Optical Misalignment Due to Temperature Gradientsin Electrooptic Modulator Crystals

Claire Loscoe and Herbert Mette

A possible source of misalignment of light in an optical communication system, utilizing electroopticmodulators, is the deflection of the light beam within the modulator crystal due to temperature gradients.The present paper investigates the light deflection resulting from a linear temperature gradient acrossvarious modulator crystals. The total effect is found to be the sum of two contributions, one due to thecrystal expansion, the other due to the index of refraction gradient, and is found to be smaller, by thefactor 10, in quartz than in KDP. The experimental method described provides also a simple way ofdetermining dn/dT in crystals for both ordinary and extraordinary rays.

Introduction

Electrooptic modulators are usually designed for usein point-to-point communication systems using coherentlight of very low beam spread (from lasers) as the car-rier. It is, therefore, important to maintain extremelyaccurate alignment of the light beam within the mod-ulator crystal in order to avoid fluctuations of the lightbeam around the target. One major source of mis-alignment that must be considered is temperaturegradients within the modulator crystal.'- 3 Such tem-perature gradients could result from the rf heating,which usually is very strong in order to achieve highmodulation indices, uneven cooling of the crystals, andalso from absorption of the strong laser light.

The present research describes the light deflectionresulting from linear temperature gradients in rightparallelepiped shaped crystals. In addition to deflec-tion, the light beam is further affected by the separationof the ordinary and extraordinary rays even for the casein which originally the optic axis is in the plane of inci-dence and parallel to the crystal surface. The experi-mental method described provides a means for meas-uring, without high precision refractometry, dn/dT forboth ordinary and extraordinary rays in crystals.

Analysis of Light Beam Deflection Effects

When a light beam passes through a crystal in whichthere exists a constant temperature gradient dT/drperpendicular to the propagation direction of the light,the direction of the light will be, in general, changed bya certain angle, a'. There have to be considered two

The authors are with the U.S. Army Electronics Command,Fort Monmouth, New Jersey.

Received 8 April 1965.

contributions to this effect: a part due to the expan-sion coefficient of the crystal which converts a previ-ously right parallelepiped into a prism, and a part dueto the change in the index of refraction with tempera-ture. First we derive an expression for the angle ofdeviation, a, for the case in which only expansion takesplace (Fig. 1). Initially the light is incident normal tothe crystal surface. However, when the gradientdT/dr is established, the crystal expands at its warmerend forming a prismlike shape. The angle of incidencenow differs from normal incidence by the angle 6, whichis equal to the angle between the old and new positionsof the sides of the crystal. For small , the followingrelation exists between the length of the crystal, S, thethermal expansion coefficient E = (dS/dT) (1/S), andthe temperature gradient dT/dr,

1 dS 1 dS dTa tana = -- = - -2 dr 2 dT dr'

1 dT5 = Se - .2 dr

(1)

From Snell's law, y = /n and a + = qpn, the geo-metrical relation so = 2 -y, and Eq. (1), it follows thata = (n - 1)SE(dT/dr), where n is the index of refractionof the crystal for the propagation direction of the lightconsidered.

Figure 2 shows the light path for the case in whichthere exists, in addition to the thermal expansion, also agradient in index of refraction along the crystal. If a'is the angle of deviation of the light beam after leavingthe crystal, a is again the angle of incidence, y and so theangles made by the normals and the tangents to thecurved path at the point of incidence and emergence,respectively, and r is the radius of curvature of the ray,it can be derived from Fig. 3 that

so = 2 -- c - , (2)

January 1966 / Vol. 5, No. 1 / APPLIED OPTICS 93

Fig. 1. Deflection of light beam in crystal due to thermalexpansion.

Fig. 2. Deflection of light beam in crystal due to thermalexpansion and refraction index gradient.

Since

a' + a = n-p; y = a/n; w = ( + As)/r - s/r,

a '= 28 (n - 1) - cn.

Thus

dT Sa' = S(n - l)e - n.

dr r

constant is determined both by the expansion coeffi-cient, E, and the refraction index gradient, dn/dT, withboth effects adding. If, however, as in most cases, theexpansion coefficient is positive and the refractiongradient negative, the two effects will partially canceland the net light deflection will be either positive ornegative, depending on whether the quantity e (n - 1) ordn/dT is larger. Equation (4) was derived for amaterial with a single, homogeneous refraction indexonly. It is, therefore, valid for both the ordinary andfor the extraordinary ray if the proper value of n for thecrystal direction in which the light travels is used. Wehave determined experimentally the angle a' in severalcrystals of quartz and KDP, and have observed bothpositive and negative values depending on the directionof light propagation, polarization, and crystal orienta-tion used.

ApparatusThe angle of light deflection, a', as a result of a ver-

tical temperature gradient dT/dr in an electroopticcrystal, was measured in the following simple experi-mental arrangement (Fig. 4). A telescope, with a smallaperture of the size of the crystal cross section understudy, was mounted in a fixed position on an opticalbench and focused on a distant target which was il-luminated with a tungsten lamp. The crystal underinvestigation was mounted between two brass blocks infront of the telescope. The upper of the two blocks,which were held together by a set of nylon screws, washeated by an attached wire resistor, while the lower oneserved as a heat sink. The size of the linear tempera-ture gradient established throughout the crostal wasmeasured by a pair of 50-A chromel-alumel thermo-couples that were cemented to the side of the crystal.When the illuminated target was then viewed throughthe telescope, establishment of a temperature gradientin the crystal resulted in an apparent upward or down-ward shift of the target scale. The deflection angle,

(3)

Equation (3) can be further simplified by applying therelation4 for the radius of curvature of a light beam ina medium having a linear refraction index gradient,1/r =- [(l/n) (dn/dr) ] and substituting

a' = (dT/dr) [(n - 1) + (dn/dT)]. (4)

Equation (4) for the light deflection of the originalbeam can now be interpreted as follows. The angle a'is proportional, as expected, to the temperature gradientand the length of the crystal, S. The proportionality

Fig. 3. Geometry for derivation of equation for light deflectionin crystals.

94 APPLIED OPTICS / Vol. 5, No. 1 / January 1966

Fig. 4. Experimental arrangement for measuring effects oftemperature gradient on light path in electrooptic material.

Table I. Experimental Values of cz'/[S(dT/dr)] for quartz andKDP.

Direction Directionof of light a'/[S(dT/dr)] X 10' deg

Material gradient ray Ordinary Extraordinary

Quartz X Y +2.59 +0.954(z cut) X Z -2.33Quartz Y X +2.22 +0.579(zcut) Y Z -2.06Quartz Z X +2.66 +1.33(z cut) Z Y +2.43 +0.605Quartz AT X +2.69 +0.96(AT cut)KDP Y X -34.2 -19.2

a', was then determined by observing the apparentshift of this scale with respect to the cross hair in thetelescope objective and dividing by the distance, d.Typical temperature gradients were 300/cm for quartz,150 /cm for KDP; lengths of the crystals were 2 cmfor quartz, 1 cm for KDP, and the distance d was 138 m.Typical deflection angles, a', were 1.15 X 10-4 rad forquartz and 5.80 X 10- rad for KDP.

During the measurements it became apparent that,upon the establishment of a temperature gradient, theimage of the illuminated target as viewed through thetelescope was split into two, corresponding to the ordi-nary and extraordinary rays even in cases where uponnormal incidence no beam split separation was observed.When a rotatable polarizing prism was placed in frontof the electrooptic cell, each image could, however, beviewed separately, and a different angle, a ', or a wasmeasured, corresponding to the ordinary and extra-ordinary ray.

ResultsIn Table I is listed the quantity a'/ [S(dT/dr)] -

E(n -1) + (dn/dT) as measured for quartz (all sampleswere natural quartz) and KDP (potassium dihydrogenphosphate) in various directions of the light and tem-perature gradient and polarizations of the incidentlight. In the present arrangement reading accuracy ofthe angle a' and, therefore, determination of dn/dT canbe considered for quartz to only two significant figuresand for KDP to three significant figures. Greateraccuracies require use of more refined optical systems.All data are valid for an average crystal temperature of350 C. As noticed, there exists in some cases a con-siderable difference in a' for the ordinary and the ex-traordinary ray, indicating that dn/dT is usually consid-erably different for the ordinary than for the extraordi-

nary ray even if n, differs from n, only by a small frac-tion.

For quartz in most cases ae' per unit gradient per unitlength is positive showing that the expansion term con-stitutes the larger contribution to the effect. However,for the case in which the light travels along the opticaxis the deflection angle is negative indicating that thedn/dT effect is predominant. In KDP, the effect isten times larger than in quartz and is negative, indicat-ing that the dn/dT term is the dominant one for thesame orientation of light ray and gradient for which inquartz the expansion term is dominant.

ConclusionsThe deflection of a light beam on passing through an

optical medium in which a linear temperature gradientis present was found to be caused by two mechanisms,i.e., expansion of the crystal and light bending in thecrystal in a refraction index gradient. The existence ofsuch linear temperature gradients is, of course, onlyapproximately true for electrooptic light modulatorcrystals when subject to rf heating by the modulatingfield, but it is realized to some degree when the crystal isin thermal contact with a metal rod, as is the case inmost traveling wave modulators. The effect of aradial temperature gradient on signal degradation hasbeen investigated in detail by Kaminow.3

The light deflection angles obtained from our datafor both electrooptic materials appear small for prac-tical temperature gradients of 100/cm to 500/cm thatmay be reached during intense rf heating and, therefore,give little cause for fluctuations or deflections of thelight beam during operation. However, since the effectis also proportional to the length of the crystal or aseries of crystal segments, such as used in traveling waveor strip line structures, light deflection may becomelarge particularly in KDP, and care therefore has to betaken in the design of modulators to reduce temperaturegradients as much as possible. Of equal seriousness isthe effect of the separation of the ordinary from theextraordinary beam direction. Since, in most electro-optic modulators, the information content of the lightbeam is contained in the degree and the state of ellipticalpolarization of the light leaving the modulator, elimina-tion of either the ordinary or the extraordinary ray, oreven their relative change in intensity, would result in aconsiderable degradation (or demodulation) of themodulated light beam.

Equation (4) can also be used to determine dn/dT,when the expansion coefficient e is known or can bemeasured more easily, by direct measurement of a smallangle by means of a telescope. This eliminates theneed for precision measurements, to one part per mil-lion, by refractometric methods, at various tempera-tures.

For example, we can test our method for quartz forwhich and dn/dT are known. With the temperaturegradient in the x direction and the light ray in the ydirection we substitute e = 14.8 X 10- 6 /deg (ex-pansion coefficient perpendicular to the optic axis) andour experimental values from Table I into Eq. (4). We

January 1966 / Vol. 5, No. 1 / APPLIED OPTICS 95

find for the ordinary ray for which n = 1.544 anda'/[S(dT/dr)] = + 2.59 X 10-6/deg that dn0/dT =-5.48 X 10-5/deg. For the extraordinary ray forwhich5 ne = 1.553 and ac'/[S(dT/dr)] = + 0.954 X10- 6 /deg we find that dn,/dT = -7.22 X 10-6 /deg.These calculated values are in good agreement withliterature values obtained by direct measurements6

which for light of wavelength 589 mg are dn,/dT =-5.39 X 10- 6 /deg and dn,/dT = -6.42 X 10-6 /deg.

The authors appreciate theduring the measurements.

help of Ralph Binder

References1. Quart. Rept. 1-6 prepared by Texas Instruments, Inc., under

U.S. Army Electronics Command Contract No. DA 36-039AMC-03250(E) (July 1963-Dec. 1964).

2. Tech. Rept. AFAL-TR-64-317 prepared by Sylvania ElectricProducts, Inc., under USAF No. AF33(657)-11383 (Feb.1965).

3. I. P. Kaminow, Appl. Opt. 3, 511 (1964).4. R. Pohl, Optik (Springer-Verlag, Berlin, 1948), 7th and 8th

eds., p. 196.5. W. G. Cady, Piezoelectricity (McGraw-Hill, New York, 1946),

pp. 412, 723.6. International Critical Tables (McGraw-Hill, New York, 1926),

Vol. 6, p. 34t.

Meeting Reports continued from page 92

Z. Sekara UCLA.

This reviewer's over-all impression of ICES-II is likely biased,for he quails before overpowering mathematics. But it is-andit has been strengthened by a quick review of the proceedings ofICES-I-that very little new and exciting has happened betweenthe two meetings, except perhaps the more powerful formulationtechniques developed by Vanmassenhove and Grosjean, the inter-sect concepts of Porod, and the work on distributions by Schmidt.In addition to the well-received tutorial papers, a good manyof ICES-II's papers were largely reviews. Several fields coveredin ICES-I were omitted in this meeting (notably light scatteringin dilute polymer solutions) presumably because of lack of majordevelopments in the interim. A few novel experiments weredescribed, as were a few new mathematical techniques and a fewadditional solutions to the Mie equations (not even tabulatednowadays, but computed on demand and then thrown away).But by and large one got the impression of much tidying-up andpolishing in a well-worked field.

Sixteenth Conference on Analytical Chemistryand Applied Spectroscopy, 1-5 March 1965,Pittsburgh, Pa.Reported by J. A. Feldman, Duquesne University

Again, the Pittsburgh Conference continues to grow in bothquantity and quality of the instrumentation presented.

There is continuation in the trend to miniaturize componentsand to adhere to the modular or building-block principle in thedesign and manufacture of sophisticated versions of proven in-struments. Techniques and equipment, formerly used as analy-tical tools, have been adapted for the preparation of sizablequantities of substances.

The push for speed and accuracy is being continued in thedevelopment of newer techniques and in newer applications ofknown techniques. Routine use of such techniques and instru-ments as neutron activation analysis, direct reading vacuumemission spectrometers, continuous analyzers, and the applica-tion of computer technology for the analysis of data have takenplace.

The advance in instrumentation used in absorption spectroscopycontinues. There are available commercial instruments forroutine work which allow the spectroscopist to scan the wave-length range for about 160 mtz in the ultraviolet to about50 ,u in the far infrared with perhaps only three changes ofinstruments. However, equipment continues to push back thelimitation on range with infrared instrumentation having beendeveloped to operate out to a wavelength of 70 1A, by means ofinterferometric techniques to 2000 , and now an instrumentcovering the X-band region of 8000-12,000 Mc/sec. Mentionshould also be made of spectrophotometric equipment beingavailable for making rapid reaction studies with reaction ratesbeing measurable to times as short as 5 msec.

From the rapid development of more sophisticated as well ascomplicated analytical instrumentation, it can readily be in-terpreted that the scientists of today and of the future must havea more complex background fully to utilize these tools. There-fore, there is need for programs that will allow the present scientistto bring his knowledge up to date either through self-study orthrough continuing education programs and that the scientist ofthe future will be prepared for the many fields, both practicaland theoretical, that were represented at the 1965 PittsburghConference.

Twelfth International Spectroscopy Colloquium,Exeter, 12-17 July 1965

Reported by E. H. S. van Someren, Cambridge, U. K.

This colloquium was held at the University of Exeter and or-ganized under the general supervision of the Institute of Physicsand the Physical Society. In reporting it I wish to dwell on theways in which it differed from other colloquia in the series, ratherthan to try to relate what was said. There were about 600 peoplepresent, from 26 countries; rather less than half of the total wereBritish, about a score were Russian; one-sixth were women, onlya few of whom were interested in the lectures. The Colloquiumwas residential in plan, i.e., about 500 participants lived inhalf-a-dozen of the Halls of Residence of the University, usedtheir lecture theatres for meetings, the physics building for anexhibition of apparatus, and the dining hall for a banquet.Because of this concentration in space the opportunities formeeting and talking with old friends, and for making new ones,were at a maximum. The organizing committee had plannedeven more time for informal talk, but the large number of papersoffered, and the wish to minimize parallel sessions, made it neces-sary to continue the working day until 6 p.m. Parallel sessionsextended from the morning coffee-break to the end of the day.(About half of the papers submitted to the Colloquium had to berejected by the technical committee which planned the program.)

Another feature of the Colloquium was simultaneous transla-tion into French, German, and Russian of the main papers of the

continued on page 125

96 APPLIED OPTICS / Vol. 5, No. 1 / January 1966