Application of Quartz Microresonators to Uncooled ...

Reprinted from Uncooled Infrared Imaging Arrays and Systems, Semiconductors and Semimetals Volume 47, Volume Editors: Paul W. Kruse&. David Skatrud; Academic Press, ©1997. The book is copyrighted, however, this chapter is not because, at the time of writing, the authors were government employees who performed this work as part of their official duties. CHAPTER

9

Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays John R. Vig and Raymond L. Filler US

ARMY CoMMUNICATIONS AND ELECTRONICS COMMAND

FoRT MoNMOUTH, NEw JERSEY

Yoonkee Kim US AllMY REsEARCH LABORATORY

FORT MONMOUTH, NEW JERSEY

L INTRODUCTION II. QUARTZ MICRORESONATORS AS INFRARED SENSORS

Ill.

QUARTZ THERMOMETERS AND THEIR TEMPERATURE COEFFICIENTS

IV. OSCILLATOR NOISE .

V.

.....

.

VI. THERMAL ISOLATION

.

. .

.

.

. .

.

.......... .

VIII. PREDICTED PERFORMANCE OF MICRORESONATOR ARRAYS

X. SUMMARY AND CONCLUSIONS . . . . . . APPENDIX. PERFORMANCE CALCULATIONS

References

. . . . . . . . . . . . .

I.

272

277

VII. INFRARED ABSORPTION OF MICRORESONATORS. IX. PRODUCIBILITY AND OTHER CHALLENGES

269 271 273 274 275

. .

FREQUENCY MEASUREMENT

.

279 281 283 284 294

Introduction

Frequency and time (which are closely related) are the physical quantities that can be determined with the highest accuracy. Most high-precision frequency control and timing devices are based on quartz resonators. A quartz crystal can act as a stable mechanical resonator, which, by its piezoelectric behavior and high Q, determines the frequency generated in an oscillator circuit. About 2 x 109 quartz resonators are manufactured annually for oscillator, clock, and filter applications. (An oscillator consists of a resonator plus feedback circuitry that can sustain the vibration of the resonator at a resonance frequency.)

269 Copyright :&; 1997 by Academic Press All rights of reproduction in any form reserved. OOS0-8784/97 $25.00

Semiconductors and Semimetals A Treatise

Edited by R. K Willardson

Eicke R. Weber

CONSULTING PHYSICIST

DEPARTMENT OF MATERIALS SCIENCE

SPOKANE, VVASHINGTON

AND MINERAL ENGINEERING

Uncooled Infrared Imaging Arrays and Systems SEMICONDUCTORS AND SEMIMETALS

UNIVERSITY OF CALIFORNIA AT BERKELEY

Volume 47 Volume Editors

In memory of Dr. Albert C. Beer, Founding Co-Editor in 1966 and Editor Emeritus of Semiconductors and Semimetals. Died January 19, 1997, Columbus, OH.

PAUL W. KRUSE INFRARED SOLUTIONS, INC. MINNEAPOLIS, MINNESOTA

DAVID D. SKATRUD DEPARTMENT OF THE ARMY PHYSICS DIVISION ARMY RESEARCH OFFICE RESEARCH TRIANGLE PARK NORTH CAROLINA

0 (-

en

-=

U5

a:l

10""

z

UJ

a:::

...._

...._

10"

l-

10'

::r:

10'

E

10""

-

0•

0 UJ

1(}"23

""

- - 600MHz 1 GHz ---- 1.8GHz

10' 10'

a_

""

""

'' ""

"

103 10

100

1000

10000

1

100

10

f [Hzj FIG. A.3. processes.

...._

'

0

1-

en

...._

~N

0

...J 0

10"'

Z10"'

00

10"' 10"'"

- - 10maec - - 100 msec

"'

'" "' "" "" '' "" "" ~.---------~,-----------,-----------_:_j 10

100

1000

10000

FREQUENCY [MHz] FIG. A.5. Thermal conductance for a response time of 10 msec, and 100 msec as a function of resonator frequency.

I '

I

290

J. R.

VIG,

R. L.

and Y.

FILLER,

9

KIM

QUARTZ MICRORESONATORS

291

TIME CONSTANT [Sec] 0.25

2.5 -

1Q10

1

I

•

0.025

II.!

.,,,

0.0025

--

I'""'

--

Radiation Loss Limit

=

~

~

Ill

~

I

10'

-N

::r:

E

..0~

o 10"

l

10"'

10"'

10""

10"'

D* versus G (and rr) at

f

=

''

"' '

"' "'

"' "'

''

''

''

',,

>--,~I 2

FIG. A.7.

J

N foo sin (nfr) [ sin (nrf Nr) N- 1 o Sy(f) (nfr)2 1- N2 sin2(m:fr) df (A16) 2

where N is the number of samples, r is the measurement time, and r is the ratio (r + rd)/r, where rd is the dead time between measurements. If there is no dead time, then r = 1. If N = oo and r = 0, we have (a;( co, r, 0)) =

Ioo Sy(f) df

(A17)

independent of r, which is the standard definition of the variance of a continuous variable. If N = co and r is finite, we have the standard variance of the averaged frequency fluctuations 2 (ay(co, r, r))

=

Joo Sy(f) sin, 2(nfr) df n?

0

Filter shape for the oo-sample variance for r

=

1 sec and r

=

0.5 sec.

1/(2nrr) for 600MHz.

2

2

''

f [Hz]

10"'

frequencies. A measure of the dispersion, that is, noise, of the measurements is the variance (or its square root, the deviation). In general, the variance of a noise process (or the sum of processes) is given by (Lesage and Audoin, 1979)

(ay(N, r, r)) =

''

0

CONDUCTANCE [Watt/K] FIG. A.6.

I

- - -.:= 1 Sec ---- -.: = 0.5 Sec

............

(A18)

which is the total power output from a filter with Sy(f) as input and transfer function H 1(j, r) where

H l(j, r) = sin2(nfr)

(A19)

(nfW

On a linear plot, the response of this filter is shown in Fig. A7. The total area under the curve is 1/(2r); therefore, a measurement of r sec looks like a bandwidth of 1/(2r) (in Hz). A problem arises in noise processes such as flicker, where the noise power increases as the frequency decreases like 1/ In that case the integral depends on N and may not converge. If N and r are finite and there is no dead time, the filter looks like a bandpass filter whose lower band edge varies as 1/(Nr). If N = 2 and there is no dead time, we have the definition of the Allan variance, which looks like a filter with transfer function H 2 (j, r ):

r.

2 Hz(j, r) = 2 sin (nfr) [ 1 _ sin~(nf2r) (nfr) 2 2 2 sm 2(nfr)

J= 2 sin(nfr?(nfr) 4

(A20)

This filter looks like Fig. A8. Both the bandwidth and center frequency of the bandpass filter move with averaging time. The decrease in the bandwidth, as the frequency decreases, is by exactly the right amount to cancel out the 1/f dependence of the flicker noise. This is why flicker of frequency is the only noise process whose Allan variance is independent of averaging time. This integral also converges for random walk (ljjl) frequency noise.

,f :t

292

J. R.

R. L.

VIG,

and Y.

FILLER,

'?:'

KIM

9

293

QUARTZ MICRORESONATORS

2 /-, I

'

I

-

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\ \

I

~

I

\

\

~

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I

\ \

I

\

I

\

\

\

I

'\

I

/

""

\

I

0

- - r=1 ---- r=10

t->

I

;

I

II

.,.:

\

I I

,,

fl

-

\ \

r

I

N

,.

,:

t=1Sec t =0.5 Sec

I

t->

I

I /

o

/

0

r,:1

.'i.

0

2

v v if

\i1. ~~. I

I''

20

10

FIG. A.9. Filter shape for the two-sample variance for '

=

1 sec and '

=

= 2 sin2(nfr) [1- sin2(njNT) N 2 sin 2 (njT)

(nfrf

2

1

50

40

J

Filter shape for the two-sample variance for r

(A21)

uN

1

60

tnr , 2

IH 3 (j,T,r)l 2 df

=

1 and r = 10.

(A23)

1o-1()·•

(A22)

'\

'

- - Ricker FM Noise ---- 11\/hite PM Noise 11\/hite FM Noise - - - Temperature Auctuations

'\

'\

''

''

10""

0

"' '

.

/

10"'2 0.001

/

' ' ....

/

' X .?

//

.

The two-sample deviation with no dead time for the noise discussed previously is shown in Fig. AlO for a 500-MHz resonator frequency. The effect of increasing the dead time is shown in Fig. A11. A frame rate of 30 Hz is equivalent to a repetition ratio of 1100 for r = 30 11-sec. Both thermal and flicker noise increase significantly when dead time is introduced.

= j(rr.u1Xr)2 + (uy)2

The noise equivalent temperature difference (NETD) is defined as (Johnson

b>- 10"'"

foo (4KBT /G),,

'f ~

The total frequency noise u N is given by a root mean square of the temperature fluctuations times the temperature coefficient and the oscillator nOise:

Figure A9 shows the filter functions for a T/r ratio of 1, and a ratio of 10 for r = 30 11-sec. The effect of the dead time is to increase the contribution of low frequencies. The two-sample variance of the spectrum of thermal fluctuations is given by (Lesage and Audoin, 1979)

2 rl&r(T,-r)=

I

0.5 sec.

Physically, the Allan variance is a measure of the fluctuations from measurement to measurement when the measurement intervals are contiguous. The filter function of the two-sample variance when the measurements are not contiguous depends on the frame interval T and the measurement interval r. The function is HJ(j, T, r)

I

f [KHz]

f [Hz] FIG. A.8.

I

30

/

''

'' '·

0.01

0.1

10

100

1000

MEASUREMENT TIMEt FIG. A.lO.

Two-sample variance of noise processes as a function of measurement time at

r = 1 for the resonator frequency of 500 MHz.

294

J. R. VIG, R. L. FILLER, andY. KIM

9

1()·1

1()•10

'0>10""

,J~

- - Ricker FM Noise ---- \Nhite PM Noise I!Vhite FM Noise - - - Temperature Auctuations 10

100

1000

10000

REPETITION RATIO r Fra. A.ll.

Two-sample variance of noise processes as a function of repetition ratio r.

and Kruse, 1993) (4F2 + 1)aN NETD = roA(AP/ATh,-;.,R

(A24)

where F is the numerical aperture of the optics, r 0 is the transmittance of the atmosphere between the target and the sensor, and (AP/AT);.,-;., is the temperature dependence of the blackbody function over the wavelength interval from 2 1 to 22 , respectively. For thermal imaging systems operating in 8- to 14-llm atmospheric window, the value of (AP/AT);.,-;. is 2.62 W/m 2 ·K (P. W. Kruse, private communication, 1994). 2

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QUARTZ MICRORESONATORS

295

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!