . . . . . . . . . . . . . . . . . . . . . . .
66
3.23 Intensity at center of aperture as a function of f . . . . . . . . . . . . . .
66
3.24 Spectral plot of ARESR as a function of f . . . . . . . . . . . . . . . . .
67
3.25 ARESR as a function of N . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.26 Spatial FWHM as a function of N . . . . . . . . . . . . . . . . . . . . . .
68
3.27 Intensity at center of aperture as a function of N
69
. . . . . . . . . . . . .
3.28 Intensity profile in aperture plane as a function of radius from the center of the aperture for different N . . . . . . . . . . . . . . . . . . . . . . . .
xiii
69
3.29 Spectral plot of ARESR as a function of N . . . . . . . . . . . . . . . . .
70
3.30 Closeup of aperture and detector for visible light setup . . . . . . . . . .
71
3.31 Lockin Amplifier Circuit for measuring IR radiation. Image modified from [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.32 Hammamatsu T11262-01 Spectral Response. Image copyright of [45] . . .
74
4.1
ARESR as a function of λ and dap−lens . . . . . . . . . . . . . . . . . . .
78
4.2
ARESR at varying dap−lens as a function of λ . . . . . . . . . . . . . . . .
79
4.3
Gas absorption profiles convoluted with ARESR for different dap−lens . .
80
4.4
ζ as a function of dap−lens for different gases . . . . . . . . . . . . . . . .
81
5.1
Mask Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.2
Lens Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.3
Backside etch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.4
Frontside etch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.5
Final 1” IR Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.1
Original signal -vs- SG filtered signal . . . . . . . . . . . . . . . . . . . .
90
6.2
Possibilities for misalignment (Not to scale) . . . . . . . . . . . . . . . .
91
6.3
Sideview of visible light setup . . . . . . . . . . . . . . . . . . . . . . . .
93
6.4
Topview of visible light setup . . . . . . . . . . . . . . . . . . . . . . . .
93
6.5
Experimental and simulation results for the 2” VIS Lens . . . . . . . . .
95
6.6
Experimental and simulation results for the 1” VIS Lens . . . . . . . . .
98
6.7
Photos of IR Experiment Apparatus . . . . . . . . . . . . . . . . . . . .
101
6.8
Detector and Aperture with Chopper . . . . . . . . . . . . . . . . . . . .
102
6.9
Infrared experiment circuit diagram. Image modified from [44] . . . . . .
102
6.10 Thermopile Reference Signal . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.11 Thermopile Vin and AD630 output voltages V3 . . . . . . . . . . . . . . .
105
6.12 Voltage outputs following AD630 at full radiation on 100µm aperture . .
107
6.13 Observed sawtooth waveform as a result of binary zone plate following background correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
109
6.14 Holographic projection from Zone Plate . . . . . . . . . . . . . . . . . . .
110
B.1 Closeup view of radial intensity distribution profile of a circular lens as a function of radial distance in focal plane . . . . . . . . . . . . . . . . . .
123
B.2 Closeup view of intensity distribution profile of a linear zone plate lens as a function of distance in the Cartesian plane from the center of the lens in the focal plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
B.3 Closeup view of error between intensity profiles of circular and linear lenses.124 B.4 Radial intensity distribution profile of a circular lens as a function of radial distance in focal plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124
B.5 Intensity distribution profile of a linear zone plate lens in the Cartesian plane as a function of distance from the center of the lens in the focal plane.125 B.6 Error between intensity profiles of circular and linear lenses. . . . . . . .
125
F.1 Spectral line integrated absorption as a function of wavenumber resolution at 2cm-1 wavenumber span . . . . . . . . . . . . . . . . . . . . . . . . . .
158
F.2 Spectral line integration error as a function of wavenumber resolution at 2cm-1 wavenumber span . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
F.3 Spectral line integrated absorption as a function of measurement span at 0.01cm-1 resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
F.4 Spectral line integration error as a function of measurement span at 0.01cm-1 resolution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
G.1 Side View of Optical Assembly . . . . . . . . . . . . . . . . . . . . . . . .
161
G.2 Top View of Optical Assembly . . . . . . . . . . . . . . . . . . . . . . . .
161
G.3 Orthogonal View of Optical Assembly . . . . . . . . . . . . . . . . . . . .
162
H.1 Detector Circuit PCB . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
H.2 Detector Circuit Schematic . . . . . . . . . . . . . . . . . . . . . . . . . .
164
I.1
Lithography Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165
I.2
AlphaStep Measurement from Back-Etch . . . . . . . . . . . . . . . . . .
166
xv
I.3
AlphaStep Measurement from Zone Plate Edges . . . . . . . . . . . . . .
166
I.4
AlphaStep Measurement from initial zones of VIS Light Zone Plate . . .
167
I.5
AlphaStep Microscope Photo of Zone Plate . . . . . . . . . . . . . . . . .
167
I.6
Detailed AlphaStep Microscope Photo of Zone Plate . . . . . . . . . . . .
168
I.7
Surface roughness of backside . . . . . . . . . . . . . . . . . . . . . . . .
168
I.8
Aluminum Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
I.9
Silicon wafers for Gas Chamber Windows . . . . . . . . . . . . . . . . . .
169
I.10 MRC 8667 Sputtering System . . . . . . . . . . . . . . . . . . . . . . . .
170
xvi
List of Acronyms Acronym
Description
AC
Alternating Current
ADC
Analog-to-Digital Converter
AR
Augmented Reality
ARESR
Aperture Restricted Effective Spectral Radiosity
BOE
Buffered Oxide Etch
CAD
Computer Automated Drawing
COTS
Commercial Off-the-Shelf
DC
Direct Current
DGA
Dissolved Gas Analysis
DI
De-ionized
DSP
Double-Sided Polished
EM
Electromagnetic
EMI
Electromagnetic Interference
FTIR
Fourier Transform Infrared Spectroscopy
FWHM
Full Width at Half Maximum
GE
General Electric
HITRAN
High-Resolution Transmission Molecular Absorption Database
HPF
High-Pass Filter
HV
High Voltage
IEC
International Electrotechnical Commission
IEEE
Institute of Electrical and Electronics Engineers
xvii
Acronym
Description
IR
Infrared
LED
Light Emitting Diode
LPF
Low-Pass Filter
Matlab
Matrix Laboratory
MEMS
Micro-Electromechanical Systems
NEP
Noise Equivalent Power
NSFL
Nanosystems Fabrication Lab
PAS
Photoacoustic Spectroscopy
PD
Partial Discharge
PLL
Phase Locked Loop
RMS
Root Mean Square
SG
Savitsky-Golay
UV
Ultraviolet
xviii
List of Commonly Used Symbols Though not an exhaustive list of all symbols used in the text, the following symbols are most commonly used. All other symbols are defined appropriately in the text where they appear. Symbols presented in the text in bold font are used to represent matrices.
Symbol
Description
Unit
ζ
Relative Absorbance
-
ηap
ARESR
-
θ
Polar Angle
rad
λ
Wavelength
nm
λi
Wavelength centered at i nm
nm
ν¯
Wavenumber
cm−1
ν¯i
Wavenumber centered at i cm−1
cm−1
π
≈ 3.141592653589793238462643383279502884197169
τ
Time constant
φ
Azimuthal Angle
rad
φap
Aperture diameter
µm
a
Absorption coefficient
ax
Slit width
m
ay
Slit length
m
A
Absorbance
-
c
Speed of light in a vacuum = 299792458
cg
Gas Concentration
rad s
m2 /mol
m/s mol/m3
xix
Symbol
Description
Unit
dap−lens
Distance between aperture and lens
m
e
≈ 2.718281828459045235360287471352662497757247
-
E
Electric field
f
Frequency
g
Line shape profile
G
Gain
h
Planck’s constant = 6.62607004 · 10−34
I
Output Spectral Radiant Intensity
W m2 ·nm·sr
Io
Incident Spectral Radiant Intensity
W m2 ·nm·sr
k
Boltzmann constant = 1.38064852 · 10−23
l
Pathlength
m
N
Number of zones in lens
-
P
Pressure
kPa
P0
1 atm = 101.3
kPa
Pin
Input power
W
r
Radius of a circle
m
Si
Spectral line centered at λi
T
Temperature
K
T0
Room temperature = 296
K
xi , yi , zi
Spatial coordinates in observation plane
m
x, y, 0
Spatial coordinates in aperture plane
m
V/m Hz 1/cm−1 m2 ·kg s
m2 · kg s2 · K
mol · cm2 nm
xx
List of Chemical Formulas Molecule
Name
C2 H2
Acetylene
C2 H4
Ethylene
C2 H6
Ethane
CH4
Methane
CO
Carbon monoxide
CO2
Carbon dioxide
H2
Hydrogen
H2 O2
Hydogen peroxide
H2 SO4
Sulfuric acid
HCl
Hydrogen chloride
HF
Hydrogen fluoride
KOH
Potassium hydroxide
N2
Nitrogen
SiO2
Silicon dioxide
xxi
Chapter 1 Introduction “I think that’s the single best piece of advice: constantly think about how you could be doing things better and questioning yourself.” – Elon Musk
1.1
Background High voltage equipment used in the power systems industry often contains
insulating mineral oil for electrical and thermal insulation [1].
For preventative
maintenance reasons, it is common in the industry to analyze the quality of the oil by measuring the concentration of certain gases produced in it [2]. The goal of this thesis is to design and test an optical instrument for measuring the concentration of one of the produced gases, acetylene, using spectroscopy. The optical system proposed in this thesis is a novel design for the above mentioned application. Many transformers, bushings, and load-tap changers utilized in the power industry use insulating mineral oil in order to electrically isolate certain conductors, as well as thermally insulate the equipment [3]. This oil is composed of hydrocarbons that include paraffins, naphatelenes, and aromatics [4].
During normal operation, electrical and
thermal stresses within the equipment may cause faults that become the source of multiple failure modes [5]. Failures resulting from such faults may have significant consequences 1
CHAPTER 1. INTRODUCTION for the power systems company and its clients [5]. Some of these consequences include large economic losses and unplanned outages of the power system affected [5]. Many of the faults that occur in high voltage equipment occur between conductors that need to be electrically isolated under high voltage with a limited physical distance between them [4]. The presence of a sufficiently high electric field between the conductors may cause a dielectric breakdown within the medium separating them [4]. This scenario can potentially create a dangerous amount of current to flow between the conductors by allowing a free path for the electrons in the respective electric fields [4]. One way of significantly reducing the possibility of a serious dielectric breakdown between two or more conductors is to have a medium with high permittivity between them [6]. Such an insulating medium is provided to maximize the breakdown strength, measured in kV/cm, between conductors [6]. A dielectric breakdown of the insulating oil and other nearby material as a result of the conductors’ strong electric fields and environmental conditions is often seen as a root cause for partial discharge [4]. This breakdown results from electrostatic forces on electric charges composing the molecular bonds in the oil [4]. These forces result in part from the conductors’ electric field, the distance separating the charge from the conductor, the bond strength of the oil molecules, humidity, temperature, and pressure [4]. Partial discharge can take on many forms of varying severity levels. Sparking and corona are two types of low energy partial discharge faults where small localized bursts of discharge occur [6]. These discharges ionize the surrounding insulator forming a conductive region around it, but do not contain enough energy to cause a complete dielectric breakdown in the insulator separating the two conductors [6]. A prolonged high energy discharge fault between conductors, known as arcing, can also occur as a result of a complete dielectric breakdown in the insulator [6]. Overheating of the insulator is another fault that is often formed as a byproduct of the energy released from the partial discharges [6]. When such faults occur in the high voltage equipment containing the mineral insulating oil, chemical reactions take place that break the oil’s hydrocarbon chains into carbon-hydrogen and carbon-carbon bonds [2]. As a result, new gases are formed
2
CHAPTER 1. INTRODUCTION and dissolve in the oil when the new hydrocarbon bonds combine [7]. These gases include acetylene (C2 H2 ), ethylene (C2 H4 ), ethane (C2 H6 ), methane (CH4 ), hydrogen (H2 ), carbon monoxide (CO), and carbon dioxide (CO2 ) [2]. They are often referred to as the key gases [2]. Knowledge about the concentration of the key gases allows for creating probabilistic predictions into the type and severity of incipient faults [2, 5]. The study of the key gases for making fault predictions is referred to as Dissolved Gas Analysis (DGA). It is worth mentioning that DGA is not the only method of predicting incipient faults. Performing a visual inspection of electrical insulation paper on the conductor windings, measuring the moisture content of the air in the transformer, and performing a power factor test are also often employed.
1.2
Existing DGA Technologies Numerous remote instruments for measuring the key gases already exist and
are used today. These instruments are based on a number of technologies including combustion, gas chromatography, and optical absorption.
1.2.1
Gas Monitoring using Combustion The first instruments for attempting to provide fault diagnosis in high voltage
equipment, developed and built in the 1970’s and 1980’s, were the Morgan Schaffer Calisto and the GE Hydran [8]. In both instruments, all dissolved key gases are passed through a special membrane permeable to gases but not liquids. In the Morgan Schaffer Calisto, a thermistor is used to measure the concentration of hydrogen gas in the system [9]. Hydrogen’s high thermal conductivity relative to the other key gases change the thermal conductivity of the thermistor elements thus creating a voltage that is further amplified [9]. This amplification is then calibrated to an absolute concentration of hydrogen measured in parts per million (ppm) [9]. The significance of measuring hydrogen is that some level of hydrogen is present in most faults thus serving
3
CHAPTER 1. INTRODUCTION as a good warning indicator [10]. A moisture sensor is also employed in the Calisto that is in direct contact with the oil [8]. Moisture is also an important variable to measure since it has been shown to deteriorate the insulating cellulose paper covering the windings, as well as reduce the insulating strength of the oil [11]. The GE Hydran instrument goes one step further than the Morgan Schaffer Calisto in that it can measure all combustible key gases, H2 , CO, C2 H2 , and C2 H4 . In the Hydran, these gases are burned in a fuel cell that produces an electric current via the thermoelectric effect [8]. By measuring the produced current, the total concentration of combustible gases is measured [8]. Later versions of the Hydran monitor also introduced the ability of measuring the moisture content of the equipment as well [8]. The main disadvantage of the GE Hydran is that it cannot provide the individual concentration of all gases, but instead only the total concentration of all combustible gases [8]. The major advantages of these instruments compared to others on the market are that they are comparatively simple, robust, and cheap [8]. Their common disadvantage is their limited ability in measuring only the combustible gases and not the full range of key gases, which thus provides only limited knowledge regarding potential faults [8]. These instruments do not allow for a full gas analysis but do provide insight into whether or not a piece of equipment should be examined immediately [8].
1.2.2
Gas Chromatography A popular method for performing DGA is gas chromatography.
Gas
chromatography is performed using an instrument called a gas chromatograph. The Serveron TM8 multigas monitor is an oft used gas chromatograph for performing DGA [12]. The principle mechanism behind gas chromatography is to measure the time, referred to as retention time, it takes for a particular gas to travel from one point to another in a special tube [13]. This operation is followed by mathematically relating the travel time of the gas to the concentration of the gas as outlined below. A gas chromatography system is shown in Figure 1.1. 4
CHAPTER 1. INTRODUCTION
Figure 1.1: A Gas Chromatograph system. Image copyright of [14] Sample gases are injected into the system with an inert gas such as nitrogen or helium, referred to as the carrier gas [14]. These gases together form what is referred to as the mobile phase [13]. The objective of the carrier gas injected at a high pressure is to allow the sample gases to successfully travel through a long winding column, known as the separation column [13]. The nature of the mobile phase’ movement perpendicular to its general direction of travel within the column is sporadic, and thus when the mobile phase travels through the column it constantly interacts with the walls of the column, known as the stationary phase [13]. Due to the specially designed chemical makeup of the stationary phase, some gases in the mobile phase will chemically interact in different ways with the stationary phase, with varying degrees of diffusion [13]. The gases that react stronger with the stationary phase travel slower than those that react less, thus creating different retention times for different gases [13]. By measuring the different times between signals at the output of the device and knowing the nature of the chemical interactions that occur between the mobile phase and stationary phase, a calculation for measuring the concentration of the samples gases is performed [13].
1.2.3
Optical absorption It is possible to measure the concentration of fault gases by analyzing how much
energy is absorbed in certain optical bands. To do this, infrared radiation from a source is passed through a chamber containing the dissolved gases, and a chamber containing no
5
CHAPTER 1. INTRODUCTION dissolved gases. By taking the differential measurement of a calibrated signal that passes through a chamber with no gasses and one that contains the gases, the concentration of the fault gases may be calculated. The overall systems block diagram for such a system, known as a spectrometer, without the calibration step is shown below in Figure 1.2.
Figure 1.2: Systems Block Diagram The spectrometer consists of a radiation source that is specified to emit radiation for the wavelengths that the key gasses absorb. This radiation is passed through a chamber that contains the key gases which absorb the radiation. A wavelength selection mechanism filters out specific spectral bands of interest corresponding to the absorption profile of the gases of interest. A detector at the end of the spectrometer measures the signal produced and a concentration estimate of the gases is calculated.
1.2.3.1
Photoacoustic Spectroscopy
A popular method for performing DGA using optical techniques is photoacoustic spectroscopy (PAS). A popular PAS instrument designed for DGA in use today is the GE Kelman Transfix monitor [8]. Photoacoustic spectroscopy is based on the principle that gas absorption causes gas molecules to vibrate, thus generating heat in a closed box containing the gases [15]. This heat causes the box to expand due to an increased pressure, thus generating acoustic vibrations in the box as a result. This acoustic vibration can then be measured using a microphone. In order to discriminate between different gases, a filter wheel is used to pass only spectral bands in which particular gases of interest absorb. The filter wheel consists of a number of windows that serve as bandpass filters for particular absorption bands corresponding to the absorption profile of gases. To reduce unwanted noise from the main signal, a chopper and lock-in amplifier is used at the source and detector. A photoacoustic spectrometer is shown in Figure 1.3.
6
CHAPTER 1. INTRODUCTION
Figure 1.3: A Photoacoustic Analyzer. Image copyright of [16] 1.2.3.2
FTIR Spectroscopy
Perhaps the most precise method for performing IR spectroscopy is Fourier Transform Infrared (FTIR) Spectroscopy. It’s principle method of operation is based on measuring differences in the pathlength of light (as a function of wavelength) between a fixed mirror, a vibrating mirror, and a detector. By taking the Fourier transform of the detector signal in the time domain, it is possible to generate a frequency response of the detector signal. The result from the frequency domain can then be used to determine the radiation absorption in different wavelength bands.
1.3
Proposed Fresnel Zone Plate A novel method for performing DGA using optical techniques on the full range
of key gases is proposed in this thesis. The goal of the proposed method is to design, manufacture, and test an optical instrument for DGA that is more informative than older monitors utilizing combustion yet cheaper, less complex, and more robust than a gas chromatograph or an FTIR spectrometer. To limit the scope of the project, the secondary goal is to design the instrument so that it can measure absorbed C2 H2 gas. The proposed instrument does not make use of a filter wheel for wavelength selection, nor does it use a microphone as a detector as in PAS. The new design is based 7
CHAPTER 1. INTRODUCTION on a custom-made Fresnel zone plate, or zone plate lens, as the wavelength selection mechanism for spectral bands of interest corresponding to the absorption profiles of the key gases. The working principle of the zone plate lens is that at particular focal lengths from the plate, constructive interference for certain wavelengths of interest occur at the focal point and destructive interference occurs for unwanted wavelengths. By adjusting the focal length of the radiation detector from the zone plate, a wide variety of different spectral bands may be selected such that they constructively interfere. Fresnel zones plates have previously been used successfully in IR spectroscopy for focusing in on the IR absorption profiles of ethylene, ammonia, and ethanol in apple warehouses; followed by analyzing how much absorption occured in those ranges [17]. This fact provides reason for developing similar systems to be used in measuring dissolved gases in high voltage equipment. The advantages of using zone plate technology compared with PAS include economic advantages in terms of manufacturing cost, and a reduction in the physical size of the final device that is otherwise taken up by a filter wheel used in PAS. Another advantage of it is enabling a framework for a realistic MEMS miniaturization of the device for future revisions of the design.
1.4
Thesis Organization This thesis is organized into seven chapters with appendices, bibliography, and an
index that follow the last chapter. • Chapter 1 presented an introduction to DGA with examples of existing instruments for performing DGA. It concluded by introducing a novel optical instrument for measuring the absorption of dissolved gases. • Chapter 2 introduces the background knowledge that is helpful in understanding the underlying theory behind the design presented in Chapter 3. • Chapter 3 presents the specific design details of the zone plate spectrometer built in this work.
8
CHAPTER 1. INTRODUCTION • Chapter 4 presents evidence for the hypothesis that one zone plate lens can be theoretically used to measure the concentrations of all key gases in DGA. • Chapter 5 discusses the MEMS fabrication process of the zone plate lens and shows the result of the lens fabrication. • Chapter 6 presents all experimental results from the experiments conducted. • Chapter 7 provides a list of future work that should be performed in order to build upon the initiated project. The chapter ends with a conclusion of the work developed in this thesis, with highlights showcasing the implications of that work.
9
Chapter 2 Background In order to design an IR spectrometer for measuring dissolved gases, the designer should have some background knowledge in the underlying theory behind spectroscopy. A review of some of this knowledge is presented in this chapter and it may be broken down as follows: • Section 2.1 presents a review of numerous technical standards used in DGA that correlate gas concentrations to common fault types. • Section 2.2 briefly reviews why optical gas absorption happens. Additionally, it shows the spectral absorption profiles of the key gases. • Section 2.3 shows some of the different radiation sources that can be used in practice. It explains the advantages and disadvantages of each technology. • Section 2.4 explains how to calculate the level of radiation that was absorbed by the key gases including losses. Further, it shows how to relate the absorption to concentrations of each gas. • Section 2.5 shows some of the background knowledge necessary for the designing the zone plate itself. Namely, it discusses the nature of diffraction through one or more apertures. • Section 2.6 discusses existing technologies commonly used in measuring infrared radiation.
10
CHAPTER 2. BACKGROUND
2.1
DGA Standards Two technical standards developed for performing DGA on the key gases include
IEEE C57.104-2008 and IEC 60599 [18]. Both standards are based on and use existing methodologies for performing DGA [2]. All methods for diagnosing the potential faults are based on measuring the concentration of key gases of interests. By measuring the concentrations of each of the key gases for a given method, particular gas ratios may be calculated, and thus providing insight into fault diagnosis. The IEEE C57.104-2008 standard uses the Rogers Ratio Method, and the Dornenburg Ratio Method as the fundamental methods upon which potential faults are diagnosed [18]. The IEC 60599 standard utilizes Duval’s Triangle Method and the Key Gas Method for diagnosing potential faults [18]. For demonstration, the Rogers Ratio Method is performed by giving a Ratio Code to particular gas ratios as shown in Table 2.1. These ratio codes are then assessed according to concentration ratio ranges under which they fall and are assigned a code number as shown in Table 2.2. By assessing the set of ratio codes and their respective code numbers, a diagnosis set forth by the standard is established as shown in Table 2.3. Table 2.1: Gas Ratio Codes [19] Gas Ratios Ratio Codes CH4 /H2 i C2 H6 /CH4 j C2 H4 /C2 H6 k C2 H2 /C2 H4 l
The Doernenburg Method also compares gas ratios, albeit different gas ratios, and assesses the ratios in numerical ranges.
The first step in the Doernenburg
method is to determine whether it is practical to continue DGA by first measuring the total concentrations of the key gases as parts per million (ppm). These minimum concentrations are referred to as L1 and are shown in Table 2.4.
If the measured
concentrations are greater than or equal to 2L1, then a fault diagnosis is made according to the ratio limits shown in Table 2.5 [19].
11
CHAPTER 2. BACKGROUND Table 2.2: Rogers Ratio Codes [19] Ratio Code Range Code ≤ 0.1 5 > 0.1, < 1.0 0 i ≥ 1.0, < 3.0 1 ≥ 3.0 2 < 1.0 0 j ≥ 1.0 1 < 1.0 0 k ≥ 1.0, < 3.0 1 ≥ 3.0 2 < 0.5 0 l ≥ 0.5, < 3.0 1 ≥ 3.0 2
i 0 5 1-2 1-2 0 0 1 1 0 0 0 5
j 0 0 0 1 1 0 0 0 0 0 0 0
Table k 0 0 0 0 0 0 0 0 1 1-2 2 1-2
2.3: Classification based on Rogers Ratio Codes [19] l Diagnosis 1-2 Normal Deterioration 1-2 Partial Discharge 1-2 Slightly overheating 1 < 0.75 > 0.4 < 0.3 Corona (Low Intensity PD) 0.4 < 0.3 Arcing (High Intensity PD) < 1, > 0.1 > 0.75 < 0.4 > 0.3
Figure 2.1: Duval’s Triangle for mineral insulating oil. Image copyright of [20]
13
CHAPTER 2. BACKGROUND
2.2
Spectral Specification All of the key gases mentioned in section 2.1 absorb photons with vibrational
energies corresponding to wavelengths in the near-infrared and mid-infrared spectral bands [21]. In order to appreciate this phenomenon, it is worthwhile to provide some background knowledge on the nature of the interaction between the gas molecules and the incoming photons. The energy E (or quanta) of any photon with frequency f is given as:
E = hf
[J] (2.1)
where h is Planck’s Constant. When a photon of frequency f strikes a gas molecule comprised of N atoms, that photon will either be absorbed or reemitted by the molecule dependant on its molecular structure [22]. If the quantum energy level of the photon is large enough to change the electric dipole moment of the molecule, then that molecule will absorb that photon in the form of vibrational motion, rotational motion, or a combination thereof [23]. These motions may take the form of symmetric and asymetric stretching, bending, rocking, twisting, or wagging [23]. These motions occur in the multiple axes where atomic bonds exist while remaining held together by “spring-like” forces [22, 24]. It is worth noting that the number of vibrational modes for a nonlinear molecule of N atoms has 3N − 6 fundamental modes of vibration, and 3N − 5 modes if it is a linear molecule [22]. The minimum quantum energy level, or fundamental modes of that photon necessary to make the molecule vibrate at different frequencies are expressed by Equation 2.2. 1 hf E = n+ 2
where n is an integer corresponding to a specific energy level.
[J] (2.2) When n = 0, the
energy is referred to the ground-state energy of that molecule, and all other energies where n > 0 correspond to vibrations of higher frequencies that can also be excited [22]. It happens that not all modes can be excited and that there are many more
14
CHAPTER 2. BACKGROUND factors at play in determining which modes will be excited [22]. Some examples include overtone and combination bands; which are frequencies analogous to the harmonics of the fundamental absorption frequency [23]. Fermi resonance, coupling of the vibrations in the molecular structure, and rotational effects during vibrational transitions, known as vibration-rotation bands are also known to affect the infrared spectra of molecules [23].
2.2.1
HITRAN Data In practice, standardized datasets are often used to observe how much infrared
radiation different molecules absorb at different wavelengths. These datasets alleviate the user from having to determine the spectral absorption properties of various molecules. One such popular dataset used is the high-resolution transmission molecular absorption database (or HITRAN). This database provides a great deal of necessary information for calculating the absorption coefficients of many molecules as a function of wavelength, temperature and pressure [21]. The absorption coefficient is a variable necessary for determining the total infrared radiation absorbed in some spectral band per unit length per unit concentration. One important variable in determining the absorption coefficient of a gas is knowing its spectral line intensity, S. The spectral line intensity is the maximum intensity of a spectral line profile centered at some wavelength. It is a variable that allows the user to separate the spectral line profile from its maximum intensity when calculating the absorption coefficient of a gas. This separation conveniently accounts for potential variations in pressure so that the line intensity does not need to be recalculated, whereas the spectral line profile should be recalculated. The absorption coefficient, a(λ), for a spectral line centered at λi and denoted Si , with a line shape profile g(λ, λi ) is provided in Equation 2.3 [25].
a(λ) = Si (T )g(λ, λi , T, P, q)
h 2i m mol
(2.3)
The spectral line intensity data from HITRAN for the key gases is shown in Figure 2.2 as a function of wavelength for a reference temperature of 297K and pressure
15
CHAPTER 2. BACKGROUND 101.3kPa [21]. Water is not a key gas but is included in Figure 2.2 anyway due to the importance of measuring humidity in oil-impregnated high voltage apparatus.
2.2.2
Spectral Line Profile As shown in Equation 2.3, it is important to know the spectral line profile of a gas
in order to calculate its absorption coefficient in a given range. The spectral line profile is dictated by numerous environmental conditions such as pressure, temperature, and gas volume mixing ratios that alter its distribution function. In this experiment, the two main line broadening functions are Doppler broadening and pressure broadening, which have Gaussian and Lorentzian distributions respectively. For convenience, wavenumbers are used in place of wavelength to calculate these broadening parameters in this thesis. Wavelength will be used in further chapters for designing the zone plate and performing analyses.
The relationship between wavenumber, ν¯ and wavelength, λ is shown in
Equation 2.4 [23]:
ν¯ =
1 λ
[cm−1 ] (2.4)
As a result of the combination of the two broadening functions, the line profile follows a Voigt profile [25]. The shape of each spectral line, g(¯ ν , ν¯i ) centered at wavenumber ν¯i as a function of wavenumber ν¯ is defined in Equation 2.5 as follows [25]: hc¯ ν ν¯ tanh( 2kT ) g(¯ ν , ν¯i ) = ν , ν¯i ) hcν¯i f (¯ ν¯i tanh( 2kT )
(2.5)
where f (¯ ν , ν¯i ) is the line broadening function centered at wavenumber ν¯i . The line broadening function is provided as [25]:
f (¯ ν , ν¯i ) =
1 √ K(xi , yi ) αDi π
where,
16
(2.6)
CHAPTER 2. BACKGROUND
Figure 2.2: Infrared Absorption Spectrum for Key Gases [21] 17
CHAPTER 2. BACKGROUND
yi K(xi , yi ) = π
ˆ
∞
−∞
αL =
αL0
(2.7)
αLi αDi
(2.8)
ν¯ − ν¯i αDi
(2.9)
yi =
xi =
2
e−t dt (xi − t)2 + yi2
P P0
T0 T
γ
αL0 = (1 − q) αL0 f + qαL0 S
(2.10)
(2.11)
and,
αDi
ν¯i = co
s
2kT m
(2.12)
In the above equations, k is Boltzmann’s constant, T is temperature in Kelvin, m is the molecular mass of the absorbed molecule, c is the speed of light, q is the volume-mixing 0 0 ratio, αLf and αLs are foreign and self-broadening parameters respectively, and T0 and P0
are the reference temperatures and pressures of 297K and 101.3kPa respectively [25]. An Example of a lineshape and how that shape may change with a parameter like pressure for a single spectral line is shown in Figure 2.3 [26].
2.3 2.3.1
Radiation Source Available Technology In order to determine the concentration of certain gases using spectroscopy, the
amount of infrared radiation that those gases will absorb must be measured. As such, an 18
CHAPTER 2. BACKGROUND
Figure 2.3: Absorption spectrum of CO2 absorption line at 667.9983cm−1 at the pressures of 0.1, 0.3 and 0.5MPa. Image copyright of [26] infrared radiation source is necessary to provide a reference signal. There are numerous infrared radiation sources to choose from including semiconductor-based technologies such as lasers, light-emitting diodes (LEDs), laser diodes, technologies based on incandescence, and others. The blackbody radiator is a popular light source used in IR spectroscopy. The primary advantage of the blackbody radiator is that it emits radiation across the entire infrared spectrum where all the key gases absorb, and it is economically cheap, robust, and readily available.
The potential disadvantage of such a source is that
its spectral power density decreases significantly with increasing wavelength beyond a temperature-dependent maximum inflection point. Also, the spectral power density at a particular band of interest for a blackbody radiator is typically far lower than that of a semiconductor-based source designed for that same band. As such, higher input powers than would be necessary for semiconductors are required for blackbodies in some particular spectral bands. However, due to the fact that blackbody sources provide infrared radiation across all infrared wavelengths, they are the only known individual source that can fulfil the needs of providing radiation that all of the key gases can absorb to some extent.
19
CHAPTER 2. BACKGROUND
2.3.2
Blackbody Radiation An ideal blackbody is defined as an object that emits electromagnetic radiation
when the temperature of that object is higher than that of its surrounding, and absorbs incoming radiation when its temperature is lower than its surroundings [27]. To create an object with such a temperature differential, current is run through a carbide wire. When current flows across a length of such wire of surface area dA1 as shown in Figure 2.4, an electric field is created across that surface that accelerates free valence electrons within the wire [28]. As many of these valence electrons collide with neighbouring atoms while drifting from one end of the surface to the other, the atoms vibrate and heat energy is released; increasing the temperature of the surface [28]. This phenomena is referred to as joule heating. The surface dA1 now emits off blackbody radiation towards an outer surface dAn as shown in Figure 2.4 as a result of the differential in temperature with its surrounding [27]. The rate of spectral energy emission per unit area, known as spectral intensity I(λ, θ, φ), passing through dAn accounts for a fraction of the total energy emission, dq emitted from dA1 . This fraction is expressed by Equation 2.13 as a function of wavelength λ, and spherical angles θ and φ that define the center of a steradian dω which contains area dA1 [27].
I(λ, θ, φ) ≡
dq dA1 cos θ · dω · dλ
h
W m2 ·nm·sr
i
(2.13)
It is worth noting that the rate of emission from each surface dA1 on the wire will most likely not be homogeneous with respect to angles θ and φ. The intensity and orientation of the emitted radiation as a function of angle and surface radiator area is known as diffusivity and is useful in calculating the spectral intensity at any given point from all surfaces dA1 on the source [27]. In reality, there are no perfect blackbody radiators since real surfaces have emissivity deviations for many wavelengths [27]. Emissivity is the intensity deviation from an ideal blackbody radiation profile from a hot surface as a function of wavelength. It represents the real emission profile of a hot surface accounting for losses. However,
20
CHAPTER 2. BACKGROUND
Figure 2.4: Radiation from a real surface. Image copyright of [27] it is practical to use ideal blackbody approximations for design purposes. The spectral intensity passing through dAn is provided by Planck’s Law defined below in Equation 2.14 [28].
Iλ,b (λ, T ) =
2hc2
hc
h
λ5 e λkT − 1
W m2
i
(2.14)
where c is the speed of light in a vacuum, h is Planck’s constant, k is Boltzmann’s constant, T is temperature, and λ is wavelength. The spectral intensity of a blackbody radiator plotted as a function of wavelength for a few separate temperatures is shown in Figure 2.5. As will be shown in chapter 3, it is important for the emanated radiation to be collimated and concentrated at large distance from the source to minimize power losses. In other words, the radiation source must have a large gain. The gain of such a source, G(θ, φ) is the defined as being the ratio of the radiation U (θ, φ) emitted in angles θ and φ to the total power Pin emanated from an ideal isotropic source [29]. For this reason, parabolic reflectors are often placed behind the radiating element to improve its gain. An expression for gain is shown in Equation 2.15 [29].
G(θ, φ) =
21
U (θ, φ) Pin /4π
(2.15)
CHAPTER 2. BACKGROUND
Spectral emissive power [W/(sr*m3 )]
8
× 10 10 T = 900°C T = 1200°C T = 1500°C
7 6 5 4 3 2 1 0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Wavelength [nm]
Figure 2.5: Blackbody radiation curves
Figure 2.6: Gas Chamber System Setup
2.4 2.4.1
Optical Absorption Measuring molecular concentrations In order to determine the concentrations of gases in a given volume, it is necessary
to determine how much radiation each gas absorbs. Assume radiation Io enters a chamber of length l and leaves out the other end of the chamber as I as shown in Figure 2.6. In this particular case, imagine C2 H2 is entered into the chamber with an inert carrier gas N2 . An inert gas is typically used to control the concentration of the gas under testing. The absorption of some incident radiation Io is a function of the absorptivity a of 22
CHAPTER 2. BACKGROUND the gases, the length l of the sample chamber, and the concentration cg of the gases. The Beer-Lambert Law, shown below in Equation 2.16, shows how the output radiation I is related to Io following the absorption of g gases [22]. PN I = e−l g=1 ag cg Io
(2.16)
where the total absorbance A of N gases is defined as [22]:
A=l
N X
ag c g
(2.17)
g=1
It is important to note some of the other variables at play in this situation. Namely, entrance and exit windows must be placed on both edges of the chamber to allow radiation to pass through the chamber. The windows must be sealed so the outside environment is not contaminated. Further, the windows should be designed such that they minimize scattering effects and losses, while maintaining high collimation.
2.4.2
Window Losses As radiation falls incident onto a window, the radiation will be divided into
transmission and reflection components. The radiation is also attenuated across the length of the window according to the Beer Lambert Law explained above. In terms of the losses that occur at the boundary of the surfaces, the intensity and directionality of the transmitted and reflected components can be determined using Fresnel’s laws of reflection [30]. Snells Law dictates the relationship between the angles of the incident and transmitted waves through some medium with the respectives indices of refraction of the two media. It is provided in Equation 2.18 [30]. n1 sin θt = n2 sin θi
(2.18)
In order to calculate the transmission and reflection components of the radiation, the wave’s polarization must be known. When an electromagnetic wave strikes the surface 23
CHAPTER 2. BACKGROUND
Figure 2.7: 2D projection of S polarized wave. Image copyright of [30]
Figure 2.8: 2D projection of P polarized wave. Image copyright of [30]
Figure 2.9: S and P polarization. Image copyright of [32] of a window, the wave’s polarization may be expressed in terms of S and P polarization as shown in Figures 2.7, 2.8, and 2.9. An S polarized electric field is defined as electric field that oscillates in a plane perpendicular to the plane of incidence, and a P polarized electric field is an electric field that oscillates in a plane parallel to the plane of incidence [30]. The amount of radiation transmitted through the medium is provided in Equations 2.19, 2.20, and 2.21 as follows [30][31]: 1 Rs (θr ) + Rp (θr ) T (θi ) = 1 − R(θr ) = 1 − 2
(2.19)
where Rs (θr ) and Rp (θr ) represent the reflection of the S and P polarization components respectively and are given as:
Er,s Rs (θr ) = Rs (θi ) = = Ei,s
cos(θi ) − cos(θi ) +
24
r r
n2 n1 n2 n1
2
− sin2 θi
2
− sin2 θi
(2.20)
CHAPTER 2. BACKGROUND
r
Rp (θr ) = Rp (θi ) =
n2 n1
2
Er,p = r 2 Ei,p n2 n1
− sin2 θi −
n2 n1
2
n2 n1
− sin θi +
2 2
cos θi (2.21) cos θi
A more in-depth derivation of these equations is provided in Appendix A.
2.5
Fresnel Zone Plate The principle idea behind a Binary Fresnel Zone Plate is that diffraction from
multiple unevenly spaced apertures creates a radiation profile in the focal plane that allows for sound spectral discrimination.
2.5.1
Zone Plate Geometry A cross section of a binary zone plate is shown in Figure 2.10. The lens is designed
such that the distance between the center of every aperture, or zone i, and the focal point, or line, P , differs by an ith multiple of λo where i ∈ Z > 1. As a result, all radiation from all slits are in phase for some particular λo while destructively interfering for most other λ at some predefined distance f from the lens. The design equations for the geometrical features are as follows:
li = f + iλo
q l1 2
[m] (2.22)
− f 2,
for i = 1 di = q P li 2 − f 2 − k=i−1 dk , for i > 1 k=1 Pk=i
θi = arctan
25
k=1 dk f
[m] (2.23)
!
[rad] (2.24)
CHAPTER 2. BACKGROUND
Figure 2.10: Ray tracing from aperture i to point P in far field
2.5.2
Diffraction The electric field distribution for wavelength λ at an observation plane lying in the
Fraunhofer1 zone a distance zi away from the aperture plane is the 2D Fourier Transform of the electric field in the aperture plane [33]. The Fraunhofer zone for an electric field of wavelength λ is defined as the region in space past a distance z = zi from the aperture where:
zi
D2 λ
(2.25)
and D is vaguely defined as the largest feature of the aperture2 [33]. Consider Figure 2.11, the more general form of the electric field distribution in the observation plane E(xi , yi , zi ) located in the Fraunhofer zone given an electric field E(x, y, 0) incident on the aperture plane, is given in Equation 2.26. This equation is known as the Fresnel-Kirchoff Diffraction Formula [33]. 1 2
The Fraunhofer zone is also sometimes referred to as the Far Field. There is no concise definition in literature of what the “largest aperture feature” means. It depends on the geometry of the aperture, and in some cases the precise definition may be open to interpretation.
26
CHAPTER 2. BACKGROUND
Figure 2.11: Illustration of geometry showing aperture and observation planes
jkejkzi E(xi , yi , zi ) = 2πzi
¨ E(x, y, 0)e
−jk
xxi +yyi zi
dxdy
(2.26)
SA
where k = 2π/λ. There are two main types of Binary zone plates: linear zone plates and circular zone plates. Solving the Fresnel-Kirchoff equation for these two geometries, the electric fields for linear and circular zone plates are shown in Equations 2.27 and 2.28 respectively [33]: 2
2
ax ay jk xi2z+yi i E(xi , yi , zi ) = E(x, y, 0) sinc kxi sinc kyi e 2zi 2zi
2
jkejkzi jk rzi E(ri , zi ) = e i 2πzi
ˆ
r2
E(r, 0)e
−j kr z
(2.27)
!
2
i
J0
r1
where ax is the width of a slit, ay is the length of a slit, r=
krri dr zi
(2.28)
√ 2 x + y 2 , r1 is the inner
radius of a ring, r2 is the outer radius of a ring, and J0 is a Bessel function of order zero [33]. If the impact of ay is neglected, the resultant intensity distributions from these two geometries are shown respectively in Equation 2.29 and Equation 2.30 [33][34].
2
I(θ) = Io sinc
πax sin θ λo
27
!
= Io sinc2 β
(2.29)
CHAPTER 2. BACKGROUND
4Io I(θ) = (1 − 2 )2
J1 (β) − J1 (β) β
!2
(2.30)
where is the ratio of the inner ring radius to the outer ring radius of some slit and J1 is a first order Bessel function [34]. The angle θ is defined as the angle between the center axis of the aperture slit and some point or line in the observation plane. In Equation 2.30, the ax in β is defined as the difference between the outer and inner radius of a ring [34].
2.5.3
Coherence To achieve the constructive interference desired in subsection 2.5.1 upon which
the zone plate design equations were based on, an assumption was made that all incident radiation throughout the whole surface SA arrives in-phase. Unfortunately this homogeneous in-phaseness is not physically possible and a measure of error for such an assumption must be taken into account. The concept of coherence provides a figure of merit for partially determining the performance of the lens. There are two types of coherence to consider: spatial coherence and temporal coherence. Spatial coherence Γ(ξ, η) is defined as the correlation between the fields at Q1 and Q2 as a function of the space between them, represented as η and ξ in Figure 2.11 [33]. Temporal coherence Γ(τ ) on the other hand is a correlation between the field at a certain point when sampled numerous times over some time period τ [33]. These degrees of variation on the electric field in time and space can be expressed by the autocorrelation function, Γ averaged out over a large number of samples expressed by the h·i symbol shown in Equation 2.31 [33].
0
0
0
0
Γ(xi , yi , xi , yi , τ ) , hE(xi , yi , t)E ∗ (xi , yi , t + τ )i
(2.31)
The normalized autocorrelation function γ describes the relationship between the autocorrelation function and the irradiance at the source accounting for both spatial and temporal coherence. It provided below as follows [33]:
28
CHAPTER 2. BACKGROUND
0
0
0
γ(xi , yi , xi , yi , τ ) ,
0
Γ(xi , yi , xi , yi , τ ) 0 0 Γ(xi , yi , xi , yi , 0)
(2.32)
where,
0
0
0 6 |γ(xi , yi , xi , yi τ )| 6 1 2.5.3.1
(2.33)
Spatial coherence
To observe how the distance between two points, Q1 and Q2 changes the spatial coherence of the signal in the observation plane, the following variables η and ξ are defined as follows:
0
ξ = xi − xi 0
η = yi − yi
(2.34)
Assume that all fields in the z = 0 plane are incoherent with respect to each other due to their random phases as expressed in Equation 2.35:
0
0
hEP 1 (x, y, 0)EP∗ 2 (x , y , 0)i = 0
(2.35)
In the circumstance that the time delay τ between the fields at EP 1 and EP 2 is zero, the spatial autocorrelation function between points Q1 and Q2 in the observation plane may be expressed in Equation 2.36 as follows [33]:
0
0
0
0
Γ xi , yi , xi , yi = Γ (ξ, η) = hE(xi , yi , zi )E ∗ (xi , yi , zi )i 0
(2.36)
0
From [33], E(xi , yi , zi ) and E ∗ (xi , yi , zi ) has been found to be: 2
2
jkejkzi jk xi2z+yi i e E(xi , yi , zi ) = 2πzi
¨
2
+y jk x 2z
EP 1 (x, y, 0)e SA
29
i
2
−jk
e
xxi +yyi zi
dxdy
(2.37)
CHAPTER 2. BACKGROUND
2 2 0
0
E ∗ (xi , yi , zi ) =
x
−jkzi
0 i
+ y
0 i
−jke −jk 2zi e 2πzi 2 2 0 0 0 0 0 0 x + y ¨ x x +y y i i 0 0 0 0 −jk jk ∗ 2zi zi × EP 2 (x , y , 0)e e dx dy
(2.38)
SA
By mathematically manipulating Equations 2.34 through 2.38, the final expression for the spatial autocorrelation function as a function of ξ and η has been shown from [33] to be: k 2πzi
Γ (ξ, η) =
!2 ¨
jk xξ+yη z
I(x, y, 0)e
i
dxdy
(2.39)
SA
Since the relationship between pathlengths from two or more points in the source screen to a single point in the viewing screen or two or more points in the viewing screen to a single point in the source screen is reciprocal, the spatial coherence between two or more points in the viewing screen also increases as area SA decreases [33]. 2.5.3.2
Temporal coherence
The temporal coherence between points P1 and P2 with respect to the observation plane is given by the Weiner-Khitchine theorem in Equation 2.40 [33]. This is also referred to as the temporal autocorrelation function. 1 Γ(xi , yi , xi , yi , τ ) = Γ(τ ) = 2 4π 0
0
ˆ
∞ jωτ ˜ I(ω)e dω
(2.40)
−∞
where τ is the time delay difference for signals of frequency ω from points P1 and P2 to arrive at point Q1 [33]. As the time delay difference τ between the signal paths increases, the signals become increasingly uncorrelated when averaged out over some period of time. The temporal coherence approaches zero as τ approaches infinity and it approaches one as τ approaches zero [33].
30
CHAPTER 2. BACKGROUND
2.6
Detector The role of the radiation detector in the spectrometer is to measure the output
power of the radiation exiting the gas chamber. There are multiple technologies for measuring this output power based on thermal, photoelectric, and photoacoustic energy conversion. This section will discuss the working principles, advantages, and limitations of each technology, as well as introduce figures of merit for quantifying the effectiveness of each detector per Watt and per Hz of radiation intensity.
2.6.1
Thermal detectors Thermal detectors convert radiation, Pin incident on them into atomic vibrations
in the crystal structure of the detector material. As a result of these vibrations in the material, electric current corresponding to Pin may be generated. All thermal detectors can be generally represented as containing a surface As of thermal capacity C that absorbs Pin , and a thermal link G that transfers some of the absorbed heat to a heatsink of constant temperature TA [35]. This radiation increases the temperature, Td of surface As and generates a voltage proportional to that change in temperature across G. The change in temperature ∆T is provided as ∆T = Td − TA . A general example of such a thermal detector is shown in Figure 2.12. The time rate of travel of the heat from As to the heat sink is expressed in Equation 2.41 as a second order differential equation [35].
It is a function of the
temperature difference ∆T between the detector and heat sink, thermal capacity C, thermal link conductance G, and the incoming radiation Pin with its respective quantum efficiency η on As [35].
C
dT (t) + G∆T (t) = ηPin (t) dt
(2.41)
The reaction speed of such a detector, or its time constant, is given by the relation of the thermal capacity of the detector As and how fast that heat can be moved through the thermal link G. This time constant, τ is shown in Equation 2.42 [36]. 31
CHAPTER 2. BACKGROUND
Figure 2.12: Block diagram of thermal detector
τ=
2.6.1.1
C G
[s] (2.42)
Thermocouples and thermopiles
A thermocouple is one such thermal detector that absorbs the radiation on a surface, referred to as the “hot junction” at the intersection of two dissimilar metals. This radiation creates a temperature gradient ∆T between the temperature of the “hot junction”, TH and the temperature, TC of another surface called the “cold junction” across a thermal link of length l. This thermal link is composed of two dissimilar metals parallel to one another that generate a voltage VN 1 between them proportional to ∆T [37]. The summation of multiple thermocouples in series is known as a thermopile and it is shown in Figure 2.13. Voltage VN i is generated due to a phenomenon known as the Seebeck effect [37]. This voltage may be expressed in Equation 2.43 where Sab is the Seebeck coefficient between metals A and B (represented as blue and green wires) shown in Figure 2.13 [37].
32
CHAPTER 2. BACKGROUND
Figure 2.13: Overview of thermopile
VN i = Sab ∆T
[V] (2.43)
Sab = Sa − Sb
h i V K
(2.44)
where Sa is the Seebeck coefficient for metal A and Sb is the Seebeck coefficient for metal B [37]. In a thermopile, the total voltage VT across the thermopile is expressed in Equation 2.45:
VT =
M X
VN i
[V] (2.45)
i=1
where M is the total number of thermocouples connected in series. The motivation for using thermopiles is to amplify the voltage signal arising from a temperature gradient. Otherwise, the output from only one thermocouple may not be sufficient to achieve satisfactory sensitivity.
2.6.1.2
The bolometer
The bolometer is a thermal detector whose surface material on As changes its total resistance as a function of temperature according to Equation 2.46 [36].
R(G) = Ro (1 + α∆T )
33
[Ω] (2.46)
CHAPTER 2. BACKGROUND where Ro is the initial resistance of the detector at its initial reference temperature, and where α is the thermal absorption coefficient of the detector material [36]. By applying a bias current across this new resistance, a new voltage proportional to the resistance can be measured. Given this differential voltage between the reference and measured signal, the temperature of the surface can be backcalculated.
2.6.1.3
The pyroelectric detector
The pyroelectric detector works on the principle of the pyroelectric effect on the sensing material surface AS arising from a change in its temperature ∆T [38]. When some materials experience a deviation in temperature, the orientation of the atoms in that material change; thus changing the material’s polarization. This change in polarization as a function of temperature is known the pyroelectric effect [38]. When an time-dependent change in temperature is noticed, an AC current iP is generated as per Equation 2.47.
iP = pAS
∆T dt
[A] (2.47)
where p is the pyroelectric coefficient of the sensor material [38].
2.6.2
Photoelectric detectors Photoelectric detectors are based on the principle of generating new free
electron carriers in the detector material proportional to the incident radiation, and producing an output current or voltage as a result [39]. This conversion is referred to as the photoelectric effect [39]. Photoelectric detectors can be either intrinsic or extrinsic semiconductors and can be generally classified into three main categories: photoconductive detectors, photovoltaic detectors, and photoemissive detectors [39].
2.6.2.1
Photoconductive detectors
The principle concept behind the photoconductive detector is measuring the change in conductivity of a detector surface of thickness tD and surface area As as a
34
CHAPTER 2. BACKGROUND
Figure 2.14: Radiation incident on pn junction in photovoltaic detector function of the net energy of photons striking the detector [39]. The conductivity changes due to electrons in the material’s atoms generating enough energy to jump from their valence to conduction bands; creating free electron carriers as a result [39]. When a bias voltage with a load resistor is connected in series with this region, more or less current can flow through that region proportional to the energy and intensity of the incoming photons [39]. The resulting voltage across the load resistor can then be used to determine the radiation intensity.
2.6.2.2
Photovoltaic detectors
Photovoltaic detectors also generate free electron carriers as a function of incident radiation, but instead of changing the conductivity of a region, a voltage signal across a pn junction is created [39]. This is shown in Figure 2.14. When incident photons of sufficient energy to break the bonds of electrons in the atoms strike the pn junction, the electrons and holes in the p and n regions recombine in the depletion region forming positive and negative ions [39]. As a result, the energy bandgap between the pn junction increases proportionally with the energy, or wavelength, of the photons. This bandgap energy generates an electric field across the junction that produces a voltage proportional to the incident radiation [39].
35
CHAPTER 2. BACKGROUND 2.6.2.3
Photoemissive detectors
Photoemissive detectors are based on the principle of a material emitting electrons outwards from its surface when a photon falls incident upon that surface [39]. This phenomenon is known as the photoemissive effect [39]. When an electron is knocked off the surface of the target material, the electric field between the electron’s original surface, the anode, and another surface some distance away, the cathode, increases [39]. If a small bias voltage is applied between the anode and the cathode, then that bias voltage will increase even further proportionally to the incident radiation. This proportionality allows for calculating the radiant intensity from the output signal.
2.6.3
Photoacoustic detectors The final type of radiation detector to be discussed is the photoacoustic detector
that was briefly introduced in subsubsection 1.2.3.1. The photoacoustic detector measures acoustic vibrations in a closed chamber that are produced by heat energy from atomic collisions within that chamber. This acoustic vibration measurement is accomplished using a microphone.
2.6.4
Sources of noise The output signal from any radiation detector will never only contain the signal
of interest. There will always be a noise component superimposed on the desired signal. One type of noise occurs when electrons moving randomly around the detector material collide with neighboring atoms in the material. This generates heat in the process [40] and the produced noise is known as Johnson noise [40]. Another variety of noise found in spectrometers is photon noise, also referred to as background noise. Photon noise consists of external signals not originating from the primary radiation source where the gas samples are absorbed but from the environment surrounding the detector [40]. If the detector output signal is modulated at some frequency f , then noise will be generated as an inverse function of this frequency, referred to as 1/f noise [40]. 1/f noise is also
36
CHAPTER 2. BACKGROUND sometimes referred to as pink noise.
2.6.5
Detector specification The desired input signal of any radiation detector will be affected by noise. In
order to quantify the effects of this noise on skewing the true value of the desired signal reading, some figures of merit will be introduced.
2.6.5.1
Responsivity
When photons fall incident upon some detector surface, the detector will provide a current output signal or a voltage output signal as a result. The ratio of the incident radiant power to the output signal in current or voltage is known as the detector’s responsivity and is shown below in Equations 2.48 and 2.49 [33]. Responsivity may be thought of as a useful figure of merit for determining a minimum level of incident radiation a theoretically lossless sensor needs to detect a signal.
2.6.5.2
RI = I/P
h
A W
i
(2.48)
RV = V /P
h
V W
i
(2.49)
Noise equivalent power
If responsivity may be thought of as a minimum amount of incident radiation a lossless detector needs to detect a signal, then the noise equivalent power (NEP) may be thought of as a figure of merit for the minimum amount of incident radiation a lossy detector needs to detect a signal [41]. This lossiness is the sum total of all noise sources present, including 1/f noise, Johnson noise, and background photon noise. The NEP is defined as the minimum irradiance on the detector area necessary to generate a current or voltage signal in the detector that is equal than the sum of all voltage or current noise sources present in the detector [41]. Two versions of the equation are provided as follows
37
CHAPTER 2. BACKGROUND for voltage and current signals.
N EP = VN /RV
(2.50)
N EP = IN /RI
(2.51)
where VN and IN are the levels of voltage and current noise respectively, and RV and RI are the voltage and current signal responsivity respectively.
2.6.5.3
Detectivity
Noise equivalent power is a good figure of merit for determining how any spot detector will behave. The problem is that a detector’s performance can fluctuate with different variables such as its surface area AS and the bandwidth BW of the incoming signal [41]. Detectivity is a useful figure of merit to resolve these considerations. It is defined in Equation 2.52 as follows [41]: √ AS BW D = N EP ∗
2.6.6
h √
m Hz W
i
(2.52)
Signal conditioning The output signal from the detector consists of two main components, a signal
corresponding to the absorbed gas and a noise component.
To mitigate the noise
component, the signal of interest can be modulated with a chopper into a square wave of some fixed frequency f1 . Assume that the average of the noise component is constant over long periods of time and is not modulated at f1 . In such cases, a device that can lock into signal components at f1 can primarily measure only the signal of interest while the averaging out the noise to zero over a long integration time. This is the principle operation of a lock-in amplifier as shown in Figure 2.15 [42]. The lock-in amplifier consists of a reference signal f2 equal to f1 , a phase shifter that locks into the same phase as the 38
CHAPTER 2. BACKGROUND
Figure 2.15: Block diagram of lock-in amplifier chopper, a low pass filter, and a DC amplifier to amplify the signal of interest to increase detector circuit’s sensitivity. After the signal is passed through the DC amplifier, an analog-to-digital converter (ADC) is used to convert the analog signal into a digital signal that can read by a microcontroller for data acquisition and further analysis. Locking in the frequency of the detector output to the same frequency as a modulated radiation source signal is known as the phase locked loop (PLL). To show the effectiveness of the PLL in radiation detection, let the input signal, I(t) from across the entire electromagnetic spectrum be expressed as shown in Equation 2.53. ˆ I(t) =
∞
A(λ, t) sin 0
2π t + φ(λ, t) dλ λ
(2.53)
where A(λ, t) is the amplitude of the signal at wavelength λ and time t. Next, assume that I(t) is multiplied by a square wave pulse train of frequency f1 and phase φ1 . Further, assume that the output signal also contains a noise component, N (t). An expression for such a function is provided in Equation 2.54. The terms expressed in the equation are derived from taking the Fourier transform of a square wave pulse train [42].
Vin (t) =
∞ 4I(t) X 1 sin ((2n + 1)(2πf1 t + φ1 )) + N (t) π n=0 2n + 1
(2.54)
If the reference reference signal R(t) at the phase locked loop at the detector is also a square wave pulse train of frequency f2 , then R(t) can be expressed as follows:
R(t) =
∞ 4B X 1 sin ((2n + 1)(2πf2 t + φ2 )) π n=0 2n + 1
39
(2.55)
CHAPTER 2. BACKGROUND where B is the amplitude of R(t). The final product Vout (t) is shown in Equation 2.56. 2 ∞ 1 16I(t)B X 1 cos (2n + 1) (2πf1 t + φ1 ) Vout (t) = Vin (t)R(t) = π2 2 n=0 2n + 1 "
−(2n + 1) (2πf2 t − φ2 ) +N (t)
∞ X
− cos (2n + 1) (2πf1 t + φ1 ) + (2n + 1) (2πf2 t − φ2 )
1 sin (2n + 1)(2πf2 t + φ2 ) n=0 2n + 1
#
(2.56)
Assuming now that f1 = f2 then Equation 2.56 can simplified to be 2 ∞ 1 16I(t)B X 1 cos (2n + 1)(φ − φ ) − Vout (t) = 1 2 π2 2 n=0 2n + 1 "
#
cos (2n + 1)(4πf t + φ1 + φ2 ) 1 sin (2n + 1)(2πf t + φ2 ) 2n + 1 # 2 " ∞ 16I(t)B X 1 1 1 = S1 + S2 + N (t) S3 2 π 2 2n + 1 n=0 2n + 1
+N (t)
(2.57)
where,
S1 = cos (2n + 1)(φ1 − φ2 )
S2 = cos (2n + 1)(4πf t + φ1 + φ2 )
S3 = sin (2n + 1)(2πf t + φ2 ) Through observation of S1 through S3 , only S1 does not contain any frequency component f and is thus not dependant on time. Further, if φ1 = φ2 then S1 will be at its maximum value. Further, notice that if the integration time t of the Vout (t) signal approaches infinity then S2 and S3 approach zero and Equation 2.57 can be further simplified to Equation 2.58. This result proves that most noise N (t) can be removed over long integration times and only the DC component, S1 is left [42]. 2 ∞ 16I(t)B X 1 1 cos (2n + 1)(φ1 − φ2 ) hVout (t)|t→∞ i ≈ π2 2 n=0 2n + 1 "
40
#
(2.58)
Chapter 3 Spectrometer Design “Scratch your own itch.” – Business Proverb Multiple aspects regarding the design of an infrared spectrometer are discussed in this chapter. They are presented as follows: • Section 3.1 introduces the reader to a systems overview of the entire spectrometer and how all components fit together to work as a single unit. • Section 3.2 discusses why C2 H2 was chosen as the key gas for which to design the lens around. • Section 3.3 justifies the selection of the radiation source used in the design. • Section 3.4 discusses the design of the gas sample chamber. • Section 3.5 presents the design details of the Fresnel Zone Plate. • Section 3.6 shows how radiation is detected in both infrared and visible light experiments. • Section 3.7 summarizes the major conclusions from the spectrometer design.
3.1
Systems Overview In this thesis, a spectrometer designed for measuring a concentration of at least
35ppm of C2 H2 gas is presented. The spectrometer consists of a radiation source, a 41
CHAPTER 3. SPECTROMETER DESIGN sample chamber, a wavelength selection mechanism, and a detector. A block diagram is shown in Figure 3.1.
Figure 3.1: Systems Block Diagram The radiation source must produce sufficient radiation in at least one of the C2 H2 absorption bands such that the absorbed C2 H2 can produce a change in signal that can be measured by the detector. To increase the sensitivity of the spectrometer, the signal from the radiation source is modulated with a mechanical chopper. Therefore, all gases that are to be absorbed by the incoming radiation will be modulated at the same frequency of the chopper. The chopper is used to minimize 1/f noise (or pink noise), as well as other background noise outside of the chopper range. Radiation entering the gas chamber will be absorbed by the gases across its length, l. Following absorption, there will be a differential in the original input signal to the new signal that absorbed the gases. To be able to determine which bands were absorbed across the whole IR spectrum, a MEMS Binary Fresnel Zone Plate is placed after the gas chamber. This zone plate increases the spectral resolution of the output signal. By measuring the output signal at a predefined distance from the lens, dap−lens the absorption in a spectral range corresponding to the absorption band of any gas can be determined. To increase spectral resolution, an aperture is inserted before the detector at the expense of reducing the signal intensity. Lastly, the absorbed radiation is detected using a thermopile. A lock-in amplifier is used in sync with the chopper to latch onto the modulated signal frequency after the detector. As a result, the influence of external noise sources of differing frequencies interfere with the absorbed signal will be reduced by the detector. The final output signal from the lock-in amplifier is then further converted to a gas concentration in software by converting the output from the amplifier through an ADC to any microcontroller. A more detailed block diagram of the spectrometer design is shown in Figure 3.2.
42
CHAPTER 3. SPECTROMETER DESIGN
Figure 3.2: Spectrometer Overview
3.2
Fault Gas Selection Recall that the goal of this thesis is to determine the concentration of incipient
fault gases in oil-impregnated high voltage equipment. Ideally, this thesis would measure multiple gases in a test environment but due to project scope constraints, its initial design purpose is to measure one gas only. As will be shown later, a design wavelength for the Zone Plate must be chosen. Therefore, an experimental setup designed for measuring C2 H2 gas will be constructed in this thesis. There are a few reasons for choosing C2 H2 over other gases. First, the minimum required concentration detection ability of any fault gas in order to take preventative maintenance measures on the equipment is for C2 H2 gas [2]. The existence of minute amounts of acetylene (C2 H2 ) gas, 35ppm, is cause for concern and its ratio to other gases allows one to determine most fault types [2]. By being able to measure such a small concentration of one of these fault gases, then the proof of concept of the Zone Plate operation for measuring the minimum required concentrations of other fault gases may also be applied. Second, C2 H2 is convenient to measure due to its spectral absorption profile shown in Figure 3.3 [21]. As will be shown in section 3.6, infrared detectors sensitive to one of its absorption bands in the 3µm range are available on the market. This band is also spectrally close to the spectral bands of many other key gases, which is helpful for future development. Luckily, they are also not so close that they can cause excessive 43
CHAPTER 3. SPECTROMETER DESIGN
Line intensity S( λ) [(cm 2/mol)/nm]
1.2
×10-18
1
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Wavelength λ [nm]
1.8
2 ×104
Figure 3.3: C2 H2 absorption spectrum [21] cross-interference. Infrared radiation sources heated to a temperature of about 1300◦C are available on the market which have strong irradiance in the 3µm wavelength range. To add, C2 H2 also has larger absorption coefficients than its closest spectral neighbor, methane. All of these reasons combined made C2 H2 a convenient choice for choosing the design parameters of the lens.
3.2.1
C2 H2 Absorption To determine how much radiation can be absorbed by C2 H2 , its absorption profile
must be known.
The spectral line intensity profile of C2 H2 consists of three main
absorption bands. The bands with the highest absorptions are from approximately 2.97µm to 3.13µm, 7.07µm to 7.98µm, and 12.29µm to 15.25µm. The average of and the total of the absorption coefficients in these bands is summarized in Table 3.1. Numerous assumptions were made when calculating these absorptions based on the resolution of each spectral line when calculating the area under them. A detailed error analysis of such calculations is presented in Appendix F. In order for some of the gas to be absorbed in some spectral range ∆λ, two conditions must be met: 1. The light source must be able to emit sufficient radiation in ∆λ. 44
CHAPTER 3. SPECTROMETER DESIGN Table 3.1: Sum absorption coefficients for C2 H2 spectral bands at 297K, 101.3kPa Spectral range Absorption coefficient [cm2 /mol] Wavelength [µm] Wavenumber [cm−1 ] Total Average per nm 2.97 - 3.13 7.07 - 7.98 12.29 - 15.25
1.04·10−17 3.19·10−18 2.81·10−17
3199 - 3370 1253 - 1415 656 - 808
6.5·10−20 3.5·10−21 9.5·10−21
2. The detector must be sufficiently sensitive to radiation in ∆λ. Therefore, all future design work for the light source selection, detector selection, and lens design depends upon first selecting one of these spectral bands as a design constraint.
3.2.2
Band Selection Of the three main C2 H2 absorption bands, it was decided that the rest of the
project design will be focused on the absorption in the 2.97 - 3.13µm band. Justification for this decision is summarized below. 1. Many commercial infrared spectrometers utilize a blackbody source such as a heated filament as a radiation source [22]. In this thesis, a heated filament will also be used. From Planck’s Law, it can be shown that the temperature of the filament is inversely proportional to the wavelength of the peak source intensity. In commercial blackbody radiators, it is easy to build or find filaments that go up to 1000◦C and higher. Recall from Figure 2.5 that close to 1000◦C, the maximum intensity of blackbody radiation is closest to the 3µm spectral band compared with other bands. Therefore, of all major C2 H2 spectral bands, the 2.97 - 3.13µm band is the most practical and optimal in terms of radiation intensity from common sources. 2. All spectrometers require having some minimum spectral resolution in order to detect individual spectral lines. Therefore, to increase the spectrometer’s detection ability, a spectral band with the highest absorption density per nm should be chosen. Table 3.1 shows that the 2.97 to 3.13µm band best satisfies this requirement. 3. There are more readily available infrared detectors on the market more sensitive to wavelengths shorter than or equal to 3µm than there are for longer wavelengths.
45
CHAPTER 3. SPECTROMETER DESIGN ×10-19
Line intensity S( ν) [(mol/cm 2)/cm-1]
9 8 7 6 5 4 3 2 1 0 3302
3304
3306
3308
3310
3312
3314
3316
Wavenumber ν [cm -1]
Figure 3.4: Sample C2 H2 spectral lines
3.2.3
Calculation of Absorption Coefficients
To determine the absorption coefficient across these spectral ranges, a Matlab simulation was conducted to determine the line profile of each spectral line S(¯ ν ) such as those shown in Figure 3.4. The line profile g(¯ ν ) was calculated using Equations 2.5 through 2.12. These equations assumed a constant temperature of 297K at standard atmospheric pressure of 101.5kPa. The total absorption is expressed in Equation 3.1. ˆ
ν¯2
A(gas, ν¯i ) =
S(gas, ν¯i )g(¯ ν , ν¯i )dν¯i ν¯1
h
cm2 mol
i
(3.1)
The total absorption from all spectral lines in the three ranges where C2 H2 absorbs the most radiation is shown in Table 3.1. The results are provided in wavenumber units as opposed to wavelength due to the simplicity in calculating through Equations 2.5 to 2.12. The Matlab code for producing these plots is shown in section C.3 of Appendix C.
3.3
Radiation Source The radiation source chosen for this project is the Hawkeye IR-Si217 as shown
in Figure 3.5. This source emits radiation from a 6mm x 4.4mm carbide surface at
46
CHAPTER 3. SPECTROMETER DESIGN a temperature of 1385◦C and emissivity of 80 percent [43]. The source includes a 1” parabolic reflector behind it to collimate the radiation with an optical gain of up to 14x [43].
Figure 3.5: Hawkeye IR-Si217 radiator with parabolic reflector Given that the filament emits blackbody radiation notwithstanding emissivity, the total energy radiated, called radiance, that may be absorbed by C2 H2 between 2.97µm and 3.12µm is calculated as being the area under the blackbody curve in that range as shown in Equation 3.2. A graph of this curve is shown in Figure 3.6 for a filament temperature of 1385◦ C. ˆ
3.12µm
LC2 H2 (λ, T ) =
2hc2
2.97µm
hc
dλ
λ5 e λkT − 1
= 4196.5
h
W sr·m2
i
(3.2)
The emitted radiation power, Prad , from the filament in 2.97 − 3.12µm may be calculated by multiplying the radiance, the surface area of the filament, and its emissivity ε [43]. This is shown below in Equation 3.3.
Prad = LC2 H2 AT ε = 4196.5 · 4.4 · 6 · 10−6 · 0.8 = 89
[mW] (3.3)
The above calculation assumes that the effective radiation that strikes the zone plate is a plane wave thus simplifying the units from W/m2 sr to W/m2 . In reality this is not the case but it is a convenient approximation for further calculations regarding the output radiation pattern of the zone plate. Dividing the total radiation of 89mW by the design spectral range of 160nm, the average spectral flux φ is shown in Equation 3.4. This is a maximum theoretical flux assuming the fully collimated plane wave emanating from the 47
CHAPTER 3. SPECTROMETER DESIGN
Spectral emissive power [W/(sr*m3 )]
6
× 10 10 T = 1385 °C
5
4
3
2
1
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Wavelength [nm]
Figure 3.6: Hawkeye IR-Si217 blackbody radiation at 1385◦C excluding emissivity and other losses source falls incident on the zone plate with no losses.
φ=
3.4
89 = 0.556 160
h
mW nm
i
(3.4)
Sample Chamber The purpose of the sample chamber is to provide a physical medium where C2 H2
can be absorbed. The design of chamber must take the following into consideration: 1. Length: The chamber must be long enough such that there is sufficient absorption of 35ppm of C2 H2 . It must also not be too long in order to minimize radiation losses within it. Such losses arise from the radiation beam’s directivity when falling incident on the inner chamber walls. 2. Transmission window: The entrance and exit transmission windows where radiation Io enters and radiation I leaves the chamber must be sealed properly so that the gas is not a safety hazard to the environment outside the chamber. They must also be made of a material of appropriate thickness that has high transmittance in the 2.97 − 3.13µm range. 48
CHAPTER 3. SPECTROMETER DESIGN 3. Gas inlets and outlets: The gas chamber should be equipped with three valves. There should be two intake valves, one for C2 H2 and one for N2 (serving as the carrier gas), and one exhaust valve to release all gases from the chamber. An optimal length for the chamber can be calculated from Beer’s Law shown below: I = e−lac Io It is important to determine at which length the absorbance drops to such a low level that the gas will almost completely absorb the incident radiation. To begin, 35ppm of C2 H2 must be converted to workable units of mol/m3 . This concentration, cg (C2 H2 ) is calculated below in Equation 3.5.
cg (C2 H2 ) =
ppm · NA · ρ 35 6.022 · 1023 = · 1.1 · 106 M 1000000 26.04 h
= 8.9035 · 1023
mol m3
i
(3.5)
where NA is Avogadro’s number, M is the molar mass of C2 H2 and ρ is the density of C2 H2 . From Table 3.1, the related average absorption coefficient a per nm of C2 H2 in the 2.97 - 3.13µm range is shown below in units of m2 /mol. " −20
a = 6.5 · 10
#
"
m2 cm2 = 6.5 · 10−24 mol mol
#
(3.6)
Finally, a plot of the absorbance as a function of l from 1cm to 15cm assuming a and c provided above is shown in Figure 3.7. It is evident that a chamber of length 10cm is long enough to absorb most radiation. A photo of the sample chamber setup is shown in Figure 3.8. The entrance and exit windows are made of two double-sided polished (DSP) 300µm thick 3” diameter silicon wafers as shown in Figure 3.9. Silicon is chosen as the material due to its high transmission around the 3µm spectral range and beyond. The DSP wafers were epoxied on to two chamber end caps with 1” holes in the center.
49
CHAPTER 3. SPECTROMETER DESIGN
0.55 0.5
Absorbance A [-]
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
Length [m]
Figure 3.7: Gas Absorbance as a function of chamber length
Figure 3.8: Gas chamber
3.5
Figure 3.9: Gas chamber windows
Fresnel Zone Plate A zone plate is a lens that focuses specific wavelengths at corresponding focii. The
purpose of the zone plate design is to provide maximum constructive interference in the wavelength range where C2 H2 absorbs the most radiation at a particular focal length. Therefore, the design wavelength λd of the lens is chosen to be at 3.045µm, the average between 2.97 and 3.12µm. The zone plate consists of an N number of open slits separated by a consecutive series of narrowing distances and slit widths. An example of a circular zone plate lens is shown in Figure 3.10.
50
CHAPTER 3. SPECTROMETER DESIGN
×10-3 3
Distance in y axis [m]
2
1
0
-1
-2
-3 -3
-2
-1
0
1
Distance in x axis [m] Figure 3.10: Sample Fresnel Zone Plate
51
2
3 ×10-3
CHAPTER 3. SPECTROMETER DESIGN 1 0.9 0.8
Intensity [W/m2]
0.7 0.6 0.5
FWHM
0.4 0.3 0.2 0.1 0 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Theta [rad]
Figure 3.11: Full Width at Half Maximum
3.5.1
Lens Design Criteria Design criteria for measuring the effectiveness of the lens must be defined and
quantified. Therefore, a number of parameters are introduced to analyze its effectiveness.
3.5.1.1
Full Width at Half Maximum (FWHM)
The full width at half maximum (FWHM) is defined as the bandwidth of the main lobe of a signal whose instantaneous power at any angle in that band is greater than or equal to the half of the maximum power of the main lobe. It will be shown later that a narrower spatial FWHM at the aperture increases the temporal coherence of the lens for the design wavelength at the aperture. It may also be used in the spectral domain to determine the spectral resolution of the zone plate. An example of the FWHM for a sample spatial signal in the focal plane at a particular wavelength is shown in Figure 3.11.
3.5.1.2
Aperture Restricted Effective Spectral Radiosity (ARESR)
The aperture restricted effective spectral radiosity (ARESR), denoted ηap , is defined as the ratio of the spectral irradiance at the aperture to the total spectral irradiance in the aperture plane.
It is useful for quantifying how well the lens
constructively interferes at some distance from the lens for some wavelength. Let θ 52
CHAPTER 3. SPECTROMETER DESIGN 1 0.9 0.8
Intensity [W/m2]
0.7 0.6 0.5
Δθ 0.4 0.3 0.2 0.1 0 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Theta [rad]
Figure 3.12: ARESR be an angle whose tangent is equal to the ratio of the aperture radius (in the aperture plane) to the line segment on the optical axis spanning from the aperture plane to some point at distance d away from it. Then ηap can be written as shown in Equation 3.7. A plot is shown in Figure 3.12 where the area under the yellow portion of the plot is the spectral power that passes through the aperture. ˆ
ˆ
1 ∆θ 2
2
Io (λo , θ)dθ ˆ
Io (λo , θ)dθ = ˆ
0
ηap (λo , ∆θ) =
0
π/2
2
Io (λo , θ)dθ 0
3.5.1.3
1 ∆θ 2
(3.7)
π/2
Io (λo , θ)dθ 0
Transmission Efficiency
The transmission efficiency of the lens is defined as the percentage of radiation that passes through the lens to the total radiation incident on the lens. It is useful for determining the minimum photon energy that must enter the lens to be able to detect it at the detector. The transmission efficiency for N zones is shown in Equation 3.8. i=N X
ai P i=1 ηP = = i=N +1 X Po di i=1
53
(3.8)
CHAPTER 3. SPECTROMETER DESIGN where P is the output power, Po is the input power, ai is the width of the slit of the ith zone and di is the distance between the (i − 1)th and ith zone. 3.5.1.4
Spectral Resolution
Spectral resolution, denoted R, is a figure of merit for determining how well some wavelength can be resolved. It is shown below in Equation 3.9 where λo is the center wavelength and ∆λ is the smallest wavelength range that can be distinguished1 . For simplicity and consistency one can choose to define ∆λ as being the FWHM of the spectral lobe centered at λo .
R=
3.5.2
Lens Design Overview
3.5.2.1
Assumptions
λo ∆λ
(3.9)
Before proceeding with showing all design details of the zone plate lens, some assumptions regarding the nature of the experiment are listed below. 1. The radiation incident upon the lens is a perfect plane wave that falls perpendicular to the zone plate surface. It is spatially coherent wavefront with zero phase delay between all zones. 2. The input radiation on the lens is coherent, quasi-monochromatic light. 3. All optical components are perfectly aligned. 4. All simulations are simulated in the Fraunhofer zone. 5. All opaque regions have zero transmission and all open regions have perfect transmission. 6. All zone plate edges are perpendicular to the zone plate surface. 7. There is no scattering from the backface of the lens. 1
There is no precise definition for determining whether a certain spectral feature is resolvable or not.
54
CHAPTER 3. SPECTROMETER DESIGN It should be mentioned that not all assumptions will be accurate in reality. Obvious examples include having perfect alignment between all components and perfect spatial and temporal coherence between all electromagnetic (EM) fields at every aperture. While perfect coherence is not possible, having a large value for the normalized autocorrelation function γ between most EM fields is possible. Mitigating coherence issues, along with achieving precise alignment, can be accomplished to a sufficient degree given a careful experimental setup.
3.5.2.2
Design Basics
When radiation of intensity Io and wavelength λo passes through a linear slit of width a, its intensity I at the output of the linear slit as a function of angle θ from the center axis of the slit is as follows [33]:
2
I(θ) = Io sinc
πa sin θ λo
!
= Io sinc2 β
(3.10)
If the slit is an annular ring as shown in Figure 3.10, then the intensity is given as [34]: 4Io I(θ) = (1 − 2 )2
J1 (β) − J1 (β) β
!2
(3.11)
where is the ratio of the inner ring radius to the outer ring radius of some slit and J1 is a first order Bessel function [34]. It was shown that using various error measurement methods that the error between modeling the intensity profiles from both annular and linear slits was less than 2%. A detailed error analysis is presented in Appendices B and C.2. This finding is also very intuitive after looking at Figure 3.10. One can see that the outer slits can be treated as a series of numerous parallel lines when integrated over short distances. For convenience, it was therefore deemed reasonable that in this thesis all lens radiation simulations designed for annular slits can be modeled as linear slits. A schematic of a binary zone plate is shown below in Figure 3.13. To achieve constructive interference from all zones for wavelength λd at focal length f , the distance di between each consecutive slit is given below. 55
CHAPTER 3. SPECTROMETER DESIGN
Figure 3.13: Ray tracing from slit to point in far field
di =
q l1 2
− f 2,
q li 2
− f2 −
for i = 1 Pk=i−1 k=1
dk , for i > 1
where:
li = f + iλ The angle θi corresponding to the ith slit is defined as the angle that corresponds to the tangent of di sums over f . Angle θi is provided as: Pk=i
θi = arctan
k=1
dk
!
f
For constructive interference to occur at P for λd , the relative amplitude and phase of the intensity I at P of the electric fields from all zones should be the same. Typically, this relative amplitude corresponds to the angle of the first mode of the signal. However, maximum constructive interference may also occur in any region between θ > 0 rad and the smallest angle whose radiation profile provides the same amplitude as that of the first 56
CHAPTER 3. SPECTROMETER DESIGN 1
Intensity I [W/m2]
0.8
0.6
0.4
0.2
0 -8
-6
-4
-2
0
2
4
6
8
Theta [rad]
Figure 3.14: Possible angles for maximum constructive interference mode. An example of such an angular region is shown in yellow in Figure 3.14 for the example function sinc(3θ). There are at least five variables to consider when designing the zone plate lens. They are as follows: 1. Design Wavelength, λd 2. Design Focal Length, fd 3. Widths of all i slits, ai 4. Diameter of the aperture in the focal plane, φap 5. Number of zones, N All variables are chosen so that the lens is optimized for optical absorption, transmission efficiency, ARESR, and spectral resolution. The main issue with trying to optimize these elements via simulation is that all design variables are interdependent upon one another.
As a result, semi-arbitrary values are chosen for the remaining
design parameters as shown in Table 3.2. Justifications for choosing these values will be presented throughout the remainder of section 3.5. Four different lenses are made in thesis. Two of which are designed to constructively interfere in the visible range at 545nm, and two to constructively interfere at 3.045µm. The reason for making two lenses in the visible range is to be able to test the proof of concept of the lens using a commercial off-the-shelf (COTS) spectrometer, the CCS175 spectrometer from Thor
57
CHAPTER 3. SPECTROMETER DESIGN
Lens 1 2 3 4
Table 3.2: Lens Design Parameters λd fd N φap aN /dN 3.045µm 50.8mm 400 100µm 1 3.045µm 25.4mm 400 100µm 1 540nm 50.8mm 137 100µm 1 540nm 25.4mm 68 100µm 1
Labs. Unfortunately this option was not immediately available for the C2 H2 IR spectral ranges.
3.5.2.3
Matlab Simulations
To shown how the intensity distribution in the aperture plane changes as a function of these variables, all Matlab simulations for the remainder of this chapter will be made using the parameters of Lens 1 as shown in Table 3.2. Simulations for the other lens can also be shown but only Lens 1 is simulated for demonstration of proof of concept. Further simulations would take up too much room without demonstrating anything particularly unique.
3.5.3
Calculation of slit widths The design considerations in determining all slit widths are as follows:
1. All slit widths must be realistic to manufacture using conventional MEMS fabrication techniques with no features smaller than 5µm. 2. Transmission efficiency should be maximized as much as possible. 3. The ARESR for the design wavelength at the focal point should be maximized. It can be observed that as θi increases, slit width ai must decrease in order to maintain the same relative phase of I at the focal length in the focal axis. For the phase of I at the aperture center to remain constant from every zone i, βi must be chosen in such a way that fulfills the relative phase requirement for every zone. A design choice is now made with respect to the maximum value of θi . Let it be required that the value of any θi will be small enough that one can apply the 58
CHAPTER 3. SPECTROMETER DESIGN small angle approximation when calculating sin θi . This therefore implies that sin θi ≈ θi for 1 ≤ i ≤ N , where N is the number of zones. The first step is to determine a1 (See Figure 3.13). Following the calculation of a1 , all consecutive ai ’s are calculated by assuming ai is inversely proportional to the radius of the lens at the ith zone. The method is as follows: Given that the phase of I should be equal for all zones, this implies from the linear slit fraunhofer diffraction equations that:
βi = β1 =
πa1 sin θ1 λd
for 1 ≤ i ≤ N
(3.12)
Furthermore, since θi is a changing variable and by using the small angle approximation that sin θ ≈ tan θ, it may be further written: k=i X
d1 πa1 πai k=1 πai tan θi f f βi = = = λd λd λd
k=i+1 X
dk πai+1 =
k=N X
dk
k=1
f λd
πaN = ... =
dk
k=1
f λd
(3.13)
Through mathematical manipulation, the following relation can be obtained:
ai
k=i X
dk = ai+1
k=1
k=i+1 X
dk
(3.14)
k=1 k=i X
dk ai+1 k=1 = k=i+1 X ai dk
(3.15)
k=1
Given that di is known for every i, it is apparent that one must first calculate a1 in order to determine all other ai ’s. 3.5.3.1
Calculation of a1
In consideration of the design requirements mentioned above, a1 is determined by considering two aspects of the lens: The transmission efficiency of the lens, and the ARESR of the lens at the focal length and design wavelength. The methodology for 59
CHAPTER 3. SPECTROMETER DESIGN 0.7 ARESR Transmission Efficiency
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
a(1) [um]
Figure 3.15: ARESR and transmission efficiency as a function of a1 determining an appropriate a1 is as follows: 1. Choose predetermined values for the focal length, design wavelength, and the number of zones of the lens. 2. Calculate all values for di . 3. Choose a predetermined value for a1 . 4. Calculate all values of ai based on a1 . 5. Using these design parameters, simulate the radiation intensity distribution in the focal plane of the lens as a function of angle. 6. Calculate the ARESR of the lens. 7. Calculate the transmission efficiency of the lens. 8. Increase the value of a1 and repeat steps 4 through 7. 9. Plot the ARESR and the transmission efficiency of the lens as a function of a1 . For Lens 1 shown in Table 3.2, the ARESR and transmission efficiency as a function of a1 are simulated as shown in Figure 3.15. It is apparent that there exists a optimal value of a1 = 70µm to maximize the ARESR. It is also shown that the transmission efficiency is linearly proportional to a1 . Since di is constant, this implies from Equation 3.15 that the ratio of ai to di is also linearly proportional to ai .
60
CHAPTER 3. SPECTROMETER DESIGN From a manufacturing standpoint, it makes sense for the ratio of open space to opaque space at any zone to be equal to one. This will maximize the size of the smallest features for an equivalent number of zones, thus optimizing the MEMS fabrication process of the lens. For this to occur, the lens should be designed such that 2a(N ) ≈ d(N ). In this thesis, the minimum value of a(N ) was chosen to be no less than 5µm and 2a(N ) ≈ d(N ). In the event that the designer is more interested in optimizing the ARESR, they should choose a1 = 70µm.
3.5.4
Calculation of aperture diameter To achieve high spectral resolution, an aperture before the detector is needed.
A 100µm aperture from Thor Labs was chosen for the experiments and remaining simulations. The criteria for deciding upon which aperture to use is based on apertures available on the market as well as simulation results.
3.5.4.1
Performance results of changing aperture size
One of the important features in quantifying the effectiveness of a spectrometer is its ability to maximize spectral resolution. In this thesis, ARESR and spectral resolution are used to judge the influence of the aperture. The ARESR as a function of aperture diameter for Lens 1 at the design wavelength is shown in Figure 3.16. The spectral ARESR of the lens as a function of wavelength is shown in Figure 3.17. It is apparent that the spectral resolution is inversely proportional to the aperture diameter. However, the transmission efficiency of the aperture is directly proportional to its diameter as well. Therefore, the designer must make an appropriate tradeoff between the required spectral resolution and an optimal power output from the aperture to the detector.
61
CHAPTER 3. SPECTROMETER DESIGN
0.44
0.42
ηap [-]
0.4
0.38
0.36
0.34
0.32
0.3 0
50
100
150
200
250
300
350
400
Aperture diameter [um]
Figure 3.16: ARESR as a function of φap
Aperture diameter = 20um
0.4
ηap [-]
ηap [-]
0.3 0.2 0.1 0 0
2000
4000
6000
8000
10000
Aperture diameter = 60um
0.6 0.4 0.2 0 0
12000
2000
Wavelength [nm]
Aperture diameter = 200um
0.6
0.4 0.2 0 0
2000
4000
6000
8000
6000
8000
10000
12000
Wavelength [nm]
ηap [-]
ηap [-]
0.6
4000
10000
0.4 0.2 0 0
12000
Wavelength [nm]
Aperture diameter = 400um
2000
4000
6000
8000
10000
Wavelength [nm]
Figure 3.17: Spectral plot of ARESR as a function of aperture diameter
62
12000
CHAPTER 3. SPECTROMETER DESIGN
3.5.5
Calculation of focal length To choose a focal length for the lens, two aspects are considered. One is choosing a
length that is practical to work with, and the other is choosing a focal length that allows for a high spectral resolution. For simplicity in testing, the focal length is chosen to be one of a multiple integer of 1” or 25mm. This corresponds directly to the spacing between holes in imperial and metric breadboards respectively. As a result, Lens’ 1 and 3 are 25.4mm and Lens’ 2 and 4 are 50.8mm. Any focal length beyond 25mm also operates in the Fraunhofer region so there is no need to worry about near field effects with regards to the expected radiation profiles. The remainder of this section shows how the performance of the lens is affected as a function of focal length. To begin, Figure 3.18 shows that as the focal length increases, the ARESR of the lens decreases exponentially for a constant aperture size. Therefor, as far as maximizing ARESR is concerned, having a larger focal length is not a good thing. One way to make up for a lower ARESR is to increase the power output of the light source. However, by noting Figure 3.19, the angular FWHM of the main lobe at the design wavelength also decreases exponentially at the same rate. The spatial FWHM as a discrete length is shown in Figure 3.20. The radiation profiles in the spatial plane as a function of angle and distance are shown in Figures 3.21 and 3.22. Figure 3.20 shows that the spatial FWHM at 200mm is still well within 100µm diameter aperture so it was a safe design choice to choose a focal length anywhere between 0mm to 200mm. A lower angular FWHM implies that the focusing properties at a particular wavelength also improves. This translates into better spectral resolution as shown in Figure 3.24. Increasing the focal length is also advantageous in increasing the power density of the main lobe as shown in Figure 3.23. In summary, it is best to maximize the focal length of the lens such that sufficient power can reach the detector, while being far away enough to achieve an acceptable level of spectral resolution. During design and testing, it is convenient to choose the focal length such that the optical components may be easily mounted on an optical breadboard.
63
CHAPTER 3. SPECTROMETER DESIGN
0.9
0.8
ηap [-]
0.7
0.6
0.5
0.4
0.3 0
20
40
60
80
100
120
140
160
180
200
160
180
200
Focal length [mm]
Figure 3.18: ARESR as a function of f
5
×10-4
4.5 4
FWHM [rad]
3.5 3 2.5 2 1.5 1 0.5 0
20
40
60
80
100
120
140
Focal length [mm]
Figure 3.19: Spatial FWHM angle as a function of f
64
CHAPTER 3. SPECTROMETER DESIGN
×10-6
16 14
10 8 6 4 2 0
20
40
60
80
100
120
140
160
180
200
Focal length [mm]
Figure 3.20: Spatial FWHM as a function of f
Focal Length = 10mm
0.15
I [W/(m2nm)]
I [W/(m2nm)]
0.06 0.04 0.02 0 -0.01
-0.005
0
0.005
0.05 0 -0.01
0.01
Focal Length = 80mm
1
0.3 0.2 0.1 0 -0.01
-0.005
0
0.005
-0.005
0
0.005
0.01
Angle in focal plane [rad]
I [W/(m2nm)]
0.4
Focal Length = 20mm
0.1
Angle in focal plane [rad]
I [W/(m2nm)]
FWHM [m]
12
0.01
Angle in focal plane [rad]
Focal Length = 200mm
0.5
0 -0.01
-0.005
0
0.005
0.01
Angle in focal plane [rad]
Figure 3.21: Intensity profile in aperture plane as a function of angle for different f
65
CHAPTER 3. SPECTROMETER DESIGN
Focal Length = 10mm
0.04 0.02 0 -3
-2
-1
0
1
I [W/(m2nm)]
I [W/(m2nm)]
0.1 0
1
×10
0
1
2
×10-4
0.5
0 -3
2
Distance in focal plane [m]
-1
Focal Length = 200mm
1
0.2
-1
-2
Distance in focal plane [m]
Focal Length = 80mm
-2
0.05
×10-4
0.3
0 -3
0.1
0 -3
2
Distance in focal plane [m]
0.4
Focal Length = 20mm
0.15
I [W/(m2nm)]
I [W/(m2nm)]
0.06
-2
-4
-1
0
1
2
×10-4
Distance in focal plane [m]
Figure 3.22: Intensity profile in aperture plane as a function of radius from the center of the aperture for different f
1 0.9
Main lobe I [W/m 2nm]
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
20
40
60
80
100
120
140
160
Focal length [mm]
Figure 3.23: Intensity at center of aperture as a function of f
66
180
200
CHAPTER 3. SPECTROMETER DESIGN f = 10mm
0.8
ηap [-]
ηap [-]
0.6 0.4 0.2 0 0
2000
4000
6000
f = 20mm
0.6
8000
10000
0.4 0.2 0 0
12000
2000
Wavelength [nm]
8000
10000
12000
10000
12000
f = 200mm
0.4 0.3
ηap [-]
0.3
ηap [-]
6000
Wavelength [nm]
f = 80mm
0.4
4000
0.2 0.1
0.2 0.1
0 0
2000
4000
6000
8000
10000
0 0
12000
Wavelength [nm]
2000
4000
6000
8000
Wavelength [nm]
Figure 3.24: Spectral plot of ARESR as a function of f
3.5.6
Calculation of N The zone plate discriminates between wavelengths by maximizing the constructive
interference at specified wavelengths while minimizing the impact from all other wavelengths. This is achieved by summing the interference profiles of multiple slits in the aperture plane. By increasing the number of zones, the “good” and “bad” interferences are intensified, thereby affecting the spatial and spectral resolution of the lens. One advantage of maximizing the number of slits is increasing the total surface area of the lens. By increasing the area, more radiation can strike the surface, thereby possibly producing more intensity. The word possibly is used since the output intensity also depends on the surface area of the radiation plane wave striking the surface. From Figure 3.25, it is apparent that the ARESR is proportional to the number of zones. The ARESR function appears to become somewhat linear after about 50 zones. There is an interesting inflection point at 220 zones where the ARESR increases more dramatically. It is not clear where this phenomena could arise from but it is suspected to be an artifact of the Matlab simulation itself. From Figures 3.26 and 3.27, it is clear that the FWHM of output radiation is inversely proportional, and the main lobe power density is directly proportional, to the number of zones. As shown, both relations are 67
CHAPTER 3. SPECTROMETER DESIGN 0.45
0.4
ηap [-]
0.35
0.3
0.25
0.2
0.15 0
50
100
150
200
250
300
350
400
Number of zones N [-]
Figure 3.25: ARESR as a function of N exponential as a function of N . Figures 3.28 and 3.29 show that as N increases, the spatial and spectral resolutions also increase. 4.5
×10-5
4
FWHM [m]
3.5 3 2.5 2 1.5 1 0.5 0
50
100
150
200
250
300
350
400
Number of zones N [-]
Figure 3.26: Spatial FWHM as a function of N In summary, it is good to maximize the number of zones in the lens in order to optimize all discussed lens parameters such as ARESR, spectral resolution, and total transmission. The only disadvantages in some circumstances are that of fabrication constraints for the smallest zones, as well as physical real estate issues in the case of the user wanting to miniaturize the lens as much as possible. 68
CHAPTER 3. SPECTROMETER DESIGN
0.3
Main lobe I [W/m 2nm]
0.25
0.2
0.15
0.1
0.05
0 0
50
100
150
200
250
300
350
400
Number of zones N [-]
Figure 3.27: Intensity at center of aperture as a function of N
×10-5
N = 10 zones I [W/(m2nm)]
4
6
2
I [W/(m nm)]
8
4 2 0 -1
-0.5
0
0.5
Distance in focal plane [m]
×10-3
-0.5
I [W/(m2nm)]
2 0.5
Distance in focal plane [m]
1
×10
0
0.5
-3
1
×10-3
N = 400 zones
0.3
4
0
1
Distance in focal plane [m]
N = 80 zones
-0.5
2
×10-3
6
0 -1
N = 20 zones
3
0 -1
1
2
I [W/(m nm)]
8
×10-4
0.2 0.1 0 -1
-0.5
0
0.5
Distance in focal plane [m]
1
×10-3
Figure 3.28: Intensity profile in aperture plane as a function of radius from the center of the aperture for different N
69
CHAPTER 3. SPECTROMETER DESIGN N = 10 zones
0.2 0.1 0 0
2000
4000
6000
8000
N = 20 zones
0.3
ηap [-]
ηap [-]
0.3
10000
0.2 0.1 0 0
12000
2000
Wavelength [nm]
ηap [-]
ηap [-]
0.2 0.1 2000
4000
6000
8000
8000
10000
12000
10000
12000
N = 400 zones
0.6
0.3
0 0
6000
Wavelength [nm]
N = 80 zones
0.4
4000
10000
0.4 0.2 0 0
12000
Wavelength [nm]
2000
4000
6000
8000
Wavelength [nm]
Figure 3.29: Spectral plot of ARESR as a function of N
3.6
Radiation Detection Radiation detection for both the visible and infrared light experiments use different
systems to achieve spectral discrimination as will be shown in the remaining subsections.
3.6.1
Visible Light Experiment For the visible light experiment, the detector is a fiber optic cable that sends
radiation data to a Thor Labs CCS175 spectrometer. A photograph of the fiberoptic probe and aperture is shown in Figure 3.30. This spectrometer has sufficient capabilities to measure the visible light spectrum that lenses 3 and 4 are based on.
3.6.2
Infrared Light Experiment For the infrared light experiment, the radiation detection mechanism is more
complex than for its visible light counterpart. The mechanism also consists of an aperture, but instead of a fiber optic probe, it is followed by a thermopile used in conjunction with a mechanical chopper and lockin amplifier to filter the signal and reduce noise. The CCS175 spectrometer cannot be used again for the infrared experiment due the spectrometer’s 70
CHAPTER 3. SPECTROMETER DESIGN
Figure 3.30: Closeup of aperture and detector for visible light setup limited spectral range from 500 − 1100nm. The advantage of using the lock-in amplifier is to filter out noise internal and external to the thermopile by mostly amplifying signals within a specific frequency band. It also reduces the 1/f noise in the system. The detector circuit is shown in Figure 3.31. Its design is modified from an lock-in amplifier application note in [44] to be applied to measuring a voltage from the thermopile sensor used, the Hammamatsu T11262-01. As shown in Figure 3.31, the radiation from the Hawkeye IR-Si217 source is first mechanically modulated at a frequency of 20Hz using an MC2000 optical chopper from Thor Labs. This relatively low frequency of 20Hz is chosen due to the large rise time of the T11262-01 thermopile. According to [45], the rise time of the T11262-01 is between 20 − 30ms as expressed in Table 3.3. The signal can be modulated at a higher frequency, but at the expense of having a higher DC bias in the signal. Such a bias affects the performance of the AD630 by diminishing its effectiveness in latching on to an equivalent square wave signal provided by the sync output of the MC2000 driving the mechanical chopper. The signal from the thermopile is first passed to a buffer U1 in order to prevent saturating the “true” thermopile voltage. This output voltage is then passed on as a voltage input to an inverting amplifier that outputs a voltage V2 . The gain G of the 71
CHAPTER 3. SPECTROMETER DESIGN
Figure 3.31: Lockin Amplifier Circuit for measuring IR radiation. Image modified from [44]
Table 3.3: Hammamatsu T11262-01 Specifications [45] Photosensitive area 1.2 × 1.2 mm Spectral response range 3 to 5 µm Photosensitivity (typ.) 50 V/W Dark resistance (typ.) 125 kΩ Output noise voltage (typ.) 45 nV/Hz1/2 Noise equivalent power (typ.) 0.9 nW/Hz1/2 Detectivity (typ.) 1.3 x 108 cmHz1/2 /W Rise time (typ.) 20 ms Rise time (max.) 30 ms Temperature coefficient of element resistance (typ.) ±0.1%/◦C Field of View (typ.) 90°
72
CHAPTER 3. SPECTROMETER DESIGN amplifier is given as:
G=
−R1 /R2 V2 (s) = Vin (s) 1 + sR1 C1
(3.16)
where s = jω and ω = 2πf . The amplified thermopile signal is then passed through the AD630 and an inverting integrator op amp that generates a signal V4 at its output. This signal V4 is in sync with the modulated signal carrier at pin 9 of the AD630. The amplitude of V4 is proportional to the amplitude of the frequency component matching the frequency of the sync signal from the thermopile. The integrator op amp U5 is used to increase the gain even more and filter out high frequency components. A Schottky diode is then used to cut out the negative portion of the waveform so that it can be passed to a low pass filter next. The low pass filter consists of R7 and C3 and is finally used in order provide a measurable value for the output of the detector as it relates to the modulated input voltage from the thermopile. One important missing factor missing in this discussion is the fact that no values were provided for the resistors and capacitors. It turned out that the justification for choosing such values largely depended on trial and error through experimentation with different values at different apparatus configurations. As such, a more detailed discussion with results for this is given in section 6.5. The Hammamatsu T11262-01 thermopile was chosen primarily due to its spectral output shown in Figure 3.32 with over 90% transmission in the 2.97 − 3.12µm spectral range [45]. The response time of 20 − 30ms is not considered very fast but sufficient for measurements that have no particular significant time restraints for the application of DGA [45]. Such a short response time may also be considered problematic when it is implemented as an input to a lock-in amplifier in order to reduce external noise. It is considered problematic since the amplitude of the noise components in 1/f noise are inversely proportional to f . When f is low and a high signal sensitivity is required to read a weak signal with strong noise components in the detector and circuit, then it may prove extremely difficult, perhaps impossible, to read that weak signal. In such cases, 73
CHAPTER 3. SPECTROMETER DESIGN
Figure 3.32: Hammamatsu T11262-01 Spectral Response. Image copyright of [45] a faster response time would need to be achieved in other ways such as reducing the radiation intensity incident on its surface.
3.7
Chapter Summary This chapter has shown that it is possible to effectively measure a key gas like
C2 H2 by choosing spectrometer components that emit and detect enough radiation at wavelengths where C2 H2 absorbs. It shows the Hawkeye IR-Si217 and the Hamamatsu T11261-01 meets these requirements. Further, it has shown that the spectral resolution of a zone plate is directly proportional to the number of zones and the focal length of a zone plate. It has also shown that there is an optimal ratio of open to opaque space in zone plates in terms of spectral resolution and transmission efficiency. Lastly, it has shown that the aperture diameter is directly proportional to transmission efficiency and inversely proportional to the spectrometer’s spectral resolution.
74
Chapter 4 Multigas Measurement 4.1
Problem Statement All design work presented so far revolved around the ability in effectively focusing
one particular wavelength of radiation in the aperture plane. Ideally, the end goal of this IR spectrometer would be to measure all 6 key gases in DGA. As such, the rest of this section will present how only one lens is sufficient in focusing all other wavelengths at varying dap−lens . It will further show how the concentrations of all other key gases can be measured as a result of this linear shift in dap−lens . First, let the following matrices be defined by their corresponding independent variables consisting of λ and dap−lens for n = 6 gases as shown in Equations 4.1, 4.2, and 4.3. Here ηap is an ARESR matrix, a is absorption coefficient, and cg is gas concentration. ηap (10, 1) ηap (20, 1) ηap (10, 2) ηap (20, 2) = . .. .. . ηap (10, 200) ηap (20, 200)
. . . ηap (12000, 1) . . . ηap (12000, 2) .. .. . . . . . ηap (12000, 200)
(4.1)
a(10, 1) a(10, 2) a(20, 1) a(20, 2) a= . . . . . . a(12000, 1) a(12000, 2)
... a(10, 6) ... a(20, 6) .. ... . . . . a(12000, 6)
(4.2)
ηap
T cg = [cg (C2 H2 ) cg (C2 H4 ) . . . cg (CO2 )]
75
(4.3)
CHAPTER 4. MULTIGAS MEASUREMENT In these equations, there are λ columns in ηap and λ rows in a that are incremented from 10nm to 12µm in steps of 10nm. Distance dap−lens is incremented from 1mm to 200mm in steps of 1mm as represented by the 200 rows in ηap . The 6 key gases are represented as a column vector in cg and as the 6 columns in a. The ranges of these variables are important to note since they are the same ranges that were implemented in the remainder of the Matlab simulations in this chapter. The total absorbance for the 6 key gases is shown in Equation 4.4:
A = lηap acg = ϕcg
(4.4)
where l is a scalar equal to the length of the chamber. A new variable ϕ is defined for simplicity in Equation 4.5 which makes it easier to determine cg later on.
ϕ = lηap a
(4.5)
The total transmission to the detector is expressed by
T = Io ηlosses exp(−A)
(4.6)
where A and T are column vectors of size (200 × 1). All losses within the system such as detector losses, window losses, and scattering losses are defined as ηlosses and the incident intensity is defined as Io . The known quantities in this system of equations are T , Io , ηlosses , A, and ϕ. The only unknown quantity that remains to be calculated is cg . The main problem that arises in finding the correct gas concentrations of cg is that there are numerous configurations of cg that can give the same value of A at different values of dap−lens . This is because many of the gas absorption profiles a overlap one another after being cross multiplied with ηap at some dap−lens . Therefore, an algorithm that can estimate cg by comparing the measured data from the detector as a function of dap−lens with the simulation results should be implemented.
76
CHAPTER 4. MULTIGAS MEASUREMENT
4.2
Spectral Discrimination To be able to measure all key gases using one lens, that lens should be able to
effectively focus in on their spectral absorption profiles. Therefor a simulation for finding the ARESR as a function of dap−lens and λ was performed for Lens 1. The simulation results are shown in Figure 4.1. The results have been presented for various values of N , namely N = 10, 80, and 400 to show how spectral resolution in general can affect the relationship between λ and dap−lens . One can notice the presence of multiple harmonics and subharmonics in Figure 4.1. These harmonics are influential in gas absorption overlapping effects. It is clear from Figure 4.1 that the wavelengths with peak ARESR are inversely proportional to dap−lens . A one dimensional view of sample neighboring dap−lens is shown in Figure 4.2. From Figure 4.2, it is clear that if the wavelengths of some of the peaks correspond to the wavelengths of peak absorption of the gases, then the lens should be successful in helping to measure all key gases. Such a simulation consisting of an cross multiplication of the gas absorption profiles with the ARESR, represented as ϕ, at different values of dap−lens for a unit length l = 1 is shown in Figure 4.3. It is apparent that over a range of 33mm to 51mm all key gases can be measured with clear peaks visible for all gases. It must be noted that there is some strong overlap for gases like ethane and ethylene due to their relatively weak absorption coefficients. In order to quantify the influence of this overlap, a new variable called the relative absorbance, ζ is introduced to determine the ratio of the absorption coefficient of each gas n to the sum of the absorption coefficients of all gases at a particular dap−lens . The relative absorbance assists in visualizing dap−lens regions where different gases absorb the most relative to all other gases. First, let ϕT be defined as the sum of all ϕ as follows:
(ϕT )z =
n=6 X
(ηap )zλ (a)λn
(4.7)
n=1
Next, let ϕT inv be defined as a matrix whose size is the same as that of ϕT , and whose every element is equal to the reciprocal of all corresponding elements in ϕT . For
77
CHAPTER 4. MULTIGAS MEASUREMENT
η ap(λ , d ap-lens) for N=10 20 0.2
40
dap-lens [mm]
60 0.15
80 100
0.1
120 140
0.05
160 180 200 2000
4000
6000
8000
10000
12000
Wavelength λ [nm]
η ap(λ , d ap-lens) for N=80 0.4
dap-lens [mm]
20 40
0.35
60
0.3
80
0.25
100
0.2
120 0.15
140 0.1
160
0.05
180 200 2000
4000
6000
8000
10000
12000
Wavelength λ [nm]
η ap(λ , d ap-lens) for N=400 0.7
20 0.6
40 0.5
dap-lens [mm]
60 80
0.4
100 0.3
120 140
0.2
160 0.1
180 200 2000
4000
6000
8000
10000
Wavelength λ [nm]
Figure 4.1: ARESR as a function of λ and dap−lens 78
12000
CHAPTER 4. MULTIGAS MEASUREMENT 0.6 dap-lens = 47mm dap-lens = 50mm
0.5
dap-lens = 53mm
ηap [-]
0.4
0.3
0.2
0.1
0 0
2000
4000
6000
8000
10000
12000
Wavelength λ [nm]
Figure 4.2: ARESR at varying dap−lens as a function of λ every element z of ϕT , let every corresponding element z of ϕT inv be described as shown in Equation 4.81 :
(ϕT inv )z =
1 (ϕT )z
(4.8)
Which can otherwise be rewritten as:
T ϕT inv = [1/(ϕT )1 1/(ϕT )2 . . . 1/(ϕT )200 ]
(4.9)
Finally, let every element zn in ζ be defined as follows:
(ζ)zn = (ϕT inv )z (ϕ)zn
(4.10)
A Matlab simulation for the ζ of each gas n is plotted for different values of N zones as shown in Figure 4.4. One can notice that as the spectral resolution increases (or N increases in this case) then ζ increases respectively for most gases. As a result it becomes easier to discriminate between the gases since there is less overlap.
1
Please note that Equations 4.8 and 4.10 are not the same as matrix multiplication.
79
×10-19
dap-lens = 33mm )/nm]
1.5
2
1
2
)/nm]
CHAPTER 4. MULTIGAS MEASUREMENT
0.5
6
dap-lens = 36mm
×10-18
C2H2: C2H4: C2H6: CH4: CO: CO2: H2O:
4 2
0 0
2000
4000
6000
8000
10000
0 0
12000
2000
×10-19
dap-lens = 46mm )/nm]
6
4000
6000
8000
10000
12000
10000
12000
10000
12000
Wavelength λ [nm]
4
2
2
)/nm]
Wavelength λ [nm]
Acetylene Ethylene Ethane Methane Carbon Monoxide Carbon Dioxide Water
2
×10-19
dap-lens = 47mm
1.5 1
2
0.5
0 0
2000
4000
6000
8000
10000
0 0
12000
2000
×10-20
dap-lens = 49mm )/nm]
8
4000
6000
8000
Wavelength λ [nm]
6
2
2
)/nm]
Wavelength λ [nm]
6
×10-19
dap-lens = 51mm
4
4 2
2 0 0
2000
4000
6000
8000
10000
0 0
12000
Wavelength λ [nm]
2000
4000
6000
8000
Wavelength λ [nm]
Figure 4.3: Gas absorption profiles convoluted with ARESR for different dap−lens
80
CHAPTER 4. MULTIGAS MEASUREMENT N=10
1 0.9
ζ [(mol/cm 2)/nm]
0.8 0.7 0.6 0.5
C 2H 2:
Acetylene
C 2H 4:
Ethylene
C 2H 6:
Ethane
CH4:
Methane
CO:
Carbon Monoxide
CO2:
Carbon Dioxide
H2O:
Water
0.4 0.3 0.2 0.1 0 0
20
40
60
80
100
120
140
160
180
200
dap-lens [mm]
N=80
1 0.9
ζ [(mol/cm 2)/nm]
0.8 0.7 0.6 0.5
C 2H 2:
Acetylene
C 2H 4:
Ethylene
C 2H 6:
Ethane
CH4:
Methane
CO:
Carbon Monoxide
CO2:
Carbon Dioxide
H2O:
Water
0.4 0.3 0.2 0.1 0 0
20
40
60
80
100
120
140
160
180
200
dap-lens [mm]
N=400
1 0.9
ζ [(mol/cm 2)/nm]
0.8 0.7 0.6 0.5
C 2H 2:
Acetylene
C 2H 4:
Ethylene
C 2H 6:
Ethane
CH4:
Methane
CO:
Carbon Monoxide
CO2:
Carbon Dioxide
H2O:
Water
0.4 0.3 0.2 0.1 0 0
20
40
60
80
100
120
140
160
dap-lens [mm]
Figure 4.4: ζ as a function of dap−lens for different gases 81
180
200
CHAPTER 4. MULTIGAS MEASUREMENT
4.3
Gas Concentration Determination If the concentrations of all gases were identical, then the shape of ζ as a function of
dap−lens would be identical to that of Figure 4.4. Unfortunately, it is highly unlikely that this would be the case in real life testing since gas concentrations often vary dramatically. Recall from section 2.1 that all DGA standards use ratios of different gases to determine fault types. Since all concentrations are measured in ppm, it is convenient to solve for the relative concentrations of all gases with respect to one another. Let these relative concentrations be referred to as crel . Further, let D be the normalized measured signal from the detector. As a result, the following system of equations must be solved for crel .
D = ϕcrel
(4.11)
Since the number of unknown variables to be solved is much smaller than the number of equations, solving Equation 4.11 constitutes an ill posed problem and may have no exact solution. An approximate solution to this problem can be found by applying least squares (LS) linear regression. In this scenario, it is desirable to minimize the residual noise (or error), J(crel ), between the measured signal and the simulation as shown in Equation 4.12 [46].
J(crel ) = kϕcrel − Dk2 =
200 X
(yi − Di )2
(4.12)
i=1
where y = ϕcrel and is a (200×1) column vector. An approximate least squares solution for crel is provided below [46].
crel ≈ (ϕT ϕ)−1 ϕT D = ϕ† D
(4.13)
The term ϕ† is known as the Moore-Penrose Pseudoinverse [46]. There are other ways of arriving at an approximate solution to Equation 4.11 but that problem is outside the scope of this thesis. The author encourages the reader to explore other existing methodologies, or invent new methodologies, for efficiently finding a solution to the equation. 82
Chapter 5 MEMS Fabrication 5.1
Introduction All four MEMS Fresnel zone plates discussed in the previous chapters were created
in the Nanosystems Fabrication Laboratory (NSFL) at the University of Manitoba. Following the Matlab simulations that defined the spacing and widths of all zones, all four devices were drawn out on one lithographic mask as shown in Figure 5.1. The lenses were designed using the Tanner L-Edit CAD program. From Figure 5.1, the two outermost zone plates are for visible light and the two center zone plates are for infrared light with focal lengths of two and one inches for both. The C code for automatically generating a series of concentric rings with appropriate widths and spacings in L-Edit is presented in Appendix D. The mask was designed with practical implementation of lithography in mind. It consists of a series of alignment features such as open slits and triangles with their vertices aligned with the centers of the lenses. Slits along which the backetch lens can be snapped off the wafer are also included. Four rectangles corresponding to the four zone plates are shown below the zone plate rings. These rectangles correspond to the lithographic profile of the back-etch of the lenses. The motivation for doing a back-etch on each lens is to maximize the power transmission efficiency through each lens. In this case, it was possible to reduce
83
CHAPTER 5. MEMS FABRICATION
Figure 5.1: Mask Design the thickness from an initial 360µm to 60µm. The tradeoff for such increased power transmission is a rougher surface finish that introduces more scattering prior to reaching the lens, thus compromising spatial coherence.
5.2
Fabrication Recipe A visual schematic of the full recipe for fabricating the zone plates is shown in
Figure 5.2. The steps for fabrication are as follows: 1. Clean wafer A double-sided polished 4” wafer was cleaned by immersing it into a 4H2 SO4 :1H2 O2 piranha solution for 10 minutes. This process removes most of the organic residues and the silicon oxide from the wafer. Afterwards the wafer was placed in a solution consisting of 1% HF for 3 minutes followed by placing in a solution consisting of 1% HCl for 1 minute. This is done to remove any residuals from the piranha solution.
84
CHAPTER 5. MEMS FABRICATION
Figure 5.2: Lens Fabrication Recipe
85
CHAPTER 5. MEMS FABRICATION 2. Thermal Wet Oxidation Following cleaning, the wafer is immediately placed in a wet oxidation furnace to grow a new layer of silicon oxide on both sides of the wafer. This oxidation is performed at a furnace temperature of 1100◦C with a water temperature of approximately 85-90◦C. It took 16 hours to grow a SiO2 layer of approximately 2µm. 3. Backside Processing To protect the polished face of the wafer where the zone plate will be fabricated, a layer of 504 photoresist is added to that polished face in order to protect it from backside fabrication processes. The photoresist is then put through a hard bake for 20 minutes at 120◦C to harden the face. After protecting the polished face, the steps for performing backside processing on the backside face are as follows: 3.1 Apply Photoresist: Spin 504 photoresist on the unpolished face. 3.2 Soft bake: Apply a soft bake at 115◦C for 60 seconds in the oven. 3.3 Mask Alignment: Align the wafer to the ABM mask aligner to expose the backside photoresist. 3.4 Lithography: Expose the photoresist to UV light for 6 seconds. 3.5 Development:
Develop the exposed wafer in developer solution for
approximately one minute until development is complete. 3.6 Clean: Rinse with DI water and dry the wafer to clean any residues. 3.7 Etch SiO2 : Remove the exposed SiO2 in a 10:1 buffered oxide etch (BOE) for 20 minutes. 3.8 Clean: Clean any residues from the wafer with de-ionized (DI) water. 3.9 Photoresist Removal: Remove the 504 photoresist in acetone solution and clean with IPA and N2 . 3.10 Etch Si: Etch the silicon wafer in a 30% KOH solution preheated to 80◦C. At an etch rate of approximately 1 - 1.2µm/minute, the wafer is etched for 240 minutes. 3.11 Etch SiO2 : Repeat step 3.7 to remove the remaining unexposed SiO2 from
86
CHAPTER 5. MEMS FABRICATION the wafer. After the backside is etched, front side processing is performed on the wafer. The steps are as follows: 4. Frontside Processing 4.1. Snap Wafer: The backside processing had grooves etched in the crystal plane along vertical and horizontal lines shown in Figure 5.1. Snapping the wafer along these lines helps in aligning the wafer backside to its frontside. 4.2. Aluminum Deposition: Sputter approximately 200nm of aluminum on the polished wafer face using the MRC 8667 Sputtering System. The base pressure for the process was Pb = 7.6·10−6 torr with a sputtering pressure Ps = 3.3·10−3 torr. The duration of the sputtering lasts 200s at 200W. The Argon flow rate used for ion bombardment during the process was 64.5sccm. 4.3. Protect Backside: Apply 504 photoresist around edges of backside face and attach to a dummy wafer. This dummy wafer acts as a shield to protect the backside from further processing. 4.4. Hard Bake: Apply a hard bake for 90 seconds so that the dummy wafer and backside face of the real wafer adhere. 4.5. Apply photoresist: Spin 504 photoresist on the polished front face. 4.6. Soft Bake: Apply a soft bake at 115◦C for 90 seconds in the oven. 4.7. Mask Alignment: Align the frontside of the wafer to the ABM Mask aligner using the snapped off wafer edges as a reference. 4.8. Lithography: Repeat step 3.4 for the frontside photoresist. 4.9. Development: Repeat step 3.5. 4.10. Etch Aluminum:
Etch the exposed aluminum in a chemical bath of
Aluminum Etchant 1960. This operation was performed for 29 seconds while being stirred with a magnetic spinner at 300 rpm. The temperature of the solution was 50◦C while sitting on a hotplate at 110◦C. 4.11. Photoresist Removal: Repeat step 3.9.
87
CHAPTER 5. MEMS FABRICATION
Figure 5.3: Backside etch
5.3
Figure 5.4: Frontside etch
Fabrication Results Following backside processing, the final
back-etch of the unpolished wafer is shown in Figure 5.3.
It consists of four recessed
valleys each pertaining to one zone plate at the total thickness of approximately 70µm with an average surface roughness of 2µm in the developed areas. Measurements pertaining to surface roughness and depth were measured using the Alpha-step Profiler. Following the
Figure 5.5: Final 1” IR Lens
frontside processing, the surface roughness of the aluminum and underlying silicon was on average less than 2µm with typical values of approximately 300Å. The front side face of the wafer before removing photoresist is shown in Figure 5.4. Unfortunately the wafer snapped while removing the final photoresist but was able to preserve the 1” IR lens as shown in Figure 5.5.
88
Chapter 6 Experimental Results and Analysis 6.1
Introduction This chapter presents empirical data resulting from the physical testing of
the one and two inch visible and infrared light lenses. The measured data is then compared to corresponding Matlab simulations to evaluate the accuracy of the designs. Potential reasons for error in all experiments are also presented in an attempt to provide explanations for discrepancies between the theoretical and measured experimental results.
6.2
Noise Reduction Data collected from the CCS175 spectrometer during the visible light experiments
along with corresponding Matlab simulation results had significant noise in its signal. As a result, signal processing in Matlab was applied to the data and simulation in order to get a better understanding of important variations in the signal. An example of this noise for a physical experiment is shown in the graph on the lefthand side of Figure 6.1. In order to reduce the impact of such noise on further analysis, the data is filtered using a Savitzky-Golay (SG) filter as shown in the graph on the right hand side of Figure 6.1. The difference between the graphs shows that it becomes much easier to distinguish relevant patterns in the signal as a result of the filter. The SG filter works by approximating a polynomial function to every datapoint 89
dap-lens = 35mm
0.2
0.03 Intensity/D [W/m2]
0.03 Intensity/D [W/m2]
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
0.15 0.1 0.05 0 500
600
700
800
900
1000
1100
dap-lens = 35mm
0.2 0.15 0.1 0.05 0 500
Wavelength [nm]
600
700
800
900
1000
1100
Wavelength [nm]
Figure 6.1: Original signal -vs- SG filtered signal in the signal based on the values of a window of neighbouring datapoints [47]. The value of the polynomial function at that datapoint becomes the new dependent variable at that datapoint. This process is repeated for every consecutive datapoint in the signal. The SG function g for a signal set of m points over intervals of nR + nL datapoints is shown below in Equation 6.1.
gm =
nR X
ck+nL sm+k
(6.1)
k=−nL
where ck+nL are the SG coefficients and sm+k are the independent variables of the original signal [47]. The SG filter is very similar to the moving average filter with the exception that it approximates datapoints using a polynomial function as opposed to calculating the average.
6.3
Reasons for Error As will be shown in the next sections, there are some discrepancies between the
Matlab simulations and the experimental results. Some possible reasons for these errors are presented in list form as shown below. These reasons for errors are valid for both the visible and infrared light experiments. 1. Alignment errors The apparatus consisted of four main optical components situated on an optical breadboard. There is a high likelihood that each of those components may have
90
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Figure 6.2: Possibilities for misalignment (Not to scale) been misaligned with respect to each other since the detector, light source, and zone plate were aligned by hand without using appropriate optical mounts as shown in Figure 6.2. In particular, 1.1. Nonzero roll and pitch in detector. 1.2. Nonzero roll in aperture. 1.3. Nonzero roll and pitch from light source. 1.4. Nonzero radial displacements of the light source, lens, aperture, and detector with respect to the optical axis. These possible displacements are represented in Figure 6.2 as rdisp1 , rdisp2 , rdisp3 and angles θ, η, and β. One observation from the experiment results was that as dap−lens was increasing, angle η was approaching zero and the experimental results started resembling the simulations much better. There was also a great deal of uncertainty in the exact alignment of rdisp1 , rdisp2 , and rdisp3 due to the fact that they were all adjusted by hand without all of the necessary optical instruments. Further, they needed to be ideally aligned to within at least 10µm of each other to have results closely matching that of simulation. The reflector on the light source for the visible light source was also adjusted manually and was discovered to have a lot of variation in pitch and yaw. This fundamentally changed the phases of the plane wave striking the zone plate which produced improper interference as a result. 91
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS 2. Fresnel -vs- Fraunhofer Diffraction All Matlab simulations assumed that the intensity at the aperture from each zone plate could be modeled as Fraunhofer diffraction as opposed to Fresnel diffraction. As mentioned in subsection 2.5.2, Fraunhofer diffraction occurs when zi
D2 . λ
As
such for a 50µm aperture at 540nm, there may have been near field diffraction effects that could have skewed the intensity pattern at the detector. 3. Coherence of Light source The radiation plane wave striking the zone plate was not perfectly coherent from either IR or visible light sources as assumed in the simulation.
This implies
that some destructive interference was caused by out-of-phase radiation. This interference compromised the reliability of the simulation results when tested in an experimental setup. 4. Heating of zone plate Given the very small features of the zone plate, it is conceivable that the radiation from the light source created enough heat to sufficiently alter the diffraction at the output. The heating of individual concentric aluminum rings could have reradiated unwanted radiation from the zone features. Further, it is possible that the heat may have been sufficient to thermally expand the zone features, thus changing the zone plate widths. 5. Manufacturing errors There may have been sufficient error in the fabrication process itself during lithography and etching that resulted in nonperfect zone widths. As a result, all zone plate features could have been scaled by some percentage value. Further, the walls of the zones might have been angled that resulted in unwanted diffraction along the walls of the zones. 6. Lens Scattering There may have been scattering from within the lens itself (soda lime glass for the mask and silicon for the IR lens) that would affect the coherence of the lens itself. There may also be scattering from within the gas chamber windows.
92
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS 7. Light bleeding from lens edges 8. Relatively large distance for dap−det 9. Detector Imperfections These imperfections include detector noise, and imperfections in spectral response and responsivity from the datasheet of the detector. It is difficult to pinpoint the full extent of the influence of each source of error due to the multiple uncertainties in the apparatus setup. It is suspected that a significant portion of this error is due to misalignment. This is due to the very tight tolerances required by the Matlab simulation and the fact that some of the optical components needed custom-made mounts that are not of industrial quality.
6.4
Visible Light Experiments The experimental setup for the visible light experiments is shown in the below
figure. It consists of a Cole Parmer Fiber-lite Model 9745-00 for the light source, a zone plate lens, an aperture, and a fiber optic cable from a Thor Labs CCS175 spectrometer, and a 50µm diameter Thor Labs aperture.
Figure 6.3: setup
Sideview of visible light
Figure 6.4: setup
93
Topview of visible light
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
6.4.1
Results for two inch lens Matlab experimental and simulation results, shown in red and blue respectively,
pertaining to the 2” visible light lens are shown in Figure 6.5. From the simulation results, it is noticeable that there is a first harmonic at 538nm, a first subharmonic at 803nm, and a second harmonic at 1080nm. Starting at dap−lens = 51mm, it can be observed that in both sets of data that as dap−lens increases, the mentioned harmonics shift towards shorter wavelengths. Table 6.1 shows approximately how much the peaks shift as a function of dap−lens . The simulation results show that the lobe moves towards shorter wavelengths in increments of between 32 and 42 nm for dap−lens ranging from 51mm to 63mm in 3mm increments. This type of inconsistent shift is similar to, and thus expected from, the simulation results presented in Figure 4.1. The only exception between this result and that of Figure 4.1 is that Figure 4.1 is for infrared light as opposed to visible light. The pattern still holds true though. Analyzing the experimental results, there appears to be no significant impact from the first harmonic as predicted in simulation. However, beginning at dap−lens = 54mm, there is a noticeable change in the spectrum from around 700 - 800nm that becomes much more evident as dap−lens increases, with a strong signal lobe noticeable at dap−lens = 60mm and 63mm. The surrounding spectrum and bandwidth of this lobe appears to correspond fairly closely to the first subharmonic predicted in simulation. This is noticeable from two small spectral “lumps” present on both ends of the lobe. It must be stressed that is not certain that the moving lobe corresponds to the first subharmonic. However, judging by how close the spectral ranges are compared to the first and second harmonics, as well as other similarities in shape, it is presumed to be that subharmonic. There is a discrepancy between the simulated and experimental results for the center wavelengths of this lobe of about 10%. There are many possible reasons for these discrepancies as section 6.3. Despite the evidence of the supposed first subharmonic in the experimental results, there is no clear evidence of the first harmonic from simulation in the experimental results. This is likely a result from misalignment and other errors. Table 6.1 also shows that there is no apparent relationship between dap−lens and the bandwidth of the lobes. 94
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Simulation Results
dap-lens = 51mm
dap-lens = 51mm
0.2
0.15
0.15
ηap [-]
0.031 Intensity/D [W/m2]
Experimental Results
0.1 B
0.05 0 500
A 600
700
800
900
0.1 A' B' C'
0.05
C 1000
0 500
1100
600
700
dap-lens = 54mm
B
0.05 A
1100
0 500
600
700
800
900
1000
1100
1000
1100
1000
1100
1000
1100
0.1 A' B' C'
0.05
C 1000
0 500
1100
600
700
800
900
Wavelength [nm]
dap-lens = 57mm
dap-lens = 57mm
0.04
0.15
0.03
ηap [-]
0.031 Intensity/D [W/m2]
1000
0.15 0.1
0.1 B A
0.05 0 500
600
700
C
800
0.02 B'
900
1000
0 500
1100
600
700
dap-lens = 60mm
0.03
ηap [-]
A
0 500
C
600
700
800
0.02
B' A'
0.01 900
1000
0 500
1100
600
ηap [-]
B 0.1 A C
700
800
900
900
dap-lens = 63mm
0.06
0.15
600
800
Wavelength [nm]
dap-lens = 63mm
0 500
C' 700
Wavelength [nm]
0.05
900
dap-lens = 60mm
0.04
B
0.05
800
Wavelength [nm]
0.15 0.1
C'
A'
0.01
Wavelength [nm]
0.031 Intensity/D [W/m2]
900
dap-lens = 54mm
0.2
0.15
Wavelength [nm]
0.031 Intensity/D [W/m2]
800
Wavelength [nm]
ηap [-]
0.031 Intensity/D [W/m2]
Wavelength [nm]
1000
0.04 0.02 0 500
1100
Wavelength [nm]
A' 600
B'
C' 700
800
900
Wavelength [nm]
Figure 6.5: Experimental and simulation results for the 2” VIS Lens
95
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Table 6.1: 2” VIS Lens Results Overview between 500nm - 1110nm
Experiment dap−lens 51mm
54mm
57mm
60mm
63mm
Harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic
≈ λc – 908nm – – 854nm – – 796nm – – 751nm – – 711nm –
≈ BW – 55nm – – 60nm – – 53nm – – 57nm – – 57nm –
Point B shift
-54nm
-58nm
-45nm
-40nm
Simulation dap−lens 51mm
54mm
57mm
60mm
63mm
Harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic
≈ λc 538nm 803nm 1080nm 510nm 761nm 1020nm – 722nm 960nm – 686nm 910nm – 654nm 870nm
96
≈ BW 80nm 85nm 80nm – 71nm 90nm – 88nm 90nm – 75nm 90nm – 75nm 80nm
Point B 0 shift
-42nm
-39nm
-36nm
-32nm
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
6.4.2
Results for one inch lens The simulation and corresponding experimental results for the 1” lens as a function
of dap−lens is shown in Figure 6.6. Similar to the 2” lens, the experimental results for the 1” VIS lens does not seem to have been influenced by the first harmonic shown in the simulation, but rather is presumed to be influenced by the first subharmonic. The reason for this is unclear but like the 2” lens, the possible reasons are discussed in section 6.3. Looking at the experimental results for dap−lens = 24mm, there appears to be a peak forming at approximately 694nm at dap−lens = 27mm. As dap−lens increases, this peak appears to move towards shorter wavelengths with peaks at 621nm, 553nm, and 527nm for dap−lens = 30mm, 33mm, and 36mm respectively. This finding is consistent with the overall theoretical prediction that as dap−lens increases, the spectrum shifts towards UV. This finding is confirmed in Table 6.2. Similar to the 2” lens, it is evident from Table 6.2 that the center wavelengths of the first subharmonics in the simulation are again approximately 10% greater than that of the experimental results. As expected, in the simulation results, the shift in A0 becomes smaller as dap−lens increases. It doesn’t seem to be the case in the experimental results where the shift in point A appears to have no effect as dap−lens increases. There are a few plausible explanations for this. One is that the lobe in question does not actually correspond to the first simulated subharmonic and therefore appropriate shifts in dap−lens cannot be compared. Another possibility is inaccuracy during the measurement of dap−lens . Finally, it is important to remember that the light source used in the experiment is not 100% collimated, and therefore has some focal point. As dap−lens increases as a result, the effects of this uneven collimation will also become apparent.
6.5
Infrared Light Experiment As an advisory to the reader, the infrared light experiment didn’t produce the
expected results in being able to measure any key gases. However, the experiment has successfully shown that the detector working in unison with a lock-in amplifier is able to
97
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Simulation Results
dap-lens = 24mm
0.2 0.15
A
0.1 0.05 0 500
600
700
800
900
dap-lens = 24mm
0.3
ηap [-]
0.03 Intensity/D [W/m2]
Experimental Results
1000
0.2 0.1
A'
0 500
1100
600
700
dap-lens = 27mm
0.2
A
0.1 0.05 600
700
800
900
1000
0.1
600
700
A
1000
1100
1000
1100
1000
1100
1000
1100
0.03
ηap [-]
0.03 Intensity/D [W/m2]
0.05
0.02
A'
0.01
0 500
600
700
800
900
1000
0 500
1100
600
dap-lens = 33mm
0.2
700
800
900
Wavelength [nm]
dap-lens = 33mm
0.04
0.15
0.03
0.1
ηap [-]
0.03 Intensity/D [W/m2]
900
dap-lens = 30mm
0.04
Wavelength [nm]
A
0.05
0.02 A' 0.01
0 500
600
700
800
900
1000
0 500
1100
600
Wavelength [nm]
dap-lens = 36mm
0.2
700
800
900
Wavelength [nm]
dap-lens = 36mm
0.04
0.15
0.03
ηap [-]
0.03 Intensity/D [W/m2]
800
Wavelength [nm]
0.1
0.1
1100
A'
0 500
1100
dap-lens = 30mm
0.15
1000
0.2
Wavelength [nm]
0.2
900
dap-lens = 27mm
0.3
0.15
0 500
800
Wavelength [nm]
ηap [-]
0.03 Intensity/D [W/m2]
Wavelength [nm]
A
0.05 0 500
0.02
A'
0.01 600
700
800
900
1000
0 500
1100
Wavelength [nm]
600
700
800
900
Wavelength [nm]
Figure 6.6: Experimental and simulation results for the 1” VIS Lens
98
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Table 6.2: 1” VIS Lens Results Overview between 500nm - 1110nm
Experiment dap−lens 24mm
27mm
30mm
33mm
36mm
Harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic
≈ λc – 742nm – – 694nm – – 621nm – – 553nm – – 527nm –
≈ BW – 35nm – – 45nm – – 47nm – – 37nm – – 35nm –
Point A shift
48nm
73nm
68nm
26nm
Simulation dap−lens 24mm
27mm
30mm
33mm
36mm
Harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic 1st subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic 1st harmonic st 1 subharmonic 2nd harmonic
≈ λc 575nm 860nm – 515nm 760nm 1020nm – 685nm 920nm – 625nm 830nm – 575nm 760nm
99
≈ BW 110nm 110nm – – 100nm 150nm – 90nm 150nm – 70nm 150nm – 75nm 120nm
Point A0 shift
100nm
75nm
60nm
50nm
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS detect a differential in the thermopile voltage signal. Further development work on the detector is required to achieve complete operation according to the specification required by the Matlab simulations. More details will be explained in this section.
6.5.1
Hardware Full system side and top view photos of the IR experiment apparatus are shown in
Figure 6.7. The system begins with the Hawkeye IR radiator connected to a power supply which illuminates the gas chamber. Following the chamber in consecutive order is the lens, chopper, aperture, and thermopile detector. All components are aligned such that they are as close as possible to being in the same optical axis. In the case of alignment for the aperture and thermopile, the thermopile was placed on a laser cut acrylic jig taped directly onto the aperture. Due to the fact that the sensitive area of the thermopile is 1.2mm2 and the aperture diameter only 100µm, it was deemed a reasonable choice to do this given the lack of choice for more appropriate optical mounts. This aperture/detector combination is shown in Figure 6.8. The IR-Si217 radiation source and AD630 lock-in amplifier is modulated with a square wave pulse train signal from an MC2000 Optical Chopper. This modulation serves to reduce 1/f noise and the influence of other external noise sources operating on inconvenient frequencies. The frequency of the chopper for different experimental conditions is chosen such that it satisfies two conditions: the frequency isn’t high enough to saturate the detector circuit, and the frequency does not interfere significantly with an existing 60Hz background noise signal VN from the surrounding environment. This VN likely originates from the electromagnetic interference (EMI) produced by the cabling and other machinery in the room operating at that frequency.
6.5.2
Detector Circuit Design The circuit used for detecting, filtering, and amplifying the heat radiation from
the thermopile is shown in Figure 6.9. This circuit begins by taking a voltage signal from the thermopile, Vin , and passing it through buffer U1 . The buffer is used to output 100
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
(a) Sideview
(b) Topview
Figure 6.7: Photos of IR Experiment Apparatus
101
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Figure 6.8: Detector and Aperture with Chopper
Figure 6.9: Infrared experiment circuit diagram. Image modified from [44]
102
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS what is essentially the open circuit of the thermopile since the input impedance of the buffer operational amplifier (op amp) is extremely large, 1TΩ [48]. The buffer is added to avoid loading the thermopile; which can compromise the signal fed to the rest of the detector circuit. Signal Vin is then amplified through an inverting op amp U2 . The op amp contains capacitor C1 to serve as a lowpass filter (LPF) with a 3dB frequency of 159Hz (R2 and C1 are constant for all scenarios of varying apertures). The signal is then passed through an AD630 Modulator/Demodulator that serves to amplify ideally only the modulated Vin and ignore other noise components. The reference signal to the AD630 is Vin , whose modulated waveform is the same square pulse train in phase with the mechanical chopper chopping Vin . The signal from the chopper is amplified through U3 in order for it to work with the AD630. The output signal from the AD630 is then passed through a non-inverting op amp U5 that serves to amplify its input signal, and further cut out high frequency components greater than 2.3kHz. Finally, a 1N5817 Schottky diode is used to cut out the negative portions of the waveform so an approximate DC value correlating to the thermopile output can be obtained from a LP filter following the diode through R7 and C3 . The final output of the LP filter is Vout which has time constant of 1 second. These values too don’t change as a function of the aperture diameter.
6.5.3
Thermopile Modeling The first step in the practical implementation of the circuit is modeling and
understanding the behavior of the thermopile. This is a necessary condition in order to design the rest of the detector circuit appropriately for the sensor in question. First, the open circuit behavior of the thermopile is observed at V2 and V3 as shown in yellow and blue in Figure 6.10 respectively with a chopper frequency of 20Hz. This figure indicates the thermopile has an open circuit DC bias of approximately 220mV RMS. Further, 20Hz noise components from the signal are amplified in the blue signal to provide a peak-to-peak voltage of 912mV. One way to mitigate the DC bias is to add a high pass filter (HPF) to the circuit by adding a capacitor just before R1 . This was attempted and worked very well for 103
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
Figure 6.10: Thermopile Reference Signal Table 6.3: Detector Circuit component values for no aperture R1 R2 R3 R4 R5 R6 R7 C1 C2 C3 10kΩ 100kΩ 1kΩ 10kΩ 10kΩ 100kΩ 100kΩ 10nF 680pF 10µF
detecting signals where there was no aperture. However, it was observed that as the aperture shrank in size, the signals at V3 and downstream became highly unstable and a direct measurement was not possible. Since the end goal of the experiment is to use a small aperture in order to maximize spectral resolution, the HPF was not implemented in the design.
6.5.4
Results with no Aperture In this subsection, no aperture was placed before the detector. The values of the
passive components used in this circuit are shown in Table 6.3. These same values are used for all future IR experiments as well since they proved to work good after multiple trial and error experiments. The signals at V2 and V3 were observed when the Hawkeye had full radiation incident on the thermopile as shown in Figures 6.11a and 6.11b for 20Hz and 100Hz chopper frequencies respectively. This experiment had a 5µF (2×10µF in series) capacitor before R1 to cut off frequencies below 3.1Hz. It is apparent from Figure 6.11 that the DC bias disappeared when no radiation was applied, and that as the frequency increased the signal SNR decreased dramatically. The quality of the signal’s
104
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
(a) Full radiation at 20Hz at Vin (yellow) and V3 (blue)
(b) Full radiation at 100Hz at Vin (yellow) and V3 (blue)
Figure 6.11: Thermopile Vin and AD630 output voltages V3 shape was also compromised due to interference from 60Hz background noise. The most probable reason why the signal suffers so much from an increased frequency is due to the long time constant of the thermopile, 20ms [45]. The important thing to note about this experiment is that the signal from the output of the lock-in amplifier closely follows that of the thermopile at 20Hz. This is proof that the lock-in amplifier works as it should.
105
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS Table 6.4: RMS voltages at key nodes for the case of a 100µm aperture V2 φap Case V4 V5 Vout ∆Vout No Radiation 412mV 362mV 57mV Approx. 1mm 17mV Full Radiation 325mV 292mV 40mV white noise No Radiation 640mV 484mV 150mV Approx. 100µm 3mV Full Radiation 637mV 482mV 147mV white noise
6.5.5
Results with Aperture In the previous section when a HPF was applied, the DC bias from the thermopile
was eliminated from the output of AD630. However, when the aperture size shrunk to less than 1mm, the HPF could not provide a steady signal so the HPF was removed. This section will provide results for an aperture sizes of 1mm and 100µm. With such a small aperture sizes, the signals at the input of the AD630, V2 looked basically like white noise even after applying full radiation to the aperture. Take the case for a 100µm aperture, there was a clear square pulse train signal at the output of the integrator circuit U5 as shown in Figure 6.12a. The Schottky diode rectifies the signal as shown in Figure 6.12b, and finally passed through a LPF to provide a nearly DC signal as shown in Figure 6.12c. Similar waveforms are found for the 1mm diameter aperture. Clearly, one can notice a strong DC bias in Figures 6.12a and 6.12b originating from the thermopile. Since it is not obvious how to eliminate the DC bias from the thermopile in this scenario without compromising the quality of the signals at the output of the AD630, the RMS voltages at V4 , V5 , and Vout are simply measured and compared for full and no radiation. The RMS voltages at V4 , V5 , and Vout for the two scenarios of full and no radiation for 1mm and 100µm diameter apertures are shown in Table 6.4. From Table 6.4, it is clear that there is a significant change in the signal for a 1mm aperture averaging around 17mV at Vout . Voltages V4 , V5 , and Vout are inversely proportional to the incident radiation due to the way the thermopile was connected giving a negative bias. As the aperture diameter drops from 1mm to 100µm, the incident radiation on the detector drops dramatically and the DC bias from the thermopile becomes the most significant portion of the signal. At this level, there is still a consistent
106
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
(a) Output of integrator U5 (V4 )
(b) Output of Schottky diode (V5 )
(c) Final DC value at final LPF (Vout )
Figure 6.12: Voltage outputs following AD630 at full radiation on 100µm aperture 107
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS voltage differential of about 2 to 3mV between full and no radiation, thus showing that this aperture can still be used to detect a change in radiation intensity. It can be observed that V4 , V5 , and Vout for the two apertures are not even close when no radiation was applied. This is due to the very geometry of the aperture and its change in emissivity and other adjustments in the apparatus setup.
6.5.6
Infrared Experiment Summary and Future Work In summary, a signal differential of approximately 2 to 3mV was observed when a
modulated thermopile detector was subject to full radiation from the Hawkeye IR source versus no radiation from the source for a 1µm aperture. In reality, when a gas like CO2 or C2 H2 is applied, and assuming 100% absorption, the reading from the detector may only change by about 5% or less. This 5% value was calculated as a ballpark approximation taking into account an average detector transmittance of 88% between 2 and 5µm, the 2.97 - 3.12µm C2 H2 spectral range, and the ratio of the integral of the blackbody curve from 2.97 - 3.12µm to the integral of the blackbody curve from 2.5 - 7.5µm. After accounting for wavelengths longer than 7.5µm and shorter than 2.5µm, 5% becomes a very reasonable estimate. Since 5% of 3mV is equal to 150µV, it becomes quickly clear that more work is needed in: 1. Reducing noise in the circuit itself by creating a PCB and shortening the leads of the thermopile. 2. Placing the detector in a cooled environment to reduce thermal noise. 3. Using a more sensitive modulator/demodulator than the AD630 IC. 4. Determine an appropriate measurement sampling time. 5. Modeling the noise present in the system so that it can be filtered out better during measurement. Of course it can be argued that strong interference from the lens should be able to precisely focus only the wavelengths of interest thus increasing the 5% value. However, it is the author’s opinion that further analysis of the results of the currently operational 108
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS visible light experiments would need to be conducted before safely making such a claim. This is due to the differences between theory and experiment.
6.6
Sawtooth Observation An unexpected, interesting phenomena was observed at the output spectra at the
aperture in various radii r away from the central axis of the lens. It turns out that it is possible to create a band-limited sawtooth spectral shape as shown in Figure 6.13. It must be noted that the waveform shown is background-corrected against its original spectrum. It is unclear to the author whether or not there are any practical applications of such a unique band-limited waveform. However, the phenomena is worth noting for future reference in case an idea for a future application of such a waveform is required.
Figure 6.13: Observed sawtooth waveform as a result of binary zone plate following background correction
109
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS
6.7
Hologram Observation Another interesting feature observed at
the output of the Zone Plate was that the lens can be used as a holographic display as shown in Figure 6.14. It is apparent from Figure 6.14 that there are two holograms of the text on the book, one above the word “Materials” and one below. Experimentation has shown that this holographic effect occurs when observing the image off-axis from the Zone Plate’s optical Figure 6.14: Holographic projection axis. Holograms have many applications in from Zone Plate the real world including Biomedical Microscopy and Augmented Reality (AR) [49][50]. The Microsoft HoloLens is but one very recent example where diffraction gratings along with appropriate lens’ and other components have been used to produce holographic images for AR environments [51]. It is the author’s opinion that the exact nature of the holographic behavior as a function of the current lens design should be studied in closer detail. Such further studies would be important in order to assess whether or not the fabricated lens in the thesis could be of use for the above mentioned applications.
6.8
Chapter Summary This chapter shows that there are significant similarities between the simulation
and experimental results for the visible light experiments.
There was disagreement
between the two in terms of center wavelengths, the amount by which the lobes shifted with dap−lens , and their approximate bandwidths. The results from the infrared light experiment has shown that it was possible to detect an approximate 2 or 3mV difference in signal when the radiation source was fully blocked versus fully open. The next step would be attempting to measure gas concentrations. A more sensitive detector with 110
CHAPTER 6. EXPERIMENTAL RESULTS AND ANALYSIS better noise modeling would be necessary before attempting to measure smaller signals. It has proven very difficult to get reliable alignment of all optical components given the fact that the main lobes in the Matlab simulations have a spatial FWHM of around ±50µm or less. Furthermore, all components were manually aligned while attempting to get to such tolerances. Better optical mounts for all optical components would be necessary to get results that would more closely match the simulation results.
111
Chapter 7 Future Work and Conclusion 7.1
Future Work There are many issues in this project that should be addressed before considering
it complete, let alone ready for commercialization. A number of proposed improvements to the project are listed below. 1. Create or find mounts for all optical components that allow for alignment tolerances to within about ±10µm between all components. 2. Application and testing of a well collimated blackbody source. This applies to the VIS light experiments as well. 3. Redo the Matlab simulations for the lens accounting for imperfect spatial and temporal coherence.
Further, spend more resources examining how the
experimental setup could have been designed better to minimize these issues. 4. Development of a linear thermal sensor that has faster response time than the Hammamatsu T11262-01 thermopile with an ideally wider IR region. Such a linear thermal sensor can then be used in conjunction with a linear zone plate. 5. Development of a thermal sensory system that can measure the radiant intensity from the lens within a 10mm radius in the aperture plane. This would be used to test the full radiation profile in the mid-IR wavelengths at the aperture plane which can then be compared with theory.
112
CHAPTER 7. FUTURE WORK AND CONCLUSION 6. Development of statistical algorithms that can predict with high confidence the concentrations of the key gases given absorption as a function of dap−lens . Linear regression is likely sufficient, but there could be better ways of solving the problem. 7. Investigate practical applications of this spectrometer in areas such as food monitoring, atmospheric monitoring, and the oil and gas industry.
This
spectrometer can also be investigated for its effectiveness in measuring gases not belonging to any of the key gases already mentioned. 8. Development of a miniaturized version of the spectrometer. This may include the development of an IR reflective white cell in place of the existing gas chamber, a smaller radiation source that can be modulated, and smaller detectors. 9. Full field experiment for measuring all of the gases inside existing oil-impregnated high voltage equipment.
The results could then be compared with other
spectrometers for reference. 10. Development of a sensor node network that consists of many spectrometers placed in different pieces of high voltage (HV) equipment. The data from the spectrometers could then be wirelessly transmitted to a ground station for further analysis. 11. Development of real-time algorithms that measure pressure and temperature in the HV equipment and send that data to a ground station. The pressure and temperature would be used to account for Doppler and pressure broadening. 12. Derive analytical solutions for many of the simulations shown in this project. This includes the function behind the ARESR curves, the FWHM curves, and the ARESR heatmap as a function of dap−lens and wavelength. Optimization techniques could then be designed and used to determine ideal values of each lens parameter as it pertains to the application at hand. 13. Development of a Java application where a user can look at the state of high voltage equipment around a geographical area. The states would be determined by automatically performing standard DGA techniques based on the measured gas concentrations. Should there be any warning signs regarding equipment failure, the software could send a flag to a maintenance crew.
113
CHAPTER 7. FUTURE WORK AND CONCLUSION 14. Conduct a full cost-benefit analysis of this spectrometer compared with existing IR spectroscopy techniques such as Fourier Transform Infrared Spectroscopy (FTIR), gas chromatography, and PAS. The criteria for the analysis could include manufacturing costs, robustness, lifetime, and spectral resolution. 15. It is apparent many of the gases are close to one another spectrally. Examples include the C2 H2 , CH4 , C2 H4 , and C2 H6 stretches. As such, an attempt should be made on developing new DGA standards based on the focusing properties of the lens and the absorption characteristics of the gases. For instance, if the detector reads a certain characteristic waveform following a dap−lens sweep, then that waveform could be associated with a specific type of fault. 16. Investigate whether or not there could be some practical application to a generated sawtooth waveform shape as a result of moving the detector off-axis across the radius of the lens. Surely the same result can be achieved using a function generator and a bandpass filter, but perhaps there could be cost savings using the new technique. 17. Investigate whether or not there could be some practical application to the holographic properties of the lens. Perform further analysis on these properties.
7.2
Conclusion This thesis has shown via simulation that it is theoretically possible to measure the
concentrations of all DGA key gases using a single linearly adjustable binary zone plate lens for both linear and circular slits. From a design perspective, the simulations have shown that as the focal length of the lens increases, the ARESR and the spectral resolution increases. They have also shown as the number of zones increases, the total power output, the ARESR and spectral resolution also increase. By adjusting the size of the aperture, the simulations have shown that the input power to the detector and the spectrometer’s spectral resolution are inversely proportional. Simulations have shown that by adjusting the distance between the lens and aperture from 20mm to 160mm, one lens can effectively focus on wavelengths ranging from 1µm to 7.5µm with a maximum ARESR of 0.71. The 114
CHAPTER 7. FUTURE WORK AND CONCLUSION focusing of the lens over this spectral range is sufficient to cover the entire spectral ranges of all key gases, whose concentrations can then be further determined using various statistical tools. Further, this thesis has presented an unorthodox method of determining the ratio of the total open space to the total area of the zone plate. This assists the designer in making a trade-off decision between the maximum ARESR and power transmission of the lens. The thesis has shown that there is a reasonable correlation between the simulated results and the experimental results for the visible light lenses. While there were some discrepancies between the two, in particular a failure to recognize the first harmonic, there was a good correlation between the first subharmonics present in both simulations and the moving lobes in the experiments. It has also been shown that the operation of a lock-in amplifier was successful in being able to measure a difference of around 3mV between full and no radiation from an IR radiation source in the IR experiment for a 100µm aperture. Compared with other competing optical technologies for performing DGA and IR spectroscopy, it is hypothesized that the product presented in this thesis may be cost effective and reliable. The justification for this hypothesis is that silicon wafers are much cheaper to manufacture than ruled IR diffraction gratings, there are no fast vibrating or extremely delicate components as are present in FTIR spectrometers, and the design presented does not require multiple spectral filter windows that are currently used in PAS spectrometers. As a result, it is reasonable to suggest that these arguments provide good reason to seriously consider further development on the initiated product.
115
Appendices
116
Appendix A Derivation of surface reflection for an S polarized wave To gain an appreciation for the origins of Equations 2.20 and 2.21, shown below again as:
Er,S = RS (θr ) = RS (θi ) = Ei,S
cos(θi ) − cos(θi ) +
r
RP (θr ) = RP (θi ) =
n2 n1
2
Er,P = r 2 Ei,P n2 n1
r r
n2 n1 n2 n1
2 2
− sin2 θi (A.1) 2
− sin θi
− sin2 θi −
n2 n1
2
n2 n1
− sin θi +
2 2
cos θi (A.2) cos θi
the derivation for the reflected wave of an S polarized wave is presented. The reflection for a P polarized wave can be derived in a similar fashion for the user’s own interest. Let a transverse electromagnetic wave travelling through a media with an index of refraction of n1 fall incident upon a flat surface media with an index of refraction of n2 at an angle θi . Upon incidence, portions of the electric field of that wave, Ei will be reflected at an angle of θr with amplitude Er , and some of it will be transmitted through the material at angle θt with amplitude Et . Snells Law dictates the relation between these waves and angles as follows:
117
APPENDIX A. DERIVATION OF SURFACE REFLECTION FOR AN S POLARIZED WAVE n1 sin θt = n2 sin θi
(A.3)
As will be shown later, it is first important to find an expression for cos(θt ). It is derived from Snell’s Law as shown in Equation A.4.
n1 sin θi = n2 sin θt n1 sin θi n2 n1 sin2 θt = sin θi sin θt n2 n1 1 − cos2 θt = sin θi sin θt n s2 n1 cos θt = 1 − sin θi sin θt n2 sin θt =
s
1−
cos θt = s
1−
cos θt =
n1 n1 sin θi sin θi n2 n2
n1 n2
2
sin2 θi
(A.4)
Since the top and bottom surfaces at the intersection of two the media have zero net electric charge, the net electric field on some closed loop covering the bottom and top portion of that surface is also equal to zero. This can be shown by Gauss’ Law which is presented in Equation A.5 ‹ EdA = S
Q =φ Eo
(A.5)
where in this case Q = φ = 0. It follows that the tangential components of the electric fields above and below the surface are equal, and it can thus be written that:
Et,S − = Et,S +
(A.6)
where Et,S − is the tangential component of the electric field below the incident surface and Et,S + is the tangential component of the electric field above the incident surface. For
118
APPENDIX A. DERIVATION OF SURFACE REFLECTION FOR AN S POLARIZED WAVE a P polarized wave, the following relation holds [33]:
Ei cos θi + Er cos θr = Et cos θt Ei cos θi + Er cos θi = Et cos θt Ei + Er = Et
cos θt cos θi
(A.7)
For an S polarized wave, it may be written that [33]:
Ei + Er = Et
(A.8)
Next, an expression for the magnetic field component of an S polarized wave in terms of its electric field is derived from Equation A.8 [33].
Er + Ei = Et Er Ei Et cos θr − cos θi = − cos θt Zo1 µ1 /n1 Zo1 µ1 /n1 Zo2 µ2 /n2 Assuming that the relative permeability and characteristic impedances of media 1 and 2 are equal, and that θr = θi , it may be further written:
Er n1 cos θi − Ei n1 cos θi = −Et n2 cos θt Ei − Er = Et
n2 cos θt n1 cos θi
(A.9)
Combining Equations A.4 and A.9, r
n2 Ei − Er = Et n1
1−
n1 n2
2
cos θi
sin2 θi (A.10)
A final expression for RS (θr ) may now be reached by combining Equations A.8 and A.10.
119
APPENDIX A. DERIVATION OF SURFACE REFLECTION FOR AN S POLARIZED WAVE
RS =
Er 2Er (Ei + Er ) − (Ei − Er ) = = Ei 2Ei (Ei + Er ) + (Ei − Er ) r
Et − Et nn12 cos1 θi 1 − = Et +
Et nn21 cos1 θi
cos θi −
n2 n1
cos θi +
n2 n1
=
cos θi − = cos θi +
r
r
1−
1−
n1 n2
1−
n1 n2
r
r
r
n2 n1 n2 n1
which is equivalent to Equation 2.20.
120
n1 n2
n1 n2
2 2
!
sin2 θi !
sin θi
2
sin2 θi
2
sin2 θi
2
− sin2 θi
2
− sin2 θi
2
Appendix B Annular Slit Modeling At the outer edges of the circular lens’, the ratio of the inner to outer radii of the slit widths to the center of the lens is very high. Due to this, when examining the slits up close, they begin to resemble parallel lines. As a result, it will be determined whether or not the intensity distribution in the aperture plane from a circular zone plate in circular coordinates can be approximately modeled as that of a linear zone plate in cartesian coordinates. The radial intensity distribution from an annular slit can be modeled as shown in Equation B.1. For a linear slit, the intensity distribution in the Cartesian plane can be modeled as shown in Equation B.2 for 1 ≤ i ≤ N . 4Io Io,c (θ) = (1 − 2i )2
J1 (βi ) − i J1 (i βi ) βi
Io,l (θ) =Io sinc2 βi
!2
(B.1)
(B.2)
where i is the ratio of the inner radius to the outer radius of the ith slit and βi for the ith slit is defined below
βi =
πai sin θi λ
121
(B.3)
APPENDIX B. ANNULAR SLIT MODELING A Matlab simulation was performed in order to analyze the extent to which the two cases are similar. First, the difference between the intensity profiles at the aperture of the circular and linear zone plates for the same ai ’s and di ’s were calculated. The intensity profiles were calculated as a function of radius for the circular zone plate and as a function of Cartesian coordinates for the linear zone plate. Next, the means and medians of the errors between the two simulations were calculated. They are shown below. Let Io,d = Io,c − Io,l . Then, Io,d = 1.65% Io,c
(B.4)
Io,d = 1.62% Io,l
(B.5)
Ig o,d = 0.35% g Io,c
(B.6)
Ig o,d = 0.36% g Io,l
(B.7)
From these results it is possible to conclude that the difference in intensity profile between a circular and linear lens is less than 2%. This difference is quite small and thus establishes a good argument for allowing circular lens to be approximately modeled as linear lens’ and vice versa. The Matlab code for performing this simulation is shown in section C.2. Figures B.1 and B.2 show closeups of the radiation profiles for the circular and linear lens respectively. Figures B.4 and B.5 show the same profiles but over a larger range. Figures B.3 and B.6 show the difference between the two profiles. It is immediately apparent that the magnitudes between the two profiles across most of their spectra are very similar, with minimal difference between them.
122
APPENDIX B. ANNULAR SLIT MODELING
0.25
I [W/(m 2nm)]
0.2
0.15
0.1
0.05
0 -1
0
1
Distance in focal plane [m]
×10-4
Figure B.1: Closeup view of radial intensity distribution profile of a circular lens as a function of radial distance in focal plane
0.25
I [W/(m 2nm)]
0.2
0.15
0.1
0.05
0 -1
0
Distance in focal plane [m]
1 ×10-4
Figure B.2: Closeup view of intensity distribution profile of a linear zone plate lens as a function of distance in the Cartesian plane from the center of the lens in the focal plane.
123
APPENDIX B. ANNULAR SLIT MODELING
×10-3
10 9 8
I [W/(m 2nm)]
7 6 5 4 3 2 1 0 -1
0
1
Distance in focal plane [m]
×10-4
Figure B.3: Closeup view of error between intensity profiles of circular and linear lenses.
0.25
I [W/(m 2nm)]
0.2
0.15
0.1
0.05
0 -0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Distance in focal plane [m]
Figure B.4: Radial intensity distribution profile of a circular lens as a function of radial distance in focal plane.
124
APPENDIX B. ANNULAR SLIT MODELING
0.3
0.25
I [W/(m 2nm)]
0.2
0.15
0.1
0.05
0 -0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Distance in focal plane [m]
Figure B.5: Intensity distribution profile of a linear zone plate lens in the Cartesian plane as a function of distance from the center of the lens in the focal plane.
12
×10-3
10
I [W/(m 2nm)]
8 6 4 2 0 -2 -0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
Distance in focal plane [m]
Figure B.6: Error between intensity profiles of circular and linear lenses.
125
0.01
Appendix C Matlab Code C.1
Calculating blackbody curves Listing C.1: LightSource.m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
% % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % CONSTANTS + e = exp (1) ; % numerical constant e h = 6.626*10^ -34; % plancks constant in Js kb = 1.381*10^ -23; % Boltzmann constant in J / K sigma = 5.67*10^ -8; % Stefan - Boltzmann constant co = 2.998*10^8; % speed of light in vacuum lambda = 100*10^ -9:1*10^ -9:10000*10^ -9; % wavelength for T = 300:200:1700 I = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./... ( lambda .* kb .* T ) ) -1) ) ; hold on E = pi * I ; plot ( lambda , E ) % perform integration over blackbody curve for % C2H2 region of interest ( ie : 2970 nm - 3120 nm ) % in [ W / m3 ]. Area under curve is intensity . % Multiply by pi since is emissive radiator I = trapz ( lambda (2871:3021) , E (2871:3021) ) hold off end % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % Draw different blackbodies for different Hawkeyes T1 = 1375+273.15; I1 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T1 ) ) -1) ) ; % [ W / m3 ] % % ---------------------------------------------------T2 = 1385+273.15; 126
APPENDIX C. MATLAB CODE 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
I2 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T2 ) ) -1) ) ; % [ W / m3 ] % % ---------------------------------------------------T3 = 1170+273.15; I3 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T3 ) ) -1) ) ; % [ W / m3 ] % % ---------------------------------------------------T4 = 1160+273.15; I4 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T4 ) ) -1) ) ; % [ W / m3 ] % % ---------------------------------------------------T5 = 1200+273.15; I5 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T5 ) ) -1) ) ; % [ W / m3 ] % % ---------------------------------------------------T6 = 1025+273.15; I6 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T6 ) ) -1) ) ; % [ W / m3 ] % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = plot ( lambda *1 e9 , I1 , lambda *1 e9 , I2 , lambda *1 e9 , I3 , ... lambda *1 e9 , I4 , lambda *1 e9 , I5 , lambda *1 e9 , I6 ) abbrv = char ( 'T = 1375 C , IR - Si207 , 24 W , 12 V ' , ... 'T = 1385 C , IR - Si217 , 37 W , 24 V ' , ... 'T = 1170 C , IR - Si253 , 20 W , 12 V ' , ... 'T = 1160 C , IR - Si272 , 30 W , 6 V ' , ... 'T = 1200 C , IR - Si295 , 40 W , 12 V ' , ... 'T = 1025 C , IR - Si311 , 70 W , 12 V ') ; ylabel ( ' Spectral emissive power [( W / sr * m3 ) ] ') xlabel ( ' Wavelength [ nm ] ') % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = trapz ( lambda (2871:3021) , I (2871:3021) ) % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = T1 = 1375+273.15; I1 = (2* h *( co ^2) ) ./( lambda .^5 .*( e .^( h .* co ./( lambda ... .* kb .* T1 ) ) -1) ) ; % [ W / m3 ] SEP1 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T1 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; A1 = 3*4.4; P1 = SEP1 * A1 ; % % ---------------------------------------------------T2 = 1385+273.15; I2 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T2 ) ) -1) ) ; % [ W / m3 ] SEP2 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T2 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; A2 = 6*4.4; P2 = SEP2 * A2 ; % % ---------------------------------------------------T3 = 1170+273.15; I3 = (2* h *( co ^2) ) ./( lambda .^5.*( e .^( h .* co ./( lambda ... .* kb .* T3 ) ) -1) ) ; % [ W / m3 ] SEP3 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T3 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; 127
APPENDIX C. MATLAB CODE 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109
A3 = 2*5; P3 = SEP3 * A3 ; % % ---------------------------------------------------T4 = 1160+273.15; I4 = (2* h *( co ^2) ) ./( lambda .^5.*( e .^( h .* co ./( lambda ... .* kb .* T4 ) ) -1) ) ; % [ W / m3 ] SEP4 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T4 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; A4 = 2.8*5; P4 = SEP4 * A4 ; % % ---------------------------------------------------T5 = 1200+273.15; I5 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T5 ) ) -1) ) ; % [ W / m3 ] SEP5 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T5 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; A5 = 3.5*12; P5 = SEP5 * A5 ; % % ---------------------------------------------------T6 = 1025+273.15; I6 = (2* h *( co ^2) ) ./( lambda .^5 .* ( e .^( h .* co ./( lambda ... .* kb .* T6 ) ) -1) ) ; % [ W / m3 ] SEP6 = integral ( @ ( lambda ) (2* h *( co ^2) ) ./( lambda .^5... .*( e .^( h .* co ./( lambda .* kb .* T6 ) ) -1) ) ,2970 e -9 ,3120 e -9) ; A6 = 4.5*17; P6 = SEP6 * A6 ;
128
APPENDIX C. MATLAB CODE
C.2
Calculating zone plate parameters Listing C.2: LensCalc.m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
% min and max wavelengths [ x10 nm ] min_wavelength = 1 max_wavelength = 1200; skip_wavelength = 1; % min and max focal lengths [ mm ] min_f = 1; max_f = 200; skip_f = 1; % min and max aperture RADIUS [ um ] min_ap = 10; max_ap = 400; skip_ap = 10; % min and max number of zones N [ -] min_N = 10; max_N = 400; skip_N = 10; % min and max a1 ' s [ um ] min_a1 = 1; max_a1 = 300; skip_a1 = 1; % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = lambda_start = 2.97 e -6; lambda_end = 3.12 e -6; Lambda_Design = 0.5*( lambda_start + lambda_end ) ; Lambda = Lambda_Design ; f1 = 50.8; f_design = 0.001* f1 - rem (0.001* f1 , Lambda_Design ) + ... 0.25* Lambda_Design ; ap = 50 e -6; ap_theta = pi /2000; N = 10; maxangle = pi /10; res = 2e -6; l (1) = f_design + Lambda_Design ; d (1) = sqrt ( l (1) ^2 - f_design ^2) ; ko = 2* pi / Lambda_Design ; load ( ' a1s . mat ') % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % ****** Uncomment to change focal length ****** % for f1 = min_f : skip_f : max_f % x = 0: res :0.001* f1 .* tan ( maxangle ) ; f_design = 0.001* f1 - rem (0.001* f1 , Lambda_Design ) +... 0.25* Lambda_Design ; f = f_design ; % ****** Uncomment to change N zones ****** % for N = min_N : skip_N : max_N % ****** Uncomment to change a (1) ****** % for a1 = min_a1 : skip_a1 : max_a1 129
APPENDIX C. MATLAB CODE 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
% %
% % %
% % % % % % % % %
clear l d sumd a Parea sumParea ****** Uncomment if changing focal length ****** a1 = a1s ( f1 /5 , 2) a1 = 137; % change depending on focal length a (1) = a1 *1 e -6; l (1) = f_design + Lambda_Design ; d (1) = sqrt ( l (1) ^2 - f_design ^2) ; Parea (1) = pi *(( d (1) +0.5* a (1) ) ^2 -( d (1) -0.5* a (1) ) ^2) ; sumParea (1) = Parea (1) ; eps (1) = ( d (1) - 0.5* a (1) ) /( d (1) + 0.5* a (1) ) ; for i = 2: N l ( i ) = f_design + i * Lambda_Design ; d ( i ) = sqrt ( l ( i ) ^2 - f_design ^2) - sum ( d ) ; a ( i ) = a (1) *( d (1) / sum ( d ) ) ; sumd ( i ) = sum ( d ) ; Parea ( i ) = pi *(( d ( i ) +0.5* a ( i ) ) ^2 - ... ( d ( i ) -0.5* a ( i ) ) ^2) ; sumParea ( i ) = sum ( Parea ) ; eps ( i ) = ( d ( i ) - 0.5* a ( i ) ) /( d ( i ) + 0.5* a ( i ) ) ; end ****** Uncomment to change aperture - lens dist ****** for f1 = min_f : skip_f : max_f x = 0: res : f1 .*0.001* tan ( maxangle ) ; x = 0: res :5000* res ; f = 0.001* f1 - rem (0.001* f1 , Lambda_Design ) + ... 0.25* Lambda_Design ; ****** Uncomment to change aperture radius ****** for ap1 = min_ap : skip_ap : max_ap ap = ap1 *1 e -6; % ---------------------------------------------------****** Uncomment one of these sections ****** SecondVar = f1 ; SecondVarSkip = skip_f ; SecondVarMin = min_f ; SecondVarMax = max_f ;
% % % %
SecondVar = N ; SecondVarSkip = skip_N ; SecondVarMin = min_N ; SecondVarMax = max_N
% % % %
SecondVar = ap1 ; SecondVarSkip = skip_ap ; SecondVarMin = min_ap ; SecondVarMax = max_ap ;
% SecondVar = a1 ; % SecondVarSkip = skip_a1 ; % SecondVarMin = min_a1 ; % SecondVarMax = max_a1 ; % % ---------------------------------------------------for LambdaRange = min_wavelength : skip_wavelength :... max_wavelength 130
APPENDIX C. MATLAB CODE 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154
clear PathLength1 Phase_test1 alpha_test1 ... beta_test1 E_test1 E_test_tot I_test_tot ... PathLength2 Phase_test2 alpha_test2 ... beta_test2 E_test2 xnew ynew Lambda = LambdaRange * 1e -8 f1 SecondVar ; 'N =10 ' ko = 2* pi / Lambda ; a(N); for i = 2: N PathLength1 (: , i ) = sqrt ( f ^2 + ( x - ... sumd ( i ) ) .^2) ; Phase_test1 (: , i ) = cos ( pi .*(( PathLength1 ... (: , i ) ./ Lambda ) - floor ( PathLength1 ... (: , i ) ./ Lambda ) ) ) ; PathLength2 (: , i ) = sqrt ( f ^2 + ( x + ... sumd ( i ) ) .^2) ; Phase_test2 (: , i ) = cos ( pi .*(( PathLength2 ... (: , i ) ./ Lambda ) - floor ( PathLength2 ... (: , i ) ./ Lambda ) ) ) ; alpha_test1 (: , i ) = atan (( x - sumd ( i ) ) / f ) ; beta_test1 (: , i ) = ko * a ( i ) * sin ( alpha_test1 ... (: , i ) ) ./2; alpha_test2 (: , i ) = atan (( x + sumd ( i ) ) / f ) ; beta_test2 (: , i ) = ko * a ( i ) * sin (... alpha_test2 (: , i ) ) ./2; E_test1lin (: , i ) = Phase_test1 (: , i ) .* sin ... ( beta_test1 (: , i ) ) ./ beta_test1 (: , i ) ; E_test2lin (: , i ) = Phase_test2 (: , i ) .* sin ... ( beta_test2 (: , i ) ) ./ beta_test2 (: , i ) ; E_test1circ (: , i ) = Phase_test1 (: , i ) .*(... besselj (1 , beta_test1 (: , i ) ) - eps ( i ) .*... besselj (1 , eps ( i ) .* beta_test1 (: , i ) ) ) ... ./( beta_test1 (: , i ) .*(1 - eps ( i ) .^2) ) ; E_test2circ (: , i ) = Phase_test2 (: , i ) .*(... besselj (1 , beta_test2 (: , i ) ) - eps ( i ) .*... besselj (1 , eps ( i ) .* beta_test2 (: , i ) ) ) ... ./( beta_test2 (: , i ) .*(1 - eps ( i ) .^2) ) ; end for i = 1: length ( x ) E_test_totlin ( i ) = sum ( E_test1lin (i ,:) ) +... sum ( E_test2lin (i ,:) ) ; I_test_totlin ( i ) = sum ( Parea ) *... ( E_test_totlin ( i ) .^2) ; E_test_totcirc ( i ) = sum ( E_test1circ (i ,:) ) ... + sum ( E_test2circ (i ,:) ) ; I_test_totcirc ( i ) = 4* sum ( Parea ) *... ( E_test_totcirc ( i ) .^2) ; end for i = 1: length ( x ) if x ( i ) =... ( Lambda *10) - 5 &&... NewGasAbsorptionArray (i , 2* Gas - 1) ... < ( Lambda *10) + 5 for f = 1:1:200 Abs_avg (f , Lambda , Gas ) = ... Abs_avg (f , Lambda , Gas ) +... NewGasAbsorptionArray (i , 2* Gas ) ; end elseif NewGasAbsorptionArray (i , ... 2* Gas - 1) == ( Lambda +1) *10 Marker = i ; flag = true ; end if flag , break , end end end end % make in nm by mulitplying 1200 x10 =12000 nm for i = 1:1200 RedrawAbs_avg (: , i *10 , :) = Abs_avg (: , i , :) ; end % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % % Calculate variables relating to absorption IntegratedTotalAbsorption = zeros (200 , 1200 , 9) ; for Gas = 1: length ( GasLengths ) EffectiveAbsorption (: ,: , Gas ) = ARESR .* Abs_avg (: ,: , Gas ) ; end for focal = 1:200 for Lambda = 1:1200 TotalAbsorption ( focal , Lambda ) =... sum ( EffectiveAbsorption ( focal , Lambda ,:) ) ; end end for focal = 1:200 for Gas = 1: length ( GasLengths ) for Lambda = 1:1200 RelativeAbsorption ( focal , Lambda , Gas ) =... EffectiveAbsorption ( focal , Lambda , Gas ) ... ./ TotalAbsorption ( focal , Lambda ) ; end end end for focal = 1:200 for Gas = 1: length ( GasLengths ) IntegratedTotalAbsorption ( focal , : , :) = ... 139
APPENDIX C. MATLAB CODE 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
IntegratedTotalAbsorption ( focal , : , :) + ... sum ( EffectiveAbsorption ( focal ,: , Gas ) ) ;
end end for focal = 1:200 for Gas = 1: length ( GasLengths ) IntegratedEf fectiveAbsorption ( LensDesign , ... focal , Gas ) = sum ( EffectiveAbsorption ( focal ,: , Gas ) ) ; IntegratedRelativeAbsorption ( LensDesign , focal ... , Gas ) = sum ( EffectiveAbsorption ( focal ,: , Gas ) ) ... / IntegratedTotalAbsorption ( focal ,1 ,1) ; end end for i = 1:1200 EffectiveAbsorption2 (: , i *10 , :) = ... EffectiveAbsorption (: , i , :) ; end % % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % % Plot graphs of abs vs wavelength at different f focals = [33 , 36 , 46 , 47 , 49 , 51]; for i = 1:6 focals ( i ) ; subplot (2 ,3 , i ) plot (10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ... ,10:10:12000 ,1) , 'k ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,2) , 'm ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,3) , 'c ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,4) , 'r ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,5) , 'g ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,6) , 'b ' , ... 10:10:12000 , EffectiveAbsorption2 ( focals ( i ) ,... 10:10:12000 ,7) , 'y ') title ([ ' d_ { ap - lens } = ' , num2str ( focals ( i ) ) , ' mm ' ]) xlabel ( ' Wavelength \ lambda [ nm ] ') ylabel ( 'S \ eta_ { ap } [( mol / cm ^2) / nm ] ') end abbrv = char ( ' C2H2 : Acetylene ' ,... ' C2H4 : Ethylene ' , ... ' C2H6 : Ethane ' ,... ' CH4 : Methane ' , ... ' CO : Carbon Monoxide ' ,... ' CO2 : Carbon Dioxide ' ,... ' H2O : Water ') ; legend ( abbrv )
140
APPENDIX C. MATLAB CODE 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181
% % = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = % % Plot integrated effective absorption figure (4) plot (1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 1) , 'k ' , ... % C2H2 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 2) , 'm ' , ... % C2H4 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 3) , 'c ' , ... % C2H6 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 4) , 'r ' , ... % CH4 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 5) , 'g ' , ... % CO 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 6) , 'b ' , ... % CO2 1:200 , IntegratedRelativeAbsorption ( LensDesign , ... 1:200 , 7) , 'y ') % H2O grid on grid minor abbrv = char ( ' C2H2 : Acetylene ' ,... ' C2H4 : Ethylene ' , ... ' C2H6 : Ethane ' ,... ' CH4 : Methane ' , ... ' CO : Carbon Monoxide ' ,... ' CO2 : Carbon Dioxide ' ,... ' H2O : Water ') ; legend ( abbrv ) xlabel ( ' Focal Length ') ylabel ( ' Average Line Intensity S ( v ) ')
141
Appendix D L-Edit C Code for generating concentric rings Disclaimer: RingsMacro.c is largely based on code from Tanner’s L-Edit Resistor.c example code. The sample Resistor.c macro code has been modified to draw concentric rings in RingsMacro.c Listing D.1: RingsMacro.c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
# include < stdlib .h > # include < math .h > # include < string .h > # include < stdio .h > # include " ldata . h " # include " demo . h " /* Input Default values */ # define LAYER1 " GDS_01_DT_00 " # define LAYER1 " Open Space " FILE * fr ; /* declare file pointer for reading from text file */ void RingsMacro ( void ) { /* Get parameters from text file */ char STR [ PAGESIZE ] , STR2 [ PAGESIZE ]; int n ; char line [80]; /* open the file for reading */ fr = fopen ( " design1c . dta " , " rt " ) ; fr = fopen ( " design2c . dta " , " rt " ) ; fr = fopen ( " design3c . dta " , " rt " ) ; fr = fopen ( " design4c . dta " , " rt " ) ; fr = fopen ( " design5c . dta " , " rt " ) ;
142
APPENDIX D. L-EDIT C CODE FOR GENERATING CONCENTRIC RINGS 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77
// * ** * * ** * ** * * ** * ** * * ** * ** * * ** * * ** * ** * * ** * ** * * */ /* Draw boxes */ LFile File ; LCell Cell ; LTechnology InitialTechSettings , tech ; LGrid GridSettings ; long InitialGridSize ; /* units variables */ float UNITS ; /* input and design variables */ LLayer layer1 ; char layer1_name [ NAMESIZE ]; /* construction variables */ LPoint cursor_location ; LCoord x , y ; /* ************************ */ /* Form default values */ /* ************************ */ Cell = LCell_GetVisible () ; // the current cell File = LCell_GetFile ( Cell ) ; // the current file /* X and Y location */ cursor_location = LCursor_GetPosition () ; /* Grid size and technology size */ InitialTechSettings = tech = LFile_GetTechnology ( File ) ; LFile_GetGrid ( File , & GridSettings ) ; InitialGridSize = GridSettings . mouse_snap_grid_size ; tech = LFile_GetTechnology ( File ) ; UNITS = tech . num / tech . denom ; /* Dont divide by 0 */ if ( UNITS == 0) UNITS = 1; /* assign default input values */ strcpy ( layer1_name , LAYER1 ) ; while (1) { layer1 = LLayer_Find ( File , layer1_name ) ; break ; /* Come out of the loop */ } double bottom ; double top ; int i =0; double LensParameters [801]; LTorusParams tParams ; for ( i =1; i < 801; ++ i ) { fgets ( line , 11 , fr ) ; sscanf ( line , " % lf " , & LensParameters [ i ]) ; sprintf ( STR , " % lf \ n " , LensParameters [ i ]) ; if ( i %2 == 0) top = LensParameters [ i ]; 143
APPENDIX D. L-EDIT C CODE FOR GENERATING CONCENTRIC RINGS 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
}
else { bottom = LensParameters [ i ]; tParams . ptCenter = LPoint_Set (0 ,0) ; tParams . nInnerRadius = bottom ; tParams . nOuterRadius = top ; tParams . dStartAngle = 0; tParams . dStopAngle = 360; LSelection_AddObject ( LTorus_CreateNew ( Cell , layer1 , & tParams ) ) ; }
fclose ( fr ) ; LDisplay_Refresh () ; }
/* End of function : Res_Macro_main */
/* * * * * * ** * * * * * * * ** * * * * * * * ** * * * * * * * ** * * * * * * * ** * * * * * * * */ int UPI_Entry_Point ( void ) /* function that happens when you hit a key */ { LMacro_BindToHotKey ( KEY_R , " Draw Rings Geometry " , " RingsMacro " ) ; return 1; }
144
Appendix E Lens Geometric Dimensions E.1
Lens 1 Zone 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Table E.1: a [µm] d [µm] 143.00 556.22 101.11 230.41 82.56 176.80 71.50 149.06 63.95 131.33 58.38 118.74 54.04 109.19 50.55 101.64 47.66 95.47 45.21 90.30 43.11 85.89 41.27 82.07 39.65 78.72 38.21 75.75 36.91 73.09 35.74 70.70 34.67 68.52 33.70 66.54 32.80 64.72 31.97 63.04 31.20 61.48 30.48 60.04 29.81 58.69 29.18 57.43 28.59 56.25 28.03 55.14
Lens 1 - λ = 3045nm, f = 50.8mm dsum [µm] Zone a [µm] d [µm] dsum [µm] 556.22 201 10.06 19.82 7909.39 786.63 202 10.03 19.77 7929.15 963.43 203 10.01 19.72 7948.87 1112.49 204 9.98 19.67 7968.55 1243.82 205 9.96 19.63 7988.17 1362.56 206 9.93 19.58 8007.75 1471.75 207 9.91 19.53 8027.28 1573.39 208 9.88 19.49 8046.77 1668.86 209 9.86 19.44 8066.21 1759.16 210 9.84 19.39 8085.61 1845.05 211 9.81 19.35 8104.95 1927.12 212 9.79 19.30 8124.26 2005.84 213 9.77 19.26 8143.52 2081.59 214 9.74 19.22 8162.73 2154.68 215 9.72 19.17 8181.91 2225.38 216 9.70 19.13 8201.03 2293.91 217 9.68 19.08 8220.12 2360.44 218 9.65 19.04 8239.16 2425.16 219 9.63 19.00 8258.16 2488.20 220 9.61 18.96 8277.11 2549.69 221 9.59 18.91 8296.03 2609.73 222 9.57 18.87 8314.90 2668.42 223 9.54 18.83 8333.73 2725.85 224 9.52 18.79 8352.52 2782.10 225 9.50 18.75 8371.27 2837.24 226 9.48 18.71 8389.97 Continued on next page 145
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
a [µm] 27.51 27.01 26.54 26.10 25.67 25.27 24.88 24.51 24.16 23.82 23.50 23.18 22.89 22.60 22.32 22.05 21.79 21.54 21.30 21.07 20.84 20.63 20.41 20.21 20.01 19.82 19.63 19.44 19.27 19.09 18.92 18.76 18.60 18.44 18.29 18.14 18.00 17.86 17.72 17.58 17.45 17.32 17.20 17.07 16.95 16.83 16.72
d [µm] 54.09 53.10 52.16 51.27 50.43 49.62 48.86 48.12 47.42 46.75 46.11 45.49 44.90 44.33 43.78 43.25 42.74 42.25 41.77 41.31 40.87 40.44 40.02 39.62 39.22 38.84 38.47 38.11 37.76 37.42 37.09 36.77 36.46 36.15 35.85 35.56 35.28 35.00 34.73 34.46 34.21 33.95 33.71 33.46 33.23 33.00 32.77
dsum [µm] 2891.33 2944.43 2996.60 3047.87 3098.30 3147.92 3196.78 3244.90 3292.32 3339.07 3385.18 3430.67 3475.57 3519.90 3563.68 3606.93 3649.68 3691.93 3733.70 3775.01 3815.88 3856.32 3896.34 3935.96 3975.18 4014.03 4052.50 4090.61 4128.38 4165.80 4202.90 4239.67 4276.12 4312.27 4348.13 4383.69 4418.96 4453.96 4488.69 4523.16 4557.36 4591.31 4625.02 4658.48 4691.71 4724.71 4757.47
Zone 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
146
a [µm] d [µm] dsum [µm] 9.46 18.67 8408.64 9.44 18.63 8427.27 9.42 18.59 8445.85 9.40 18.55 8464.40 9.38 18.51 8482.91 9.36 18.47 8501.37 9.34 18.43 8519.80 9.32 18.39 8538.19 9.30 18.35 8556.55 9.28 18.31 8574.86 9.26 18.28 8593.13 9.24 18.24 8611.37 9.22 18.20 8629.57 9.20 18.16 8647.74 9.18 18.13 8665.86 9.16 18.09 8683.95 9.14 18.05 8702.01 9.12 18.02 8720.02 9.10 17.98 8738.00 9.08 17.94 8755.95 9.07 17.91 8773.86 9.05 17.87 8791.73 9.03 17.84 8809.57 9.01 17.80 8827.37 8.99 17.77 8845.14 8.97 17.73 8862.88 8.96 17.70 8880.57 8.94 17.67 8898.24 8.92 17.63 8915.87 8.90 17.60 8933.47 8.89 17.56 8951.03 8.87 17.53 8968.57 8.85 17.50 8986.06 8.83 17.46 9003.53 8.82 17.43 9020.96 8.80 17.40 9038.36 8.78 17.37 9055.73 8.77 17.33 9073.06 8.75 17.30 9090.36 8.73 17.27 9107.63 8.72 17.24 9124.87 8.70 17.21 9142.08 8.68 17.18 9159.26 8.67 17.15 9176.40 8.65 17.11 9193.52 8.64 17.08 9210.60 8.62 17.05 9227.65 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
a [µm] 16.61 16.49 16.38 16.28 16.17 16.07 15.97 15.87 15.77 15.68 15.58 15.49 15.40 15.31 15.22 15.14 15.05 14.97 14.89 14.81 14.73 14.65 14.57 14.50 14.42 14.35 14.28 14.21 14.14 14.07 14.00 13.93 13.87 13.80 13.74 13.67 13.61 13.55 13.49 13.43 13.37 13.31 13.25 13.20 13.14 13.09 13.03
d [µm] 32.55 32.33 32.12 31.91 31.70 31.50 31.30 31.11 30.92 30.73 30.55 30.37 30.19 30.02 29.85 29.68 29.51 29.35 29.19 29.04 28.88 28.73 28.58 28.43 28.29 28.15 28.01 27.87 27.73 27.60 27.46 27.33 27.20 27.08 26.95 26.83 26.71 26.59 26.47 26.35 26.24 26.12 26.01 25.90 25.79 25.68 25.58
dsum [µm] 4790.02 4822.35 4854.46 4886.37 4918.07 4949.57 4980.87 5011.98 5042.90 5073.63 5104.18 5134.55 5164.74 5194.76 5224.61 5254.29 5283.80 5313.16 5342.35 5371.39 5400.27 5429.00 5457.58 5486.01 5514.30 5542.45 5570.45 5598.32 5626.05 5653.64 5681.11 5708.44 5735.64 5762.72 5789.67 5816.50 5843.21 5869.80 5896.27 5922.62 5948.86 5974.98 6000.99 6026.89 6052.69 6078.37 6103.95
Zone 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
147
a [µm] d [µm] dsum [µm] 8.60 17.02 9244.68 8.59 16.99 9261.67 8.57 16.96 9278.63 8.56 16.93 9295.56 8.54 16.90 9312.46 8.53 16.87 9329.34 8.51 16.84 9346.18 8.50 16.81 9362.99 8.48 16.78 9379.78 8.46 16.76 9396.53 8.45 16.73 9413.26 8.43 16.70 9429.96 8.42 16.67 9446.63 8.41 16.64 9463.27 8.39 16.61 9479.88 8.38 16.58 9496.47 8.36 16.56 9513.03 8.35 16.53 9529.55 8.33 16.50 9546.06 8.32 16.47 9562.53 8.30 16.45 9578.98 8.29 16.42 9595.40 8.28 16.39 9611.79 8.26 16.37 9628.15 8.25 16.34 9644.49 8.23 16.31 9660.81 8.22 16.29 9677.09 8.21 16.26 9693.35 8.19 16.23 9709.58 8.18 16.21 9725.79 8.16 16.18 9741.97 8.15 16.15 9758.12 8.14 16.13 9774.25 8.12 16.10 9790.36 8.11 16.08 9806.43 8.10 16.05 9822.49 8.08 16.03 9838.51 8.07 16.00 9854.52 8.06 15.98 9870.49 8.05 15.95 9886.45 8.03 15.93 9902.37 8.02 15.90 9918.28 8.01 15.88 9934.15 7.99 15.85 9950.01 7.98 15.83 9965.84 7.97 15.81 9981.64 7.96 15.78 9997.42 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
a [µm] 12.98 12.92 12.87 12.82 12.77 12.72 12.67 12.62 12.57 12.52 12.47 12.42 12.38 12.33 12.28 12.24 12.19 12.15 12.10 12.06 12.02 11.98 11.93 11.89 11.85 11.81 11.77 11.73 11.69 11.65 11.61 11.57 11.53 11.50 11.46 11.42 11.39 11.35 11.31 11.28 11.24 11.21 11.17 11.14 11.11 11.07 11.04
d [µm] 25.47 25.37 25.27 25.16 25.06 24.96 24.87 24.77 24.68 24.58 24.49 24.40 24.30 24.21 24.12 24.04 23.95 23.86 23.78 23.69 23.61 23.53 23.45 23.37 23.29 23.21 23.13 23.05 22.97 22.90 22.82 22.75 22.67 22.60 22.53 22.46 22.39 22.32 22.25 22.18 22.11 22.04 21.98 21.91 21.84 21.78 21.71
dsum [µm] 6129.42 6154.79 6180.05 6205.22 6230.28 6255.24 6280.11 6304.88 6329.56 6354.14 6378.62 6403.02 6427.32 6451.54 6475.66 6499.70 6523.65 6547.51 6571.29 6594.98 6618.59 6642.12 6665.57 6688.93 6712.22 6735.43 6758.55 6781.60 6804.58 6827.48 6850.30 6873.05 6895.72 6918.32 6940.85 6963.31 6985.70 7008.01 7030.26 7052.44 7074.55 7096.59 7118.57 7140.48 7162.32 7184.10 7205.81
Zone 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367
148
a [µm] d [µm] dsum [µm] 7.94 15.76 10013.18 7.93 15.73 10028.92 7.92 15.71 10044.63 7.91 15.69 10060.31 7.89 15.66 10075.97 7.88 15.64 10091.61 7.87 15.62 10107.23 7.86 15.59 10122.82 7.85 15.57 10138.39 7.83 15.55 10153.94 7.82 15.52 10169.46 7.81 15.50 10184.96 7.80 15.48 10200.44 7.79 15.46 10215.90 7.77 15.43 10231.33 7.76 15.41 10246.74 7.75 15.39 10262.13 7.74 15.37 10277.50 7.73 15.34 10292.85 7.72 15.32 10308.17 7.70 15.30 10323.47 7.69 15.28 10338.75 7.68 15.26 10354.01 7.67 15.24 10369.24 7.66 15.21 10384.46 7.65 15.19 10399.65 7.64 15.17 10414.82 7.63 15.15 10429.97 7.62 15.13 10445.10 7.60 15.11 10460.21 7.59 15.09 10475.30 7.58 15.07 10490.37 7.57 15.05 10505.41 7.56 15.03 10520.44 7.55 15.01 10535.44 7.54 14.98 10550.43 7.53 14.96 10565.39 7.52 14.94 10580.34 7.51 14.92 10595.26 7.50 14.90 10610.16 7.49 14.88 10625.05 7.48 14.86 10639.91 7.47 14.84 10654.75 7.45 14.82 10669.58 7.44 14.80 10684.38 7.43 14.78 10699.17 7.42 14.77 10713.93 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
E.2
a [µm] 11.01 10.97 10.94 10.91 10.88 10.84 10.81 10.78 10.75 10.72 10.69 10.66 10.63 10.60 10.57 10.54 10.51 10.48 10.46 10.43 10.40 10.37 10.35 10.32 10.29 10.26 10.24 10.21 10.18 10.16 10.13 10.11 10.08
d [µm] 21.65 21.59 21.52 21.46 21.40 21.34 21.28 21.22 21.16 21.10 21.04 20.98 20.92 20.87 20.81 20.75 20.70 20.64 20.59 20.53 20.48 20.43 20.37 20.32 20.27 20.22 20.17 20.12 20.06 20.01 19.96 19.91 19.87
dsum [µm] 7227.46 7249.05 7270.57 7292.03 7313.43 7334.77 7356.05 7377.27 7398.42 7419.52 7440.56 7461.55 7482.47 7503.34 7524.15 7544.90 7565.60 7586.25 7606.84 7627.37 7647.85 7668.28 7688.65 7708.98 7729.24 7749.46 7769.63 7789.74 7809.81 7829.82 7849.79 7869.70 7889.57
Zone 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
a [µm] 7.41 7.40 7.39 7.38 7.37 7.36 7.35 7.34 7.33 7.32 7.31 7.30 7.29 7.28 7.28 7.27 7.26 7.25 7.24 7.23 7.22 7.21 7.20 7.19 7.18 7.17 7.16 7.15 7.14 7.13 7.13 7.12 7.11
d [µm] 14.75 14.73 14.71 14.69 14.67 14.65 14.63 14.61 14.59 14.57 14.56 14.54 14.52 14.50 14.48 14.46 14.44 14.43 14.41 14.39 14.37 14.35 14.34 14.32 14.30 14.28 14.27 14.25 14.23 14.21 14.20 14.18 14.16
dsum [µm] 10728.68 10743.41 10758.11 10772.80 10787.47 10802.12 10816.75 10831.36 10845.95 10860.53 10875.08 10889.62 10904.14 10918.64 10933.12 10947.58 10962.03 10976.46 10990.86 11005.25 11019.63 11033.98 11048.32 11062.64 11076.94 11091.22 11105.49 11119.74 11133.97 11148.19 11162.38 11176.56 11190.72
Lens 2 Zone 1 2 3 4 5 6 7
Table E.2: Lens 2 - λ = 3045nm, f = 25.4mm a [µm] d [µm] dsum [µm] Zone a [µm] d [µm] dsum [µm] 103.00 393.31 393.31 201 7.22 14.14 5609.50 72.83 162.93 556.25 202 7.20 14.10 5623.60 59.46 125.03 681.28 203 7.19 14.07 5637.67 51.50 105.42 786.70 204 7.17 14.04 5651.71 46.06 92.88 879.58 205 7.15 14.00 5665.71 42.04 83.98 963.56 206 7.13 13.97 5679.68 38.92 77.24 1040.80 207 7.12 13.94 5693.62 Continued on next page 149
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
a [µm] 36.41 34.33 32.56 31.05 29.72 28.56 27.52 26.58 25.74 24.97 24.26 23.62 23.02 22.46 21.95 21.46 21.01 20.59 20.18 19.81 19.45 19.11 18.79 18.48 18.19 17.91 17.65 17.39 17.15 16.91 16.69 16.47 16.27 16.07 15.87 15.69 15.51 15.33 15.17 15.00 14.85 14.69 14.55 14.40 14.26 14.13 13.99
d [µm] 71.90 67.53 63.88 60.76 58.06 55.69 53.60 51.72 50.03 48.49 47.09 45.80 44.61 43.52 42.50 41.54 40.65 39.82 39.03 38.29 37.59 36.93 36.30 35.71 35.14 34.60 34.08 33.58 33.11 32.66 32.22 31.80 31.40 31.01 30.64 30.28 29.93 29.60 29.27 28.96 28.66 28.36 28.08 27.80 27.53 27.27 27.01
dsum [µm] 1112.69 1180.22 1244.10 1304.86 1362.93 1418.62 1472.22 1523.93 1573.96 1622.45 1669.54 1715.34 1759.95 1803.47 1845.96 1887.51 1928.16 1967.98 2007.01 2045.31 2082.90 2119.83 2156.14 2191.84 2226.98 2261.58 2295.66 2329.24 2362.35 2395.01 2427.23 2459.04 2490.44 2521.45 2552.09 2582.37 2612.30 2641.90 2671.17 2700.13 2728.79 2757.15 2785.23 2813.02 2840.55 2867.82 2894.84
Zone 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254
150
a [µm] d [µm] dsum [µm] 7.10 13.91 5707.52 7.08 13.87 5721.40 7.06 13.84 5735.24 7.05 13.81 5749.05 7.03 13.78 5762.83 7.01 13.75 5776.57 7.00 13.72 5790.29 6.98 13.68 5803.97 6.96 13.65 5817.63 6.95 13.62 5831.25 6.93 13.59 5844.84 6.92 13.56 5858.41 6.90 13.53 5871.94 6.88 13.50 5885.44 6.87 13.47 5898.92 6.85 13.45 5912.37 6.84 13.42 5925.78 6.82 13.39 5939.17 6.81 13.36 5952.53 6.79 13.33 5965.86 6.78 13.30 5979.16 6.76 13.28 5992.44 6.75 13.25 6005.69 6.73 13.22 6018.91 6.72 13.19 6032.10 6.70 13.16 6045.26 6.69 13.14 6058.40 6.67 13.11 6071.51 6.66 13.08 6084.60 6.64 13.06 6097.65 6.63 13.03 6110.69 6.62 13.01 6123.69 6.60 12.98 6136.67 6.59 12.95 6149.62 6.57 12.93 6162.55 6.56 12.90 6175.45 6.55 12.88 6188.33 6.53 12.85 6201.18 6.52 12.83 6214.01 6.51 12.80 6226.81 6.49 12.78 6239.58 6.48 12.75 6252.33 6.47 12.73 6265.06 6.45 12.70 6277.76 6.44 12.68 6290.44 6.43 12.65 6303.10 6.41 12.63 6315.73 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
a [µm] 13.87 13.74 13.62 13.50 13.39 13.27 13.16 13.06 12.95 12.85 12.75 12.65 12.56 12.47 12.37 12.29 12.20 12.11 12.03 11.95 11.87 11.79 11.71 11.64 11.56 11.49 11.42 11.35 11.28 11.21 11.14 11.08 11.01 10.95 10.89 10.83 10.77 10.71 10.65 10.59 10.54 10.48 10.43 10.37 10.32 10.27 10.22
d [µm] 26.77 26.53 26.29 26.07 25.85 25.63 25.42 25.21 25.01 24.82 24.63 24.44 24.26 24.08 23.91 23.74 23.57 23.41 23.25 23.09 22.94 22.79 22.64 22.49 22.35 22.21 22.08 21.94 21.81 21.68 21.56 21.43 21.31 21.19 21.07 20.95 20.84 20.73 20.62 20.51 20.40 20.30 20.19 20.09 19.99 19.89 19.79
dsum [µm] 2921.61 2948.13 2974.43 3000.50 3026.34 3051.97 3077.39 3102.61 3127.62 3152.44 3177.07 3201.51 3225.77 3249.85 3273.76 3297.49 3321.06 3344.47 3367.71 3390.80 3413.74 3436.52 3459.16 3481.65 3504.01 3526.22 3548.30 3570.24 3592.05 3613.73 3635.29 3656.72 3678.02 3699.21 3720.28 3741.24 3762.07 3782.80 3803.42 3823.93 3844.33 3864.62 3884.81 3904.90 3924.89 3944.78 3964.58
Zone 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
151
a [µm] d [µm] dsum [µm] 6.40 12.61 6328.34 6.39 12.58 6340.92 6.38 12.56 6353.48 6.36 12.54 6366.02 6.35 12.51 6378.53 6.34 12.49 6391.02 6.33 12.47 6403.49 6.31 12.44 6415.93 6.30 12.42 6428.36 6.29 12.40 6440.76 6.28 12.38 6453.13 6.27 12.35 6465.49 6.25 12.33 6477.82 6.24 12.31 6490.13 6.23 12.29 6502.42 6.22 12.27 6514.69 6.21 12.25 6526.93 6.20 12.22 6539.16 6.18 12.20 6551.36 6.17 12.18 6563.54 6.16 12.16 6575.70 6.15 12.14 6587.84 6.14 12.12 6599.96 6.13 12.10 6612.06 6.12 12.08 6624.13 6.10 12.06 6636.19 6.09 12.04 6648.22 6.08 12.02 6660.24 6.07 12.00 6672.24 6.06 11.97 6684.21 6.05 11.95 6696.17 6.04 11.94 6708.10 6.03 11.92 6720.02 6.02 11.90 6731.91 6.01 11.88 6743.79 6.00 11.86 6755.64 5.99 11.84 6767.48 5.98 11.82 6779.30 5.97 11.80 6791.10 5.96 11.78 6802.88 5.94 11.76 6814.64 5.93 11.74 6826.38 5.92 11.72 6838.10 5.91 11.70 6849.80 5.90 11.69 6861.49 5.89 11.67 6873.16 5.88 11.65 6884.81 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148
a [µm] 10.17 10.12 10.07 10.02 9.97 9.93 9.88 9.83 9.79 9.74 9.70 9.66 9.61 9.57 9.53 9.49 9.45 9.41 9.37 9.33 9.29 9.25 9.22 9.18 9.14 9.11 9.07 9.03 9.00 8.96 8.93 8.90 8.86 8.83 8.80 8.76 8.73 8.70 8.67 8.64 8.61 8.58 8.55 8.52 8.49 8.46 8.43
d [µm] 19.70 19.60 19.51 19.42 19.33 19.24 19.15 19.06 18.98 18.89 18.81 18.73 18.65 18.57 18.49 18.41 18.33 18.26 18.18 18.11 18.03 17.96 17.89 17.82 17.75 17.68 17.61 17.55 17.48 17.42 17.35 17.29 17.22 17.16 17.10 17.04 16.98 16.92 16.86 16.80 16.74 16.68 16.63 16.57 16.52 16.46 16.41
dsum [µm] 3984.27 4003.88 4023.39 4042.80 4062.13 4081.37 4100.52 4119.58 4138.56 4157.45 4176.26 4194.99 4213.63 4232.20 4250.69 4269.10 4287.43 4305.69 4323.87 4341.98 4360.01 4377.97 4395.87 4413.69 4431.44 4449.12 4466.73 4484.28 4501.76 4519.18 4536.53 4553.82 4571.04 4588.20 4605.30 4622.34 4639.32 4656.23 4673.09 4689.89 4706.63 4723.31 4739.94 4756.51 4773.03 4789.49 4805.89
Zone 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348
152
a [µm] d [µm] dsum [µm] 5.87 11.63 6896.44 5.86 11.61 6908.05 5.85 11.59 6919.64 5.84 11.58 6931.22 5.84 11.56 6942.77 5.83 11.54 6954.31 5.82 11.52 6965.84 5.81 11.50 6977.34 5.80 11.49 6988.83 5.79 11.47 7000.30 5.78 11.45 7011.75 5.77 11.43 7023.18 5.76 11.42 7034.60 5.75 11.40 7046.00 5.74 11.38 7057.38 5.73 11.37 7068.75 5.72 11.35 7080.10 5.71 11.33 7091.43 5.70 11.32 7102.74 5.69 11.30 7114.04 5.69 11.28 7125.32 5.68 11.27 7136.59 5.67 11.25 7147.84 5.66 11.23 7159.07 5.65 11.22 7170.29 5.64 11.20 7181.49 5.63 11.18 7192.67 5.62 11.17 7203.84 5.61 11.15 7214.99 5.61 11.14 7226.13 5.60 11.12 7237.25 5.59 11.10 7248.35 5.58 11.09 7259.44 5.57 11.07 7270.51 5.56 11.06 7281.57 5.56 11.04 7292.61 5.55 11.03 7303.64 5.54 11.01 7314.65 5.53 11.00 7325.65 5.52 10.98 7336.63 5.51 10.97 7347.59 5.51 10.95 7358.54 5.50 10.94 7369.48 5.49 10.92 7380.40 5.48 10.91 7391.30 5.47 10.89 7402.19 5.46 10.88 7413.07 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
a [µm] 8.40 8.37 8.34 8.32 8.29 8.26 8.24 8.21 8.18 8.16 8.13 8.10 8.08 8.05 8.03 8.00 7.98 7.96 7.93 7.91 7.88 7.86 7.84 7.81 7.79 7.77 7.75 7.72 7.70 7.68 7.66 7.64 7.61 7.59 7.57 7.55 7.53 7.51 7.49 7.47 7.45 7.43 7.41 7.39 7.37 7.35 7.33
d [µm] 16.35 16.30 16.25 16.19 16.14 16.09 16.04 15.99 15.94 15.89 15.84 15.79 15.75 15.70 15.65 15.60 15.56 15.51 15.47 15.42 15.38 15.33 15.29 15.25 15.20 15.16 15.12 15.08 15.04 14.99 14.95 14.91 14.87 14.83 14.79 14.76 14.72 14.68 14.64 14.60 14.56 14.53 14.49 14.45 14.42 14.38 14.35
dsum [µm] 4822.25 4838.55 4854.79 4870.98 4887.13 4903.22 4919.26 4935.25 4951.19 4967.08 4982.92 4998.71 5014.46 5030.16 5045.81 5061.41 5076.97 5092.48 5107.95 5123.37 5138.75 5154.08 5169.37 5184.62 5199.83 5214.99 5230.11 5245.18 5260.22 5275.21 5290.17 5305.08 5319.96 5334.79 5349.58 5364.34 5379.06 5393.73 5408.37 5422.98 5437.54 5452.07 5466.56 5481.01 5495.43 5509.81 5524.16
Zone 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395
153
a [µm] d [µm] dsum [µm] 5.46 10.86 7423.93 5.45 10.85 7434.78 5.44 10.83 7445.61 5.43 10.82 7456.43 5.43 10.80 7467.23 5.42 10.79 7478.02 5.41 10.77 7488.79 5.40 10.76 7499.55 5.39 10.75 7510.30 5.39 10.73 7521.03 5.38 10.72 7531.75 5.37 10.70 7542.45 5.36 10.69 7553.14 5.36 10.68 7563.82 5.35 10.66 7574.48 5.34 10.65 7585.13 5.33 10.63 7595.77 5.33 10.62 7606.39 5.32 10.61 7616.99 5.31 10.59 7627.59 5.30 10.58 7638.17 5.30 10.57 7648.74 5.29 10.55 7659.29 5.28 10.54 7669.83 5.27 10.53 7680.36 5.27 10.51 7690.87 5.26 10.50 7701.37 5.25 10.49 7711.86 5.25 10.47 7722.33 5.24 10.46 7732.80 5.23 10.45 7743.24 5.22 10.44 7753.68 5.22 10.42 7764.10 5.21 10.41 7774.51 5.20 10.40 7784.91 5.20 10.38 7795.30 5.19 10.37 7805.67 5.18 10.36 7816.03 5.18 10.35 7826.37 5.17 10.33 7836.71 5.16 10.32 7847.03 5.16 10.31 7857.34 5.15 10.30 7867.64 5.14 10.29 7877.92 5.14 10.27 7888.20 5.13 10.26 7898.46 5.12 10.25 7908.71 Continued on next page
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 196 197 198 199 200
E.3
a [µm] 7.31 7.30 7.28 7.26 7.24
d [µm] 14.31 14.28 14.24 14.21 14.17
dsum [µm] 5538.47 5552.75 5566.99 5581.19 5595.36
Zone 396 397 398 399 400
a [µm] 5.12 5.11 5.10 5.10 5.09
d [µm] 10.24 10.22 10.21 10.20 10.19
dsum [µm] 7918.94 7929.17 7939.38 7949.58 7959.77
Lens 3 Zone 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
a [µm] 58.70 41.51 33.89 29.35 26.25 23.96 22.19 20.75 19.57 18.56 17.70 16.94 16.28 15.69 15.16 14.67 14.24 13.84 13.47 13.13 12.81 12.51 12.24 11.98 11.74 11.51 11.30 11.09 10.90 10.72 10.54 10.38 10.22 10.07 9.92
Table E.3: Lens 3 - λ = 540nm, f = 50.8mm d [µm] dsum [µm] Zone a [µm] d [µm] dsum [µm] 234.23 234.23 70 7.01 14.06 1960.08 97.02 331.25 71 6.97 13.96 1974.03 74.45 405.70 72 6.92 13.86 1987.89 62.76 468.47 73 6.87 13.76 2001.66 55.30 523.76 74 6.82 13.67 2015.32 49.99 573.75 75 6.78 13.58 2028.90 45.97 619.73 76 6.73 13.49 2042.39 42.79 662.52 77 6.69 13.40 2055.79 40.19 702.71 78 6.65 13.31 2069.10 38.01 740.72 79 6.60 13.23 2082.32 36.16 776.88 80 6.56 13.14 2095.47 34.55 811.42 81 6.52 13.06 2108.53 33.14 844.56 82 6.48 12.98 2121.51 31.88 876.44 83 6.44 12.90 2134.41 30.76 907.21 84 6.40 12.83 2147.24 29.75 936.96 85 6.37 12.75 2159.99 28.84 965.80 86 6.33 12.67 2172.66 28.00 993.80 87 6.29 12.60 2185.26 27.24 1021.04 88 6.26 12.53 2197.79 26.53 1047.57 89 6.22 12.46 2210.25 25.87 1073.44 90 6.19 12.39 2222.64 25.26 1098.70 91 6.15 12.32 2234.96 24.70 1123.40 92 6.12 12.25 2247.21 24.16 1147.56 93 6.09 12.19 2259.40 23.67 1171.23 94 6.05 12.12 2271.52 23.20 1194.43 95 6.02 12.06 2283.57 22.76 1217.19 96 5.99 11.99 2295.57 22.34 1239.52 97 5.96 11.93 2307.50 21.94 1261.47 98 5.93 11.87 2319.37 21.57 1283.04 99 5.90 11.81 2331.18 21.21 1304.25 100 5.87 11.75 2342.93 20.87 1325.12 101 5.84 11.69 2354.62 20.55 1345.67 102 5.81 11.63 2366.25 20.24 1365.91 103 5.78 11.58 2377.83 19.95 1385.86 104 5.75 11.52 2389.35 Continued on next page 154
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
E.4
a [µm] 9.78 9.65 9.52 9.40 9.28 9.17 9.06 8.95 8.85 8.75 8.65 8.56 8.47 8.38 8.30 8.22 8.14 8.06 7.99 7.91 7.84 7.77 7.71 7.64 7.58 7.51 7.45 7.39 7.34 7.28 7.22 7.17 7.12 7.07
d [µm] 19.66 19.39 19.13 18.88 18.64 18.41 18.19 17.97 17.76 17.56 17.37 17.18 17.00 16.82 16.65 16.49 16.33 16.17 16.02 15.87 15.73 15.59 15.45 15.32 15.19 15.06 14.94 14.82 14.70 14.59 14.48 14.37 14.26 14.16
dsum [µm] 1405.52 1424.91 1444.04 1462.92 1481.56 1499.97 1518.16 1536.13 1553.89 1571.45 1588.82 1606.01 1623.00 1639.83 1656.48 1672.97 1689.29 1705.47 1721.48 1737.35 1753.08 1768.67 1784.12 1799.44 1814.63 1829.70 1844.64 1859.46 1874.16 1888.75 1903.23 1917.60 1931.86 1946.02
Zone 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137
a [µm] 5.73 5.70 5.67 5.65 5.62 5.60 5.57 5.54 5.52 5.50 5.47 5.45 5.43 5.40 5.38 5.36 5.33 5.31 5.29 5.27 5.25 5.23 5.21 5.19 5.17 5.15 5.13 5.11 5.09 5.07 5.05 5.03 5.01
d [µm] 11.47 11.41 11.36 11.31 11.25 11.20 11.15 11.10 11.05 11.00 10.95 10.91 10.86 10.81 10.77 10.72 10.68 10.64 10.59 10.55 10.51 10.46 10.42 10.38 10.34 10.30 10.26 10.22 10.19 10.15 10.11 10.07 10.04
dsum [µm] 2400.82 2412.23 2423.59 2434.89 2446.15 2457.35 2468.50 2479.60 2490.65 2501.66 2512.61 2523.52 2534.38 2545.19 2555.96 2566.69 2577.37 2588.00 2598.59 2609.14 2619.65 2630.11 2640.54 2650.92 2661.26 2671.56 2681.83 2692.05 2702.23 2712.38 2722.49 2732.56 2742.60
Lens 4 Zone 1 2 3 4 5 6
a [µm] 41.50 29.34 23.96 20.75 18.56 16.94
Table E.4: Lens 4 - λ = 540nm, f = 25.4mm d [µm] dsum [µm] Zone a [µm] d [µm] dsum [µm] 165.63 165.63 35 7.01 14.11 980.04 68.61 234.23 36 6.92 13.91 993.95 52.64 286.88 37 6.82 13.72 1007.66 44.38 331.26 38 6.73 13.53 1021.19 39.10 370.36 39 6.64 13.36 1034.55 35.35 405.71 40 6.56 13.19 1047.73 Continued on next page 155
APPENDIX E. LENS GEOMETRIC DIMENSIONS Zone 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
a [µm] 15.69 14.67 13.83 13.12 12.51 11.98 11.51 11.09 10.71 10.37 10.06 9.78 9.52 9.28 9.06 8.85 8.65 8.47 8.30 8.14 7.99 7.84 7.71 7.58 7.45 7.34 7.22 7.12
d [µm] 32.51 30.26 28.42 26.88 25.57 24.43 23.43 22.55 21.76 21.04 20.39 19.80 19.26 18.76 18.30 17.87 17.47 17.09 16.74 16.41 16.09 15.80 15.52 15.25 15.00 14.76 14.53 14.32
dsum [µm] 438.22 468.48 496.90 523.78 549.35 573.78 597.21 619.76 641.52 662.56 682.96 702.76 722.02 740.78 759.08 776.95 794.41 811.50 828.24 844.65 860.74 876.54 892.06 907.32 922.32 937.08 951.62 965.93
Zone 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
156
a [µm] 6.48 6.40 6.33 6.25 6.19 6.12 6.05 5.99 5.93 5.87 5.81 5.75 5.70 5.65 5.59 5.54 5.50 5.45 5.40 5.36 5.31 5.27 5.23 5.19 5.15 5.11 5.07 5.03
d [µm] 13.02 12.86 12.71 12.56 12.42 12.29 12.15 12.03 11.90 11.78 11.66 11.55 11.44 11.33 11.23 11.13 11.03 10.93 10.84 10.75 10.66 10.57 10.49 10.40 10.32 10.24 10.17 10.09
dsum [µm] 1060.76 1073.62 1086.33 1098.90 1111.32 1123.61 1135.76 1147.78 1159.68 1171.46 1183.13 1194.68 1206.12 1217.45 1228.67 1239.80 1250.83 1261.76 1272.60 1283.34 1294.00 1304.57 1315.06 1325.46 1335.78 1346.02 1356.19 1366.28
Appendix F Error measurement for calculating Voigt absorption profiles Numerous assumptions were made when calculating the absorption coefficient of each spectral line. From the HITRAN database, the number of spectral lines for C2 H2 is 20371 [21]. Each spectral line follows a Lorentzian profile and has an area that spans over a very wide spectral range as shown in Figure 3.4. The value of the function decreases exponentially as the test wavenumber deviates further from the center wavenumber. It is important for the calculated value of the area to be accurate. Therefore, the number of data points per spectral line must be chosen such that the computation time, but also the accuracy in the calculated area to the true area, under the line is reasonable. Also, one must choose an appropriate spectral range under which to calculate the area. This is because it can be computationally unreasonable to measure over very wide ranges for the advantage of minuscule accuracy improvements. It was necessary to determine an appropriate number of data points per some spectral line profile under which to integrate. For testing purposes, it was decided to choose a span of one wavenumber for calculating the area under such area. The area per wavenumber was calculated and plotted using MATLAB as shown in Figure F.1. For the simulation, a sample spectral line located at 729cm−1 was chosen. It was chosen for its relatively strong line intensity, thereby serving as a worst case sample for all other
157
APPENDIX F. ERROR MEASUREMENT FOR CALCULATING VOIGT ABSORPTION PROFILES
1.35
×10-18
Absorption [mol/cm2]
1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Wavenumber step resolution [cm-1]
Figure F.1: Spectral line integrated absorption as a function of wavenumber resolution at 2cm-1 wavenumber span
0.18 0.16 0.14
Error []
0.12 0.1 0.08 0.06 0.04 0.02 0 10-3
10-2
10-1
Wavenumber measurement range [cm-1]
Figure F.2: Spectral line integration error as a function of wavenumber resolution at 2cm-1 wavenumber span
158
APPENDIX F. ERROR MEASUREMENT FOR CALCULATING VOIGT ABSORPTION PROFILES 12
×10-19
Absorption [mol/cm2]
11
10
9
8
7
6 0
1
2
3
4
5
6
7
8
9
10
Wavenumber measurement range [cm-1]
Figure F.3: Spectral line integrated absorption as a function of measurement span at 0.01cm-1 resolution measurements. The error in calculating the area with respect to 1000 datapoints as a function of the total number of datapoints is shown in Figure F.2. As a result, it is apparent that the error in calculating the area with only 100 datapoints per wavenumber is well below 1%.
Therefore, all integration measurements are calculated with 100
datapoints. 0.45 0.4 0.35
Error []
0.3 0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber measurement range [cm-1]
Figure F.4: Spectral line integration error as a function of measurement span at 0.01cm-1 resolution Next, the wavenumber span under which to integrate was chosen. Since the 159
APPENDIX F. ERROR MEASUREMENT FOR CALCULATING VOIGT ABSORPTION PROFILES Lorentz function upon which the spectral line profile is derived decays exponentially, a wavenumber span from 0.2cm−1 to 10cm−1 was tested. The number of datapoints per wavenumber was chosen to be 100. The total absorption as a function of span is shown in Figure F.3. The error with respect to the longest span at 10cm−1 is shown in Figure F.4. As shown in Figure F.4, for an integration span of 2cm−1 at a sample spectral line located at 729cm−1 , the error is 4.7%. This error was deemed acceptable and as such all spectral lines were integrated over a 2cm−1 wavenumber span.
160
Appendix G Solidworks Model
Figure G.1: Side View of Optical Assembly
Figure G.2: Top View of Optical Assembly
161
APPENDIX G. SOLIDWORKS MODEL
Figure G.3: Orthogonal View of Optical Assembly 162
Appendix H Detector Circuit PCB
Figure H.1: Detector Circuit PCB
163
APPENDIX H. DETECTOR CIRCUIT PCB
Figure H.2: Detector Circuit Schematic 164
Appendix I Cleanroom Photos
Figure I.1: Lithography Alignment 165
APPENDIX I. CLEANROOM PHOTOS
Figure I.2: AlphaStep Measurement from Back-Etch
Figure I.3: AlphaStep Measurement from Zone Plate Edges
166
APPENDIX I. CLEANROOM PHOTOS
Figure I.4: AlphaStep Measurement from initial zones of VIS Light Zone Plate
Figure I.5: AlphaStep Microscope Photo of Zone Plate
167
APPENDIX I. CLEANROOM PHOTOS
Figure I.6: Detailed AlphaStep Microscope Photo of Zone Plate
Figure I.7: Surface roughness of backside
168
APPENDIX I. CLEANROOM PHOTOS
Figure I.8: Aluminum Etching
Figure I.9: Silicon wafers for Gas Chamber Windows
169
APPENDIX I. CLEANROOM PHOTOS
Figure I.10: MRC 8667 Sputtering System
170
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Index 1/f noise, 36
Boltzmann’s constant, 18
1N5817 Schottky diode, 103
Buffer Circuit, 100
504 Photoresist, 87
Bushing, 1
ABM Mask Aligner, 86
Carrier Gas, 5
Absorbance, 49
CCS175 Spectrometer, 57
Absorption, 14
Chopper, 6
Absorption coefficient, 44
Coherence, 28
AD630 Modulator/Demodulator, 100
Cole Parmer Fiber-lite Model 9745-00, 93
Alpha-step Profiler, 88
Combination bands, 15
Annular ring, 55
Combustible Gas Monitoring, 3
Aperture, 26
Conduction band, 35
Arcing, 2
Corona, 2
ARESR, 52 Augmented Reality, 110 Autocorrelation function, 28 Avogadro’s Number, 49
Depletion region, 35 Detectivity, 38 DGA, 11 Dielectric, 2
Backside Processing, 86
Dielectric Breakdown, 2
Beer-Lambert Law, 23
Diffusion, 5
Bending (molecular), 14
Diffusivity, 20
Bessel Function, 28
Dissolved Gas Analysis, 11
Binary Zone Plate, 50
Doppler broadening, 16
Blackbody radiation, 19
Dornenburg Ratio Method, 11
Bolometer, 33
DSP wafer, 49 176
INDEX Duval’s Triangle, 11 Electric dipole moment, 14 Electrical conductivity, 35 Electromagnetic Interference, 100 EMI, 100
Harmonic, 94 Hawkeye IR-Si217, 46 Heatsink, 31 HITRAN database, 15 Holographic, 110
Emissivity, 20
IEC 60599, 11
Energy bandgap, 35
IEEE C57.104-2008, 11
Etching, 86
Infrared absorption, 14
Fermi resonance, 15
Johnson noise, 36
Foreign-broadening, 18
Joule heating, 20
Fraunhofer Diffraction, 92
Key Gas Method, 11
Fraunhofer zone, 26 Free electron carrier, 35
Laser, 19
Fresnel Diffraction, 92
Laser diode, 19
Fresnel Zone Plate, 25
Least Squares Linear Regression, 82
Fresnel’s law of reflection, 23
LED, 19
Fresnel-Kirchoff Diffraction Formula, 26
Light emitting diode, 19
Frontside Processing, 87
Line broadening function, 16
FTIR Spectroscopy, 7
Line profile, 46
FWHM, 52
Linear slit, 55 Lithography, 86
Gain, 21 Gas Chromatography, 4 Gas Permeable Membrane, 3
Load-Tap Changer, 1 Lock-in amplifier, 6 Losses, 76
Gauss’ Law, 118 GE Hydran, 3, 4
MC2000 Optical Chopper, 71
GE Kelman Transfix, 6
MEMS, 8 Microscopy, 110
Hammamatsu T11262-01 Thermopile, 71
Microsoft HoloLens, 110
Hard Bake, 87 177
INDEX Mobile phase, 5
Pink noise, 36
Moisture, 3, 4
Piranha Clean, 84
Molar mass, 49
Planck’s Law, 21
Morgan Schaffer Calisto, 3
PLL, 39
MRC 8667 Sputtering System, 87
PN junction, 35
NEP, 37 Noise (detectors), 36 Noise equivalent power, 37 Normalized autocorrelation function, 28 NSFL, 83 Operational Amplifier, 103
Power Factor Test, 3 Pressure broadening, 16 Pyroelectric detector, 34 Pyroelectric effect, 34 Quantum efficiency, 31 Quantum energy state, 14
Overheating, 2
Radiation Detector, 31
Overtone bands, 15
Responsivity, 37
P polarization, 24 Parabolic Reflector, 21 Partial Discharge, 2 PAS, 6
Retention time, 4 Rocking (molecular), 14 Rogers Ratio Method, 11 Roll, pitch, yaw, 91
Phase locked loop, 39
S polarization, 24
Photoacoustic detector, 36
Savitsky-Golay, 89
Photoacoustic spectroscopy, 6
Sawtooth waveform, 109
Photoconductive detectors, 34
Scattering, 92
Photoelectric detectors, 34
Seebeck coefficient, 32
Photoelectric effect, 34
Seebeck effect, 32
Photoemissive detectors, 36
Self-broadening, 18
Photoemissive effect, 36
Semiconductor, 34
Photon, 14
Semiconductor radiation sources, 19
Photon noise, 36
Serveron TM8, 4
Photovoltaic detectors, 35
Signal conditioning, 38 178
INDEX Snell’s Law, 117
Wagging (molecular), 14
Soft Bake, 86
Wavenumber, 16
Sparking, 2
Weiner-Khitchine theorem, 30
Spatial autocorrelation function, 30
Wet Oxidation, 86
Spatial coherence, 28, 29
Window, 48
Spectral intensity, 21
Zone Plate, 50
Spectral line, 46 Spectral Resolution, 54 Spectrometer, 41 Sputtering, 87 Stationary phase, 5 Stretching (molecular), 14 Tanner L-Edit, 83 Temporal autocorrelation function, 30 Temporal coherence, 28, 30 Thermal absorption coefficient, 34 Thermal capacity, 31 Thermal detectors, 31 Thermal link, 31 Thermistor, 3 Thermocouple, 32 Thermopile, 32 Time constant, 31 Transformer, 1 Transmission Efficiency, 53 Twisting (molecular), 14 Valence band, 35 Vibration-rotation bands, 15 Vibrational mode, 14 179
Colophon This thesis was typeset in LATEX based on the open source template developed by Mauricio Lobos. The YouTube video explaining his template in detail is provided on his YouTube channel below: https://www.youtube.com/watch?v=Qjp-a2uZWZc The Google Drive folder where one may download Mauricio’s TEX templates is found in: https://drive.google.com/drive/u/0/folders/0B4HppN1nEJ6KSTl5ekp1M2t4U00 The IDE used for writing this document was TEXstudio version 2.11.2.
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