dispersive laser system, which produces a multiwavelength, ultrabroadband, ..... an ultrabroadband output spectrum spreading over a large spectral region, .... of the lens, OO âoptical axis, knGânormal to the diffraction grating, βâangle ..... represents the active element; an intracavity thin lens of focal length f; and a plane.
MULTIWAVELENGTH AND ULTRABROADBAND SOLID-STATE AND SEMICONDUCTOR SPATIALLY-DISPERSIVE LASERS
by IGOR S. MOSKALEV
A DISSERTATION Submitted to the graduate faculty of The University of Alabama at Birmingham, in partial fulfillment of the requirements for the degree of Doctor of Philosophy BIRMINGHAM, ALABAMA 2004
ABSTRACT OF DISSERTATION GRADUATE SCHOOL, UNIVERSITY OF ALABAMA AT BIRMINGHAM Degree
PhD
Program
Physics
Name of Candidate
Igor S. Moskalev
Committee Chair
Sergey B. Mirov
Title
Multiwavelength and Ultrabroadband Solid-State and Semiconductor Spatially-Dispersive Lasers This thesis is devoted to the experimental investigation of a flexible, spatially-
dispersive laser system, which produces a multiwavelength, ultrabroadband, tunable laser radiation. The principle of operation of these lasers is based on an efficient suppression of the mode competition, natural for conventional laser sources, by means of the spatial separation of different frequency components in the laser gain medium. The present work is devoted to the investigation of a continuous-wave, spatiallydispersive laser, where the spatial dispersion of the frequency components is achieved with a dispersive system, based on an intracavity focusing lens and a Littrow-mounted diffraction grating. Several new spatally-dispersive laser systems of this type are demonstrated: (1) a semiconductor multiwavelength, spatially-dispersive laser, based on a single-chip laser diode, operating in the spectral region of 650–670 nm; (2) a semiconductor multiwavelength, spatially-dispersive laser, based on a multi-stripe laser diode, operating in the spectral region of 1540–1580 nm; and (3) a solid-state, multiwavelength, ultrabroadband, tunable spatially-dispersive laser, based on the polycrystalline Cr2+ :ZnSe, operating in the spectral region of 2100–2850 nm.
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DEDICATION To my Mother, Ludmila G. Moskaleva
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ACKNOWLEDGMENTS I wish to express my deep gratitude to my advisor, Dr. Sergey B. Mirov, for his professional guidance through this work and during my entire study at the Department of Physics at UAB. I would like to personally thank Dr. Vladimir V. Fedorov for his invaluable help in the theoretical analysis and experimental realization of the laser systems presented in this work. I am greatly thankful to Dr. Tasoltan T. Basiev, one of the inventors of the laser system under consideration, for his invaluable help in developing the theoretical approach and experimental methods of this work. I wish to express my thanks to the members of my graduate committee, Drs. David Shealy, Gary Grimes, Chris Lawson, and Richard Fork. Without their unyielding assistance this work would not have been be possible. I would like to personally thank Dr. Jerry Sewell for giving me the unique opportunity to make extensive use of the facilities of his Mechanical Workshop: the practical realization of the laser systems, presented in this work, would not have been possible without his invaluable assistance. I wish to thank Professors G. Belenky and D. Garbuzov for providing the 1.55 µm and 1.85 µm diode lasers used in this work. I am also grateful to Dr. B. Badikov for providing Cr2+ :ZnSe crystals, which were used as the gain medium for the Cr2+ :ZnSe polycrystalline spatially-dispersive lasers. I thank Dr. Dmitri V. Martyshkin, whose cogent suggestions were extremely important in the preparation of this thesis. I am also very thankful to the Department of Physics for giving me the unique opportunity to study at UAB.
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TABLE OF CONTENTS Page ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
CHAPTER 1 INTRODUCTION . . . 1.1 Problem Statement 1.2 Historical Overview 1.3 Motivations . . . .
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1 1 3 6
2 THEORETICAL MODEL . . . . . . . . . . . . . . . . . . 2.1 Principle of Operation of the Spatially-Dispersive Laser 2.2 Theoretical Methods and Approximations . . . . . . . 2.3 The Laser Model in the Gaussian Beam Approximation 2.4 Spectral Resolution of the Spatially-Dispersive Cavity . 2.5 Maximum Number of Oscillating Wavelength Channels 2.6 Calculation of the Spatially-Dispersive Cavity . . . . . 2.7 Spatial Distribution of the Output Radiation . . . . . . 2.8 Complex Intracavity Focusing System . . . . . . . . . .
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10 10 15 19 28 31 37 42 45
3 EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cr2+ :ZnSe Polycrystalline Spatially-Dispersive Laser . . 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . 3.1.2 Available Optical Elements and Equipment . . 3.1.3 Preliminary Experiments . . . . . . . . . . . . 3.1.4 Dual-Wavelength, Grating-Tunable Cr2+ :ZnSe Dispersive Laser . . . . . . . . . . . . . . . . .
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50 50 50 52 56
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TABLE OF CONTENTS (Continued) Page Dual-Wavelength, Motion-Tunable Cr2+ :ZnSe SpatiallyDispersive Laser . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Dual-Wavelength, Spacing-Tunable Cr2+ :ZnSe SpatiallyDispersive Laser . . . . . . . . . . . . . . . . . . . . . . . 3.1.7 Spatial Distribution of the Output Radiation. Intracavity Mode Structure . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 Ultrabroadband, Multiwavelength Cr2+ :ZnSe SpatiallyDispersive Laser . . . . . . . . . . . . . . . . . . . . . . . 3.1.9 Modulation of the Ultrabroadband Spectrum by a SelfFormed Intracavity Fizeau Interferometer . . . . . . . . . 3.1.10 Tuning of the Multiwavelength Spectrum Across the Entire Gain Curve of the Cr2+ :ZnSe Active Medium. . . . . . . . 3.1.11 Summary: Cr2+ :ZnSe Spatially-Dispersive Laser . . . . . . 3.2 Semiconductor Multiwavelength Spatially-Dispersive Lasers . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Semiconductor Single-Chip Spatially-Dispersive Laser . . . 3.2.3 Semiconductor Multi-Stripe Spatially-Dispersive Laser . . 3.2.4 Summary: Semiconductor Spatially-Dispersive Lasers . . . 3.1.5
74 80 88 96 103 109 112 113 113 114 122 128
4 CONCLUSIONS AND FUTURE OUTLOOK . . . . . . . . . . . . . . 129 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 APPENDIX A THE KNIFE EDGE METHOD . . . . . . . . . . . . . . . . . . . . . . 145 B CHARACTERIZATION OF THE GAUSSIAN BEAMS
. . . . . . . . 148
C ALIGNMENT OF THE SPATIALLY-DISPERSIVE CAVITY . . . . . 152
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LIST OF TABLES Table
Page
1
Example calculation: parameters of gain elements and available optics . .
39
2
Available optics for the Cr2+ :ZnSe spatially-dispersive laser . . . . . . . .
53
3
Spatial properties of the output beams . . . . . . . . . . . . . . . . . . .
90
4
Major cavity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5
Measurement of the wavelength spacing . . . . . . . . . . . . . . . . . . . 107
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LIST OF FIGURES Figure
Page
1
Spatially-dispersive cavities proposed by Danailov and Christov . . . . . .
4
2
The spatially-dispersive laser proposed by Basiev, Mirov, et al. . . . . . .
5
3
Basic configuration of the spatially-dispersive laser . . . . . . . . . . . . .
11
4
Schematic diagram of the spatially-dispersive laser in the Gaussian beam and the paraxial approximations . . . . . . . . . . . . . . . . . . . . . . .
20
5
Transmission of a Gaussian beam through an arbitrary optical system. . .
25
6
Transmission of the Gaussian beam by a single thin lens. . . . . . . . . .
26
7
The TEM00 Gaussian mode of the axial wavelength channels incident on the diffraction grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
8
Overlap of adjacent channels in the gain medium. . . . . . . . . . . . . .
31
9
Simultaneous operation of three highly overlapped wavelength channels. .
35
10 Calculation of cavity parameters for obtaining the muximum number of spectral channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
11 Spatial distribution of the output radiation. . . . . . . . . . . . . . . . . .
42
12 Spatially dispersive laser with a complex multi-lens focusing system. . . .
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13 Schematic diagram of the single-wavelength tunable Cr2+ :ZnSe laser based on the spatially-dispersive cavity. . . . . . . . . . . . . . . . . . . . . . . .
57
14 Tuning the output spectrum of the single-wavelength spatially-dispersive laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
15 Schematic diagram of the dual-wavelength spatally-dispersive laser. . . . .
62
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LIST OF FIGURES (Continued) Figure
Page
16 Calculation of the grating angular orientation. . . . . . . . . . . . . . . .
64
17 Tuning of the dual-wavelength spectrum by the diffraction grating. . . . .
67
18 Experimental setup for measurements of the intensity profile of the dual pump beam with the knife edge method. . . . . . . . . . . . . . . . . . .
69
19 The results of the measurements of the intensity profile of the dual pump beam with the knife edge method. . . . . . . . . . . . . . . . . . . . . . .
70
20 Experimental measurements and the laser model predictions of the output wavelength spacing vs the grating angular orientation. . . . . . . . . . . .
71
21 Non-parallelism of the pump beams. . . . . . . . . . . . . . . . . . . . . .
72
22 Experimental measurements and the laser model predictions of the output wavelengths vs the grating angular orientation. . . . . . . . . . . . . . . .
74
23 Schematic diagram of the motion-tuning of the dual-wavelength spatiallydispersive laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
24 Tuning of the dual-wavelength spectrum by simultaneous transverse motion of the pump beams across the laser crystal. . . . . . . . . . . . . . . . . .
76
25 Experimental measurements and the laser model predictions of the output wavelength vs the transverse coordinates of the laser modes. . . . . . . .
78
26 Experimental measurements and the laser model predictions of the output wavelength spacing vs the average position of the dual pump beam. . . .
80
27 Schematic diagram of the dual-wavelength, spacing-tunable spatallydispersive laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
28 Tuning of the dual-wavelength spectrum by simultaneous transverse motion of the pump beams across the laser crystal. . . . . . . . . . . . . . . . . .
83
29 Experimental measurements and the laser model predictions of the output wavelength spacing vs the spatial spacing of the beams. . . . . . . . . . .
85
30 Experimental measurements and the laser model predictions of the output wavelength vs the coordinates of the laser modes. . . . . . . . . . . . . .
86
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LIST OF FIGURES (Continued) Figure
Page
31 The spherical aberration in the misaligned spatially-dispersive cavity.
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87
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33 Measurements of the intensity profiles of the output beams with the knife edge method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
34 Knife edge method measurement errors . . . . . . . . . . . . . . . . . . .
91
35 Dependencies of the beam waists of the laser modes on the distance between the crystal and the intracavity lens. . . . . . . . . . . . . . . . . . . . . .
94
36 Schematic diagram of the multiwavelength Cr2+ :ZnSe spatally-dispersive laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
32 Setup for measurements of the spatial profiles of the output beams.
37 Ultrabroadband multiwavelength output spectra of the multiwavelength Cr2+ :ZnSe spatally-dispersive laser. . . . . . . . . . . . . . . . . . . . . . 100 38 Spatial intensity profile of the pump beam near the output facet of the laser crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 39 Difference between the measured pump beam size in front of the laser crystal and the actual width of the pumped region of the crystal. . . . . . 102 40 Ultrabroadband, tunable multiwavelength spectra . . . . . . . . . . . . . 105 41 Dependence of the wavelength spacing of the multiwavelength spectra on the longitudinal displacement of the Cr2+ :ZnSe crystal. . . . . . . . . . . 108 42 Tuning of the ultrabroadband, multiwavelength spectrum as a whole by rotation of the diffraction grating in its dispersion plane. . . . . . . . . . 110 43 Experimental setup of the single-chip spatially-dispersive diode laser . . . 114 44 The output spectrum of the visible semiconductor laser in the free-running refime of operation superimposed with its luminescence spectrum. . . . . 117 45 Multiwavelength spectra of the visible spatially-dispersive diode laser. . . 119 46 Continuous tuning of a dual-wavelength output spectrum of the semiconductor multiwavelength spatially-dispersive laser. . . . . . . . . . . . . . . 121 x
LIST OF FIGURES (Continued) Figure
Page
47 Experimental setup of the multi-stripe spatially-dispersive diode laser. . . 123 48 Ultrabroadband multiwavelength spectra of the infrared, multi-stripe, spatially-dispersive diode laser. . . . . . . . . . . . . . . . . . . . . . . . . 125 49 Multiwavelength spectra of the infrared, multi-stripe, spatially-dispersive diode laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 50 Measurement of the intensity profiles of the Gaussian beams with the “knife edge method”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 51 Characterization of the Gaussian beam by measuring its intensity distributions in two arbitrary planes . . . . . . . . . . . . . . . . . . . . . . . . 149 52 Cavity alignment—aligning the pump and HeNe beams. . . . . . . . . . . 154 53 Aligning the diffraction grating. . . . . . . . . . . . . . . . . . . . . . . . 155 54 Aligning the input mirror and Cr2+ :ZnSe laser crystal. . . . . . . . . . . . 156 55 Aligning the pump lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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1
CHAPTER 1 INTRODUCTION 1.1
Problem Statement
Lasers, being oscillators at optical frequencies, produce optical radiation that is characterized by an extremely high degree of monochromaticity, coherence, directionality, and brightness. These unique properties of laser light inspired widespread use of laser sources in numerous applications. Monochromaticity, or narrow spectrum output, which is usually much smaller than the gain profile of the laser amplification material, is considered to be one of the inherent properties of lasers. Even media with a very broad, homogeneously broadened gain profile (e.g., laser dyes, Ti-sapphire, alexandrite, forsterite, color center), when placed in non-dispersive cavities, provide significant spectral narrowing of the output radiation. The physical mechanism of the line narrowing is a consequence of the spectral dependence of the gain profile of the amplification media: frequency modes with highest gain grow in the laser cavity faster than others, lower the total inversion population, and eventually suppress other frequency components with lower gain [1–4]. However, there are many practical applications that require sources of radiation combining the spatial coherence, high intensity, and continuous or discreet ultrabroadband multiwavelength spectra. Among such applications are coherent-transient spectroscopy [5], Fourier optical processing [6, 7], all-optical image transfer [8, 9], multiwavelength spectroscopy [10–12], optical memory [13–15], and information processing [16–21].
2 Since the 1970’s there have been many attempts to build multiwavelength and ultrabroadband laser sources suitable for these applications. All of these attempts have been based on the idea of the suppression of the mode competition in the laser resonator, which is responsible for the narrowing of the laser output spectrum. There are two possible ways for implementation of this idea: mode separation in the temporal domain and mode separation in the spatial domain. Lasers where the frequency mode separation is realized in those domains are denoted as “temporally-dispersive” and “spatially-dispersive” lasers, respectively. The key idea of the temporally-dispersive lasers is a mode-locked operation in which the output contains a multiplicity of temporally shifted, ultrashort optical pulse trains, each pulse train operating at a different wavelength [22–28]. All wavelengths are consequently generated in the same region of the laser gain element. The ultrafast, mode-locked lasers provide broadband output radiation, a large number of output spectral lines, high average powers, and a low-coherent output beam. These lasers, however, can be used only in those applications where the continuous-wave lasing is not required. While temporally-dispersive lasers are limited to the pulsed operation, the spatially-dispersive lasers are flexible in terms of regime of lasing: pure CW [9], pulsed nano– [23, 29–34], pico-, and femtosecond [24, 35] regimes are feasible. Spatiallydispersive cavities provide spatial separation of different wavelength channels in the laser gain medium, which leads to an effective suppression of the intracavity mode competition. The construction of the cavities is flexible and allows utilization of a variety of active media, including crystals [9, 29–31, 33, 34, 36–39], dyes [32, 40–42], as well as multistripe diode chips [17–19, 43–45], and laser diode arrays [46–48].
3 Spatially-dispersive lasers are capable of generating arbitrary spectral structure of the output radiation, such as superbroadband continuous- and discreet multiwavelength spectra [29, 41].
1.2
Historical Overview
The technology of the spatially-dispersive lasers was introduced in 1971 by Ashkin and Ippen [49], where a dispersion apparatus inside the laser cavity was used to disperse different emitting wavelengths along distinct feedback paths in the resonator. It was proposed basically for narrowband tunable oscillation, where frequency tuning was achieved by scannable pumping beams without any realignment of intracavity elements. In 1989, Danailov and Christov [40] proposed to utilize Ashkin’s principle of the spatially dispersive cavity for the generation of laser radiation with ultrabroadband spectrum and further developed the principle of ultrabroadband lasing for dyes [41, 42, 50, 51]. The principle of operation of this spatially-dispersive laser is based on the use of the well-known effect of “lateral walk-off” of spectral components of a light beam that passes through a pair of conjugate dispersive elements, in this case two diffraction gratings or two prisms. The schematic diagrams of the lasers are shown in Fig. 1. The grating (or prism) conjugate pair, along with the intracavity aperture, forces each wavelength to propagate along a unique path and, therefore, each wavelength is amplified in its own unique region of the gain medium. This process eliminates frequency mode competition and gives rise to a ultrabroadband, multifrequency operation of the laser with continuous output spectrum. Installation of an optional intracavity spatial mask or an interferometer allows for the obtaining of a discreet multiwavelength output spectrum [41].
4
Gain
Pump
M1
λ1
G1
λ2 G2
Pump
A
λ1 , λ 2
M2
Gain
M1
λ1 λ2
P2 P1
A
λ1 , λ 2
M2
Fig. 1. Spatially-dispersive cavities proposed by Danailov and Christov. The dispersive element consists of a pair of conjugate diffraction gratings G1 and G2 (a) or prisms P1 and P2 (b). The wavelengths in the range [λ1 , . . . , λ2 ] are amplified in different regions of the gain element. The optional spatial mask SM splits otherwise continuous ultrabroadband output spectrum into a discreet multiline spectrum. The intracavity aperture transmits only the TEM00 modes of the wavelength components.
The choice between the grating and prism-based conjugate diffractive pairs depends on the requirements of the laser output characteristics: gratings provide much higher spectral resolution than prisms, but have significantly larger optical loss and much lower damage threshold. Therefore, the prism pair is more appropriate for high power lasers or lasers based on low-gain laser media. However, the grating pair is more suitable for such efficient laser media as, e.g., semiconductors, or for moderate output power lasers with broadband output spectrum that consists of a large number of ultranarrow spectral lines. Since the 1990s this spatially-dispersive laser and its modifications have been used by many research groups to create multiwavelength lasers for different purposes: three-wavelength Nd:Y3 Al5 O12 lasers [23], ultrabroadband DCM dye lasers [32], ul4+ trabroadband LiF:F+ 2 color center lasers [37], ultrabroadband Cr : forsterite lasers
[38], ultrabroadband Ti3+ : sapphire lasers [39], [52], and diode-pumped broadband Cr:LiSGAF and Cr:LiSAF lasers [9].
5 In early 1990s, a new spatially-dispersive resonator was introduced by Basiev, et al. [29]. This spatially-dispersive laser scheme is based on the combined operation of the intracavity lens and the Littrow-mounted diffraction grating. These cavities were utilized for realization of the first solid state multiwavelength and ultrabroadband lasers [30, 31, 36, 53]. A schematic diagram of the laser is depicted in Fig. 2. PSfrag replacements λ2 λ1
SM
A λ1
Pump
λ2 Min
Gain
Lens
Littrow grating
Fig. 2. The spatially-dispersive laser proposed by Basiev, Mirov, et al. in 1993 [29]. For every wavelength in the range [λ1 , . . . , λ2 ] the diffraction grating retroreflects the first order of diffraction back into the cavity. Each wavelength is amplified within a unique region of the gain medium, and the mode competition is significantly reduced, which allows multiwavelength operation of the laser. Only TEM00 modes are transmitted through the intracavity aperture A. The zeroth order of diffraction is used as the laser output. A spatial mask SM is used for obtaining a discreet multiwavelength output spectrum.
The lens and the grating, which is used as the output coupler, enforce each wavelength from a spectral region [λ1 , . . . , λn ] to propagate along its own trajectory. As a result, every wavelength is amplified in a unique region of the gain element and, therefore, the mode competition between different wavelength components is eliminated. The intracavity aperture A is used as a high-order mode suppression element: only the TEM00 modes of the wavelength channels are transmitted. The continuous
6 ultrabroadband output spectrum, generated by the laser, can be transformed to an arbitrarily shaped discreet multiwavelength spectrum by a spatial mask, introduced into the wide pump beam, as was demonstrated in Refs. [30, 31]. In 1994, Basiev, et al. published the first experimental study of a roomtemperature LiF:F+ 2 superbroadband laser [30]. In 1995 they demonstrated a novel, “white-color” laser, where the superbroadband IR output spectrum from the LiF:F− 2 color-center laser, operating in the range of 1.1–1.24 µm, was frequency doubled with a simple, single-lens phase-matching optical system to obtain a “white” light in the visible spectral interval of 0.55–0.62 µm [31]. In 1997 the authors performed an experimental and theoretical investigation of the temporal, spectral, and spatial features of the pulsed spatially-dispersive LiF:F− 2 color-center laser and studied simultaneous frequency doubling of the ultrabroadband, infrared (IR) output radiation in a single nonlinear crystal [33]. Later, in 2001, they reported on the simultaneous frequency doubling and frequency quadrupling of the superbroadband laser radiation − from the LiF:F+ 2 and LiF:F2 color center lasers. This yielded laser radiation with
an ultrabroadband output spectrum spreading over a large spectral region, from the infrared (∼ 1.2 µm) to the ultraviolet (∼ 0.2 µm) [34].
1.3
Motivations
As mentioned earlier, the spatially-dispersive lasers are very useful for many applications. Among those are the multiwavelength laser spectroscopy and optical communications. Both applications have similar requirements: an ultrabroadband, multiwavelength laser source, capable of generating the laser radiation with a preassigned spectral structure.
7 In the field of laser spectroscopy, this problem is currently solved with tunable laser sources, where the laser operating wavelength is tuned from one desirable value to another [54]. However, those lasers have a number of disadvantages that limit their usage and often lead to a design that is quite complex and sensitive to misalignment. First of all, such lasers can generate only one wavelength at a time. As a result, for multiwavelength spectroscopy applications the wavelength must be sequentially tuned. Secondly, a continuous tuning of the wavelength in many widely used tunable lasers is not always possible due to wavelength jumps from one longitudinal mode to another, which is known as the “mode-hoping” [55–77]. The latter problems can be partially solved by simultaneous tuning of the laser selective element (such as diffraction grating or prism system) and changing the length of the cavity. A drawback of this method is that it limits the continuous tuning range (without mode-hoping), and makes the laser design very complex and sensitive to small misalignment. Similar problems are common for laser sources used in the optical data networks [78,79]. Among optical network technologies Dense Wavelength Devision Multiplexing (DWDM) techniques provide the highest data transmission capacity because a large number of optical data channels are simultaneously transmitted through a single optical fiber. Viable DWDM systems require low cost, reliable, and compact multifrequency laser sources that are flexible in tuning the output spectrum to the desirable DWDM band while maintaining accurate adherence to the specifications of particular wavelength spacing. Most widely used telecom laser sources are based on semiconductor laser gain media due to a number of unique properties. Commonly used short-cavity DWDM laser sources, such as distributed feedback (DFB) lasers, distributed Bragg reflector (DBR) lasers, and vertical cavity surface emitting lasers
8 (VCSELs), have to be externally wavelength stabilized to control the temperaturedependent output wavelength drift. The production technology and management of those lasers are quite expensive, which limits their mass production and use in telecom systems. One of the most competitive alternatives to those devices is the semiconductor, long-external-cavity, tunable lasers with grating-stabilized comb of frequencies. Examples of these lasers include multichannel grating cavity lasers with bulk-optics [17, 18, 45–48, 80–84] and lasers with integrated design [19, 43, 44, 85–87]. However, those systems lack flexibility in the selection of wavelengths. There are also limitations in the total amount of available channels, as well as the available wavelength spacing. In addition to that, compensation for optical cross talk can be difficult in some cases. Therefore, it is necessary to create an alternative multiwavelength DWDM laser source based on cheap, commercially available semiconductor lasers and a well developed multiwavelength cavity technology. In this context, the Littrow-mounted grating spatially-dispersive laser appears to be a very attractive choice due to the following advanced properties of this laser: • Great flexibility in choice of the gain media: dyes, solid-state crystals, and single-chip and multi-stripe laser diodes, as well as diode laser arrays; • Simplicity and compactness of the cavity; • Possibility of obtaining any pre-assigned structure of the output spectrum; • Dynamic tunability of the output spectrum without readjustment of the optical elements of the laser; and • Fast and easy wavelength spacing tuning. Despite a relatively long history for these Littrow-mounted grating spatiallydispersive lasers, a thorough investigation of their operation has not been performed,
9 and no detailed theoretical model of this class of laser is available. Some important aspects of the laser operation were considered in the original papers of the laser inventors: in Ref. [31] the general principle of the laser cavity operation is described in geometrical optics approximation; Ref. [33] is devoted to the theoretical analysis of the temporal characteristics of a nanosecond version of the laser; in Ref. [34] a very simplified model of the laser cavity in the Gaussian beam approximation is presented. Up to now, all studies of this laser have been limited to the pulsed operation of the LiF:F+∗∗ and LiF:F−∗∗ color-center spatially-dispersive lasers. The continuous2 2 wave (CW) operation of the spatially-dispersive laser has not been demonstrated or considered theoretically. This research work represents the first attempt of a systematic theoretical and experimental investigations of the continuous-wave, multiwavelength, and ultrabroadband lasers, based on the Littrow-mounted grating spatially-dispersive cavity.
10
CHAPTER 2 THEORETICAL MODEL 2.1
Principle of Operation of the Spatially-Dispersive Laser
The basic idea of the spatially-dispersive laser is as follows. A special spatiallydispersive laser resonator splits the gain medium of the laser into a number of nonoverlapping regions, called active channels, in such a way that each channel amplifies only a single wavelength from the gain curve of the laser active medium. Each lasing wavelength is determined by the position of the channel in the active medium and parameters of the laser resonator. As a result, the competition among different wavelengths is eliminated, and the laser operates at many wavelengths simultaneously. Thus, the spectrum of the output radiation consists of a large number of spectral lines and covers a wide spectral range, unlike the conventional single-wavelength lasers. Moreover, the radiation at each wavelength propagates along its own, unique path, so that different output wavelengths are spatially separated. The basic configuration of the laser is shown in Fig. 3. The laser resonator consists of the following major elements: a plane input mirror, a rectangular gain element, an intracavity focusing lens, and a diffraction grating operating in the Littrow mount scheme. In addition, in some situations a spatial mask and an intracavity aperture, installed in the focal plane of the intracavity lens, might be necessary. The spatial mask could be used to select only desirable wavelengths and suppress others, and it is equivalent to multi-beam pumping by spatially dispersed pump beams.
11 x α>0 α0 z α f ⇒ f (1 − g1 ) > f ⇒ 1 − g1 > 1 ⇒ g1 < 0 ⇒ g2 < 0 .
(16)
This condition sets some limitations on the position of the intracavity lens and the length of the resonator. The physical meaning of this condition, besides the stability of the cavity, is that it selects all cavity configurations for which the mode size on the diffraction grating is larger than the mode size on the input mirror. As it will be shown later in Section 2.5, this is required for obtaining the maximum possible number of spectrally resolved wavelength channels. For further analysis of the laser it is necessary to find the laser mode diameters in the major planes of the cavity. The most important planes of the resonator are the following: (1) the plane of the input mirror, because the laser mode in this plane determines the size of the holes in the spatial mask or the diameters of the multiple
23 pump beams if the pump beam shaping is used for obtaining the multiwavelength discrete output spectrum; (2) the plane of the output facet of the laser crystal, since the mode size in this plane determines the mutual overlap of adjacent wavelength channels; (3) the focal plane of the intracavity lens, because the mode size in this plane sets the diameter of the intracavity aperture (when it is required for out-filtering of the high-order transverse modes and obtaining the multiwavelength output spectrum); and (4) the plane of the output mirror, because this is essentially the plane of the diffraction grating, and the mode size on the grating determines its spectral resolution and, therefore, the spectral resolution of the spatially-dispersive resonator. The beam radii of the fundamental (TEM00 ) mode at the mirror wm and the diffraction grating wg are calculated via the ABCD law for a self-reproducing Gaussian mode (see [3], Section 5.2.1 for more details):
2 wm
wg2
Lλ0 π
Lλ0 π
=
=
21
,
(17)
21
.
(18)
g2 g1 (1 − g1 g2 )
g1 g2 (1 − g1 g2 )
Dividing Eq. (17) by Eq. (18), one obtains a simple but very useful relation between the beam waists on the input and the output laser mirrors:
wg = w m
g1 g2
21
.
(19)
In these three equations the radius of the Gaussian beam is defined as the radius where its intensity is decreased to 1/e2 of its maximum value.
24 The mode radii on the crystal output facet wc and on the aperture wA can be found by tracing the Gaussian beam through the cavity, starting from the input mirror. Since the input mirror is plane, the waist w0 of the Gaussian beam is located at the mirror surface and is equal to wm ; the beam confocal parameter then reads
z0 =
2 πwm . λ0
(20)
The beam radius is changed with the distance z from the beam waist according to the following law of propagation of the Gaussian beams:
w 2 (z) = w02 1 +
z z0
2 !
.
(21)
When the Gaussian beam passes through an optical system described by a matrix A B C D , its radius obeys the same law, with the distances z and z0 transformed ac-
cording to the following equations (see [3], p. 80):
−(Az + B)(Cz + D) − ACz02 z = , C 2 z02 + (Cz + D)2 0
z00 = z0
Cz 0 + A . Cz + D
(22) (23)
In these equations z is the distance from the location of the beam waist w0 to the entrance principal plane of the optical system, z 0 is the distance from the output principal plane of the system to the location of the new beam waist, and z00 is the confocal parameter of the transformed Gaussian beam (note that Eq. (2.63) of [3] contains an error; Eq. (22) shows its correct form).
25 The transmission of a Gaussian beam through an arbitrary optical system, PSfrag replacements described by Eqs. (22) and (23), is illustrated in Fig. 5.
Optical system
z0
w0
A B C D
z00
w00
z
z0 Output Input Principal Plane Principal Plane
Fig. 5. Transmission of a Gaussian beam through an arbitrary optical system.
In this model of the spatially-dispersive cavity the laser crystal is represented by a dielectric slab of a length d1 and a refractive index n. Thus, the crystal is described by the following beam transmission matrix:
1 MSL = 0
d1 n
1
.
(24)
The waist of the Gaussian beam, incident on the laser crystal, is located on the input mirror and, therefore, w0 = wm and z = d0 . Then, using Eqs. (21), (22), and (23), one obtains the following expressions for the new beam confocal parameter z00 , the position of the new beams waist z 0 , and the new beam waist radius w00 :
z00
= z0 ,
d1 z = d0 + , n 0
w00
=
z00 λ0 π
12
= wm .
(25)
26 It is now straightforward to find the beam radius on the output facet of the crystal by substitution of Eq. (25) into Eq. (21): λ20 (d0 + d1 /n)2 . 2 π 2 wm
2 wc2 = wm +
(26)
The beam radius in the focal plane of the intracavity lens can be found by tracing the beam from the input mirror to the focal plane. The transmission of a Gaussian beam by a single thin lens is illustrated in Fig. 6. The thin lens with
PSfrag replacements θ0
w0
wf
w00 z0 − f
z
z0
Fig. 6. Transmission of the Gaussian beam by a single thin lens.
a focal length f is described by the following beam transmission matrix (see [3], Section 1.2.1):
1 0 ML = . 1 −f 1
(27)
The position of the beam waist of the transformed Gaussian beam is calculated using Eq. (22): z0 = f ·
z02 + z(z − f ) . z02 + (z − f )2
(28)
27 The confocal parameter of the transformed beam is obtained from Eq. (23) using the result of Eq. (28): z00 =
z0 f 2 . z02 + (z − f )2
(29)
The radius of the beam waist of the transformed beam, calculated with Eq. (20) for new value of z00 , is given by the following expression:
w00 =
w0 f z02 + (f − z)2
21 .
(30)
The Gaussian beam radius in the focal plane of the thin lens is obtained by propagation of the beam from the plane of the new beam waist into the focal plane using Eq. (21), where the beam waist is given by Eq. (30), the distance of propagation is z 0 − f (see Fig. 6), and the distance z 0 is found from Eq. (28):
f wA = w 0 = f z0
λ0 πw0
≡ f · θ0 .
(31)
As one can see from Eq. (31) the radius of the refracted Gaussian beam in the focal plane of a thin lens equals the product of the lens focus and the divergence θ0 of the Gaussian beam, incident on the lens. It is now straightforward to find the beam radius of the axial laser mode in the focal plane of the intracavity lens. The Gaussian beam, incident on the lens, is the one emerging from the laser crystal and, therefore, its beam waist is given by Eq. (25). Thus, the beam radius in this plane is
wA =
λ0 f . πwm
(32)
28 The notation of the beam radius in Eq. (32) reflects the fact that this is the radius of the laser mode in the plane of the intracavity aperture (see Fig. 4), and within the considered model of the laser, the mode diameter in this plane determines the aperture size.
2.4
Spectral Resolution of the Spatially-Dispersive Cavity
The spectral separation of two adjacent channels, spatially separated in the gain medium by a distance δx, is found using Eq. (9):
δλ = δx
2t cos β . f
(33)
In order to achieve this spectral separation, the cavity must provide spectral resolution better than or equal to that given by Eq. (33). The cavity spectral resolution is determined by the resolving power of the diffraction grating,
R=
λ0 , δλg
(34)
which, in turn, is determined by the number of illuminated grooves [88]:
R=
λ0 = Ngrooves , δλg
(35)
where λ0 is the wavelength of the axial spectral channel, and δλg is the minimum wavelength interval resolved by the grating. The number of illuminated grooves Ngrooves can be calculated in terms of the beam diameter on the grating. The situation is illustrated in Fig. 7.
29
g
β 2W
TEM00 mode
2wg
t
PSfrag replacements
Fig. 7. The TEM00 Gaussian mode of the axial wavelength channels incident on the diffraction grating. The width of the projection of the mode into the grating plane Wg determines the number of illuminated grooves and the grating resolving power.
The plane of the beam waist is projected onto the plane of the grating at the angle β and, therefore, the number of illuminated grooves is proportional to the width of the illuminated region Wg , which is given by the following expression, as one can see from Fig. 7: 2Wg =
2wg . cos β
(36)
The number of the illuminated grooves Ngrooves is proportional to the illuminated width of the grating surface and is inversely proportional to the distance between adjacent grooves, given by the grating spacing constant t:
Ngrooves =
2Wg 2wg = . t t cos β
(37)
Therefore, as it follows from Eq. (35), the resolution of the diffraction grating λg (the minimum wavelength spacing, resolved by the grating) is given by the following equation: δλg =
λ0 Ngrooves
=
λ0 t cos β . 2wg
(38)
30 The requirement on the minimum spectral resolution of diffraction grating means that the condition δλg ≤ δλ must be satisfied. Otherwise two arbitrary wavelength channels, spatially separated by the distance δx in Eq. (33) will be spectrally indistinguishable. Applying this condition to Eqs. (33) and (38), one obtains the following requirement as to the minimum diameter of the laser modes on the diffraction grating required to spectrally resolve two wavelength channels, spatially separated in the gain medium by a transverse distance δx:
wg ≥
λ0 f . 4δx
(39)
Inequality (39) connects the spectral properties of the spatially-dispersive laser with the physical parameters of its optical elements. Indeed, the mode size wg is determined by the positions and parameters of the elements of the laser resonator, as it follows from Eq. (18). Thus, to obtain the required mode size on the grating, one has to choose the appropriate configuration of the spatially-dispersive cavity. Therefore, at this point the geometrical optics approach, described in Section 2.1, is united with the Gaussian model of the laser, considered in this section, and one can see that these two approaches represent mutually complementary parts of the theoretical model of the spatially-dispersive laser. It is now possible to describe the process of the multiwavelength operation in detail and obtain a method for the calculation of the parameters of the laser cavity.
31 2.5
Maximum Number of Oscillating Wavelength Channels
The whole concept of the ultrabroadband, multiwavelength operation of the laser is based on the idea of eliminating the competition between different frequency components by spatially separating them in the gain medium. Consequently, the overlap between the wavelength channels in the gain medium must be minimized in order to avoid the crosstalk and competition between the modes of adjacent channels. Using this requirement one can calculate the maximum number of simultaneously oscillating independent wavelength channels. The situation is illustrated in Fig. 8, PSfrag where the replacements modes of two adjacent wavelength channels in the gain medium are shown.
n
2wm d0 Input Mirror
λ1
∆x
2wc
2wc
λ2
d1 Laser Crystal
Fig. 8. Overlap of adjacent channels in the gain medium. Two adjacent wavelength channels in the laser crystal are shown. The mutual overlap of the channels is minimized when they are separated by their diameters on the output facet of the crystal.
The minimum overlap between two adjacent channels is realized when they are separated by a transverse distance δx, equal to or larger than their diameters on the output facet of the laser crystal 2wc :
δxmin ≥ 2wc .
(40)
32 Therefore, the minimum channel overlap and the maximum amount of oscillating spectral lines within a wavelength range ∆λ can be obtained within the transverse region of the gain medium ∆x by minimizing the beam radius wc . The minimization of the radius is performed using Eq. (26). This equation can be considered as a functional dependence of the beam radius wc on the beam waist wm , with all of the other variables (d0 , d1 and λ0 ) being parameters. Since all terms in Eq. (26) are positive, it is enough to minimize wc2 , which significantly simplifies the algebraic calculations. 2 and A2 ≡ ((d0 + d1 /n)λ0 /π)2 , By introducing the auxiliary variables ζ ≡ wc2 , η ≡ wm
Eq. (26) is reduced to the following expression:
ζ =η+
A2 . η
The minimum of ζ is given by
0=
dζ A2 = 1 − 2 ⇒ ηmin = A ⇒ ζmin = 2ηmin = 2A , dη η
where ηmin denotes the value of η for which ζ takes its minimum value ζmin . Consequently, the minimum value of wc is achieved when
2 2 wc,min = 2wm =
2λ0 (d0 + d1 /n) . π
(41)
In many cases the input mirror is located very close to the input facet of the laser crystal: in semiconductor lasers the mirror is usually deposited directly onto the rear facet; in optically-pumped solid-state lasers the input plane mirror is located as close to the crystal as possible to achieve the maximum pump intensity.
33 Therefore, for the purposes of this estimation one can assume that d0 d1 /n. Consequently, Eq. (41) for the minimum of the mode radius wc on the output facet of the laser crystal can be reduced to the following simple form:
2 wc,min =
2λ0 d1 , nπ
(42)
which, according to Eq. (41), corresponds to the following optimum mode waist on the input mirror wm : 2 wm =
λ0 d 1 . nπ
(43)
The maximum number of independent wavelength channels can be estimated as the ratio of the width of the laser crystal to the minimum beam diameter on the output facet of the crystal:
Nchannels,max
12 ∆x nπ = = ∆x , 2wc,min 8λ0 d1
(44)
which is a function of the parameters of the laser crystal only. The laser can always be designed and tuned in such a way that the beam radius wc is equal to that given by Eq. (42) and the maximum number of oscillating spectral channels, given by Eq. (44), is obtained. In all other cases the number of simultaneously working channels will be reduced. When the maximum number of spectral channels is achieved, their spectral separation will be minimal and can be calculated using Eqs. (33) and (9):
δλmin
1 4wc t ∆λ ∆λ 8λ0 d1 2 = cos β = = . f Nchannels,max ∆x nπ
(45)
34 Experiments show, however, that in certain spatially-dispersive laser systems the number of simultaneously oscillating channels is much larger than that given by Eq. (44), and their minimum spectral separation is much smaller than that calculated with Eq. (45). For example, the number of simultaneously oscillating spectral lines generated by the Cr2+ :ZnSe spatially-dispersive laser (which will be described in detail in Section 3.1.8) is an order of magnitude larger than that predicted by Eq. (44). In that case, the mode diameter in the laser crystal 2wc is about 100 µm, and the total width of the pumped region ∆x is approximately 600 µm, which would give, according to Eq. (44), at most six spectral channels, spectrally separated by about 24 nm, as can be found from Eq. (45). However, the laser can produce up to 50 well distinguished spectral channels, spectrally separated by about 2 nm. These results suggest that the laser is oscillating at a large number of overlapping channels. In order to analyze the simultaneous operation of partially overlapped wavelength channels, let us consider the situation shown schematically in Fig. 9. Consider the operation of the axial wavelength channel corresponding to the wavelength λ0 . This channel overlaps with two adjacent channels λ1 and λ2 , and the spatial separation of the channels is smaller than their radius wc on the output facet of the gain element. The mode competition occurs only in the zones of overlap of the channels. The rest of the mode volume of the axial channel in the gain medium belongs solely to this channel, where the light is amplified without cross talk with the neighboring wavelength channels. On the other hand, the spatial separation between the adjacent channels is small. According to Eq. (33) they are located very close to each other on the gain spectral curve. Consequently, the gain for the overlapping channels will be approximately the same.
35 No competition
Competition
d0 Input mirror
d1 Laser crystal
(a)
λ1 λ0 λ2
δx
δx
2wc
Competition
n
No competition Cross sections at the input facet of the laser crystal
Cross sections at the output facet of the laser crystal
(b)
(c)
Fig. 9. Simultaneous operation of three highly overlapped wavelength channels. (a) top view on the gain element with the oscillating overlapping channels; (b) cross sections of the channels at the input facet of the gain element; and (c) cross sections of the channels on the output facet of the gain element.
The axial channel is still amplified even in the competition zones of the gain element, although its gain in those zones is reduced. Therefore, one can conclude that the considerable overlap of the adjacent channels is equivalent to their attenuation which, in turn, is equivalent to an increase of the intracavity loss. Thus, the higher the gain of the active media, and the smaller the intracavity loss, the more significant the overlap of the adjacent wavelength channels that is allowed. The mode competition will couple the adjacent channels with each other and they will not be completely independent, but it will not necessarily suppress them. Such behavior of the spatially-dispersive laser is indeed observed in the experiments. For instance, the low-feedback, external cavity, spatially-dispersive semiconductor laser, described in Subsection 3.2.2, has a very high intracavity loss and only a limited number of non-overlapping wavelength channels, given by Eq. (44), can operate simultaneously, despite the high gain of the semiconductor.
36 On the other hand, the Cr2+ :ZnSe spatially-dispersive laser, described in Section 3.1, is less sensitive to intracavity loss and can operate at a large number of overlapped wavelength channels, despite the comparatively low gain of this laser material. In order to generalize Eqs. (44) and (45) for the description of the operation of partially overlapping wavelength channels, it is convenient to express the requirement of their minimum spatial separation (40) in the following way:
δxmin ≥ 2ξwc,
(46)
where the parameter ξ characterizes the channels overlap. When ξ ≥ 1 the channels do not overlap in the gain element and the condition (40) is satisfied. When ξ < 1, the overlap of adjacent channels grows, and it reaches its maximum when ξ = 0. It must be noted here that the notion of the radius of the Gaussian beam is a matter of accepted convention. In this analysis the radius of the Gaussian beam corresponds to the 1/e2 intensity level. Therefore, the definition of the channels overlap agrees with the standard convention. The introduction of the ξ-parameter is thus generally equivalent to a redefinition of the notion of the Gaussian beam radius. Therefore, the analysis presented by Eqs. (40) to (45) is generalized by the following substitution: wc −→ ξwc . As a result, the maximum number of simultaneously oscillating channels (44), the minimum spectral separation of two adjacent spectral lines (45), and the condition on the required spectral resolution of the diffraction grating (39) are generalized to the following equations:
37
r ∆x nπ ∆x Nchannels,max = = , 2ξwc,min ξ 8λ0 d1 r ∆λ 8λ0 d1 ∆λ =ξ , δλmin = Nchannels,max ∆x nπ wg2 ≥
λ20 f 2 . 2 64ξ 2 wc,min
(47)
(48)
(49)
As was described above, the ξ-parameter is determined by many factors, most important of which are the gain of the active medium and the total intracavity loss. As a result, the exact value of ξ can only be found experimentally. However, one can derive a general approach for calculation of the parameters of the spatially-dispersive laser based on the physical and the spectral properties of the gain element and the requirements on the spectral structure of the output radiation.
2.6
Calculation of the Spatially-Dispersive Cavity
It is convenient to begin with expressing the requirements on the major cavity properties Eqs. (47)–(49) in terms of the cavity stability parameters g1 and g2 . From Eqs. (12), (17), and (43) it follows that:
2 wm
λ0 d 1 = = nπ
r Lλ0 g2 π g1 (1 − g1 g2 ) 2 g2 d1 = (1 − g1 g2 ) fn g1
g1 =
g22
⇒ ⇒
d21 g2 ⇒ = L2 2 n g1 (1 − g1 g2 ) 2 d1 g1 = g2 − g1 g22 ⇒ nf
g2 , + A2
(50)
where A ≡ d1 /(nf ). Equation (50) determines all cavity configurations for which the mode diameter on the output facet of the gain element is minimum.
38 From Eqs. (19), (42), and (49) one obtains the following inequality:
2 wm
g1 λ2 f 2 ≥ 02 2 g2 64ξ wc
wc2
⇒
g1 λ2 f 2 ≥ 02 2 g2 32ξ wc
2λ0 d1 g1 λ20 f 2 nπ ≥ nπ g2 32ξ 2 2λ0 d1
g1 ≤ g 2
⇒
⇒
π2 , 128ξ 2 A2
(51)
where the constant A is the same as in Eq. (50) and it is taken into account that the g-numbers are both negative as given by Eq. (16). Inequality (51) determines all spatially-dispersive cavities that have sufficient spectral resolution to resolve the minimum wavelength interval (48) corresponding to the maximum number of oscillating channels (47). Finally, the stability requirement (15) for the negative g1 - and g2 -numbers can be rewritten as follows: g1 >
1 . g2
(52)
For a given laser crystal all possible cavity configurations are determined by these three expressions, Eqs. (50), (51), and (52). The only independent variable is the focal length of the intracavity lens, which is found using Eq. (9) and by applying requirements on the overall output spectral bandwidth ∆λ and the transverse size of the gain element ∆x. Once the appropriate values of the g-numbers are determined, the positions of the grating and the lens are found from Eqs. (13) and (14), respectively. Let us consider two examples of such calculations of the spatially-dispersive cavities: a single-chip semiconductor spatially-dispersive laser (Subsection 3.2.2) and a solid-state Cr2+ :ZnSe spatially-dispersive laser (Subsection 3.1.8).
39 The parameters of the gain elements and the available optical components for these lasers are shown in Table 1.
Table 1. Example calculation: parameters of gain elements and available optics Laser
λ0 , µm
n
d1 , mm
∆x, µm
∆λ, nm
t, mm
f , mm
Diode Cr2+ :ZnSe
0.65 2.55
3.6 2.44
0.5 4.0
100 600
10 700
1/1800 1/324
4.5–16 25
The calculation of the spatially-dispersive cavity is performed by building and analyzing the stability diagrams (50), (51), and (52), and auxiliary dependencies of ∆λ versus ∆x and Nchannels,max versus ξ-number. The calculation begins with finding the appropriate focal length of the intracavity lens for the given output spectral bandwidths ∆λ and the transverse sizes of the gain elements ∆x. The calculation process is illustrated in Fig. 10. As shown in Fig. 10(a), the focal length of the lens for the diode laser, corresponding to the required output spectral bandwidth of 10 nm (Table 1), is about 9 mm. In the case of the solidstate laser, the available 25 mm intracavity lens will allow obtaining of the spectral bandwidth of about 135 nm. Using these results one can calculate the constant A and plot the stability diagrams (50), (51), and (52), as depicted in Fig. 10(b), where the example diagrams for the considered laser materials are shown. The most appropriate configurations of the spatially-dispersive cavity are then found from the stability diagrams using the following constraints.
40
Cr2+ :ZnSe
14 13 12 11 10 9 8 7 6
∆λ, nm
∆λ, nm
Diode
7
6
8
9
10
11
12
170 160 150 140 130 120 110 100
(a)
20
22
24
f , mm -50
28
30
-8 -30
g1
ξ
.0 =1
g1
0. 4 ξ=
0.2
-10
ξ=
ξ=
0.8
ξ = 0. 1
ξ=
0. 6
ξ=0 .3 ξ= 0.4
-12
-40
PSfrag replacements
26
f , mm
ξ=
-6
6 0.
0.8 ξ= 1.0 ξ=
(b)
-20 -4 -10 0
-2 0
-0.02
-0.04
-0.06
0
-0.08
0
-0.1
-0.2
g2
-0.3
-0.4
g2 60
Nchannels,max
Nchannels,max
20 16 12 8 0.3
0.4
0.5
0.6
0.7
ξ
0.8
0.9
1
50 40
(c)
30 20 10 0
0.2
0.4
0.6
0.8
1
ξ
Fig. 10. Calculation of cavity parameters for obtaining the maximum number of spectral channels. The graphs in the left column correspond to the diode laser; the graphs in the right column represent the Cr2+ :ZnSe laser. Raw (a): dependencies of the overall spectral bandwidths ∆λ on the lens focal length for the given transverse sizes ∆x of the gain elements; Raw (b): stability diagrams (50) (bold solid line), (51) for different ξ-numbers (dashed lines), and (52) (thin solid line); Raw (c): maximum number of spectral channels as a function of the ξ-number.
41 First, all stable cavities lie below the curve given by inequality (52), shown by thin solid lines in Fig. 10(b). Second, all cavities appropriate for obtaining the maximum number of oscillating spectral channels belong to the curve given by Eq. (50), shown by bold solid lines in Fig. 10(b). Third, all cavities that provide high enough spectral resolution are located above the lines given by Eq. (51), shown by dashed lines in Fig. 10(b), which are plotted for several different values of the ξ-number, that characterizes mutual overlap of adjacent wavelength channels. In addition, it is convenient to calculate the maximum number of wavelength channels as a function of the overlap ξ-number, as shown in Fig. 10(c). As one can see from Fig. 10(b), the minimum value of the ξ-number for the diode laser is about 0.3 which, according to Fig. 10(c), corresponds to the maximum number of spectral channels of about 20. When the overlap is minimum (ξ = 1), at most six wavelength channels will oscillate simultaneously. One of the most important conclusions about the behavior of this laser is that it will always produce a discrete multiwavelength spectrum. Actual experiments show that due to high intracavity losses, caused by the absence of the appropriate AR-coating on the output facet of the diode chip, the minimum value of the ξ-number is about 0.8 and only up to 8 output spectral lines can be observed, as will be demonstrated in Subsection 3.2.2. In the case of the solid-state laser, however, the maximum overlap of the spectral channels corresponds to the ξ-number of about 0.1, as can be seen in Fig. 10(b). In that case up to 60 simultaneously operating wavelength channels are observed. Experiments show that the contribution of the intracavity losses due to the mode competition is relatively small in this case, and this regime of the multiwavelength laser operation can be realized, as will be demonstrated in Subsection 3.1.9.
42 2.7
Spatial Distribution of the Output Radiation
The output radiation of the laser is formed by the reflection of the zeroth order of diffraction of the wavelength channels from the diffraction grating, as shown in Fig. 3 and discussed in Section 2.1. The situation is illustrated in more detail in Fig. 11, where three output wavelength channels are shown.
z
PSfrag replacements
λ0
w(z )
λ1 λ2
α+β wA
wg ϕ = π − (2β + α)
O α
β
l−f
O0
F Fig. 11. Spatial distribution of the output radiation.
As one can see in Fig. 11, from the point of view of a distant external observer, the wavelength channels emerge from a point F , which is a reflection image of the focal point of the intracavity lens. However, the output beams are continuations of the corresponding laser modes with their beam waists located at the surface of the diffraction grating. Consequently, the waists of the output beams are equal to the waists of the laser modes on the diffraction grating wg given by Eq. (18).
43 Therefore, the propagation of each output beam is governed by Eq. (21), where the beam waist w0 ≡ wg :
w 2 (z) = wg2 1 +
z z0
2 !
.
(53)
In Eq. (53) the confocal parameter z0 = πwg2 /λ and the distance z is measured along the beam optical axis from the location of the beam waist at the grating, as shown in Fig. 11. The divergence of each laser beam is calculated from the confocal parameter:
θ(λ) =
λ wg = . z0 πwg
(54)
Usually the laser is built to provide the highest possible spectral resolution. For this reason, the diameters of the laser beams on the grating wg are relatively large, as follows from Eq. (39). Consequently, the divergence of each laser beam is usually very small. For instance, in the case of the Cr2+ :ZnSe laser, the divergence is less than 2 mrad (see Section 3.1.7). Corresponding z-axes of the laser beams propagate at angles ϕ with respect to the cavity optical axis OO 0 given by the following equation (Fig. 11): ϕ = π − (2β + α) .
(55)
It is useful to identify the beams by their wavelengths. In that case Eq. (54) is transformed to one of the following equations with the help of Eqs. (2) and (3):
λ
, or ϕ(λ) = π − 2β + arcsin 2t λ λ 0 + arcsin . ϕ(λ) = π − 2 arcsin 2t 2t
(56)
44 Therefore, the output radiation of the laser consists of multiple low-divergent laser beams that have a fan-shaped angular distribution ϕ(λ) and propagate from the imaginary point F located behind the grating. It can be collimated with a single lens or with a more complex multi-lens focusing system. The choice between one of the expressions (56) depends on what is more convenient to use—the central wavelength λ0 or the grating angular orientation β. The latter is preferable when the grating is fixed and the control of the output spectrum is done by the spatial shaping of the pump radiation. One reason that the information about the spatial structure of the output radiation is essential is its applications, such as coupling the output channels to singlemode fibers in telecom devices or designing a spectroscopic measurement system that uses the laser as the multiwavelength light source. However, the parameters of the output beams can also be used for reconstruction of the intracavity mode structure, which can be useful in the design process of the laser. Indeed, a Gaussian beam is fully characterized by its wavelength λ and its confocal parameter z0 , which can be derived from the experimental measurements of the Gaussian beam profiles in at least two planes located at a known distance from each other. Thus, from the measurements of the intensity profiles of the output laser beams one can find the beam waist on the diffraction grating wg . Using this result and the parameters of the spatially-dispersive cavity, one can fully characterize the laser modes of different wavelength channels by applying Eqs.(17) and (18) and the supporting formulas (11)– (14). An example of such characterization of the laser modes is considered in detail in Section 3.1.7. A detailed description of the characterization of the Gaussian beams from the measurements of their intensity profiles is given in Appendix B.
45 2.8
Complex Intracavity Focusing System
In some situations the original laser scheme, shown in Fig. 4, needs to be modified in order to obtain the desirable characteristics of the output radiation. In particular, the single intracavity focusing lens sometimes needs to be replaced with a complex multi-lens optical system. For instance, if the semiconductor gain media is used, then it is first necessary to collimate the highly divergent fast axis of the diode laser output beam (perpendicular to the plane of the spatially-dispersive cavity). Only after that can the diode be used in the spatially dispersive laser scheme. Another very common example is the case when the intracavity lens cannot be considered as a thin lens, because its thickness is comparable to its focal length (this often occurs when the lens material has a low refractive index, such as CaF2 lenses used in the middle infrared spectral region). In this section general corrections to the Gaussian beam model of the laser are introduced for the spatially dispersive cavity scheme in which the intracavity lens is replaced with an arbitrarily complex multi-lens focusing system. Such a modified spatially dispersive laser scheme is shown in Fig. 12. Note that the intracavity aperture is not shown in Fig. 12 because it is located inside the focusing system in one of its internal focal planes. In the paraxial approximation and the framework of the matrix formalism, the focusing system is described by the following transformation matrix:
A T B T MT = , C T DT
(57)
where the subscript “T ” in the notation of the matrix elements stands for “telescope” because some lasers described in this work contain intracavity telescopes.
46 λ1
def f
LT
O
Gain Medium
AT BT C T DT
λ2
l
xt
d2 x
Input mirror
d1
λ0
α
Multi-lens focusing system
O0
Grating
Fig. 12. Spatially dispersive laser with a complex multi-lens focusing system.
A light ray, located at a coordinate x from the optical axis OO 0 , incident onto the focusing lens system parallel to this axis, is transformed by the system according to the following transformation low (see [3], Section 1.2):
xT A T = α CT
B T x ⇒ DT 0
α = x · CT ≡ −
x , fT
(58)
where fT ≡ −1/CT is the effective focal length of the multi-lens system, xT is the new coordinate of the ray, and α is the new angle of propagation. Comparison of Eq. (58) with Eq. (1) shows that in the geometrical optics approximation this complex laser system can be treated in exactly the same way as the basic laser scheme in Section 2.1, with the only difference being that the focal length of the single intracavity lens is replaced with an effective focus of the multi-lens focusing system. Consequently, the Gaussian beam model of this complex laser system will be developed in the same way as in Section 2.3.
47 Indeed, the transmission matrix of the new laser system is calculated as follows:
1 l A T M = − f1T 0 1
g1∗
BT 1 def f = 0 1 − f1T DT
L , ∗ g2 ∗
(59)
where the matrix elements are given by the following expressions:
l fT def f g2∗ = DT − fT g1∗ = AT −
(60) (61)
L∗ = BT + def f AT + lDT −
def f l . fT
(62)
From Liouville’s theorem [3] it follows that the determinant of any ray transmission B matrix CA D is equal to the ratio of the refractive indexes of the media n1 and n2
between which the ray is transmitted, i.e., the following expression is always true:
det M = AD − BC =
n1 . n2
(63)
For detailed treatment of this theorem see Ref. [3]. In the laser system under study the rays are always transformed from air to air, even when the input mirror is located very close to the rear facet of the laser gain crystal. Therefore, the determinant of the ray transmission matrix of the laser from the input mirror to the grating always equals 1, regardless of the cavity internal complexity. Thus, one can assume that the following condition always holds:
AD − BC = 1 .
(64)
48 Consequently, one obtains the following equation for the matrix (59):
g1∗ g2∗ +
L∗ =1 fT
⇒
L∗ = fT (1 − g1∗ g2∗ ) ,
(65)
which coincides with the right hand side of Eq. (12). Therefore, the ray transmission matrix (59) for the complex resonator is mathematically equivalent to the transmission matrix (10) of the basic laser scheme with new stability parameters g1∗ and g2∗ and the new effective length of the laser resonator L∗ . Thus, the whole analysis represented by Eqs. (15) to (56) can be applied for this complex cavity with the new g-numbers g1∗ and g2∗ . The general algorithm for finding the cavity parameters, described in Section 2.6, can be used for the new laser scheme. However, when the appropriate values of the g ∗ –numbers are found, one needs to solve Eqs. (60), (61), (62), and (65) simultaneously for AT , BT , and DT in order to find the appropriate configuration of the multi-lens focusing system. One can notice, however, that these four equations contain six unknowns: the three matrix elements, the effective length L∗ , and the distances def f and l. More equations are required for the system to be solvable. However, one of the unknown distances, usually def f , is specified by additional requirements on the placement of the optical elements, and only five unknowns are left. On the other hand, one can obtain one more equation, the fifth, by applying Liouville’s theorem to the multi-lens focusing system itself:
A T DT − C T B T = A T DT +
BT =1. fT
(66)
49 The system of five equations (60), (61), (62), (65), and (66) is thus complete and is solvable for all unknown variables. Once the parameters of the focusing system are determined, one needs to find the most convenient focal point, where the fundamental modes of all wavelength channels intersect. The intracavity aperture size, that might be used for out-filtering of the high-order transversal modes, can then be calculated using the method represented by Eqs. (28) to (31). It is noteworthy that from the theoretical point of view there is an infinite number of appropriate cavity configurations. In practice, however, this number is highly limited by the parameters of the available set of the optical elements, requirements of the overall complexity of the optical system and, possibly, the constraints of the cavity misalignment sensitivity that require the laser to be as simple as possible. In the next chapter several new spatially-dispersive laser systems will be studied experimentally, and all conclusions of the developed theoretical model will be thoroughly tested.
50
CHAPTER 3 EXPERIMENTS 3.1
Cr2+ :ZnSe Polycrystalline Spatially-Dispersive Laser
3.1.1 Introduction This section describes a new cw middle-infrared (mid-IR) polycrystalline Cr2+ :ZnSe, widely tunable, multiwavelength, ultrabroadband laser, based on the Littrow-mounted grating spatially-dispersive cavity. The demand for multiwavelength room-temperature mid-IR sources of the laser radiation is stimulated by such fields as multiwavelength spectroscopy, trace gas detection, differential absorption light detection and ranging, free-space optical communications, optical coherence tomography, white light Schleiren imaging, and numerous wavelength-specific military applications. In 1996 and 1997 DeLoach, et. al. [89] and Page, et. al. [90] conducted extensive investigations of the spectral properties of metal-doped zinc chalcogenides. Since then, continuous-wave, gain-switch, and mode-locked lasing of Cr2+ :ZnS and Cr2+ :ZnSe has been demonstrated [91–95]. These studies have shown the excellent properties of Cr2+ :ZnS and Cr2+ :ZnSe for generation of mid-IR laser radiation in a broad (2–3 µm) spectral range. The Cr2+ :ZnSe spatially-dispersive laser demonstrated in this section is not only of great scientific interest, but also is an excellent candidate for a cost-effective and reliable mid-IR laser source for generation of the multiwavelength, ultrabroadband, tunable laser spectrum for the applications mentioned.
51 First, a dual-wavelength operation of the spatially-dispersive laser, achieved by a dual-beam pumping, tunable over a spectral range of 600 nm (2200–2800 nm), is described. Three sets of experiments are considered in detail and analyzed in the framework of the laser model: simultaneous tuning of the dual-wavelength output spectrum by rotation of the Littrow grating in the diffraction plane; simultaneous tuning of the dual-wavelength output spectrum by transverse shift of the dual-pump beam; and tuning of the wavelength spacing of the dual-wavelength output spectrum by changing the transverse distance between the components of the dual-pump beam. Second, a multiwavelength ultrabroadband operation of the spatially-dispersive laser, pumped by a highly elliptical, horizontally stretched pump beam, is demonstrated. The laser produces a continuous ultrabroadband output spectrum with the overall spectral bandwidth of up to 135 nm. Third, a spectral modulation of the ultrabroadband spectrum by means of a self-formed tunable intracavity interferometer is shown. It is demonstrated that the interferometer not only transfers the continuous ultrabroadband spectrum to a discrete multiwavelength spectrum, but also increases the overall output spectral bandwidth to more than 200 nm (2400–2600 nm). The multiwavelength output spectrum in this case consists of a large number of close spectral lines. The wavelength spacing between the lines can be continuously tuned by changing the free spectral range of the intracavity interferometer. Finally, tuning of the entire multiwavelength ultrabroadband spectrum (20– lines, 200 nm) over a spectral range of 600 nm (2200–2800 nm) by means of rotation of the Littrow grating in its dispersion plane is shown.
52 3.1.2 Available Optical Elements and Equipment The optical elements for the laser cavity, available for the experiments, were not specially designed for the spatially-dispersive laser. As a result, it was not possible to first design the laser cavity in accordance with its theoretical model, and then build the laser. Instead, the best possible configuration of the laser resonator was calculated for the available optical elements, the properties of the output radiation were estimated for this cavity, and then the laser was built from the available optical components. The laser was assembled and adjusted to its best performance, and the results were compared to the theory. The parameters of the available optical elements are summarized in Table 2. As one can see, the most noticeable problem is the absence of anti-reflection (AR) coatings on the surfaces of the laser crystal and the intracavity lens, which cause huge intracavity losses. Indeed, the Fresnel reflections from two surfaces of the laser crystal (n = 2.44 at 2.6 µm) and on the surfaces of the intracavity lens (n = 1.42 at 2.6 µm) give a single-pass loss of about 40%. This results in a small output power (up to 15 mW), despite the high pump power (the maximum incident pump power was about 6 W, and the absorption of the Cr2+ :ZnSe crystal was 70% at 1.56 µm). In addition to that, the available diffraction gratings reflect more than 98% into the first order of diffraction when it is installed at the blaze wavelength, which lowers the laser threshold, thus increasing the laser spectral tuning range, but it also reduces the output power, since most of the generated optical energy stays inside the cavity. It is clear that in these conditions the output power could not be optimized, and these experiments were concentrated on the investigation of the spectral properties of the polycrystalline Cr:2+ ZnSe spatially-dispersive laser.
53
Table 2. Available optics for the Cr2+ :ZnSe spatially-dispersive laser Optical Element
Parameters
Values
Plane input mirror
Substrate Thickness AR coating HR coating
Glass 4 mm < 0.1% at 1.56 µm > 99.5% at 2.2–2.8 µm
Laser crystal
Material Size (L×W×H) Refractive index Cr concentration AR coating
Polycrystalline Cr2+ :ZnSe 4 × 8 × 1 mm3 2.44 at 2.6 µm 2 × 1019 cm−3 none
Intracavity lens
Material Refractive index Type Surface radii Thickness Focal length AR coating
CaF2 1.42 at 2.55 µm Biconvex ≈ 9.74 mm 10 mm 25 mm none
Diffraction Gratings
Groove densities
600 g/mm 324 g/mm Au 2500 nm > 98% into 1st order
Coating Blaze wavelength Efficiency Pump Lenses
Material AR coating Type Focal lengths
Glass < 0.1% at 1.56µm Planoconvex 50 mm 65 mm 70 mm 80 mm
54 It was decided to conduct experiments that demonstrate the concept of the multiwavelength spatially-dispersive laser, corroborate the laser theoretical model, and provide an experimental basis for future development of the lasers of this type. Therefore, an emphasis was placed on the optimization and measurements of the spectral and spatial properties of the laser output radiation, and on a comparison of the experimental results with the theoretical predictions. For detection of the output radiation in the mid-IR spectral region a sensitive PbS detector was used. The detector can only measure a pulsed signal and, therefore, the output radiation needs to be modulated. A mechanical chopper was used for that purpose. Initially the chopper was installed in the output laser beam and the multiwavelength laser was operated in a true CW regime. However, the thermal lensing effects in the laser crystal due to the intense pumping caused an instability of the multiwavelength operation of the laser, making it difficult to perform precise measurements. To reduce the power of the thermal lens, which is proportional to the average pump power (see e.g. [3]), the chopper was moved to the pump beam. The chopper modulation frequency was about 125 Hz, and its on/off ratio was 1/2. Consequently, the average pump power was reduced by 50%, significantly decreasing the undesirable thermal lensing in the laser crystal. For the measurements of the laser output spectrum, a scanning spectrometer was used. The PbS detector was installed on the output slit of the spectrometer. The spectrometer controller was synchronized with the modulated detector signal using a lock-in amplifier. The spectrometer requires up to a minute to acquire a single spectrum, which has a spectral width of several hundred nanometers, so there was some time-averaging of the laser spectrum. However, fast measurements of small
55 portions of the output spectrum revealed that the laser is sufficiently stable to assume that the measured average and an instant spectra coincide with each other. The pump source of the spatially-dispersive laser is a CW Er-doped fiber laser operating at a wavelength of 1.56 µm, which is close to the maximum of the Cr2+ :ZnSe absorption band. The maximum output power of the laser is 10 W. The laser produces a collimated Gaussian beam with a diameter of about 8 mm, which is larger than is optimum. For that reason, a self-made two-lens beam compressor was used to reduce the beam diameter to approximately 1 mm. The Er-doped pump laser is extremely sensitive to back reflections: a retroreflection of the pump beam by an uncoated optical surface with the average power of as low as 100 mW can damage the laser fiber. To avoid possible fiber damage, the laser has a built-in protection system that turns off the laser when a critical level of the back reflected signal is detected. The pump laser beam was focused into an uncoated Cr2+ :ZnSe crystal with a plane input facet. Consequently, due to the back reflections the Er laser switched off long before the lasing threshold of the spatiallydispersive laser was achieved. A Faraday rotator was not available at the time of the experiments, and some other measures to reduce the influence of the back reflections had to be found. One solution is the installation of the laser at a large distance from the retroreflecting surfaces. Preliminary investigations, however, revealed that this method did not adequately reduce the back reflection. The partial solution was the installation of the Cr2+ :ZnSe crystal at some angle with respect to the optical axis of the incident pump beam and shifting the crystal slightly from the focal plane of the pump lens. Still, the residual back reflections limited the maximum usable pump power to about 6 W.
56 3.1.3 Preliminary Experiments Despite the fact that CW and pulsed single-wavelength Cr2+ :ZnSe lasers were investigated quite thoroughly in previous works [110, 113, 122, 123], it was not clear if the laser would operate effectively with such huge intracavity losses. Therefore, it was necessary to perform some preliminary study of the laser, based on the spatiallydispersive cavity. The purpose of the preliminary experiments was to find the best pump conditions (such as the pump beam profile, threshold conditions, and the best pump lens) and the cavity adjustment to obtain the maximum possible spectral tuning range. To achieve this goal, a single-wavelength, tunable laser, based on the same cavity configuration, was built and studied. The transformation of the single-wavelength laser into the multiwavelength spatially-dispersive laser is straightforward: the pump beam is reshaped in accordance with the desired output spectrum and the cavity parameters, as described in the previous chapter (see also [109,116,117]). The singlewavelength laser scheme, together with the pump and the complete measurement systems, is shown in Fig. 13. As described above, the original laser beam from the Er laser head passes through a 10× two-lens beam compressor, where it is compressed to approximately 1 mm in diameter. Then it goes through the chopper to two folding glass prisms that are used to direct the beam along a chosen optical axis. The beam is then focused by the pump lens into the Cr2+ :ZnSe laser crystal. As discussed earlier, the crystal is slightly tilted to avoid the back reflections of the pump radiation. It must be noted here that the crystal is tilted in the horizontal plane only (as shown in the figure) due to the following reasons.
57
8 mm pump beam from 1.56 µm, 10W Er-fiber laser f1 = 80 mm Scanning spectrometer
f2 = 8 mm
Signal
ism Pr
Chopper 125 Hz, 1/2
Beam compressor
Sync signal
Lock-in Amplifier
PbS detector
Diffuser
PC and spectrometer controller Collector lens
λ CaF2 lens Cr2+ :ZnSe f = 25 mm Crystal
fp
ism
Tuning
600 g/mm
Pr
Grating 75 mm
10
20 mm
Input mirror
Pump lens fp = 65 mm
Fig. 13. Schematic diagram of the single-wavelength, tunable Cr2+ :ZnSe laser based on the spatially-dispersive cavity.
The Cr2+ :ZnSe crystal has some deviation of Cr concentration across its height due to the thermal diffusion method of doping the zinc selenite crystal by chromium (see [110,113,122,123] for a details). It was found that the best regions of the crystal, where the concentration of Cr is optimal for lasing, are located close to the top and the bottom surfaces of the crystal (at a distance of approximately 1/3 of its height from the surfaces). Thus, tilting the thin (1 mm) and long (4 mm) crystal in the vertical plane causes both the pump beam and the laser mode to be attenuated by the crystal surface (top or bottom), which introduces additional intracavity losses. Therefore, only a small vertical tilt could be used.
58 The generated mid-IR laser radiation is focused by the collector lens onto a diffuser installed in front of the entrance slit of the spectrometer. The diffuser was used for two purposes: (1) to reduce the signal level entering the spectrometer, to prevent saturation of the PbS detector, and (2) to make sure that all wavelengths enter the spectrometer equally. The lasing wavelength is tuned by rotation of the diffraction grating in its dispersion plane. The tuning of the output spectrum of the laser is shown in Fig. 14.
PSfrag replacements
Measured intensity, a.u.
Pump power, W 1
3.5 3.0
2.0
2.0
2.0
2.0
3.0
3.5
2300
2400
2500
2600
2700
2800
0.8 0.6 0.4 0.2 0 2100
2200
Wavelength, nm Fig. 14. Tuning the output spectrum of the single-wavelength spatially-dispersive laser. The top x-axis shows the corresponding pump power at every wavelength.
Although the Cr2+ :ZnSe amplification band would allow the wavelength tuning within a range of 2000–3000 nm [110,113,122,123], the tuning of the laser was limited to the spectral range of 2150–2800 nm. The major reason for this limitation is that this is the spectral range of the input mirror. There is no doubt that a wider tuning range could easily be achieved with a more appropriate input mirror.
59 During the tuning of the laser, the pump power was set to overcome the lasing threshold and obtain a stable laser operation. The corresponding values of the pump power are shown in Fig. 14 on the top x-axis above every spectral line. At some wavelengths it was necessary to increase the pump power to obtain a stable operation of the laser. This explains why the spectral line at 2300 nm, located closer to the center of the laser amplification band, is less intense than the line at 2200 nm. When the lasing threshold was reached, the laser cavity was adjusted to obtain the maximum output signal. The best cavity configuration was found in the experiment shown in Fig. 13, where the positions of the cavity elements are indicated by the distances between them. It must be mentioned here that these positions are approximate (±2 mm for the lens position and ±10 mm for the position of the diffraction grating). First of all, it is not possible to measure their values exactly. Second, the positions are slightly different for different wavelengths. The latter is explained by the differences in the laser mode size for different wavelengths. It was also found in the experiments that the lowest lasing threshold and the widest tuning range are obtained when the pump lens with the focal length of 65 mm is used. All the other available pump lenses gave inferior results. The measured diameter of the pump laser beam near the input facet of the Cr2+ :ZnSe crystal is about 160 µm (the measurement technique of the Gaussian beam radius is described in detail in Appendix A). This suggests that for the central wavelengths of the tuning range, the threshold pump power density is about 100 W/mm2 , and for the side wavelengths it is about 170 W/mm2 . Therefore, when the single-wavelength laser is transformed into a multiwavelength spatially-dispersive laser, the number of the output wavelengths and the
60 overall output spectral bandwidth will be limited. One can estimate that for a discrete multiwavelength output spectrum one can expect up to 3 to 4 equally-spaced spectral lines that cover the entire tuning range (∼ 2 W of pump power per line) and up to 5 to 6 lines if the laser operates between 2300–2600 nm. In the case of a continuous output spectrum, when the pump beam is reshaped into a highly elliptical laser beam with the thickness of ∼ 50 µm, the total width of the pump beam cannot be larger than approximately 0.8 mm. For the existing cavity parameters, this would give the output spectral bandwidth of approximately 70 nm with the high-resolution diffraction grating (t = 1/600 mm) and about 180 nm with the low-resolution grating (t = 1/324 mm), as follows from Eq. (9). Consequently, it generally does not make any sense to split the pump beam into more than six parts to obtain a discrete multiwavelength spectrum or stretch the beam to the width of more than approximately 1 mm. Note also that in order to obtain a wider output spectral bandwidth the low-resolution diffraction grating should be used. In the following subsections two types of Cr2+ :ZnSe spatally-dispersive lasers will be presented and analyzed in detail: (1) a dual-wavelength spatially-dispersive laser and (2) a multiwavelength spatially-dispersive laser with a quasi-continuous or discrete shape of the output spectrum. In the first case, the pump beam is split into two equal parts, and each part pumps its own region of the Cr2+ :ZnSe crystal. It will be shown how the laser spectrum can be tuned as a whole by means of rotation of the diffraction grating or by simultaneous transverse shift of the dual pump beam. Then the individual tuning of the components of the output spectrum, by independent transverse shift of each of the two pump beams, will be demonstrated.
61 In the second case, the pump beam is reshaped into a highly elliptical, horizontally stretched laser beam to continuously pump a wide region of the Cr2+ :ZnSe crystal to obtain a broadband, continuous output spectrum. In addition, a new method of shaping the output spectrum (unexpectedly found during the experiments) by means of a self-formed tunable intracavity Fizeau interferometer will be demonstrated. Finally, grating tuning of the entire ultrabroadband spectrum across the laser amplification band will be shown. In both cases, the experimental results will be analyzed using the theoretical model of the laser developed in Chapter 2 to verify the validity of the model. Before proceeding with the presentation and analysis of the experimental data, it must be mentioned that the alignment procedure of the spatially-dispersive cavity is not trivial due to the complexity of the cavity and also because here one works entirely with the laser light outside of the visible spectrum, which makes it impossible to use the pump beam and the luminescence light for the cavity alignment. The most successful alignment procedure, which was developed during the preliminary experiments, is described in detail in Appendix C.
3.1.4 Dual-Wavelength, Grating-Tunable Cr2+ :ZnSe Spatially-Dispersive Laser A schematic diagram of the dual-wavelength spatally-dispersive laser, the pump system, and the measurement system is shown in Fig. 15. The original pump beam is split into two components by a self-made beamsplitter that consists of two mirrors M1 and M2 , and is directed to the laser by a moving and rotating folding prism P2 . The pump beams propagate symmetrically with respect to the optical axis OO 0 and intersect in the front focal point of the pump lens.
62 Chopper 125 Hz, 1/2
Beam compressor 8 mm pump beam from 1.56 µm, 10W Er-fiber laser
P
f2 = 8 mm
Signal
Sync signal
Lock-in Amplifier
PC and spectrometer controller
R = 50% 1
λ1 ∼ 1 mm CaF2 lens Cr2+ :ZnSe f = 25 mm Crystal
M2
M1
PbS detector
Collector lens
R = 100% Two-mirror beamsplitter
45 mm
fp
O
170 mm
2 270 mm
Scanning spectrometer
λ2
ism
1
f1 = 80 mm
Diffuser
Pr
O0
Grating 600 g/mm
75 mm
10
20 mm
Input Pump lens Auxiliary Moving and rotating folding prism P2 mirror fp = 65 mm aperture
Fig. 15. Schematic diagram of the dual-wavelength spatally-dispersive laser.
After refraction on the pump lens, the beams propagate parallel to the optical axis OO 0 and at the same time they are simultaneously focused into the Cr2+ :ZnSe crystal. A combined adjustment of the mirrors M1 and M2 and the folding prism P2 , which are installed on tilting mirror mounts, allows changing of the angle between the pump beams, at the same time keeping their intersection point unchanged. This method can be used to obtain the desirable spatial separation of the pump beams in the crystal. To facilitate the alignment of the pump beams, an iris aperture is installed in the front focal point of the pump lens. The aperture position is fixed, and during the tuning of the spatial spacing between the pump beams it is only necessary to adjust the beams so that they are transmitted without loss.
63 The alignment procedure of the cavity, described in Appendix C, was initially performed for the pump beam 1 with beam 2 blocked. After that beam 1 was shifted in the crystal and beam 2 unblocked. The transverse positions of the beams were controlled by observation of the output spectrum of the laser. The goal was to obtain two distinguished spectral lines with a relatively small wavelength spacing, located symmetrically near the center of the laser amplification band. The hardware used in the experiments limited the minimum spectral separation between the lines to approximately 40 nm. It noteworthy that this method of focusing multiple pump beams into the laser crystal with a single lens can be used for any number of beams. It is also possible to arrange a control of individual beams, as will be demonstrated in Subsection 3.1.6, in which case one can obtain any desired spectral distribution of the output radiation by positioning the pump beams in the laser crystal and controlling their intensity. The output radiation, which consists of two laser beams, is collected by a large CaF2 lens and focused onto a diffuser, installed on the entrance slit of the scanning spectrometer. The spectral tuning is performed by a gradual rotation of the diffraction grating in its dispersion plane. In this case the pair of the output wavelengths is moved across the tuning interval of the laser. The angular orientation of the diffraction grating β cannot be measured directly, because there is no method of direct measurement of the angle between the grating normal and the optical axis of the cavity of the working laser. One might note that since the dual-wavelength lasing is achieved by a reshaping of the pump beam of initially aligned single-wavelength laser, for which β can easily be found from the Littrow condition (2), it is only necessary to measure the change of β while the
64 laser is tuned. However, the problem of this method is that conventional rotation stages do not provide high enough accuracy to perform the measurements correctly (usually, non-expensive stages give an error of angle measurements of 1–2 arc deg). In addition, the laser cavity might be slightly misaligned during this transformation and a subsequent fine adjustment. Moreover, in order to test the laser model, the angle should be measured as accurately as possible. Therefore, a more reliable method of the measurement of β is desirable. One such method can be derived as follows. The exact value of β, as well as the angles α between the laser modes and the cavity optical axis (see Fig. 4), can be extracted for every pair of the tuned spectral lines from their spectrum, independently from the laser construction. This can be done with a few general assumptions about how these spectral lines are obtained. Let PSfrag replacements us consider a misaligned spatially-dispersive cavity shown in Fig. 16.
hλi
λ2 α2
O
λ1 α1 1 Pump 1
α1 OL
OL0
α2 2
O β
~ nG
O0 Pump 2
Fig. 16. Calculation of the grating angular orientation. OL OL0 –optical axis of the intracavity lens; OO 0–optical axis of the cavity; α1 , α2 –angles of propagation of the laser modes with respect to the optical axis OO 0 ; ~ nG –normal to the diffraction grating; and hλi = (λ1 + λ2 )/2–average wavelength for the pair λ1 , λ2 .
65 The cavity misalignment is caused by horizontal tilt and shift of the intracavity lens (and, therefore, its optical axis OL OL0 ) with respect to the cavity axis of symmetry OO 0 . The axis of symmetry is defined as a line that goes exactly in the middle between the two laser modes before they enter the lens, when the output spectral pair is tuned to the center of the laser tuning curve. By this definition, the laser modes propagate spatially and angularly symmetrically with respect to OO 0 before they reach the lens. It is easy to show that in the paraxial approximation the refraction on the lens does not break the symmetry and, therefore, α1 = −α2 . Thus, in order to find the exact values of β and α ≡ |α1 | = |α2 |, the only necessary assumption is that the wavelengths satisfy the Littrow grating equation (2), i.e., λ1 = 2t sin(β − α)
(67)
λ2 = 2t sin(β + α),
where t is the grating spacing constant. From Eq. (67) one can easily find that hλ i h λ i 1 2 1 arcsin + arcsin , β= 2 2t 2t 1 α= 2
arcsin
hλ i 2
2t
− arcsin
h λ i 1
2t
.
(68)
(69)
It must be emphasized that, from this point of view, there might be any system of any complexity that generates the radiation with the wavelengths λ1 and λ2 . In this sense, the determination of the angles is independent from the laser model, and the only requirement on its properties is the compliance with the system (67). Moreover, this method does not require the paraxial approximation. Instead, if from the analysis of
66 the values of the determined angles it follows that the approximation is fulfilled, then one can conclude that the two laser beams are symmetric inside the rest of the laser system. In what follows, it will be always assumed that the experimental values of β and α are obtained from the output spectrum as described above. The tuning of the dual-wavelength spectrum, obtained in the experiment, is shown in Fig. 17(a). The total pump power was 6 W (higher powers caused unacceptable increase of back reflections that switched off the pump laser, as discussed before). The corresponding angular orientation of the grating is indicated on the top x-axis. Graphs Fig. 17(b) and Fig. 17(c) show the measured angles β and α versus the average wavelength hλi = (λ1 + λ2 )/2. In general case, this average wavelength is not equal to the wavelength λ0 = 2t sin β, which corresponds to the axial spectral channel OO 0 :
1 hλi = (λ1 + λ2 ) = t(sin(β + α) + sin(β − α)) = 2t sin β cos α . 2
(70)
However, in the paraxial approximation cos α ≈ 1 and hλi ≈ λ0 . It is clear that the paraxial approximation is valid here, since cos α = cos(0.54◦ ) ≈ 0.99995. The dependence of α on the wavelength is approximated by a linear function with a slope ∂α/∂λ ≈ 3.6×10−6 (arc deg)/nm, which indicates that the angle remains constant during the wavelength tuning. The random character of small (less than ±6 %) deviations of the angle from its average value of about 0.53 arc deg shows that there is no systematic change in α during the laser tuning. The random error is explained by contributions of several factors, besides possible errors in determination of the wavelengths from the recorded output spectra (±1 nm).
67
β, arc deg 42.1
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β, arc deg
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Experiment Linear fit: β = −15.5 + 0.026λ
50
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Wavelength, nm α, arc deg
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Experiment Linear fit: α = 0.53 + 3.6 × 10−6 λ
(c)
0.56 0.54 0.52 2100
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Wavelength, nm
Fig. 17. Tuning of the dual-wavelength spectrum by the diffraction grating. The top x-axis of Graph (a) shows corresponding angular orientation of the grating β. Graphs (b) and (c) show the angles β and α vs average wavelength and corresponding linear fits.
68 First, there might be a mismatch between the diameters of the laser modes in the Cr2+ :ZnSe crystal and the pump beams, i.e., the pump beams are larger than the laser modes. As a result, the laser modes can oscillate slightly within the pumped regions as the wavelengths are tuned. Since the angles α are proportional to the transverse positions of modes, they will also oscillate randomly. The second source of the random oscillations of α is the thermal lensing effect in the laser crystal. The thermal lensing effect depends on the power level of both the laser mode and the corresponding pump beam. As the wavelength is tuned, the lasing power of the modes and the corresponding thermal lenses change. This effect contributes to the random oscillations of the mode positions in the laser crystal, which cause the small oscillations of the α-angles. In order to examine the laser theoretical model, it is necessary to measure the exact transverse positions of the lasing channels in the laser crystal. Such a measurement cannot be performed directly in the working laser. However, it is reasonable to assume that the laser modes are approximately coaxial with the pump beams inside the Cr2+ :ZnSe crystal. Thus, it is enough to measure the x-coordinates of the pump beams in front of the crystal input facet. It is also important to make this measurement under the same conditions as those in the working laser (e.g., the pump power level). There is probably no laser beam analyzer that could do this kind of measurement in the present system; this is why the “knife edge method” of measurement of the Gaussian beam profile was used (see [3], p. 612). The positions of the beams and their diameters are extracted from the measured beam intensity profiles. The application of this method in the present experiments is illustrated in Fig. 18. The measurement is performed in the following way.
PSfrag replacements
69 1.5 mm
Laser crystal
Pump lens
Pump beam 1
Pump beam 2
Power meter
Input mirror
Blade on a motion stage with a micrometer screw
Fig. 18. Experimental setup for measurements of the intensity profile of the dual pump beam with the knife edge method.
A vertical metallic blade is installed on a motion stage with a micrometer screw. The blade is moved in small steps (10 µm) across the measured beam, and the transmitted power as a function of the blade position is recorded. The recording begins when the measured beam is completely open and ends when the transmitted power is negligible compared with its maximum. The two pump beams are measured separately, i.e., one of them is blocked while the other is measured. Unfortunately, the approximate formula (23.17) of [3] cannot be used here, since it is necessary to measure not only the pump beam diameters, but also the exact transverse distance between them, which requires knowledge of the exact transverse coordinates of the axes of the beams. In addition to that the measurement should be performed with the highest possible accuracy. Therefore, after the measurement was done, the experimental data was processed using a numerical procedure described in Appendix A. The results of the measurements of the pump beam profiles in front of the input mirror, together with corresponding theoretical fits, are shown in Fig. 19. The fitted values of the beam centers and the beam radii are indicated in the figure as w1 , x01 and w2 , x02 for the pump beams 1 and 2, respectively.
70 1
Intensity, a.u.
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w2 ≈ 114 µm x02 ≈ 630 µm
w1 ≈ 113 µm x01 ≈ 165 µm
0.8 0.6 0.4
Experiment Fitting integral
0.2 0 0
100
200
300
400
500
600
700
800
Coordinate, µm Fig. 19. The results of the measurements of the intensity profile of the dual pump beam with the knife edge method. The determined beam radii and beam positions are indicated in the figure as w1 , x01 and w2 , x02 , respectively.
From these measurements one finds that the beam spatial separation is δx ≈ 466 µm. One can also estimate the radii of the pump beams in the laser crystal using Eq. (88), where the distance z is an effective distance from the measurement plane to the crystal input facet, and w is the measured pump beam radii. This calculation leads to the pump beam radii in the crystal of approximately 111 µm, which is practically the same as the measured radii. It will be shown in Subsection 3.1.7 that the radii of the pump beams in the laser crystal are larger than the radii of the laser modes, indicating a mode mismatching between the pump beams and the laser modes, which leads to the small oscillations of the output wavelengths and angles α. The expression for the wavelength separation of the oscillating channels (5) as a function of the grating angle β can be written in the following form:
δλ = A cos β ,
(71)
. For the current cavity parameters the measured spatial where the constant A ≡ δx 2t f
71 spacing δx between the pump beams, one obtains that A ≈ 62 nm. Experimentally measured wavelength spacing δλ versus β, together with the theoretical function (71), is shown in Fig. 20.
50
δλ, nm
45
PSfrag replacements
40 35 30 40
Experiment Fit: δλ = Ae cos β, Ae ≈ 63 nm Theoretical model
42
44
46
48
50
52
54
56
β, arc deg Fig. 20. Experimental measurements and the laser model predictions of the output wavelength spacing vs the grating angular orientation.
The experimental data is fitted by a function δλ(β) = Ae cos β, which is also shown in the figure. The fitting parameter Ae ≈ 63 nm (which gives the channel spacing δx ≈ 469 µm) is very close to the theoretical value of the constant A ≈ 62 nm in Eq. (71), with the difference of only 0.64 %, which shows a very good agreement of the theory and the experiment. The small systematic error is most likely caused by the method of measurement of the beam spacing δx. Indeed, the spatial separation is measured in front of the input mirror at a distance of approximately 1.5 mm from its front surface, as shown in Fig. 18. The mirror substrate has a thickness of about 4 mm, and there is also an approximately 1 mm separation between the mirror surface and the input facet of the Cr2+ :ZnSe crystal, as shown in Fig. 15.
72 Therefore, taking into account the mirror glass refractive index n ≈ 1.5, the pump beams propagate a distance of about 5 mm between the measurement plane and the laser crystal. Consequently, if the beams are not perfectly parallel, the distance between their axes in the crystal is not the same as in the measurement plane, as shown in Fig. 21. PSfrag replacements
δxc Laser crystal
Pump lens
+F
δxm
Measurement plane
Input mirror
Intersection point
Fig. 21. Non-parallelism of the pump beams. When the pump beams are not parallel, the beam spacing in the measurement plane δxm is different from the spacing in the laser crystal δxc . This leads to the difference of the theoretical and the experimental values of the constant A in Eq. (71). The dashed lines show an ideal case when the incident pump beams intersect exactly in the front focal point of the pump lens.
The parallelism of the pump beams, in turn, is very sensitive to the position of their intersection point in front of the pump lens: if there is a slight misalignment of this point from the front focal point of the pump lens, the beams easily become non-parallel. Since the experimental value of δx is slightly larger than its theoretical value, calculated on the basis of the measurements of the pump beams spacing, one can conclude that the pump beams are slightly divergent. The random errors in δλ are caused by the random oscillations of the angles α, as was discussed earlier in this section.
73 As follows from the laser theoretical model, the grating tuning of the laser changes the output wavelengths according to Eq. (4). The transverse coordinates of the wavelength channels in the symmetric case, considered here, are equal to 1/2 of the channel spacing δx and are opposite in sign, i.e. (see Fig. 3): δx , 2 δx . x(λ2 ) ≡ x2 = − 2
x(λ1 ) ≡ x1 =
(72)
Thus, the dependencies of the output wavelengths on the angle β during the laser tuning are governed by the following expressions, derived from Eq. (4): δx λ1 (β) = 2t sin β − , 2f δx λ2 (β) = 2t sin β + . 2f
(73)
The experimental and theoretical dependencies λ1,2 (β) are shown in Fig. 22. The experimental data are obtained from the output spectrum. The theoretical curves are obtained from equations (73) where the spatial spacing δx is the measured transverse distance between the pump beams. The experimental dependencies are fitted by a function λ(xe ) = 2t sin(β − xe /f ), also shown in the figure, with the fitting parameter xe , which corresponds to the transverse coordinates of the channels. Fig. 22 shows a very good agreement of the theory and the experiment. The determined transverse coordinates of the wavelength channels xe suggest that the channel spacing δx equals approximately 469 µm. This result is in a perfect agreement with earlier analysis of the wavelength spacing δλ versus angle β, and with the direct measurements using the knife edge method.
74 2800 2700
λ1,2 , nm
PSfrag replacements
2600
Experimental λ2 Fit: λ2 = 2t sin(β + xe /f ), xe ≈ 235 µm Theoretical λ2
2500 2400 2300
Experimental λ1 Fit: λ1 = 2t sin(β − xe /f ), xe ≈ 235 µm Theoretical λ1
2200 2100 40
42
44
46
48
50
52
54
56
β, arc deg Fig. 22. Experimental measurements and the laser model predictions of the output wavelengths vs the grating angular orientation.
In the following subsection another method of the output wavelength tuning, which represents one of the unique properties of the spatially-dispersive laser, will be demonstrated. The method is called “motion-tuning” which reflects the fact that the entire output spectrum can be tuned without any readjustment of the laser resonator by a simple transverse motion of the pump beams across the gain crystal.
3.1.5 Dual-Wavelength, Motion-Tunable Cr2+ :ZnSe Spatially-Dispersive Laser As follows from Eq. (4) and Eq. (7), the output wavelengths are functions of the transverse positions of the pump beams in the laser crystal. Therefore, if the dual pump beam is shifted horizontally (in the direction perpendicular to the laser optical axis), the entire dual-wavelength output spectrum will be tuned. From the previous experiments it follows that one could expect a tuning range of about 620 nm (Fig. 17). In order to perform such a tuning of the output spectrum without altering its structure, it is necessary to move the pump beams simultaneously without changing
75 the distance between them. This was performed in the following way without any modifications of the experimental setup shown in Fig. 15, including the parameters of the laser resonator. The auxiliary aperture was removed, and the pump lens and the folding prism P2 , installed on motion stages, were moved simultaneously in the transverse direction. Such a motion of the optical elements results in a parallel transverse shift of the pump beams, supposedly without any change of the distance between them. This PSfrag replacements wavelength tuning procedure is shown schematically in Fig. 23.
λ02
λ2
λ01
λ1
x
1 2 O
O0 0 10 20
Fig. 23. Schematic diagram of the motion-tuning of the dual-wavelength spatiallydispersive laser. The shifted optical elements, pump beams, and the laser modes are outlined by dashed lines. The zero x-coordinate corresponds to the optical axis OO 0 .
The coarse control of the motion of the optical elements was done with the micrometer screws of the motion stages. The exact required positions of the pump beams were controlled by observing the output spectrum. After each move of the pump beams, which resulted in the desired output spectrum, their exact positions were measured with the knife edge method described in the previous subsection. The experimental results of this wavelength tuning experiment are presented in Fig. 24.
76
hxi = 21 (x1 + x2 ), mm 2.0
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3 2 1 0 -1 -2 -3 -4 2100
x1 x2
(b)
λ ) − βe ), βe ≈ 47.2 arc deg Fit: x1 = −f (arcsin( 2t λ Fit: x2 = −f (arcsin( 2t ) − βe ), βe ≈ 47.2 arc deg
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Wavelength, nm 10 8 6 4 2 0 -2 -4 -6 2100
α1 α2
(c)
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Wavelength, nm
Fig. 24. Tuning of the dual-wavelength spectrum by simultaneous transverse motion of the pump beams across the laser crystal (a). The top axis shows corresponding average transverse position of the pump beams. Graph (b) shows the coordinates x1 and x2 vs the wavelengths of the corresponding spectral channels. Graph (c) shows the angles α1 and α2 vs corresponding wavelengths of the tuned spectral lines.
77 The graph Fig. 24(a) shows the successive spectral pairs of the tuned output spectrum. The top x-axis indicates the corresponding average coordinates of the pump beams hxi = (x1 + x2 )/2, which show the displacements of the beams relative to their initial positions and are convenient to indicate the displacements of the dual pump beam as a whole. The zero position of the pump beams corresponds to the wavelength pair, which is centered at approximately the middle of the tuning curve (the 5th spectrum in Fig. 24(a)). The grating angular orientation β ≈ 47 arc deg corresponds to the spectrum of the central spectral pair, and it was found using Eq. (68). The diffraction grating was untouched during the entire tuning experiment. The exact positions of the pump beams, measured with the knife edge method, are shown in Fig. 24(b). As in the previous subsection, it is assumed that the laser modes are coaxial with the pump beams. The experimental data are fitted by a function x(λ) = −f (arcsin(λ/2t) − βe ) with the fitting parameter βe . This function follows from Eq. (4) and describes the dependence of the channel coordinate x on the output wavelength λ. The values of βe , found by the fitting procedure, are approximately 47.15 arc deg and 47.21 arc deg for the channels 1 and 2, respectively, which is very close to the measured value of the grating angular orientation β = 47 arc deg. These results show that the paraxial formula (4) for the dependence λ(x) describes the experimental data very well. In contrast to the grating-tuning experiments, where the angles α undoubtedly conformed to the paraxial approximation, the present experimental data allow testing of the laser model at large values of the x-coordinates of the spectral channels, at which the angles α approach the limit of validity of the paraxial approximation.
78 The dependence of the output wavelength on x is shown in Fig. 25, where the experimental data from the two lasing channels are merged together to obtain a single function λ(x).
2800
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λ, arc deg
2700 2600 2500 2400
Experimental λ vs x
2300
Fit: λ = 2t(sin βe −
2200
Fit: λ = 2t(sin β −
2100
-4
-3
x f
x f
cos βe ), βe ≈ 46.9 arc deg
cos β), β ≈ 47.1 arc deg
-2
-1
x, mm
0
1
2
3
Fig. 25. Experimental measurements and the laser model predictions of the output wavelength vs the transverse coordinates of the laser modes.
The laser theoretical model predicts that this dependence is described by Eq. (7), which is also shown in the figure. The experimental data are fitted by the following function:
λ(x) = 2t sin(βe ) −
x cos(βe ) , f
(74)
where the fitting parameter is βe , which corresponds to the constant angular orientation of the diffraction grating. The value of βe , found by the fitting procedure is 46.93 arc deg, which is almost equal to the measured angle β ≈ 47 arc deg. As can be seen in Fig. 25, theoretically calculated wavelengths for the measured x-coordinates
79 and angle β coincide with almost all of the experimental data points. However, one can see a growing difference between the theoretical calculations and the experimental data at large values of x, which is explained as follows. As can be seen from Fig. 24(c), which shows the dependencies of angles α1 and α2 on the output wavelengths, the angles approach the paraxial limit (approximately 15 arc deg) at large values of the coordinates of the channels x1 and x2 . This leads to a growing discrepancy between the paraxial formula (7) and the exact equation (2) for the dependence of λ(α). As it follows from the laser model, when the grating angular orientation β is unchanged, the wavelength spacing between two tuned channels does not depend on the absolute values of their coordinates x1 and x2 but is determined only by their spatial separation δx = x2 − x1 . The experimental and theoretical dependencies of δλ on the average coordinate of the channels hxi, which characterizes the absolute values of x, is shown in Fig. 26. The theoretical curves were calculated for the measured x-coordinates of the channels using the approximate paraxial formula (5). The linear fit of the experimental data shows that the wavelength spacing increases with the absolute values of the x-coordinates. However, the theory shows a constant δλ with small random oscillations due to a non-constant spatial separation of the channels δx. The experimental data contain a random noise due to the oscillations of α discussed previously. Nevertheless, the differences between the laser model predictions for δλ and the experimental data do not exceed 4% for small angles α, corresponding to the central spectral pairs, and 8% for the angles approaching the paraxial limit and, therefore, the model provides quite an acceptable approximation.
80 hαi = 21 (α1 + α2 ), arc deg
δλ, nm
8.7
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50 48 46 44 42 40 38 36 -4
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Linear fit of experimental data Experiment Approximate paraxial formula
-3
-2
-1 0 hxi = 21 (x1 + x2 ), mm
1
2
3
Fig. 26. Experimental measurements and the laser model predictions of the output wavelength spacing vs the average position of the dual pump beam.
Since the wavelength spacing is determined by the spatial separation of the pump beams, one can tune δλ by changing δx. This method of spectral tuning is referred to here as “spacing-tuning”, and reflects the general idea of such a manipulation of the output spectrum. This experiment was done by changing the transverse positions of the pump beams by shifting the beams individually, and it is described in the following subsection.
3.1.6 Dual-Wavelength, Spacing-Tunable Cr2+ :ZnSe Spatially-Dispersive Laser In order to perform the tuning of the spectral separation between the output wavelengths it is necessary to change the spatial separation between the pump beams. However, due to hardware limitations the pump system allows changing δx within a very limited range. Thus, the pump system was slightly modified to obtain an independent control of each pump beam and allow a large change of the corresponding x-coordinates. The modified experimental setup is shown in Fig. 27.
81 PSfrag replacements
λ1
λ2
Dual pump beam
P2
Auxiliary aperture
P3
Fig. 27. Schematic diagram of the dual-wavelength, spacing-tunable spatallydispersive laser. The positions of the pump beams can be changed independently of each other by motion and rotation of the folding prisms P2 and P3 . The auxiliary aperture is used to indicate the front focal point of the pump lens.
The tuning of the output wavelengths of the spectral channels is performed independently of each other in the following way. First, the corresponding folding prism is rotated slightly to obtain a new wavelength. During such a rotation, the tuned pump beam becomes partially blocked by the auxiliary aperture and, therefore, the prism is moved until the beam is transmitted again. This tuning is done in small steps until the tuned spectral channel works at the desired output wavelength. The same procedure is repeated for the other channel. Thus, it is possible to obtain any desired wavelength spacing between the lasing channels. The minimum wavelength spacing is limited to about 50 nm because at smaller spatial separations of the pump beams the folding prisms become so close to each other that they block the opposite beams. This problem can be solved by increasing the distance between the pump lens and the folding prisms, and by decreasing the diameters of the pump beams. However, this could not be performed in these experiments due to a limited space on the optical table and the absence of appropriate lenses for a stronger pump beam compressing telescope.
82 Nevertheless, the achievable range of the spatial separation δx between the pump beams was quite enough for the purposes of this experiment and allowed a large tuning range of δλ, from about 50 nm to almost 550 nm. The experimental results of the wavelength spacing tuning are depicted in Fig. 28. Graph Fig. 28(a) shows the successive spectral pairs of the tuned output spectrum. The top x-axis indicates the coordinates of the pump beams. The zero x-coordinate and the grating angular orientation β ≈ 48.85 arc deg correspond to the average wavelength hλi of the central spectral pair corresponding to the 1st spectrum in Fig. 28(a). The exact positions of the pump beams vs corresponding output wavelengths are shown in Fig. 28(b). The beam x-coordinates were measured with the knife edge method. As in the previous subsections, it is assumed that the laser modes are coaxial with the pump beams and, therefore, the transverse spacings of the lasing modes are equal to those of the pump beams. The experimental dependencies x1,2 (λ) are fitted by a function x(λ) = −f (arcsin(λ/2t) − βe ) with the fitting parameter βe , as in the previous experiment. The values of βe , found by the fitting procedure, are approximately 48.4 arc deg and 48.9 arc deg for the channels 1 and 2, respectively. This is very close to the measured value of the grating angular orientation, which shows a very good agreement between the paraxial model and the experimental results. Corresponding angles α1,2 , retrieved from the measured spectra with Eq. (69), are shown in Fig. 28(c). The graphs demonstrate a linear dependence of α on x, which is expected in the paraxial approximation according to Eq. (7) (to see this, one simply needs to replace the fraction x/f with α in that equation to find that α(λ) is a linear function).
83
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2700
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x1 , x2 , mm
4
PSfrag replacements
2 0 -2
α1 , α2 , arc deg
-4 2200 8 6 4 2 0 -2 -4 -6 2200
(b)
Experimental x1 Experimental x2 Fit: x1 = −f (arcsin(λ1 /2t) − βe ), βe ≈ 48.4 arc deg Fit: x2 = −f (arcsin(λ2 /2t) − βe ), βe ≈ 48.9 arc deg
2300
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2500 2600 Wavelength, nm
2700
2800
Experimental α1 Experimental α2 Theoretical: α1 = −x1 /f , α2 = −x2 /f
(c)
2300
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2500 2600 Wavelength, nm
2700
2800
Fig. 28. Tuning of the dual-wavelength spectrum by individual transverse motion of the pump beams across the laser crystal. Graph (a) shows the successive output spectra during the tuning; the top axis shows the corresponding transverse positions of the pump beams. Graph (b) shows the coordinates x1 and x2 vs the wavelengths of the corresponding spectral channels. Graph (c) shows the angles α1 and α2 vs the corresponding wavelengths of the tuned spectral lines.
84 As was shown in the previous experiments on the grating and motion tuning of the output spectrum, the experimental dependencies of the output wavelength on the transversal coordinate of the laser modes demonstrated a very good agreement with the paraxial model. However, it was difficult to directly verify equation (5) for calculation of channels wavelength spacing due to the random measurement errors, whose values were about several percent of the measured δλ itself. As discussed previously, those errors are the result of several factors, such as the measurement errors in the determining of the output wavelengths and the spatial spacing between the laser modes due to a non-perfect mode matching between the laser modes and the pump beams. In contrast to those results, in this experiment the values of δλ are very large and change significantly during the spectral tuning (from about 50 nm to approximately 550 nm). Consequently, the cumulative relative error in determining the wavelength spacing is much smaller than in the previous cases and, therefore, the theory can be examined directly with a higher certainty. The experimental measurements of δλ as a function of the absolute value of δx, and the theoretical calculations of the wavelength spacings for the measured spatial separations of the pump beams, are shown in Fig. 29. As can be seen from Fig. 29, the dependence of δλ on the modulus of δx shows a linear behavior, as expected from the laser model. The calculated values of the wavelength spacings almost coincide with the experimental data points, with the largest relative error of less than 5%, which demonstrates a very good agreement of theory and experiment even at angle α, approaching the limits of the paraxial approximation.
85 600 Experiment Paraxial approximation
δλ, nm
500
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400 300 200 100 0
0
1
2
3
4
5
6
δx, mm Fig. 29. Experimental measurements and the laser model predictions of the output wavelength spacing vs the spatial spacing of the beams. The theoretical values are calculated with the paraxial formula (5) for the measured spatial spacings of the pump beams.
The dependences of the output wavelengths on the transverse coordinates of the channels are shown in Fig. 30. As in the previous subsection, the data from both wavelength channels are combined together to obtain a single function λ(x). The top x-axis shows corresponding α-angles. In addition to the experimental data, the figure also shows the theoretically calculated wavelengths for the measured x-coordinates using equation (4). As before, the experimental data are fitted with the theoretical function λ(x) = 2t sin(βe − x/f ), derived from Eq. (4), with the fitting parameter βe , which corresponds to the grating angular orientation. The value of the angle of 48.7 arc deg, found by the fitting procedure, is very close to that determined from the central spectral pair, which once again demonstrates a very good agreement between the experimental data and the predictions of the laser model. As in the previous experiments, the theoretical and the measured output wavelengths almost coincide with each other for almost all of the data points.
86 α, arc deg 7.5
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-2.3 -3.6 -4.8 -6.0 -7.1
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λ, nm
2700 2600 2500 2400
Experimental λ vs x Fit: λ = 2t sin(βe − x/f ), βe ≈ 48.7 Theoretical λ vs x
2300 2200
-4
-3
-2
-1
x, mm
0
1
2
3
Fig. 30. Experimental measurements and the laser model predictions of the output wavelength vs the transverse coordinates of the laser modes. The top axis shows the experimental values of the corresponding α angles.
However, one can see that the discrepancies between the theoretical and experimental dependencies increase for large positive values of x. A very similar behavior of λ(x) was observed in the previous experiment (Fig. 25): for large negative and the central (close to zero) x-coordinates the theoretical and experimental data points are very close, but as the coordinate becomes large positive, the data begin to diverge. This intuitively contradicts with the expected behavior of the function: one would expect that the differences between the theoretical and experimental data points would depend only on the absolute values of the x-coordinates. The errors should be the same for large negative and large positive distances between the lasing channels and the laser optical axis, and a symmetric picture should be observed. The apparent contradiction, however, is easily explained by the effect of the spherical aberration that occurs due to a misalignment of the intracavity lens from the chosen optical axis. This effect is illustrated in Fig. 31.
87
F1
α1
λ2 OL
α2
O0 z
x
x=0
−δx
F0
δx
1
OL0
2
O F2
λ1
Fig. 31. The spherical aberration in the misaligned spatially-dispersive cavity. OO 0 – cavity optical axis in the chosen coordinate system; OL OL0 –lens optical axis; F0 , F1 and F2 are the focal points of refracted beams. The focal points do not coincide due to the spherical aberration and |α1 | > |α2 |.
The intracavity lens is slightly shifted in the horizontal direction and its optical axis OL OL0 does not coincide with the optical axis of the cavity OO 0, as shown in the figure. As a result, beam 1, located at a distance +δx from the laser optical axis, is farther from the lens axis than beam 2, which has a transverse coordinate −δx. Consequently, beam 1 experiences a stronger refraction than beam 2 and intersects the cavity axis closer to the lens. As a result, the angle of refraction α1 is larger than α2 and is closer to the paraxial limit, and the actual value of α1 will differ from its theoretical paraxial value more than the second angle α2 . This explains the non-symmetrical behavior of the differences between the theoretical and experimental measurements of the dependencies λ(x), shown in Fig. 25 and Fig. 30. At this point, only the spectral properties of the laser output radiation and the corresponding part of the theory model were analyzed. The three sets of experiments on the wavelength tuning of the laser unambiguously confirm the geometrical part of the laser model.
88 However, in order to examine the Gaussian beam approximation, it is necessary to obtain the intracavity mode structure. Unfortunately, there are no methods for a direct measurement of the laser mode intensity profiles inside a working laser. Therefore, it is necessary to reconstruct the intracavity mode structure based on the measurements of the spatial distribution of the output radiation and knowledge of the parameters of the laser resonator, as was mentioned in Section 2.7. This problem is addressed in the following subsection.
3.1.7 Spatial Distribution of the Output Radiation. Intracavity Mode Structure The intensity profiles of the output beams of the dual-wavelength laser were
PSfrag replacements
measured with the knife edge method in two planes along the beams as shown in Fig. 32.
δz = 270 mm w2
w1
λ1
z1
Measurement Plane 2
z2
m
α
Folding Al mirror 20 m
λ2
50 mm
wg
Laser system
Measurement Plane 1
Fig. 32. Setup for measurements of the spatial profiles of the output beams. The distances z1 and z2 are measured from the surface of the diffraction grating.
The measurement planes are located at a distance of approximately 270 mm from each other. The absolute distances from planes 1 and 2 to the grating are
89 approximately 70 ± 5 mm and 340 ± 5 mm, respectively. The distance between the planes is measured with a precision of about ±1 mm, and the distances from the planes to the grating are measured with worse precision of about ±5 mm because it was not possible to precisely trace the beam paths from the grating to the folding mirror. The measurements were performed for the central wavelength pair located near 2450 nm (Figs. 17 and 24). The micrometer screw allows for moving the measuring blade with an accuracy of 5 µm; however, the measurements were done with the blade motion steps equal to 50 µm and 100 µm in the planes 1 and 2, respectively. The results of the profile measurements are shown in Fig. 33.
PSfrag replacements Intensity, a.u.
1 0.8
w1 ≈ 539 µm
w2 ≈ 537 µm
x01 ≈ 522 µm
x02 ≈ 2971 µm
0.6
Experiment Fitting integral
0.4
(a)
0.2 0 0
1000
2000
3000
4000
Coordinate, µm
PSfrag replacements Intensity, a.u.
1 0.8
w1 ≈ 1714 µm
w2 ≈ 1431 µm
x01 ≈ 1667 µm
x02 ≈ 9767 µm
0.6
Experiment Fitting integral
0.4
(b)
0.2 0 0
2000
4000
6000
8000
10000
12000
Coordinate, µm
Fig. 33. Measurements of the intensity profiles of the output beams with the knife edge method. Graph (a) shows the measurements in plane 1; Graph (b) corresponds to the measurements in plane 2 (see Fig. 32). The beam centers and radii are indicated as x01 , x02 and w1 , w2 for beams λ1 and λ2 , respectively.
90 The measured beam radii were used to determine the waists radii of the output beams and their locations using the complex method, described in Appendix B. The results of this analysis are summarized in Table 3. The notations in the table are the same as in Fig. 32.
Table 3. Spatial properties of the output beams Parameter λ w1 w2 z1 z2 z0 wg θ0 α
Beam 1
Beam 2
Notes
2428 nm 538.6 µm 1714.4 µm 39.4 mm 309.4 mm 93.6 mm 496.5 mm 1.56 mrad —
2473 nm 536.8 µm 1431.4 µm 66.7 mm 336.7 mm 115.7 mm 465.0 µm 1.69 mrad —
Fig. 22 Fig. 33 Fig. 33 Eq. (84) Eq. (85) Eq. (86) Eq. (86) λ θ0 = πw g 0.72◦
The results demonstrate a certain discrepancy between the values of the beam waist wg for the output beams. However, this discrepancy cannot be explained by the differences of the intracavity mode structures of the wavelength channels. Indeed, the angle α is very small and, consequently, the waists of the laser modes on the grating are the same. Therefore, the error comes from the measured parameters: the beam radii and the distance between the measurement planes. The error in ∆z is less than 0.4% of its value and, therefore, can be ignored. The errors come from the measurements of the beam radii. The values of the beam radii w1 in the plane 1, located closer to the grating, are almost the same, with the difference of less than the positioning accuracy of the measurement blade. The latter also supports the proposition that the waists of the beams wg are equal, as follows
91 from Eq. (53). Therefore, it is most likely that these radii are measured correctly. Consequently, the major source of the error should be contained in the measurements of the beam profiles in plane 2. It was found that the error in measuring the beam profiles in this plane is a consequence of two factors: (1) the knife edge method measurement is a very slow procedure (more than an hour is required to scan plane 2), and (2) the output beams slowly drift in the transverse direction due to the thermal lensing effects in the laser crystal. As a result, during the long scanning process of the beam profiles they can either “run away” from the measuring blade, in which case the measured diameter is larger than its actual value, or move towards the blade, in which case the measured size is smaller than the actual one. This effect is illustrated in Fig. 34.
PSfrag replacements
drift
w0 w
(a)
w0
Blade
w0
Blade
w w0 drift
(b)
Fig. 34. Knife edge method measurement errors. Notations: w, actual beam diameter; w 0 , measured beam diameter; w0 , beam waist; solid lines indicate the undisturbed beam. (a) the beam “runs away” from the blade and w 0 > w; (b) the beam moves towards the blade and w 0 < w. The larger the distance between the measurement plane and the beam waist, the larger the measurement error.
This explains why the measured diameters have much smaller differences in measurement plane 1 than in plane 2. In addition, as one can see from Fig. 32 and
92 Table 3, the calculated values of the beam waist positions z1 and z2 are much closer to their measured values for beam 2. Therefore, the measurements of the diameters of the beam 2 are more accurate than that of beam 1. Consequently, it will be assumed in the following that the value of wg ≈ 465 µm is an acceptable experimental estimation of the channel beam radii at the diffraction grating. This value will be used below to determine the intracavity mode structure. Since the intracavity CaF2 lens has a thickness of 10 mm, comparable to its focal length of 25 mm, it is not possible to use the thin lens approximation in the laser model, and so the cavity must be treated as described in Section 2.8. The transformation matrix of a biconvex thick lens, with a focal length f , a thickness T , the refractive index n, and the radius of curvature of its surfaces R, reads:
(1−n)T nR
1 + MT L = − f1
T n
1+
(1−n)T nR
.
(75)
Using Eqs. (60), (61), (65), and (18), and the lens parameters given in Table 2, one can calculate the beam waist radius on the grating wg and compare it to the measured value. The waist on the input mirror wm can be calculated using Eq. (17) and compared to the measured pump beam radius (see Fig. 19) to determine how well the laser modes are matched to the pump beams. This method can be used for testing how accurately the Gaussian beam model describes the laser mode parameters and, thus, determine if the model is applicable here. However, in order to conduct this calculation the major cavity parameters, d 0 , d1 , d2 and l (Fig. 15) must be known. The approximate values of the parameters, together with their uncertainties, are summarized in Table 4.
93
Table 4. Major cavity parameters Parameter
Value
d0 d1 d2 l
1 mm 4 mm 15 mm 75 mm
Uncertainty ±0.2 ±0.1 ±1.5 ±5.0
mm mm mm mm
The most important parameters, d2 and l, are measured with the largest errors because it is not possible to measure them exactly along the invisible beam paths. Therefore, instead of calculating the beam waists for the fixed values of the cavity parameters, it is more convenient to plot the dependencies of the waists on the them. The dependencies of wm , wc , and wg on the distance d2 for the maximum and the minimum values of l are shown in Fig. 35. Fig. 35 shows the beam waists calculated for one wavelength channel only, because the differences between the channels are within the measurement errors. As one can see, for the lens positions between approximately 15.17 mm and 15.26 mm and the grating positions between 70 mm and 80 mm, the calculated beam waist on the grating wg equals to its experimentally measured value. Since these positions are practically equal to their experimental values, this demonstrates a very good agreement of the Gaussian beam model of the laser and the experimental data. This also indicates that the wavelength channels are indeed represented by the fundamental TEM00 Gaussian modes, as predicted by the model. Moreover, the higher order modes are not generated even though the laser operates without the intracavity aperture (Fig. 4), because the high order modes are efficiently suppressed due to high intracavity losses.
94
(a) Beam waist on the grating 700
wg µm
600 500
Measured wg
400 300 14.8
15
15.2
15.4
15.6
15.8
16
16.2
16.4
d2 , mm (b) Beam waists on the input mirror and in the crystal
wm , wc , µm
PSfrag replacements
120
Pump beam radius
100 80
Crystal output facet (wc )
60 Input mirror (wm )
40 20 14.8
15
15.2
15.4
15.6
15.8
16
16.2
16.4
d2 , mm Fig. 35. Dependencies of the beam waists of the laser modes on the distance between the crystal and the intracavity lens. Graph (a) shows the beam waists on the diffraction grating. The horizontal dashed line shows the measured value of wg . Graph (b) shows the beam waists on the input mirror and the output facet of the crystal. The horizontal dashed line shows the pump beam radius. The vertical dashed lines in both graphs indicate the range of d2 for which the calculated wg equals its experimental value. The graphs contain the plots calculated for the wavelength channel λ1 .
95 On the other hand, one can see that the mode radii wm on the mirror and wc in the laser crystal are almost twice as small as the radius of the pump beam. This indicates that the mode matching between the pump beams and the laser modes is far from optimal. However, this configuration of the cavity parameters (that defines the mode radii) was found experimentally to be the most appropriate for obtaining the widest tuning range and, therefore, is the most efficient. The explanation of this apparent contradiction is that during the calculation of the laser modes the following three factors were not taken into account: (1) the laser experiences significant intracavity losses (for both the pump and the lasing wavelengths) due to the absence of the AR-coatings on the surfaces of the optical elements; (2) the pump beams are detuned along the cavity optical axis to avoid back reflections of the pump radiation (see Subsection 3.1.3), so that the pump beam waist is outside the laser crystal; and (3) the thermal lensing is not taken into account, although this effect might cause distortion of both the pump beams and the laser modes inside the gain medium. These factors change the mode matching conditions significantly from the simple rule of the pump beam and the mode radii equality to some much more complex configuration. Unfortunately, these effects could not be studied in detail here and will be addressed in one of the author’s future studies. Based on the experiments, described in the last three subsections, one can conclude that the theoretical model of the laser is in a very good agreement with the experimental results. Practically all propositions, discussed in Section 2.2, were analyzed and verified by the experiment. However, it is important to note that the agreement that was obtained between the theory and experiments is a consequence of not only a great consistency of the laser model and the experimental data themselves,
96 but is also in part a result of a relative simplicity of the spatially-dispersive dualwavelength laser. Indeed, the spatially-dispersive laser operates at only two wavelengths, and the spectral channels are spatially separated by the pumping of two distinct, nonoverlapping regions of the gain element with two separate, independent pump beams. Any interaction between the lasing channels is thus eliminated by the laser design, and one of the most important propositions of the theoretical model, that the spectral channels can be considered as independent tunable lasers, definitely holds (it must be mentioned here, however, that the spectral channels are still coupled with each other to some extent because of the thermal lensing, which distorts the gain element). However, in order to improve our understanding of the ultrabroadband, multiwavelength operation of the spatially-dispersive laser, it is necessary to consider the case when a large number of closely located spectral channels are generated simultaneously. Such an attempt is described in the next subsection, where a multiwavelength and ultrabroadband Cr2+ :ZnSe spatially-dispersive laser is studied.
3.1.8 Ultrabroadband, Multiwavelength Cr2+ :ZnSe Spatially-Dispersive Laser One method of obtaining a multiwavelength output spectrum consists in pumping the laser with a large number of pump beams. However, in that case the output beam of the Er-doped fiber laser would have to be split into multiple portions with a complex multi-beam beamsplitter, which was not available at the time of the experiments. Also, in that case a limited number of output wavelengths, determined by the number of the pump beams, would be observed. Moreover, with such a pumping method, it would be difficult to obtain an ultrabroadband continuous out-
97 put spectrum, because the number of the pump beams would be very limited, due to a limited available pump power, as discussed in Subsection 3.1.3. An alternative approach consists of pumping the laser with a wide plane pump beam, as was done in previous works with the LiF:F+∗∗ superbroadband 2 lasers [29–31, 33, 34, 53]. Those lasers are capable of producing ultrabroadband continuous output spectra, as well as the multiwavelength spectra, consisting of a large number of spectral lines, when the wide pump beam is split into several parts by a spatial mask. This pumping technique is much simpler than the multi-beam pumping, since it only requires a reshaping of the pump beam into a highly elliptical, horizontally stretched pump laser beam. Such a beam shaping can be done with a single cylindrical lens, compressing the original pump beam of a large diameter in the vertical plane, as was done in works, referenced above. However, as shown in the case of the Cr2+ :ZnSe laser, in Subsection 3.1.3, the pump beam cannot be stretched larger than approximately 0.8 mm due to the limited available pump power. Since the incident pump beam diameter is about 1 mm, the pump beam should also be slightly compressed in the horizontal plane. In these experiments the highly elliptical, horizontally stretched pump laser beam was obtained with a long focal length spherical lens and a short focal length cylindrical lens: the horizontal compression of the original circular pump beam is done by the spherical lens, whose focal length is 200 mm, and the vertical compression is performed with the cylindrical lens, whose focal length is 30 mm. This configuration of the pump lenses was found to be optimal in a series of experiments with the goal to obtain the widest output spectrum. A schematic diagram of the complete experimental setup is shown in Fig. 36.
98
8 mm pump beam from 1.56 µm, 10W Er-fiber laser
f2 = 8 mm
f1 = 80 mm Signal
Scanning spectrometer
Sync signal
Lock-in Amplifier
PbS detector
Diffuser fp
PC and spectrometer controller
Collector lens
λ1
ism Pr
Chopper 125 Hz, 1/2
Beam compressor
λn
1 mm
FI 20 mm
35 mm
0.6 mm
Pr
ism
CaF2 lens Cr2+ :ZnSe f = 25 mm Crystal
Grating 324 g/mm 57 mm
10
20 mm
Input Cylindrical lens mirror f = 30 mm
Spherical lens f = 200 mm
Fig. 36. Schematic diagram of the multiwavelength Cr2+ :ZnSe spatally-dispersive laser. The spherical and cylindrical pump lenses create a highly elliptical, horizontally stretched pump laser beam in the Cr2+ :ZnSe crystal providing a broadband pumping. The abbreviation FI denotes a Fizeau interferometer formed by the working surface of the input mirror and the uncoated input facet of tilted laser crystal (see details in the text).
As one can see in Fig. 36, the low-resolution diffraction grating, with a grating spacing constant of 1/324 mm, was used in this laser. The reason for that was to obtain the maximum possible output spectral range, which is limited due to the limited pump power. As follows from Eq. (9), the output spectral bandwidth is proportional to the width ∆x of the pump beam and the diffraction spacing constant t. Since the beam width is limited by the available pump power, the output spectral bandwidth can be enlarged by increasing the grating spacing constant.
99 The longitudinal position of the spherical pump lens determines the width of the pump beam in the laser crystal. The lens is installed on a motion stage and one can tune the beam width by the longitudinal displacement of the lens. This allows for the maximum possible output spectral bandwidth by finding such a position of the lens along the laser optical axis that provides the optimal width of the pump beam in the laser crystal. The optimal beam width is the largest allowable width, which provides a sufficiently high intensity for the pump beam to overcome the lasing threshold for the maximum possible spectral range. The laser was initially aligned for a single-wavelength regime of operation as described in Appendix C. In this case the laser was pumped by the spherical pump lens only, which was initially installed at a distance of approximately 10 cm from the input mirror. However, due to a low intensity of the pump radiation, no lasing was initially observed. The lasing was obtained immediately after installation and adjustment of the cylindrical lens. After that the pump system and the cavity were fine tuned to obtain the maximum possible spectral width of the ultrabroadband output radiation. Typical examples of the ultrabroadband spectra, obtained during the laser adjustment, are shown in Fig. 37. As mentioned above, the cavity and the pump system were adjusted to obtain an optimal configuration that provides the widest possible output spectrum. With a given width of the pump beam, the maximum output spectral bandwidth is mostly limited by the achievable maximum intensity of the pump radiation and significant intracavity losses, which are mainly caused by the absence of the AR coatings on the surfaces of the intracavity lens and the facets of the Cr2+ :ZnSe laser crystal. If the intracavity losses were negligible, the output bandwidth would be much larger.
100
∆λ ≈ 109 nm
Intensity, a.u.
∆λ ≈ 112 nm
∆λ ≈ 135 nm
∆λ ≈ 128 nm PSfrag replacements
2450
2475
2500 2525 Wavelength, nm
2550
2575
Fig. 37. Ultrabroadband multiwavelength output spectra of the multiwavelength Cr2+ :ZnSe spatally-dispersive laser. The approximate overall spectral bandwidths at the zero level are indicated on the graphs as ∆λ.
101 On the other hand, if it were possible to use a more powerful pump laser, the output spectral bandwidth could be enlarged by increasing the transverse size of the pump beam. Further increase of the pump beam width does not lead to the increase of the output bandwidth in the present system, due to the limited pumping power. The obtained optimal spatial distribution of the pump radiation was measured with the knife edge method. The measurements were done in front of the output facet of the laser crystal, approximately 1 mm from the surface. The results of the measurements are depicted in Fig. 38.
PSfrag replacements
Intensity, a.u.
1 Experiment Fitting integral
0.8
2w ≈ 557 µm
0.6 0.4 0.2
0 200 250 300 350 400 450 500 550 600 650 700 750 800 850
Coordinate, µm Fig. 38. Spatial intensity profile of the pump beam near the output facet of the laser crystal, as measured with the knife edge method. The measured total width of the beam is indicated as 2w in the figure.
Note that the spatial profile of the pump beam is still Gaussian because it was obtained from a circular Gaussian beam with two coaxial lenses, one of which is spherical. After refraction on the cylindrical lens the pump beam profile becomes elliptical and, therefore, astigmatic. However, since the spectral distribution of the
102 output radiation is determined by the pump beam transverse size, one is interested in measuring the pump beam width only. It is important to note that the measured beam width 2w is slightly smaller than the actually pumped region of the laser crystal due to the following two reasons. First, the measurement is done at a distance of approximately 1 mm from the crystal output facet and, since the pump beam is horizontally converging, the beam width inside the crystal is slightly larger than 2w. Second, the parameter 2w represents the beam width at the 1/e2 intensity level, and is smaller than the actually pumped width of the crystal. This situation is illustrated in more detail in Fig. 39. PSfrag replacements
Gaussian profile
Spherical pump lens
δxc
2w
Measurement plane
Laser crystal
Input mirror
Cylindrical pump lens
Fig. 39. Difference between the measured pump beam size in front of the laser crystal and the actual width of the pumped region of the crystal. Due to the horizontal convergence of the pump beam, the measured width 2w is smaller than the beam size inside the laser crystal. Since the measurement of the beam profile gives the beam width at the 1/e2 intensity level (solid lines), the actual pumped width δxc (dotted lines) is even larger than the measured size 2w; to illustrate the latter, the corresponding Gaussian intensity profile is also shown with the vertical dashed line showing the 1/e2 intensity level.
Such measurement errors of the pump beam size result in some discrepancy between the theory predictions and experimental results for the overall output spec-
103 tral bandwidth. Indeed, from Eq. (9) it follows that the output spectral bandwidth ∆λ for the measured beam width 2w ≈ 557 µm is about 124 nm. However, the ultrabroadband spectra demonstrate the maximum bandwidth of about 135 nm, which requires a wider pump beam. Moreover, as will be demonstrated in the next subsection, under certain conditions the output spectral bandwidth can be further increased to about 200 nm with the same width of the pump beam. On the other hand, some of the ultrabroadband spectra have a smaller spectral width than follows from the theory. These differences among the ultrabroadband spectra are due to slightly different adjustments of the vertical angular orientation of the Cr2+ :ZnSe crystal in the resonator, which slightly changes the intracavity total loss.
3.1.9 Modulation of the Ultrabroadband Spectrum by a Self-Formed Intracavity Fizeau Interferometer Since the spatial profile of the pump radiation is Gaussian, one would expect to obtain a Gaussian-like, smooth output spectrum. However, as seen in Fig. 37, the ultrabroadband spectra are irregularly modulated. It was found that the shape of the spectra is mostly determined by the angular orientation of the laser crystal. Eventually it was determined that the input facet of the crystal and the working surface of the input mirror form an intracavity interferometer, which performs such a spectral modulation. When the crystal is slightly tilted in the horizontal plane, the laser generates a discrete, regularly spaced, multiwavelength ultrabroadband spectrum. The outline of such spectrum is in agreement with the Gaussian spatial distribution of the pump radiation. Moreover, the wavelength spacing between the spectral lines,
104 the number of the spectral lines, and their width can easily be changed by tuning the distance between the laser crystal and the input mirror. Such tuning of the multiwavelength spectra is shown in Fig. 40. The spectra in Fig. 40 correspond to different longitudinal positions of the laser crystal along the cavity optical axis. The crystal was mounted on a translational stage that provided the positioning accuracy of about ±1 µm. The micrometer screw of the motion stage could be used to measure its displacement with the accuracy of ±5 µm. During this tuning experiment, no other adjustments of the laser resonator or the pumping system were required. The interferometer formed by two reflecting surfaces separated by a slightly wedged air gap is known as the Fizeau interferometer (see [88], p. 179), whose free spectral range is given by the following equation: λ2 , δλ = 2s cos γ
(76)
where s is the thickness of the air gap, γ is the angle of refraction of the incident radiation. In this case a normal incidence is always realized, since the laser radiation is always perpendicular to the mirror surface. Therefore, the spectral lines are separated by the wavelength spacing given by Eq. (76), with cos γ ≈ 1. The first spectrum in Fig. 40 was taken when the crystal was located at the smallest possible distance from the mirror (though not in contact with it, to avoid scratching the mirror surface), which was estimated to be less than 10 µm. This spectrum is the same as in the last graph of Fig. 37. In this case the entire ultrabroadband spectrum is located within a single transmission peak of the interferometer.
105
s ≈ 10 µm
#0
s ≈ 670 µm
#7
s ≈ 70 µm
#1
s ≈ 770 µm
#8
s ≈ 170 µm
#2
s ≈ 870 µm
#9
s ≈ 270 µm
#3
s ≈ 970 µm
#10
s ≈ 370 µm
#4
s ≈ 1.1 mm
#11
s ≈ 470 µm
#5
s ≈ 1.2 mm
#12
s ≈ 570 µm
#6
s ≈ 1.3 mm
#13
Intensity, a.u.
PSfrag replacements
2400 2450 2500 2550 2600
2400 2450 2500 2550 2600
Wavelength, nm
Wavelength, nm
Fig. 40. Ultrabroadband, tunable multiwavelength spectra. The successive tuning of the wavelength spacing is done by changing the air gap between the laser crystal and the mirror from approximately 10 µm to 1.3 mm. The corresponding interferometer lengths s = s0 + δs a indicated in the graphs (see text for details)
106 Then the distance between the mirror and the crystal was slightly increased until it could be measured more precisely. After that the crystal was moved along the laser optical axis with a 100 µm step (with the accuracy of 10 µm) until a quasicontinuous spectrum was observed, which corresponded to the air gap thickness of approximately 1.3 mm. The minimum measurable separation of the crystal and mirror surfaces s0 was estimated to be between approximately 50 µm and 100 µm (through direct measurements using pieces of paper with the thickness of approximately 50 µm), with the average value of 75 µm. Since only the displacement δs of the laser crystal was measured exactly, the interferometer free spectral range (and, therefore, the wavelength spacing of the multiwavelength spectra) is given by the following expression:
δλ =
λ2 . 2(s0 + δs) cos γ
(77)
As one can see in Fig. 40, the spectral lines in the multiwavelength spectra are approximately equally spaced, apart from some random deviations due to suppression of some spectral lines when the spectrum approaches the continuous distribution. The average wavelength spacing δλ for each spectrum was measured by counting the number of the spectral peaks M in those spectral regions ∆λ which contain a continuous train of well distinguished, approximately equally spaced spectral lines. Sometimes two spectral intervals ∆λ were used and the obtained wavelength spacings were averaged. The wavelength spacing was then computed as δλ = ∆λ/(M − 1). The spectral intervals ∆λ, the determined δλ, and corresponding displacements of the laser crystal δs during the laser tuning are summarized in Table 5.
107
Table 5. Measurement of the wavelength spacing Spectrum #
δs, mm
λ1 , nm
λ2 , nm
∆λ, nm
M
δλ, nm
0 1 2 3 4 5 6 7 8 9 10 11 12
∼ −0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
13
1.2
14
1.3
— 2423 2408 2445 2446 2446 2400 2391 2418 2439 2432 2438 2442 2497 2442 2521 2456
— 2555 2606 2608 2610 2579 2599 2508 2572 2528 2558 2540 2492 2521 2516 2557 2503
— 132 198 163 164 133 199 207 154 89 126 102 50 24 74 36 47
— 4 12 15 20 21 37 45 39 26 40 36 20 10 32 15 22
— 44 18 11.6 8.63 6.65 5.53 4.7 4.05 3.56 3.23 2.91 2.63 2.67 2.39 2.57 2.24
The measured wavelength spacing versus the displacement of the laser crystal is shown in Fig. 41. The experimental data are fitted by the function (77) with two fitting parameters: the initial air gap thickness s0 , and the refraction angle γ (to verify that it is indeed small). One can see in Fig. 41 that the experimental data lie precisely on the theoretical curve. The best estimate of the refractive angle γ is about 10−8 rad, which confirms the assumption of the normal light incidence. The best estimate for the initial interferometer spacing, s0 , is approximately 71 µm, which is practically equal to its measured value. In should be noted here that the multiwavelength spectra were obtained with the same pump beam as the continuous ultrabroadband spectra shown in Fig. 37.
108 50 Experimental measurements
δλ, nm
40
PSfrag replacements
Theoretical fit: δλ =
λ2 2s cos γ
30 20 10 0 -0.1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 δs, mm
1
1.1 1.2 1.3
Fig. 41. Dependence of the wavelength spacing of the multiwavelength spectra on the axial displacement of the Cr2+ :ZnSe crystal.
One can see that the maximum bandwidth of the discrete spectra, shown in Fig. 40, is almost 70 nm larger than the bandwidth of the continuous spectra which, according to the laser model, requires a wider pump beam. However, the spectral bandwidths of the spectra in Fig. 37 and the spectrum #0 in Fig. 40 are limited by the bandwidth of the intracavity interferometer. Indeed, the finesse of the interferometer, estimated from the reflection coefficients of the input mirror (99.9%) and the input facet of the crystal (17.5%), is about 3.5. Thus, when the crystal is located at a distance of less than 10 µm from the mirror, the interferometer bandwidth is about 130 nm and the entire ultrabroadband spectrum is located within a single transmission peak of the interferometer, as mentioned above. On the other hand, when the interferometer is tuned to a smaller free spectral range, the energy from multiple interference minima is transferred into its maxima, increasing the field intensity in those regions where the maxima are located.
109 This effect is equivalent to increasing the positive feedback of the laser resonator, and decreasing the intracavity losses, for the transmitted spectral lines. Therefore, the lasing threshold is lowered for all lasing wavelengths, including the side wavelengths located closer to the boundaries of the pumped region of the laser crystal where the pump intensity is smaller. As a result, more side wavelengths reach the lasing threshold and the output spectral bandwidth is increased. The results of the experiments with the intracavity Fizeau interferometer stimulate the following important conclusion: when it is necessary to obtain a discrete, ultrabroadband, multiwavelength spectrum from the spatially-dispersive laser, one could use this interferometric method of spectrum modulation and tuning instead of multi-beam pumping or the use of a spatial mask. The first approach requires complex beam-shaping equipment, and the second one causes the loss of the pump energy, while the interferometric approach is very easy to realize and might even improve the result when pump power is limited.
3.1.10 Tuning of the Multiwavelength Spectrum Across the Entire Gain Curve of the Cr2+ :ZnSe Active Medium. As was demonstrated in Section 3.1.4, the multiwavelength output spectrum can be tuned as a whole by rotation of the diffraction grating in its dispersion plane. Such a wavelength tuning experiment was also done with one of the multiwavelength spectra shown in the previous subsection.
The results of this experiment are
demonstrated in Fig. 42, where the successive measurements of the multiwavelength spectrum during the tuning are shown.
110
20 lines, ∆λ ≈ 113 nm
27 lines, ∆λ ≈ 180 nm
Intensity, a.u.
26 lines, ∆λ ≈ 179 nm
18 lines, ∆λ ≈ 150 nm
13 lines, ∆λ ≈ 122 nm
7 lines, ∆λ ≈ 75 nm
Absorbance
PSfrag replacements
0.6 0.5 0.4 0.3 0.2 0.1
2200
Atmospheric water absorbance
2300
2400
2500
2600
2700
2800
Wavelength, nm
Fig. 42. Tuning of the ultrabroadband, multiwavelength spectrum as a whole by rotation of the diffraction grating in its dispersion plane. The last graph (at the bottom) shows the absorption spectrum of the atmospheric water vapor [96].
111 As can be seen in the Fig. 42, when the spectrum is close to the center of the laser tuning range (2500 nm), the number of lasing wavelengths, their strength, and the overall spectral bandwidth are maximized. When the spectrum is tuned towards smaller wavelengths, i.e., to the beginning of the tuning range, the gain is reduced and some side spectral lines disappear. For the same reasons, the same behavior should be expected also when the ultrabroadband spectrum is tuned to the end of the tuning curve. However, some spectral lines are randomly suppressed as the spectrum passes 2550 nm and nears 2800 nm. This suppression of the spectral lines is caused by the intracavity absorption of the mid-IR laser light by the atmospheric water vapor, whose absorption spectrum is shown in the last graph of Fig. 42. One can see that the peaks of the absorption spectrum are complementary to the suppressed spectral lines of the laser radiation. This effect, generally undesirable, opens an interesting possibility for designing a sensitive intracavity laser gas sensor for detecting low concentrations of gases. For that purpose it is necessary to build the spatially-dispersive laser that produces a continuous ultrabroadband output spectrum that has the shape of the top-hat function. Such spectrum can be obtained by reshaping the elliptic pump beam to compensate for the non-constant gain curve of the laser material. A gas flow cell should then be installed in the cavity at the focal point of the intracavity lens, where all wavelength channels intersect with each other. The positions and strength of the distortions of the output spectrum due to the absorption of the flowing gases will allow their identification [96, 97]. Such a spatially-dispersive laser sensor is not limited to the mid-IR spectral range and could be build for any other desirable spectral region with the appropriate laser material.
112 3.1.11 Summary: Cr2+ :ZnSe Spatially-Dispersive Laser In this section, a multiwavelength, ultrabroadband, continuous-wave, broadly tunable mid-IR (2200–2800 nm) laser source, based on a 1.56-µm Er-fiber-laser pumped, polycrystalline Cr2+ :ZnSe gain medium, utilized in the Littrow-mounted grating spatially-dispersive cavity, was described. The feasibility and applicability of the Littrow-mounted grating spatiallydispersive laser, based on the polycrystalline Cr2+ :ZnSe, for generation of multiwavelength, ultrabroadband laser radiation in the middle infrared spectral region, was demonstrated experimentally. Several different methods for tuning the laser output spectrum were demonstrated: changing the spatial distribution of the pump radiation in the gain medium, rotation of the Littrow-mounted diffraction grating, and tuning a self-formed intracavity interferometer. The theoretical model, developed in the geometrical and the Gaussian beam approximations, was found to be appropriate for the description of the generation of the continuous-wave multiwavelength and ultrabroadband spectra by the Littrowmounted grating spatially-dispersive lasers.
113 3.2
Semiconductor Multiwavelength Spatially-Dispersive Lasers
3.2.1 Introduction Dense wavelength division multiplexing (DWDM) promises to greatly expand the capacity of optical fibers, both those yet to be installed, and those already deployed in optical networks. Low cost laser sources, maintaining accurate adherence to the particular channel wavelength spacing, are required for viable DWDM systems. As the channel spacing of DWDM network decreases, short-cavity lasers, such as distributed feedback (DFB) lasers, distributed Bragg reflector (DBR) lasers, and vertical cavity surface emitting lasers (VCSELs), have to be externally stabilized to control the temperature-dependent output wavelength drift.
To improve the
wavelength stability, long-external-cavity lasers can be used, such as the multichannel grating cavity lasers with bulk-optic [17, 18, 46, 48] and integrated design [19, 85, 86]. However, most of those systems lack the flexibility in selection of output wavelengths, total amount of available spectral channels, and wavelength spacing. In this section a prototype of a new semiconductor laser source for DWDM systems, based on the Littrow-mounted grating spatially-dispersive cavity, is demonstrated. Two spatially-dispersive diode lasers are described: a single-chip spatiallydispersive diode laser, operating in the visible spectrum (660 nm), and a multistripe spatially-dispersive diode laser, operating in one of the DWDM spectral bands (1570 nm). The spatially-dispersive semiconductor lasers produce multiwavelength output spectra, tunable over the amplification bandwidth of the semiconductor media. The feasibility of conventional laser diodes as the gain media for the spatiallydispersive lasers is demonstrated.
114 3.2.2 Semiconductor Single-Chip Spatially-Dispersive Laser The semiconductor multiwavelength laser is based on a conventional AlGaAs single-chip diode laser operating in the wavelength region of 657–667 nm. The size of the active p-n junction is 500 × 100 × 1 µm3 . The spectral bandwidth of the diode amplification band is approximately 10 nm (at FWHM). The rear facet of the diode chip is uncoated and the front facet is AR–coated with a residual reflection of PSfrag replacements approximately 0.5%. Fig. 43 is schematic diagram of the laser showing two views of the spatially-dispersive laser cavity.
Variable cylindrical telescope
λ2
Collimating microobjective f = 3 mm f = 4.5 mm Diode
f = 24 mm
Grating
0.5 mm
Variable slit
1800 g/mm ∼ 40 mm
∼ 27 mm
∼ 5 mm
4.5 mm
100 µm
λ1
(b) Top view (horizontal plane) Cylindrical lens Grating
Cylindrical lens
40 arc deg P-N-junction
(a) Side view (vertical plane) Fig. 43. Cavity configuration of the single-chip spatially-dispersive diode laser. The top view (a) shows the plane of the spatially-dispersive cavity. The side view (b) shows the vertical plane of the resonator corresponding to the fast axis of the diode chip. The laser radiation in this plane must be collimated before the diode can be used in the spatially-dispersive cavity.
115 The laser cavity consists of the diode chip, a diode laser collimator, twolense cylindrical telescope, a variable slit, and a diffraction grating. The focal length of the microobjective is 4.5 mm, and the focal lengths of the cylindrical lenses of the telescope are 3 mm and 24 mm. The diffraction grating spacing constant is 1/1800 mm. All lenses are AR-coated for an optimal wavelength of 650 nm. The cavity is more complex than the basic design of the spatially-dispersive laser (Fig. 4). The main reason for such complexity is that the radiation from the laser diode is highly divergent in the vertical plane, as shown in Fig. 43(b), and must be collimated before it can be used, which is done here with the collimating microobjective. In general, the combination of the microobjective and the diode chip can be considered as a complex gain element of the spatially-dispersive laser. In order to obtain the desired focal length of the intracavity focusing system (about 9 mm, as shown in Section 2.6) and gain some flexibility in tuning the parameters of the laser cavity, a variable telescope is used instead of the single intracavity lens. This telescope allows one to change the intracavity mode structure and, thus, tune the number of oscillating spectral channels and the overall output spectral bandwidth. However, as was shown in Section 2.8, this cavity is equivalent to the basic laser scheme. The diode collimator is mounted on a three-dimensional motion stage to allow fine tuning of its position. The telescope lenses are mounted on rotation stages with the rotation axes coinciding with the optical axis of the laser cavity. These rotation stages are used to adjust the angular orientation of the lenses to avoid any twist between them and to orient the lenses exactly perpendicular to the plane of the spatially-dispersive cavity. The rotation stages, in turn, are mounted on twodimensional motion stages for fine tuning of the horizontal and the axial positions
116 of the lenses. The diffraction grating is mounted on a laser mirror mount, which is installed on a one-dimensional axial motion stage. The diode chip is fixed on a heat sink, which is installed on a thermo-electric (TE) cooler. The laser diode and its thermal stabilization system are driven by an all-purpose diode laser driver SDL-840. The uncoated rear facet of the laser crystal works as a 30% input mirror of the laser resonator. The residual radiation from the rear facet of the diode laser is used for the measurements of the output spectrum: the radiation from the diode rear facet is collected with a spherical lens and focused onto a diffuser, installed in front of a multi-mode waveguide that delivers the light to a spectrograph with a CCD camera. This method of detection of the output radiation is used to avoid any readjustments of the measurement system during the tuning of the multiwavelength laser. The diode laser, available for these experiments, does not have effective ARcoating on the output facet. The residual 0.5% reflection causes a strong coupling of the cavities formed by the diode facets and the external spatially-dispersive cavity, thus reducing the coupling efficiency between the external cavity and the diode chip and increasing the total intracavity loss. As a result, the residual reflection of the output facet provides strong optical feedback without any additional output couplers. This results in an unstable operation of the multiwavelength laser, spectral line broadening, and limitations on the overall output spectral bandwidth. Such a cavity coupling limits the number of the output spectral channels and causes a low output power. In general, the required reflection coefficient of the output facet of the semiconductor laser chip for narrowlinewidth, highly efficient, and stable operation of the external cavity diode laser is on the order of 10−4 or smaller, as was demonstrated in a
117 significant number of investigations of external cavity semiconductor lasers [98–108]. Consequently, the diode chip itself can achieve the lasing threshold even without any external cavity, which is known as the free-running operation of the laser diode. The output spectrum of the diode laser used in this experiments in the free-running regime of operation, together with the diode luminescence spectrum, is shown in Fig. 44.
1.0
Intensity, a.u.
0.8
PSfrag replacements
Lasing Luminescence
0.6 0.4 0.2 0.0 640
645
650
655
660
665
670
675
680
Wavelength, nm
Fig. 44. The output spectrum of the visible semiconductor laser in the free-running refime of operation superimposed with its luminescence spectrum.
The threshold pump current of the diode in the free-running mode was about 0.85 A (which varies slightly for different laser diode samples). The lasing threshold with the spatially-dispersive cavity was approximately 0.68 A. The operating current of the multiwavelength laser was about 0.75 A, which demonstrates that the laser was always working within a very small window of the pump currents and the operating current was close to the lasing threshold.
118 For those reasons, any additional intracavity loss caused an instability in the multiwavelength regime of operation. In addition to that, a residual superluminescence of the diode was still present due to the laser operation near the lasing threshold. The external cavity coupled this luminescence signal back to the diode chip, causing additional instability of the laser. To reduce this effect, a variable slit was installed in the telescope, near the conjugate focal points of the telescope cylindrical lenses, as shown in Fig. 43. The exact position of the slit and its width were found experimentally. The slit width was between 100 and 150 µm. Even in these, relatively severe conditions, it turned to be quite possible to obtain a stable, highly reproducible regime of multiwavelength operation of the semiconductor spatially-dispersive laser. Moreover, the multiwavelength output spectrum was tunable in terms of the number of the oscillating spectral lines, and the ultrabroadband spectrum could be tuned as a whole. The tunable multiwavelength output spectra, produced by the semiconductor spatially-dispersive laser, are shown in Fig. 45. The tuning of the number of the output spectral lines was performed by changing the effective focal length of the intracavity telescope, which was done by moving the second cylindrical lens along the cavity optical axis. The maximum overall output bandwidth is approximately 8 nm, which constitutes about 80% of FWHM of the luminescence bandwidth of the laser diode (10 nm). The maximum number of simultaneously oscillating channels is 8, as one can see from the Graph 9 in Fig. 45, which shows that the maximum allowable mutual overlap of the adjacent spectral channels corresponds to the ξ-number of 0.8, as was discussed in Section 2.6.
PSfrag replacements
Intensity, a.u.
119
656
660
664
Wavelength, nm
#1
#2
#3
#4
#5
#6
#7
#8
#9
656
660
664
Wavelength, nm
656
660
664
Wavelength, nm
Fig. 45. Multiwavelength spectra of the visible spatially-dispersive diode laser. The tuning is performed by changing the effective focal length of the intracavity telescope.
The shift of the wavelength combs from the central position is caused by the transverse misalignment of the cylindrical lenses of the intracavity telescope, which changes the angle of incidence of the wavelength channels on the diffraction grating and, therefore, shifts the whole frequency comb. This effect gave rise to the interesting idea of tuning the entire multiwavelength spectrum across the amplification band of the gain element, similar to the multiwavelength tuning experiment of the Cr2+ :ZnSe spatially-dispersive laser described in Subsection 3.1.10. However, the gain bandwidth of the semiconductor is almost the same as a typical bandwidth of the multiwavelength spectra.
120 Therefore, it was necessary to obtain a multiwavelength spectrum with relatively small overall bandwidth. Consequently, a dual-wavelength output spectrum, similar to the spectrum #1 in Fig. 45 (but with a smaller bandwidth), was obtained by adjusting the effective focal length of the intracavity telescope and was tuned across the laser amplification band. The tuning was performed by transverse shift of the large lens of the intracavity telescope, which is equivalent to a rotation of the diffraction grating in its dispersion plane, since it simply changes the angle of incidence of the channels on the diffraction grating. It was much easier to move the lens than to rotate the grating, because the precision one-dimensional motion stage works much better, and provides much smoother motion, than the mirror mount that was used to adjust the angular orientation of the grating. The dual-wavelength spectrum was tuned over almost the entire amplification band of the diode laser. During the tuning the intracavity slit was also moved slightly, in the same direction as the tuning lens. That is necessary because when the lens is shifted, so does the optical axis of the cavity. The experiment was successfully repeated several times, demonstrating an excellent reproducibility of the result, which is shown in Fig. 46. Despite the successful realization of the single-chip semiconductor spatiallydispersive laser, it should be noted that the telecom applications require a more reliable device. Indeed, the spectral channels in a single diode chip will always interact with each other even if their mutual overlap is small. Such an interaction of the spectral channels will cause an increase of the noise in the transmitted information signals, highly undesirable in telecom devices.
Intensity, a.u.
121
PSfrag replacements
656
657
658
659
660
661
662
663
664
665
666
Wavelength, nm
Fig. 46. Continuous tuning of a dual-wavelength output spectrum of the semiconductor multiwavelength spatially-dispersive laser.
For this reason it seems more convenient to use a multi-stripe semiconductor or a laser diode array instead of the single-chip gain element. The spatially-dispersive cavity will enforce each diode stripe to operate at a specific wavelength, and all spectral channels will be completely independent of each other. If such a diode array consists of a large number of single transversal mode diodes, each diode can be enforced to operate at a single transversal mode and produce a diffraction-limited, narrow linewidth output beam, which can be coupled to a single-mode telecom fiber. A prototype of such a spatially-dispersive multi-stripe diode laser is demonstrated in the following subsection.
122 3.2.3 Semiconductor Multi-Stripe Spatially-Dispersive Laser The semiconductor IR multiwavelength laser is based on a multi-stripe AlGaAs laser diode operating near 1570 nm. The diode chip consists of five stripes with the sizes of the active p-n junctions of 2000 × 100 × 1 µm, separated from each other by 500 µm. The rear facet of the laser is uncoated and as in the previous case serves as the laser input coupler. The front facet has a reflection of about 4%. It must be mentioned here that this laser is initially designed as a high power semiconductor laser source and is generally not appropriate for the spatially-dispersive laser due to the absence of the AR-coatings on the output facets of the diode stripes. However, for the purposes of demonstration of the feasibility and applicability of such diode chip as a gain medium of the spatially-dispersive laser, this diode chip is quite appropriate. Therefore, in this set of experiments the goal was to utilize the whole ∼ 40 nm gain spectral bandwidth of the diode chip, obtain the maximum possible number of the output spectral channels, which must be equal to the number of the diode stripes, and demonstrate the possibility of using a multi-stripe semiconductor laser as a gain medium for the multiwavelength spatially-dispersive laser. A schematic diagram of the multi-stripe spatially-dispersive semiconductor laser is shown in Fig. 47.
The cavity consists of the multi-stripe diode chip, a
laser diode collimator, a focusing cylindrical lens, and a diffraction grating. The microobjective focal length is 4.5 mm; the focal length of the cylindrical lens is 25 mm. The diffraction grating spacing constant is 1/1200 mm. Unfortunately, all lenses are AR-coated for 650 nm and, therefore, they introduce significant additional intracavity losses. The diode collimator is mounted on a tree-dimensional motion stage to allow a fine tuning of its position.
Variable cylindrical telescope
123
Cylindrical lens f = 25 mm
Collimating microobjective f = 4.5 mm
Diode 2 mm
λ1 λ2 λ3 λ4 λ 5
Grating 1200 g/mm
2 mm ∼ 50 mm
∼ 35 mm
4.5 mm
(b) Top view (horizontal plane) Grating
Cylindrical lens 40 arc deg
0.5 mm
P-N-junction
100 µm
(a) Side view (vertical plane) Fig. 47. Cavity configuration of the multi-stripe spatially-dispersive diode laser. The top (a) and side views (b) of the laser resonator are shown.
The cylindrical lens is mounted on a rotation stage (to allow its vertical angular adjustment), which is installed on a two-dimensional motion stage for fine tuning of the horizontal and the axial positions of the lens. The diffraction grating is mounted on a laser mirror mount, which is installed on a one-dimensional, axial motion stage. The diode chip is fixed on a heat sink, which is installed on a thermo-electric (TE) cooler. The laser diode and its thermal stabilization system are driven by an allpurpose diode laser driver SDL-840. In the same way as in the case of the single-chip diode laser, the rear facet of the laser crystal works as a 30% input mirror of the laser resonator. The residual radiation from the rear facet of the diode laser is used for the measurements of the output spectrum: the radiation from the diode rear facet is collected with a spher-
124 ical lens and focused onto a diffuser, installed in front of a multi-mode waveguide that delivers the light to a scanning spectrometer. It takes the spectrometer up to a minute to acquire a single spectrum. Moreover, since the radiation from the laser had a very low intensity, an IR visualizer could not be used. Therefore, the only method to control the laser operation was the continuous observation of its output spectrum during any readjustment of the cavity, which made the procedure of obtaining the multiwavelength spectrum and its manipulation very long. Nevertheless, with a certain amount of patience, very good experimental results could be obtained. For the same reasons as in the previous case, the laser cavity design is more complex than the basic configuration of the spatially-dispersive laser. In the limits of the laser model it is equivalent to the basic laser scheme, as was demonstrated in Section 2.8. The radiation from the diode must be collimated in the vertical plane, as shown in Fig. 47(b), and only after that can the diode be used in the spatially-dispersive cavity. However, due to the absence of appropriately AR-coated lenses, it was not possible to use an intracavity telescope, as in the previous case, because an addition of any extra optics into the cavity increases the intracavity losses to such a high level that the lasing threshold for the external cavity becomes higher that the threshold for the free-running operation of the laser. As a result, obtaining the multiwavelength operation becomes impossible and the diode chip itself operates at the wavelength closest to the center of its amplification band. Therefore, only a single cylindrical tuning lens, shown in Fig. 47(a), was used to adjust the angle of incidence of the wavelength channels on the diffraction grating to obtain the multiwavelength output spectrum. Consequently, the laser output spec-
125 trum could be tuned in a very limited spectral range. Nevertheless, it was possible to obtain two types of the multiwavelength spectra: ultrabroadband quasi-continuous output spectra (similar to those obtained with the Cr2+ :ZnSe, Fig. 37), modulated by the stripes of the diode chip, and relatively narrow-linewidth multiwavelength spectra, also modulated by the semiconductor stripes. The ultrabroadband output spectra obtained in the experiments are shown in Fig. 48. In this case the laser initially operates at only two central diode stripes, because the effective focal length of the intracavity lens system is relatively large.
1 0.8
δλ ≈ 6 nm ∆λ ≈ 11 nm
#1
δλ ≈ 7 nm ∆λ ≈ 12 nm
#2
0.6 0.4 0.2
PSfrag replacements Intensity, a.u.
0 1 0.8
δλ ≈ 9 nm ∆λ ≈ 13 nm
#3
δλ ≈ 10 nm ∆λ ≈ 15 nm
#4
0.6 0.4 0.2 0 1 0.8
δλ ≈ 6 nm ∆λ ≈ 17 nm
#5
δλ ≈ 5.5 nm ∆λ ≈ 16 nm
#6
0.6 0.4 0.2 0 1560 1565 1570 1575 1580 1585 1590
Wavelength, nm
1565 1570 1575 1580 1585 1590 1595
Wavelength, nm
Fig. 48. Ultrabroadband multiwavelength spectra of the infrared, multi-stripe, spatially-dispersive diode laser. The average spectral separations of the spectral lines and the overall output bandwidths are indicated in the graphs as δλ and ∆λ, respectively.
126 The spatially-dispersive cavity spreads the radiation, reflected by the grating, across two central diode stripes and enforces their broadband operation. Because the linewidths of the spectral lines are comparable to their spectral separation, the output spectrum has a quasi-continuous shape, as one can see in Graph 1 of Fig. 48. However, since the diodes are separated by a free space, the spectrum is split into two components. By decreasing the focal length of the intracavity lens system, the spectral separation of the two broad spectral lines is increased, as one can see in Graphs 2–4, Fig. 48. Further decrease of the focal length of the intracavity lens system leads to the oscillation of one more diode stripe, and three spectral lines are observed, as shown in Graphs 5 and 6 of Fig. 48. One can see in Graphs 1–4 of Fig. 48 that during the tuning of the dualwavelength spectrum, the entire spectrum is shifted to the lower end of the spectral gain profile. This is a consequence of a slight angular misalignment of the motion axis of the stage, on which the intracavity cylindrical lens is moved in the axial direction, and the optical axis of the laser. To enforce the operation of the third stripe, this misalignment was slightly corrected by a small transverse shift of the lens, which led to the opposite shift of the three-line output spectrum, as one can see in Graphs 5 and 6 of Fig. 48. Due to the high intracavity losses, further decrease of the focal length of the intracavity lens system did not lead to the simultaneous operation of all five stripes of the diode. Consequently, the laser was completely realigned by changing the longitudinal positions of both intracavity lenses and the diffraction grating, and by adjusting the focal length of the intracavity telescope to reduce the overall spectral bandwidth. In this case all stripes would work near the maximum of the amplification band of
127 the laser and, therefore, they would reach the lasing threshold. The multiwavelength output spectra, obtained in this experiments, are shown in Fig. 49.
PSfrag replacements
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δλ ≈ 2.5 nm ∆λ ≈ 9 nm
#1
δλ ≈ 2.5 nm ∆λ ≈ 10 nm
#2
0.8 0.6 0.4 0.2 0
1565
1570
1575
Wavelength, nm
1580
1565
1570
1575
1580
Wavelength, nm
Fig. 49. Multiwavelength spectra of the infrared, multi-stripe, spatially-dispersive diode laser. The average spectral separations of the spectral lines and the overall output bandwidths are indicated in the graphs as δλ and ∆λ, respectively. The nonuniform outline of the five-wavelength spectrum is caused by strong intracavity loss due to spherical aberration of the collimating objective.
It was relatively easy to obtain the multiwavelength operation with four diode stripes with the output spectrum depicted in Graph 1 of Fig. 49. However, because the clear aperture of the diode laser collimator was comparable to the diode chip transverse size, the side stripes experienced significant spherical aberrations, which led to additional intracavity losses. This is why it was difficult to obtain a five-line uniform output spectrum (as could be done with four stripes). Consequently, the five-line spectrum, shown in Graph 2 of Fig. 49, is so irregular, and two side spectral lines have low intensities.
128 Despite the many difficulties in conducting these experiments and building this laser, very good results have been obtained. The behavior of the output spectrum is in a very good agreement with the theory and the principle of operation of the laser. The most important result is the demonstration of the feasibility of applications of the multi-stripe laser diodes as a gain medium in the spatially-dispersive laser resonator. Due to a number of reasons, discussed in detail in Chapter 4, the multistripe semiconductors and diode laser arrays seem to be the best choices for building inexpensive and highly reliable telecom transmitters based on the Littrow-mounted grating spatially-dispersive laser resonator.
3.2.4 Summary: Semiconductor Spatially-Dispersive Lasers In this section, new multiwavelength, ultrabroadband semiconductor laser sources, based on the Littrow-mounted grating spatially-dispersive laser resonator, have been demonstrated. A multiwavelength oscillation has been achieved with a single broad-stripe laser diode, operating at 660 nm, and with a multi-stripe laser diode, working in one of the telecom spectral bands near 1.55 µm. It was shown that one of the major advantages of this laser scheme is that a large class of active media can be used: a single diode chip, a diode array, or a multi-striped diode. With a properly prepared multistripe diode chip one can design a DWDM laser producing hundreds of spectral lines. This work demonstrates a promising and successful attempt to realize an alternative multiwavelength laser source for the optical DWDM systems.
129
CHAPTER 4 CONCLUSIONS AND FUTURE OUTLOOK 4.1
Conclusions
This thesis is devoted to the experimental and theoretical investigation of the continuous-wave multiwavelength, ultrabroadband lasers, based on the Littrowmounted grating spatially-dispersive laser resonator. In the framework of this research the following results have been obtained: 1. A consistent theoretical model of the Littrow-mounted grating spatiallydispersive laser has been developed. 2. The first single-chip visible (660 nm) multiwavelength semiconductor laser, based on the Littrow-mounted grating spatially-dispersive cavity, has been demonstrated. 3. The first multi-stripe, near infrared (1550 nm) multiwavelength semiconductor laser, based on the Littrow-mounted grating spatially-dispersive cavity, has been demonstrated. 4. The first multiwavelength and ultrabroadband, widely tunable, polycrystalline Cr2+ :ZnSe laser, based on the Littrow-mounted grating spatially-dispersive cavity, has been demonstrated Selected results of this research have been submitted for publication as a chapter in the book, “Laser Beam Shaping Applications”, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., (Marcel & Dekker, 2005) [109]. The major results
130 have been published in several peer-reviewed journals: “Optics Letters” [110], “Optics Communications” [111], “Optics Express” [112], and “IEE Optoelectronics” [113] (in 2004 this article was awarded the Snells Premium by the Institute of Electrical Engineering, UK). In addition, the results were presented to the scientific community at a number of international conferences on laser physics and optics: “SPIE Annual Meeting” [114], “In-Plane Semiconductor Lasers V” [115], “Laser Beam Shaping III” [116], “Laser Beam Shaping IV” [117], “Annual meeting/18th Laser Science Conference” [118, 119], “Advanced Solid State Lasers” [120], “Conference on Lasers and Electro-Optics” [121, 122], “Mid-IR Solid-State Lasers” [123], “5th International Conference on Mid-Infrared Optoelectronic Materials and Devices” [124], and “International Quantum Electronics Conference and Conference on Lasers, Applications, and Technologies” [125] The theoretical model of the laser was developed in the framework of a certain set of simplifying assumptions about the generation of the multiwavelength, ultrabroadband laser light by the considered spatially-dispersive laser system. However, the conducted experiments have shown that even such a simplified model of the laser, based on the paraxial and the Gaussian optics approximations, gives quite an acceptable description of the operation of the laser and provides a consistent and reliable method for calculation of the laser cavity parameters for obtaining the desirable properties of the multiwavelength output spectrum. In the experimental part of this work, several new multiwavelength laser systems, based on the Littrow-mounted grating spatially-dispersive cavity, have been demonstrated experimentally: (1) a single-chip multiwavelength spatially-dispersive diode laser, operating in the visible spectral region (660 nm); (2) a multi-stripe, mul-
131 tiwavelength spatially-dispersive diode laser, operating in one of the DWDM spectral bands (1570 nm); and (3) a solid-state multiwavelength, ultrabroadband, continuouswave, broadly tunable mid-IR (2200–2800 nm) laser source, based on 1.56 µm Erfiber-laser pumped, polycrystalline Cr2+ :ZnSe gain medium.
4.2
Future Research
Despite the quite successful results, this work must be considered only as the first attempt of a systematic investigation of the continuous-wave spatially-dispersive multiwavelength, ultrabroadband laser of this type, and it should be understood that significant additional research work lies ahead. Concerning the theoretical analysis of the laser, it is necessary to address such questions as the calculation and control of the linewidths of individual wavelength channels, the spectral and the spatial distributions of the output power of the laser, the mutual influence and the interaction of the spectral channels, the frequency stability of the laser, the methods for obtaining a truly single transverse mode operation of the wavelength channels, and the coherence of different spectral lines. Each of those subjects deserves separate, extensive research. In the short-term, one could see several directions for this research, directly related to the experimental results of this study. First of all, the laser systems described here were built with less than optimal optical components and, therefore, the results could be significantly improved and double-checked by designing and building the same laser systems with more appropriate optics. Regarding the design of the prototype laser source for the DWDM applications, it is necessary to obtain a diode laser array that consists of a large number of regularly
132 spaced mid-IR single-transverse-mode laser diodes that have high quality AR-coatings on their output facets. An alternative to the AR-coated laser diodes, whose major disadvantages are relatively short life times and expensive coatings, is the utilization of new bended-stripe laser diodes that provide a residual reflections on the output facet as low as 10−8 without any AR-coatings. Such devices would allow for the building of a multiwavelength laser source capable of producing a large number of spectral lines controllable by the spatially-dispersive cavity. The appropriately designed spatially-dispersive laser forces each channel to operate at a single transverse mode, which results in output radiation, consisting of a large number of diffraction-limited Gaussian beams, operating at different wavelengths. Such laser beams can be coupled to single-mode telecom fibers for further use in optical communication systems. A single spatially-dispersive multiwavelength laser could replace a large number of the single-frequency or tunable laser transmitters currently used in telecom. Therefore, the development of the spatially-dispersive DWDM laser is not only of a great scientific interest but is also important for practical applications. As regards mid-IR multiwavelength lasers based on the metal-doped chalcogenides, such as the polycrystalline Cr2+ :ZnSe multiwavelength laser demonstrated here, the present experimental results can be significantly improved in terms of the overall output spectral bandwidth and the maximum output power. This will be performed by utilizing the optical elements with high quality AR-coatings and, possibly, with a more powerful pump laser. As discussed earlier, single-pass losses were on the order of 40% in the Cr2+ :ZnSe lasers. At the same time, as was demonstrated in previous studies [110, 113], it is possible to obtain an efficiency as high as 53% with
133 this material. Therefore, with the maximum pump power of 10 W provided by the available Er-fiber laser, the output power for the Cr2+ :ZnSe laser of up to 4.5 W is definitely achievable. The overall output spectral bandwidth of the laser was also significantly limited by the intracavity loss. However, the polycrystalline Cr2+ :ZnSe laser material is capable of generating a much broader output spectrum (2000–3000 nm). Moreover, the laser can further be improved in terms of its portability and efficiency by using the semiconductor optical pump at 1.85 µm, as was earlier demonstrated for a singlewavelength laser, based on the polycrystalline Cr2+ :ZnSe [123]. It is also necessary to study the possibility of generation of the mid-IR multiwavelength spectra by the spatially-dispersive lasers with monocrystalline Cr2+ :ZnS and Cr2+ :ZnSe gain media. There are numerous scientific and practical applications in which the laser systems demonstrated are of a great interest. Among those are Fourier optical processing, all-optical image transfer, multiwavelength spectroscopy, sensitive intracavity multiwavelength spectroscopy and sensing, optical memory, information processing, and optical computing and optical communications. One of the greatest advantages of this spatially-dispersive laser system is its independence of the laser material, as it was demonstrated in this work by the examples of two types of semiconductor gain media and the solid-state laser material. Therefore, with the appropriate choice of gain medium, the multiwavelength laser can be build for any spectral region and provide any desirable structure of the multiwavelength output radiation, thus making this laser system available for any of the practical applications discussed.
134
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145
APPENDIX A THE KNIFE EDGE METHOD
146 This appendix is devoted to a description of the knife edge method for measuring and reconstruction of Gaussian beam intensity spatial profiles. A schematic diagram of such a measurement is shown in Fig. 50 (see also Fig. 18).
PSfrag replacements
x
Gaussi
an b ea m
wx Moving blade (knife)
Large detector
Fig. 50. Measurement of the intensity profiles of the Gaussian beam with the “knife edge method”. The detector must be much larger than the beam diameter.
The intensity profile of a two-dimensional Gaussian beam in the plane of the measurements is described as by the following equation: 2(x − x0 )2 I(x, y) = I0 exp − wx2
!
2(y − y0 )2 exp − wy2
!
≡ I0 I(x)I(y),
(78)
where wx , wy are the beam radii, and x0 , y0 are the beam centers in the x and y coordinates, respectively. When the beam is completely open, the total power, measured by the detector, is an integral of the intensity taken over all space
P = P0
Z
∞
I(x)dx −∞
where P0 is some proportionality coefficient.
Z
∞
I(y)dy, −∞
(79)
147 However, when the beam is partially closed in the x direction, the total power is calculated by the following integral:
P = P0
Z
∞
I(x)dx x0
Z
∞
I(y)dy,
(80)
−∞
where x0 is the current position of the blade edge. The integral in the y coordinate is some unknown constant and, therefore, Eq. (80) can be rewritten as follows:
P =C
Z
∞ x0
! 2(x − x0 )2 exp − dx, wx2
(81)
where C is a constant. The beam radius wx does not depend on the power level and, therefore, one can always normalize the measured power to its maximum (i.e., divide by C), so that the power values are within the interval [0, 1], and then fit the data numerically by the following function:
f (x0 , wx ) =
Z
x∞ x0
! 2(x − x0 )2 exp − dx, wx2
(82)
where x∞ is some reasonably large x coordinate that approximates the infinite value of x. The fitting parameters here are the beam center x0 and its radius wx . This method allows a full characterization of the Gaussian beam intensity profile in the plane of measurement. If necessary, the y coordinate can be processed in the same way. In this work, the fitting procedure was done with the help of a self-written computer program that uses a simple adaptive algorithm to find the best estimates of x0 and wx .
148
APPENDIX B CHARACTERIZATION OF THE GAUSSIAN BEAMS
149 This appendix describes a method for characterization of the spatial properties of the Gaussian beams by measuring their spatial intensity profiles in different planes along the beam propagation direction. Suppose that two beam radii w1 and w2 have been measured at unknown positions z1 and z2 , located at a distance ∆z from each other, as shown in Fig. 51. PSfrag replacements z2
z1
Gaussi
an b ea m
z=0 w0 (0)
w1 (z1 )
w2 (z2 ) ∆z
Fig. 51. Characterization of the Gaussian beam by measuring its intensity distributions in two arbitrary planes.
The zero z coordinate is chosen to be at the center of the beam waist, in which case the waist position will be determined directly by either z1 or z2 , which will be derived later. Using Eqs. (20), (21) and the results of these measurements, one obtains the following system of equations for the propagation of the Gaussian beam:
2 ! z1 2 w (z ) = w 1 + 1 1 0 z0 2 ! z 2 w2 (z2 ) = w02 1 + z0 z2 − z1 = ∆z 2 z0 = πw0 . λ
(83)
150 After cumbersome but simple calculations one obtains the following solution:
z1 =
b+D 2a
(84)
z2 = ∆z + z1 2 2 1 w1 z2 − w22 z12 2 z0 = w22 − w12
(85)
w0 =
(87)
z0 wj
z02 + zj2
12 ,
(86)
where j = 1 or 2 and the auxiliary variables are defined as follows:
a=
4λ 4 π 2 ∆z + (w2 − w12 )2 π λ
b = 2∆z 2 (w12 + w22 ) c=
λ∆z 4 π
D = b2 − 4ac
21
.
The Gaussian beam is now fully characterized by its confocal parameter z0 , the beam waist w0 , and the position of the waist, which is found from the coordinates z1 or z2 . The method can be simplified if the location of the beam waist is known. In that case, the beam confocal parameter is found directly from Eq. (53):
1 ! π 4λ2 2 2 2 4 z0 = wj + wj − 2 zj , 2λ π
(88)
where j = 1 or 2. The beam waist w0 is found from the confocal parameter z0 :
w0 =
r
λz0 . π
(89)
151 The accuracy of both methods depends strongly on the accuracy of the measurements of the beam radii. In general, the first method should always give better results, especially if the beam radii are measured in more than two planes and evaluated in pairs. However, this will work only if the beam profile is stable and can be considered to be constant in time compared to the length of the measurements. If the characteristic time of random fluctuations in the beam profile are comparable to the measurement time (e.g., when the method of the profile measurement is very slow, such as the “knife edge method” described in Appendix A), the error might be significant, especially at large distances from the beam waist. Alternatively, and more preferable, one can use a fast laser beam analyzer.
152
APPENDIX C ALIGNMENT OF THE SPATIALLY-DISPERSIVE CAVITY
153 This appendix describes the alignment procedure of the Cr2+ :ZnSe spatiallydispersive laser. The procedure is performed with the help of a low-power HeNe laser. The procedure consists of the following major steps: 1. Horizontal alignment of the pump beam; 2. Alignment of the HeNe laser beam with the pump beam; 3. Preliminary alignment of the diffraction grating; 4. Alignment of the input mirror and the Cr2+ :ZnSe crystal; 5. Alignment of the pump lens; 6. Alignment of the intracavity lens; 7. Final alignment of the diffraction grating; For the alignment procedure one needs four iris apertures and an IR visualizer, sensitive to 1.56 µm wavelength. First, the pump beam is aligned so that it propagates horizontally at a height, determined by available optical hardware (mirror mounts, motion stages and their combinations). The appropriate height is found by preliminary assembling of the spatially-dispersive cavity. When the pump beam is set up, the iris apertures are centered on the beam as shown in Fig. 52. The apertures are roughly centered on the beam using the beam visualizer and then fine tuned with a powermeter (to obtain maximum transmission, when the aperture being aligned is almost closed). After the alignment the apertures are closed to approximately 0.5 mm (the diameter of the He-Ne beam). After the apertures are centered on the pump beam, the pump laser is switched off and a removable plane mirror M2 is installed in the beam path, as shown in Fig. 52. The HeNe beam is then aligned by the mirrors M1 and M2 so that it propagates through all four apertures (the folding prisms remain untouched).
PSfrag replacements 154
He-Ne beam
ism Pr
M2
A1
0.3–0.5 m
Pump beam
He-Ne Laser M1 Plane of diffraction grating
Plane of intracavity lens
A2
∼ 0.3 m
Pr
∼ 1.5 m
ism
A3
A4 Plane of input mirror
Plane of pump lens
Fig. 52. Cavity alignment—aligning the pump and HeNe beams. A1 –A4 are iris apertures; M1 , folding Al mirror; M2 , removable Al mirror. The distances shown are approximate.
The purpose of the next step is to set up the diffraction grating in such a way that its grooves are exactly vertical and the first order of diffraction for the central lasing wavelength of the Cr2+ :ZnSe laser is reflected exactly back. For that purpose the grating is installed on its mount and temporarily introduced into the HeNe beam path, as shown in Fig. 53. The first task is easily done in two steps: (1) the grating is rotated in its plane until all orders of diffraction of the HeNe beam lie at the same horizontal line, and (2) the grating is tilted vertically until all orders of diffraction propagate exactly horizontally, which is done by retroreflecting the zero order of diffraction into the aperture A3 . The second task is more difficult because one cannot use a laser beam with the wavelength of 2500 nm.
155 However, since the wavelength of the HeNe is about 632 nm, its forth order of diffraction coincides with the first order of 2528 nm, which is very close to the center of the amplification band of the Cr2+ :ZnSe. Thus, it is simply necessary to direct the HeNe forth order back to the aperture A3 to align the grating approximately, as shown in Fig. 53. PSfrag replacements
-1
0
1 He-Ne diffraction orders
2 3
4
He-Ne beam A3
Fig. 53. Aligning the diffraction grating. All diffraction orders of the HeNe beam must lie in the same plane and propagate horizontally to guarantee that the grating grooves are exactly vertical. The forth order must be retroreflected.
On the next step of the alignment, the position of the mount of the diffraction grating on the optical table is noted and the grating (together with its mount) is removed from the HeNe beam. The input mirror is installed and aligned to retroreflect the incident HeNe beam back into the aperture A3 , as shown in Fig. 54. The mirror has a negligible wedge, and the reflections from both the front and the back mirror surfaces are indistinguishable. The working surface of the mirror has a higher reflection than the AR-coated back surface and, therefore, is easily identified. It was mentioned earlier that the pump laser is very sensitive to back reflections, and it might seem here that the residual reflections from the mirror could affect its operation. However, it was verified experimentally that the available mirror has very
156 efficient AR coatings on its surfaces and does not affect the operation of the pump PSfrag replacements laser even at the highest (10 W) pump power.
Input mirror
Aperture shift Refracted beam
He-Ne beam
Undisturbed beam A4
Reflected bea m Laser crystal
A3
Fig. 54. Aligning the input mirror and Cr2+ :ZnSe laser crystal.
After the mirror is alighted, the Cr2+ :ZnSe crystal is installed. The alignment of the crystal is done in two steps: (1) vertical alignment and (2) tilting to misalign back reflections from the path of the incident beam. At the first step, the alignment of the crystal is done in exactly the same way as the alignment of the input mirror. Then the crystal is tilted in the horizontal plane until the reflected beam goes out of the A3 hole. It was found that the tilt of less than 1 arc deg is enough for the laser operation. When the crystal is tilted, the initial path of the HeNe beam is disturbed due to refraction on the crystal. The crystal is plane-parallel with a very good accuracy and, therefore, the beam is only shifted horizontally from its path but not tilted. Consequently, the aperture A4 must be realigned to indicate the new position of the beam, as shown in Fig. 54. It is now possible to align the pump lens. The lens is mounted on a standard tilting mirror mount, which is mounted on a 3-D motion stage. The lens alignment procedure is illustrated in Fig. 55. First, the pump lens is installed at the desired position: centered on the HeNe beam and at a
157 distance of its focal length from the Cr2+ :ZnSe crystal. The lens is then tilted so that the residual reflection of the HeNe beam from the lens input surface is retroreflected to the center of the aperture A3 . The lens is moved in the vertical and the horizontal planes until the transmitted HeNe beam is centered on the aperture A4 . The ARcoating of the pump lens is good enough to prevent the influence of back reflections on the operation of the pump laser. PSfrag replacements
Input mirror Undisturbed beam
He-Ne beam
Transmitted beam
A4
Required shift
Reflected b eam
Laser crystal
A3 Required tilt
Fig. 55. Aligning the pump lens. A strongly misaligned lens is shown to demonstrate the method.
It might be necessary to repeat the procedures of tilting and shifting the lens several times one after another to precisely align the lens. This happens because the lens cannot be installed exactly at the center of rotation of the mirror mount, and every time the lens is tilted it is also slightly shifted in the vertical and the horizontal planes. The alignment of the intracavity lens is very similar to the alignment of the pump lens. However, the intracavity lens is biconvex, and its surfaces have relatively small radii of curvature. This makes it impossible to observe a weak, highly divergent beam reflected from the input surface of the lens. Therefore, the only method to align
158 it is to shift it in the vertical and the horizontal planes until the transmitted beam is centered on the aperture A4 . If necessary, the lens can be fine tuned later, when the lasing is obtained. The final step of the alignment is installation of the diffraction grating back to its place. When the grating is installed, one can try to observe the reflected HeNe 4-th order of diffraction through the top facet of the Cr2+ :ZnSe crystal to verify the grating alignment. Once the cavity is aligned, mirror M2 is removed to unblock the pump beam (Fig. 52). To obtain the lasing, the pump power is initially set to a high level (about 5 W). It worthy to note that the alignment procedure was done with a 632 nm laser beam; however, the pump beam wavelength is 1560 nm and the lasing wavelength is about 2500 nm. Therefore, the refractive index of the optical components is different for these wavelengths. Consequently, the cavity is generally slightly misaligned, which is why there is no lasing when the pump beam is unblocked. To obtain the lasing, it is necessary to fine tune the cavity components. The fine-tuning is usually made in the following cyclic order: 1. Adjust the grating in the vertical plane; 2. Adjust the longitudinal position of the pump lens; 3. Adjust the transverse position of the pump lens; 4. Repeat the steps 1–3 until the lasing is obtained; 5. Adjust all optical elements for the maximum power; The pump power is gradually reduced and the fine alignment procedure is repeated until the new maximum output power is obtained. The whole process is iterated until the lowest possible lasing threshold is obtained.
159 It must be mentioned here that the cavity alignment for the minimum lasing threshold is different from the alignment for the maximum output power. The reason is that the laser crystal experiences a thermal lensing, and the thermal lens becomes an additional optical element of the cavity. The strength of the thermal lens is determined by both the pump power (due to heating the crystal) and the lasing field intensity (due to a change in the index of refraction of the laser material in the region of the laser mode). Therefore, the cavity alignment for low pump powers (low lasing threshold, low output power) is different from the alignment for high pump powers (high lasing threshold, high output power).
