PROCEEDINGS LETTERS 1751

Fig. I . Upper trace: transmltted mlcrowake pattern: vertical scale: 5 mV per dlvlslon. Lower trace: applied voltage pulse: vertical scale: 40 V per division. Horizontal scale: 0.5 0 s per dikislon. Microwave uavelength: 8.1 mm. Incident CW pouer: 6 mW.

microwaves. The voltages needed for modulation were very high (about 4 kV for complete transparency), as their slabs were about 1 cm long.

To reduce the voltages needed, a modulator for Q-band microwaves using short chips of both n- and p-type germanium of resistivity between 4.6 and 6.2 R'cm was constructed in this laboratory. The modulator con- sisted of a Q-band waveguide section with a ridge 0.1 inch wide in the center, the distance between the ridge and the top of the waveguide being 0.1 cm. The transitions before and after the ridge were accomplished by tapered sections down to the bottom of the waveguide. A chip of Ge sat in a slight depression on the top of the ridge; the chips used were about 0.1 by 0.1 cm in cross section. Contacts were made to the chip by evaporat- ing Sn onto opposite ends. With the chip in the depression, pulses could be applied to the top Sn contact by a spring-loaded screw pressing down on the top of the chip through an insulated hole in the waveguide roof. The pulses were applied via a miniature Conhex connector soldered to the screw contact.

Pulses of 0.5 to 2 p duration were applied to the chip at a rate of 50 per second from a Spencer-Kennedy type 503A fast-rise pulse generator, with which pulse amplitudes of up to 150 volts were available. Two n-type Ge chips of lengths 0.175 and 0.122 cm and one p-type Ge chip of length 0.115 cm were used. The VSWRs due to the reflection losses from their front surfaces were 1.33, 1.38, and 1.15, and their standing absorptions were 3.8, 5.6, and 7.25 dB, respectively. When pulses were applied to a chip in the modulator, the observed transmitted microwave power through the chip increased during the pulses. This increase became larger as the applied voltage was increased. The transmitted microwave pattern for the shorter n-type Ge chip is shown together with the applied pulse in Fig. 1. It was found that the transmission pattern varied somewhat with the setting of an E-H tuner before the modulator. This is thought to be due to phase changes of the transmitted microwaves arising from the change in carrier concentration as injected carriers from the contacts pass through the chip in the waveguide. The risetime of the modulator with each chip was of the order of hundreds of nanoseconds, and at best about 100 11s with the shorter n-type Ge chip as shown in Fig. 1.

It was found that up to the highest fields applied, which were about 1 kV km, the increase in microwave transmission increased monotonically with the applied field. The increase was fairly linear for the longer n-type and the p-type Ge chips but not for the shorter n-type Ge chip, the slope of the microwave transmission increase against applied voltage curve being greater at lower voltages. The values of microwave conductivity at 1 kV/cm were calculated from the graphs and normalized with respect to the zero field value. The values of normalized microwave conductivity were 0.75 and 0.675 for the long and short n-type Ge chips, and 0.80 for the p-type Ge chip, respectively. These values are comparable to previously obtained values of normalized microwave conductivity at 1 k V / ~ m . ~

Thus the possibility of decreasing the required voltages for hot carrier modulators by decreasing the semiconductor length has been demon- strated. In the case of the short chips, the required voltages were decreased

Solid-Srare Necrronics, vol. 11, pp. 2 W 2 1 8 . February 1968. B. R. Nag "Hot-carrier mkcrowave conductlvlty of elemental semiconductors,"

by an order of magnitude less than for previous that by careful engineering. the chip length coulc of magnitude further, so that about 40 to 50 \ complete transparency. This would make hot practical.

ACKNOWLEDGMENT The author wishes to thank Drs. D. Walsh a

tinuing interest and advice.

modulators. It is believed i be decreased by an order ~olts would be needed for carrier modulators more

.nd H. Motz for their con-

B. T. G. TAN' Dept. of Engrg. Sci.

University of Oxford Oxford, England

Currently with the Department of Physics, University of Singapore, Singapore IO

Temperature Dependence of Resonant Frequencies of LiNb03 Plate Resonators

A bstract--The temperature d c i e n t of the resouant frequeucy and the coupling constant of the Lam, single crystal were f d for varim crys- tallographic orientatiors from plate-shaped resonators in tbe thickness vibrational mode. The temperature coefficient varied between 7.1 and 9.3 x in any crystal orientation.

In recent years extensive studies on the piezoelectric properties of LiNbO, single crystals have been made by several authors. Yamada et al. reported the experiment on bar-shaped samples in length extensional modes of vibration,' while Warner et al. investigated the properties for plate-shaped samples in thickness extensional and thickness shear modes.2 The temperature dependence of the parameters, however, has been re- ported only for bar-shaped samples in length modes in the resonant fre- quency range of a few hundred kHz. In the present experiment. the electro- mechanical coupling constant and the variation of the resonant frequency against the temperature were measured as functions of the orientation of the crystal for the plate-shaped LiNbO, resonator in the thickness vibra- tional mode.

Plate-shaped samples of various crystallographic orientations were cut from the single crystal grown by the Chockralski method. The samples were nearly square, with a side of about 0.5 cm and a thickness of 50 to 250 microns. Gold electrodes were evaporated on the major surfaces of the plate, so the sample was to be excited by the perpendicular field.

The effective electromechanical coupling constant k is defined, accord- ing to Warner et as

k = [X.tan XI1

x = 4 2 ' &if,

where f, is the resonant frequency and f, the antiresonant frequency. The effective electromechanical coupling constant was obtained by measuring the resonant frequency and its higher harmonic^.^ The value thus found is plotted in Fig. 1. The curves in Fig. l(c) are in good agreement with the data reported by Warner et al.' except at the vicinity of k20' from the Z-cut. These disagreements would be due to the difficulty of the observa- tion of weak resonances.

The temperature coefficient of the resonant frequency was measured with the sample put in a Gebriider Haarke Ultrathermostat for a tempera- ture range of 5" to 95'C. The variation of the resonant frequency with the

Manuscript received April 15, 1968; revised July 8, 1968. ' T. Yamada, N. Niizeki, and H. Toyoda, "Piezoelectric and elastic properties of lithium

niobate single crystals," Japan 1. Appl. Phys., vol. 6, pp. 151-155, February 1967. ' A. W. Warner, M. Onoe, and G. A. Coquin, "Determination of elastic and piezoelec-

tric constants for crystals in class (3m)," 1. Acousr. SOC. Am., vol. 42. pp. 1223-1231. Decem- ber 1967.

' M. Onoe, H. F. Tiersten, and A. H. Meitzler. "Shift in the location of resonant fre- quencies caused by large electromechanical coupling in thickness-mode resonators." J. Acousr. SOC. Am., vol. 35, pp. 3 6 4 2 , January 1963.

1752 PROCEEDINGS OF THE IEEE. OCTOBER 1968

I- Y- CUT X-CUT Y-CL I I I *

ANGLE FROM X-AXIS

X-CUT Z

-TRANSVERSE MODE +LONGITUDINAL MODE

j o t \

0 ' 30' 60' ANGLE FROM X-AXIS

0 \

I 0 '

I 30' 60'

ANGLE FROM X-AXIS

UT

Z-CUT Y-CUT Z- CUT

ANGLE FROM Y-AXIS Fig. I . Experimental values of effectme coupling constants k for LiNbO, c r y s t a l s , with

plates rotated around (a) Z-axis, (b) Y-axis, and (c) X-axis.

--j Y-CUT X-CUT Y- CUT

5 4

-30' 30' ANGLE FROM X-AXIS

X-CUT Z- CUT x io" I I

10 ( b )

-TRANSVERSE MODE -LONGITUDINAL MODE 1

01 I 0 '

I 30' 60' 90'

ANGLE FROM X-AXIS

w E

5- 3 + w a 2

I 1 I I I -60' -30' 0' 30' 60' 90'

ANGLE FROM Y-AXIS Fig. 2. Experimental values of temperature coefficients of resonant freweIICieS for LiNbO,

plate resonators, with plates rotated around (a) Z-axis, (b) Y-axis, and (c) X-axis.

PROCEEDINGS LETTERS 1753

temperature was linear in this temperature range. The result is plotted in Fig. 2. The values of the temperature coefficients lie between 7.1 and 9.3 x T-' for any orientation. It is noted that the X-cut plate generates a shear wave with the largest effective coupling constant, while the temper- ature coefficient of its resonant frequency is larger than that of most of the other cuts.

ACKNOWLEDGME~T The authors wish to express their appreciation for the technical assist-

AKIRA FUKUMOTO AKINORI WATANABE

Central Research Lab. Matsushita Elec. Industrial Co., Ltd.

Kadoma, Osaka, Japan

ance given by F. Muramatsu.

Comments on "Distortion and Crosstalk of Linearly Filtered,

order modulation terms in the distortion. This relationship was only touched upon in our paper' and, in view of Bucher's comment. a further discussion may be of interest. The quantity u used in the following para- graphs plays much the same role as Aup does in Bucher's original work.3

In our original notation,'

e(r) = Im log fiT)ej4c'-r'dr s: where + ( t ) is the input phase angle and Y ( T ) is a normalized form of the impulse response of the linear filter. The desired series for O(t) is

where

x , .

Angle-Modulated Signals" = 1 + 5 5 ( j u ) m

In Bedrosian and Rice's interesting paper,' the series expansion (1 1) m! is closely related to an expansion for dejdr previously ~ s e d ~ . ~ for dealing with the distortion of FM pulses passed through a linear network. In the work on which an earlier paper3 was based, the following expression was developed : The coefficient K. is related to Bucher's function K,(t) by

log M(u) = C(U) = : 5 ( j u ) m . m = l m!

K , ( t ) , From statistical t h e ~ r y , ~ the c,'s are related to the pm's in exactly the

K3U) ' K'W 1 0 . ' K2(t ) ~ same way as the cumulants of a probability distribution are related to its

K 2 ( t ) K,(Z) moments. The sequence of equations 1 -

2! 1

2!

, . . . . . . . .

' K m ( r ) K m - ~ ( t ) K m - z ( f ) L + ' ( r J __ ___ m! (m - I)! (m - 2)! m! ~

where K,( t ) are defined by Bucher's equation In the notation of Bedrosian and Rice,' (Aop)mIC,( t )=S~y(r)[+(t - r)]"ds. It turns out that for m z 1, (AW,)'"~ 'D,(t)/(m+ 1) is equivalent to their f,+ J(m+ I)! . ' It might be that in the case of higher-order terms, it would be easier to write down the result from the determinant above than from their recursion formula.' This is doubtless an academic point since the higher-order terms are un- likely to find much use.

T. T. N. BUCHER Radio Corporation ok America

Princeton, N. J.

Author's Reply4 We wish to thank Bucher for bringing to our attention the interesting

work done by himself3 and Ditl.' His series for the time-dependent por- tion O ( r ) of the output phase angle serves to bring out the relation between the cumulants (or "semi-invariants") used by statisticians and the higher-

- + o + o + ' " = - C' PI

O ! O !

- P'25+!51+++ P3 2! O! l ! l ! 2! 2!

can be obtained by equating coefficients of powers ofju in the relation ob- tained by differentiating

with respect to u. The procedure used by us' to obtain the series for O ( t ) is somewhat

analogous to calculating the c,'s step by step from (4). On the other hand, the determinant for D,(r) in Bucher's series for O ( t ) may be obtained from (4) by replacing c,+ Jn!, p. by (Amp)"+ 'DJ t ) , (AoJ'KJr), respectively, dividing out Amp, and solving the first m + 1 equations for D,(t) by Cramer's rule. Kendall and Stuart5 give explicit expressions for the first ten cumu- lants in terms of the moments. These may be used to express Do(t), Dl(?), . ' , D9( t ) as polynomials in the K,,,(t)'s. In effect, this gives the ex- pansion of Bucher's determinant D,( t ) when m=O, 1, . . . ,9.

In the development just given, the coefficients c, in the series (1) for O ( t ) are computed in terms of the pm's. It turns out that K. given by (3) is analogous to the nth moment about 0. The development given in our original paper' is analogous to working with moments about the mean. There the quantity corresponding to the nth moment about the mean is

Manuscript received May 13, 1968: revised June 21, 1968. F, = j:dr)[c&f - T ) - @(t)]"'dr (6) I E. Bedrosian and S . 0. Rice, Proc. I€€€, vol. 56, pp. 2-13, January 1968 .

modulated systems], Horhfrequenzrech. und Elekrroukusr., vol. 65, p. 136 er seq., 1957.

Commun. ondElecrronics. vol. 46. pp. 1017-1022. January 1960.

A. Ditl. "Verzermngen in frequenz rnodulierten Systemen" [Distortions in frequency-

' T. T. N. Bucher, "Network response 10 transient frequency modulation inputs."

' Manuscript received July 8, 1968. Griffin and Co.. 1963. p. 70. M. G . Kendall and A. Stuart. The Adconced Theory of Srarisrics, vol. I . London: