JOURNAL OF APPLlED PHYSiCS
VOLUME 38, NUMBER 13
DECEMBER 1961
Photo elastic Properties of Selected Materials and Their Relevance for Applications to
Acoustic Light Modulators and Scanners
R. W.
DIXON
Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey
(Received 31 July 1967)
Measurements of the photoelastic tensor components and figures of merit of several materials selected
for possible application in acoustic light modulators and scanners are presented. The most important
results are that GaP is superior to fused quartz by nearly two orders of magnitude for modulation and
scanning of visible light of wavelength longer than 0.58 IJ., and that GaAs is similarly superior at wavelengths longer than about 1 IJ..
In addition to the criteria of high optical quality,
availability of large single crystals, and tolerable acoustic attenuation, the usefulness of diffracting materials
for application in ultrasonically driven light modulators
and scanners obviously increases when materials are
selected which produce large refractive-index variations
per unit strain, i.e., materials which can diffract appreciable portions of incident light with as little acoustic
driving power as possible. Unfortunately, this selection
cannot be intelligently made until the components of
the photoelectric (strain-optic) tensors of these materials are known. This paper, therefore, presents measurements of the magnitudes of the photoelastic tensor
components of several materials selected for their
possible applicability in acoustic light modulators and
scanners. Figures of merit and other information pertinent to their use in such devices are also given.
The experimental technique was the same as that
used previously in measuring the photoelastic properties
of lithium niobate.I The intensity of light Braggdiffracted from an acoustic wavepulse propagating in a
fused-quartz buffer rod is compared with the light
diffracted from a similar pulse propagating in a sample
bonded to the fused quartz.
If an ultrasonic strain Ski propagates in a crystal
the induced change in the optical indicatrix is I:::.B ij =
PijHSkl, where Piikl are the components of the photoelastic (strain-optic) tensor.2 The resulting modulation of
the refractive index can diffract a portion of an incident
optical beam. S Consider an acoustic pulse which
diffracts light of intensity II as it crosses an optical
beam in the quartz prior to entering the sample, and
diffracts light of intensity Is after having traveled into
the sample, been reflected from the free end, and
traveled back into the quartz. If a similar acoustic
pulse, before and after being reflected from the free
end, diffracts light of intensity 14 and 15 as it traverses
a light beam in the sample, then the ratio (14 I 5/I tIs) !/2
I R. W. Dixon and M. G. Cohen, App!. Phys. Letters 8, 205
(1966). Incorrect values of the photoelastic constants of fused
quartz were unfortunately used in this reference. The corrected
data appear in Tables I and II of this paper.
2 ] . F. Nye, Physical Properties of Cyrstals (Clarendon Press,
Oxford, England, 1960).
3 M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693
(1965).
is equal to (n 6 p2/ pv3 ) sample/ (n6 p2/ p'lfl) fus£d quartz, where
n is the refractive index, P the photoelastic component
appropriate for the directions and polarizations of the
optical and acoustic waves, p the density, and v the
acoustic velocity.! The bond properties (assuming
that the bond is uniform over the acoustic beam cross
section, as can readily be checked experimentally),
acoustic attenuation, and reflection loss at the free end
have canceled from this expression.
Acoustic wave pulses in the frequency range 200-400
MHz, with durations of order 10-7 sec, were generated
piezoelectrically in thin-film cadmium sulfide transducers deposited on the fused quartz buffer rod. Phasesensitive detection improved the accuracy with which
the intensity of the diffracted light pulses could be
measured.! By choosing the directions and polarizations
of the acoustic and optical waves appropriately, accurate (within a few per cent) high-frequency values
of the independent photoelastic tensor coefficients,
relative to those of fused quartz, could be obtained.
The complexity of a complete analysis in crystals of
low symmetry will be appreciated. (The tensor-transformation properties of Pijkl are very similar to those
of the elastic-stiffness tensor.) In addition, any optical
anisotropy which is present complicates the measurements by greatly increasing the necessary acoustic
frequencies. 4 For this reason some of the tensor components have not yet been obtained.
The magnitudes of the photo elastic components are
presented in Table 1. Materials listed include some
having low ultrasonic losses, LiNbOs5 ,6 LiTaOs,6 YAG,7
YIG,1 a-AbOj,8,9 Ti0 2,9,1° as well as a number whose
large refractive indices make their light-scattering
abilities potentially large.
The wavelength dispersion of many of the photo• R. W. Dixon, IEEE J. Quant. Electron QE-3, 85 (1967);
Proc. Symp. on Modem Optics, The Polytechnic Institute of
Brooklyn, 1967 (to be published).
6 E. G. Spencer, P. V. Lenzo, and K. Nassau, App!. Phys.
Letters 7, 67 (1965).
6 E. G. Spencer and P. V. Lenzo, J. App!. Phys. 38, 423 (1967)
7 E. G. Spencer, R. T. Denton, T. B. Bateman, W. B. Snow,
and L. G. Van Uitert, ]. App!. Phys. 34, 3059 (1963).
8 J. de Klerk, Phys. Rev. 139, A1635 (1965).
9 H. J. Shaw, D. K. Winslow, A. Karp, and R. A. Wilson, App!.
Phys. Letters 4, 28 (1964).
10 T. A. Midford and S. Wanuga, J. App!. Phys. 36,3362 (1965).
5149
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5150
R. W. DIXON
TABLEr!. Magnitudes of photoelastic constants.B,b
Material
Fused quartz
GaPd
GaAsd
Ti0 2
LiNbO,·
YAGf
YIGg
LiTa03
AS2S3h
AS,S3
SF-4
a-AI.03 i
CdS;
,B-ZnSk
ADpm
KDpn
Te
H 20 0
c
>'(f./o)
Pu
P I2
PH
0,63
0,63
1.15
0.63
0.63
0.63
1.15
0.63
0.63
1.15
0.63
0.63
0.63
0.63
0.63
0.63
10.6
+0.121
-0.151
-0.165
0.011
0.036
-0.029
0.025
0.0804
+0.277
+0.308
+0.232
,....,0.20
0.142
+0.091
0.302
0.251
0.155
,....,0.31
+0.270
-0,082
-0,140
0.172
0.072
+0.0091
0.073
0.0804
+0.272
+0.299
+0.256
,....,0.08
0.066
-0.075
-0.074
-0,072
~-0.01
-0.0615
0.041
0.022
0.085
~0.054
P'I
PI'
P aa
P 41
0,0965
0,178
0,168
0.092
0,058
0.088
0.155
0.086
0.094
0.150
0.024
~O
~O
P I4
PssE
0.031
0.252
0.041
+0.075
0.246
0.249
0.130
0.195
0.225
0.236
0.246
0.263
0,221
0.075
0.058
• Absolute signs were measured for GaAs and GaP by noting the sign of
the retardation for light polarized at 45° with, and propagating normally
to, a [100J axis along which a dc stress was applied. Relative signs were
measured using ultrasonic techniques. Knowledge of these signs. by use of
the usual tensor transformations, allows computation of the photoelastic
interaction for arbitrary directions and polarizations of the interacting
waves.
b Other compilations of photoelastic data appear in. e.g .• LandoltBornstein. New Series. Group III. Vol. I. K.-H, Hellwege, Ed. (SpringerVerlag, Berlin. 1966). Vol. 1. p. 138; and R. Krishnan. Progress in Crystal
Physics (Madras. Central Art Press. 1958).
c W. Primak and D. Post. J. Appl. Phys. 30, 779 (1959).
d Figures of merit for diffraction from longitudinal acoustic waves propagating along [111 J directions with optical polarization in the scattering
plane are within a few per cent of those for acoustic waves in the [110J
direction, while for acoustic waves in the [1001 direction the values are
reduced by 30% to 40%.
e See Ref. 1.
f Tbe measurements presented here determine the magnitudes of pa.
P21. and p.. and the relative sign PIl/P21 <0. The independent dc stressoptic measurements of KarT (private communication), when reduced to
strain-{)ptic coefficients. yield PIl-P21 = -0.036 and p.. = -0.059. values
5% higher than predicted by the data presented here. This agreement is
considered satisfactory in view of the experimental uncertainties in both
experiments and provides a check. in addition to the measurements of
Primak and Post on fused Quartz. on the absolute accuracy of the present
technique. The measurements are consistent with the assumption that
isothermal and adiabatic photoelastic components differ very little. It
should be noted that P21 in Y AG is anomalously small (compare. e.g.• P21
in YIG). As noted by Karr. this makes strain-induced refractive index
variations in Y AG small and increases the desirability of the material as a
laser host crystal. Contrast the case of GaP in which the induced birefringence per unit strain is very large.
g See R. W. Dixon and H. Matthews. Appl. Phys. Letters 10, 195 (1967).
h Arsenic trisulfide glass absorbed strongly at 0.63 f./o. produced some
depolarization of the transmitted optical beam. and showed rather high
acoustic losses (~10 dB/cm at 150 MHz). At 1.15 f./o. however. the optical
properties were much improved. The material should be very useful where
its high ultrasonic attenuation can be tolerated.
i D. E. Caddes and C. D. W. Wilkinson (IEEE J. Quant. Electron.
QE-2. 330 (1966)) previously noted that P13 in sapphire is very small.
The present work shows that Pal is similarly small. The results are anomalous because both components are allowed by symmetry.
; The values of pu and P3l reported here are approximately twice as
large as those reported by Carleton and Soref in Ref. (15).
k Considerable confusion exists in the Iiterature b concerning the value
of p .. in ,B-ZnS. The value I often Quoted of p.. =0.84. allegedly the largest
of any known material. is surely inaccurate. The crystal used here was
natural spalerite obtained through the generosity of Professor C. Frondel
of Harvard University.
I C. D. West and A. S. Makas. J. Chern. Phys. 16,427 (1948).
m The value of p"E measured here agrees well with a previous de measurement. p"E = -0.077. by R. O·B. Carpenter thesis. Harvard University.
Cambridge. Massachusetts. 1951 (unpublished). but disagrees with the
value. p",E = -0.1107. measured by C. D. West and A. S. Makas. Am.
Mineralogist 35, 130 (1950).
n Previous dc measurements by C. D. West and A. S. Makas. Am. Mineralogist 35, 130 (1950) gave p",E = -0.0685 and -0.007 <! (PIl-P12) <0.
o The photoelastic constant of water was estimated using the LorentzLorenz formula. See. T. M. Smith and A. Korpel. IEEE J. Quant. Electron
QE-I. 283 (196$).
elastic tensor components has also been investigated.
In particular, the work of Tell et aU! in CdS and ZnO
has been confirmed. In the present context the important observation is that photoelastic dispersion appears
to be a much less important consideration for modulator
design than was initially expected. Most materials show
less dispersion than CdS and ZnO, and in addition the
dispersion tends to be strongest well into the absorption
region where the material cannot be used for modulation or scanning purposes. Details of these experiments
will be reported in a separate publication.
There are at least three different criteria for judging
a material's usefulness in optical information processing
applications. Gordon!2 has shown that for an acoustic
modulator the center frequency jo, the dynamic bandwidth I1j, and the scattering parameter 'YJ (the fractional
light power scattered is sin217!/2) are under a wide variety
of circumstances related by
11 B. Tell, J. M. Worlock, and R.
6, 123 (1965).
jol1j17~9(n7f/pv)
(X03h COsBO)-IPa ,
(1)
in which Xo is the optical wavelength in air, flo the Bragg
angle, P a the acoustic power, and h the acoustic-beam
height. Assuming that the acoustic-beam height is held
J. Martin, App!. Phys. Letters
12
E. I. Gordon, Appl. Opt. 5, 1629 (1966).
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TABLE II. Figures of merit for acousto-optic devices.
Material
X(Il)
Fused quartz
0.63
Fused quartz
0.63
GaP
0.63
GaP
0.63
GaAs
1.15
GaAs
1.15
n
1.46
3.31
3.37
p(g/cma)
2.2
4.13
5.34
v(1()6 em/sec)
0kt. wave
po arization
and direction"
M 1 (n7 p2/pv)d
M 2 (n 6p2/pv3)
Ma(n7p2/pv 2)
Pa/fo
(mW/MHz)
long.
5.95
J.
7.89XIQ-7
1. 51 X1Q-18
1. 29X 10-12
5.4
trans.
3.76
0.963
0.467
0.256
long. in [110J
6.32
II
590
44.6
93.5
0.074
0
trans. in [1ooJ
4.13
liar J. in [OlOJ
137
24.1
33.1
0.21
r
long. in [110J
5.15
II
925
104
179
0.24
trans. in [100J
3.32
liar J. in [OlOJ
155
46.3
49.2
0.86
Acoustic wave
polarization
and direction
Ti02
0.63
2.58
4.6
long. in [11-20J
7.86
LiNbOa
0.63
2.20
4.7
long. in [11-20J
6.57
YAG
0.63
1.83
4.2
long. in [looJ
YAG
0.63
YIG
1.15
2.22
LiTaO a
0.63
A52Sa
0.63
27.1
0
>-'l
t<j
;;p.
r.r.
>-'l
H
n
3.93
7.97
0.87
(b)
66.5
6.99
10.1
0.69
8.53
II
0.16
0.012
0.019
370
0
long. in [110J
8.60
.l
0.98
0.073
0.114
61
t<j
5.17
long. in [100J
7.21
J.
3.94
0.33
0.53
80
>-'l
2.18
7.45
long. in [oolJ
6.19
\I
11.4
1.37
1.84
3.8
2.61
3.20
long.
2.6
J.
762
433
293
0.024
en
~
"'0
~
......
l'1
619
347
236
0.179
J.
1.83
4.51
3.97
1.75
r.r.
11 in [oolJ
24.3
3.41
4.41
1.58
t<j
\I or J. in [oolJ
\I in [11-20J
10.6
0.57
4.9
1.42
l'1
7.32
0.34
0.66
51.8
12.1
12.4
0.56
II in [OlOJ
16.0
2.78
2.62
2.65
1.83
liar
[oolJ
3.34
6.43
1.83
3.8
5.50
II in [OlOJ
!I or .l in [oolJ
8.72
1.91
1.45
4.8
1.57
3.83
0.95
7.3
43.6
160
29.1
0.24
10200
4400
4640
2.46
1.616
3.59
long.
3.63
p-ZnS
0.63
2.35
4.10
long. in [110J
5.51
p-ZnS
0.63
trans. in [110J
2.165
a·A12Oa
0.63
1. 76
4.0
long. in [oolJ
CdS
0.63
2.44
4.82
long. in [11-20J
4.17
ADP
0.63
1.58
1.803
long, in [1ooJ
6.15
ADP
0.63
trans. in [looJ
KDP
0.63
long. in [looJ
KDP
0.63
H2O
0 163
long.
2.34
"'0
II
1.15
0.63
1.51
.lin [oolJ
0
SF-4
10.6
.L
62.5
AS<~3
Tee
liar
"'0
~
11.15
trans. in [1ooJ
1.33
1.0
long.
1.5
4.8
6.24
long. in [11-20J
2.2
• The optical-beam direction actually differs from that indicated by the magnitude of the Bragg
angle. The polarization is defined as parallel or, perpendicular to the scattering plane formed by the
acoustic and optical k-vectors.
b See Ref. 1.
"
.L in
II in [ooolJ
10.5
7.14
C An ordinarily polarized wave is assumed. If the increased optical absorption (due to impurities in
the currently available samples) of the extraordinary polarization can be tolerated. then M 1 = 5.7 X to-',
M,=2.02X1o-a, Ma=2.59X1o-a, and Pa/f.-1.28 mW/MHz.
d Figures of merit are given in cgs uults.
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"%j
r
n
>-'l
t<j
ti
a::
;;p.
>-'l
t<j
~
......
;;p.
r
en
<n
.....
<n
.....
5152
R.
W.
constant, the combination of material parameters
M I =n7p2/pv constitutes a "figure of merit" by which
materials for use as acoustic modulators may be compared in the usual situation in which both bandwidth
and diffracted intensity are important. Since the parameter TJ is given b y 12
TJ;:::::'5(n6p2/p'lfl) (A02h coS200)-lWOPa ,
(2)
(where Wo is the acoustic beam width), if only the
scattered light intensity were important an appropriate
figure of merit would be M 2=n6p2/pv3. Finally, in an
acousto-optic device in which the acoustic-beam height
is not otherwise constrained, and may be made as
small as the optical beam in the region of the lightsound interaction, it can be shown that h is approximately hmin=v/Af. Using this relation Eq. (1) may be
written
thus defining a third generally useful figure of merit,
viz., Ms=n7p2/p7?
Figures of merit are presented in Table II, along with
other relevant data. Water, which is sometimes useful
at low acoustic frequencies, is included for reference.
M I , M 2, and Ms are given for several typical acousticand optical-beam polarizations and propagation directions and may be directly compared in assessing a
material's applicability for a given application. The final
column shows values of Pa/jo from Eq. (3), for the
condition of 100% deflection (TJ= 2.5) and assuming
cosO= 1, computed at the indicated optical wavelengths.
It should be emphasized that the entries in this table
are appropriate for lossless optical and acoustic waves.
It will be seen from Table II that GaP and GaAs are
exceptionally good modulating materials, their merit
deriving largely from their high refractive indices. The
room-temperature absorption edge in GaP begins near
0.58/J. but the undoped crystal is essentially transparent
at longer visible and near-ir wavelengthsP It is therefore appropriate for modulation of the 0.63 /J. transition
of the He-Ne laser. When cooled, the band edge in
GaP begins at about 0.53 fJ.,t3 coincidentally the second
harmonic of the Nd:YAG laser, a wavelength potentially useful in device applications. It is also noteworthy
that transverse elastic waves along the [100J direction
in GaP have figures of merit Mt, M 2 , and M s, which are
17.4, 16.0, and 25.6 times larger, respectively, than the
figures of merit for diffraction from longitudinal waves
in fused quartz. The rotation of the plane of polarization which can be made to accompany light diffraction
from transverse waves in this material makes the
crystal potentially useful as a polarization switch or
permits the use of a crossed polarizer after the modulator, q ~onvenience in some experiments. Large high
optic4l quality single crystals of GaP have previously
la
p. ],
p~an
and
p. g.
r~pmi!-5,
Fhys. Rev. 150, 690 (1966).
DIXON
been rare, but crystals are now sufficiently available14
that it is realistic to consider their use in light deflectors
and modulators.
GaAs is equally efficient for modulation of the 1.06
and 1.15-/J. transitions of the Nd:YAG and He-Ne
lasers and has previously been shown useful for modulating the 10.6-/J. output of the C0 2 laser.l 5 For the latter
purpose tellurium should also be considered.l6
The high-frequency acoustic attenuations in GaP17
and GaAs18 are not extremely low, making the materials
questionable for room-temperature applications at
microwave frequencies, but the materials have been
utilized by the author for light modulation at frequencies near 500 MHz without great difficulty. When
very high acoustic frequencies must be used, low acoustic losses make rutile and (except for problems of
optical damage) 19 lithium niobate appear most promising. Sapphire and lithium tantalate can also be obtained in very high quality samples and possess low
high-frequency acoustic losses.
The acoustic power necessary for a given modulator
requirement may be estimated using Eq. (2). Results
using longitudinal waves in GaP at 0.63 /J. and in GaAs
at 1.15 /J. arejo6!TJ=2.12XI04h-I P a andfo6!TJ=6.42X
103h-I P a , respectively, wherejo and 6j are measured in
MHz, the acoustic-beam height h in em, and the
acoustic power Pa in watts. The acoustic-beam height
may be made small without undue diffraction loss or
transducer loading by focusing the acoustic energy into
the interaction region with a cylindrical transducer.2o
The amount of focusing which is possible in a particular
case may be limited by sample fracture, acoustic
harmonic generation, or by the more fundamental
requirement that the height of the focused beam be
larger than the diameter of the optical beam over the
entire width of the interaction region. Values of h of the
order of 0.015 em have been obtained at 400 MHz using
3-mm-high CdS transducers deposited on cylindrically
ground surfaces. 2o For an optical wavelength of 0.63 /J.
14 C. Frosch (private communication). The author wishes to
thank Mr. Frosch for furnishing the GaP crystals used in the
experiments reported here.
15 H. R. Carleton and R. A. Soref, App!. Phys. Letters 9, 110
(1966). The p values for acoustic waves along [111J [P ... = }(Pn+
2P21-2P.. ) and PII=i(Pn+2hl+4P44)J are 0.100 and 0.244
according to the date measured here at 1.15 }L, but are 0.015 and
0.355 according to the data in this reference at 10 }L. Significant
photoelastic dispersion has been measured near the band edge and
may account for these differences.
16 R. W. Dixon and A. N. Chester, App!. Phys. Letters 9, 190
(1966) .
17 No acoustic-loss measurements for GaP were found in the
literature, but attenuation measured by optical-beam probing
techniques in the samples used here was less than 1 dB/em at 400
MHz for [110J longitudinal waves.
18 Attenuations of 1.8 dB/em at 200 MHz and 5.5 dB/cm at 425
MHz for [100J longitudinal waves were measured in the samples
used here. See also, V. K. Belyaev and 1. 1. Reshetnyak, Soviet
Phys. Acoust. 12, 315 (1967); M. Pomerantz, Phys. Rev. 139
A501 (1965).
19 A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. C. Smith, A. A.
Ballman, H. J. Levinstein, and K. Nassau, App!. Phys. Letters 9,
72 (1966).
20 M. G. Cohen, J. App!. Phys. 38,3821 (1967).
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PHOTOELASTIC
PROPERTIES OF SELECTED
5153
two smaller than predicted by theory. More refined
experiments are in progress. On the basis of these
preliminary experiments it is, however, possible to be
very enthusiastic concerning the application of GaP
to acousto-optic devices. GaAs and AS2S3 will work
equally well for modulation and scanning at the important 1.06- and 1.15-/-1 wavelengths. Construction of
optical scanners using these techniques should be
especially important because of the simplicity of these
scanners compared with existing electro-optic deflectors.
It is a pleasure to acknowledge the expert experimental assistance of R. L. Field, Jr., in performing the
photoelastic measurements. Conversations with M. G.
Cohen, F. M. Smits, and E. I. Gordon have been very
helpful.
in GaP, jo=t:.j= 400 MHz, and h=0.015 cm, one predicts 7J~8.8Pa (W), or about 0.71 mW/MHz. Thus
100% deflection (71=2.5) should be obtained over 400
MHz of bandwidth for P a.=285 mW, corresponding to
2-3 W of rf power with available transducers.
This particular prediction has not been checked in
detail, although preliminary results involving crude
bandwidth measurements, a focused acoustic beam
height of 0.019 cm,21 and a center frequency of 300
MHz show experimental values of 71/ P a about a factor
21 This value of h is conservative by the criterion used in deriving
Eq. (3), the modulator efficiency could theoretically be increased
about another order of magnitude by focusing the acoustic power
more strongly. The value of h=0.019 cm was chosen because this
value was available experimentally, but there appears to be no
serious impediment to somewhat stronger focusing.
JOURNAL OF APPLIED PHYSICS
MATERIALS
VOLUME 38, NUMBER 13
DECEMBER 1967
Microwave Rectification using Piezoelectric Quartz and Zinc Oxide
PAUL
H.
CARR
Air Force Cambridge Research Laboratories, Office of Aerospace Research, L. G. Hanscom Field, Bedford, Massachusetts
AND
ANDREW J. SLOBODNIK, JR.
Air Force Cambridge Research Laboratories, Office of Aerospace Research, L. G. Hanscom Field, Bedford, Massachusetts, and Department of
Electrical Engineering,* Massachusetts InstitlJte of Technology, Cambridge, Massac11usetts
(Received 7 April 1967; in final form 12 July 1967)
A theoretical and experimental study was made of the rectification of microwave energy by a thin piezoelectric disk placed in a "-'3-GHz reentrant cavity. Using a one-dimensional approximation, an equivalent
circuit consisting·of a frequency-dependent voltage source in series with a capacitance was derived for the
rectified output. The voltage is the sum of a coherent part, proportional to the nonlinear coefficients of the
constitutive relations and to the Maxwell-Faraday stress, and a thermal part, proportional (for quartz)
to the coefficient of thermal expansion. The risetime for the coherent voltage at an overtone acoustic
resonance frequency was about 1 ILSeC, while that for the thermal heating was of the order of milliseconds.
The rectified voltage is proportional to the microwave power and is typically mV/W when polished X-cut
disks are excited at acoustic resonance frequencies and one to two orders of magnitude smaller at other
frequencies. No acoustic resonance was observed for C-cut zinc oxide, and a nearly frequency-independent
value of 20 mV/W was measured.
1. INTRODUCTION
This paper contains a theoretical and experimental
treatment of the rectification of microwave energy with
a thin piezoelectric disk. When a sinusoidally varying
excitation is applied to a solid, nonlinearities can produce both second harmonic and static terms. Previously,
second harmonic generation has been used to measure
the nonlinear or third-order elastic coefficients of
solids.t-3 Buck and Thompson4 have suggested the
possibility of using the static term for measuring these
coefficients.
* Part of this paper is based on a thesis submitted in partial
fulfillment of the degree of Master of Science in the Department of
Electrical Engineering at MIT on 26 May 1966.
1 M. A. Breazeale and J. Ford, J. App!. Phys. 36, 3486 (1965).
2 A. Hikata and C. Elbaum, Phys. Rev. 144, 469 (1966).
3 P. H. Carr, IEEE Trans. Sonics and Ultrasonics SU-13, 103
(1966) .
4 O. Buck and D. O. Thompson, J. Metals 17, 1022 (1965).
Our detection of the static term was made by measuring the static voltage that accompanies the static
strain in a piezoelectric disk. The theoretical treatment
includes the electric field and strain dependence of the
elastic, piezoelectric, and dielectric coefficients; the
Maxwell-Faraday stress; and the thermalization of the
energy. Quartz was chosen as a test case since the
third-order elastic coefficients have recently been measured,5 and some measurements of the other nonlinearities have also been made. 6
The experimental apparatus and method are briefly
described in the next section and the theory, which is
worked out in detail in Appendix A, is summarized in
Sec. III. The experimental results are described and
compared with theory in Sec. IV, and conclusions are
:; R. N. Thurston, H. J. McSkimin, and P. Andreatch, Jr., J.
App!. Phys. 37, 267 (1966).
6 W. G. Cady, Piezoelectricity, (Dover Publications, Inc., New
York, 1964) 2nd ed., pp. 220, 700.
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