Group Title: investigation of the linear electro-optic effect of epitaxial zinc sulfide films for integrated optics
Title: An investigation of the linear electro-optic effect of epitaxial zinc sulfide films for integrated optics
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Title: An investigation of the linear electro-optic effect of epitaxial zinc sulfide films for integrated optics
Physical Description: x, 227 leaves. : ill. ; 28 cm.
Language: English
Creator: Ebersole, John Franklin, 1946-
Copyright Date: 1974
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Subjects / Keywords: Electrooptics   ( lcsh )
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Statement of Responsibility: by John Franklin Ebersole.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 225-226.
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General Note: Vita.
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Full Text

AN INVESTIGATION OF THE LINEAR ELECTRO-OPTIC EFFECT OF EPITAXIAL

ZINC SULFIDE FILMS FOR INTEGRATED OPTICS














By

JOHN FRANKLIN EBERSOLE


















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF

THE UNIVERSITY OF FLORIDA IN PARTIAL

FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA

1974





Copyright



By



John Franklin Ebersole

1974














































































I


This is a triple-exposure photograph showing a 10X microscope

objective (to the left), a gallium arsenide rectangular substrate with a

semi-circular film of zinc sulfide, and a penny for size comparison.

Also shown is a red (X = 6328 A) laser beam in two positions. The upper

beam (in air) is visible due to scatter from surface dust particles on

the ZnS-GaAs sample as well as from smoke used to help indicate the

path of the beam. This beam (slightly off-axis with respect to the 10X

objective axis) represents light rays undeviated by the film. The lower

beam represents light refracted away from the undeviated beam according

to Snell's Law into the zinc sulfide when the film is placed at the focal

point of the objective. The beam is guided by the film and propagates

until it reaches the other end of the semi-circle, at which point the

light radiates brightly. That this lower beam is visible may be

attributed to surface irregularities as well as to internal scattering

from ZnS crystal imperfections.





To my wife, Ingrid





ACKNOWLEDGMENTS


I wish to express my appreciation to Professor Stanley S. Ballard,

Supervisory Committee Chairman, for his valuable professional advice

and guidance during my years as a graduate student. I am very grateful

to Itek Corporation for providing the opportunity, facilities, and

equipment for this research. Special appreciation is expressed to

Dr. Ralph E,. Aldrich, Dr. Richard H. Hudgin, and Dr. James C. Wyant for

their many helpful suggestions concerning the various phases of my re-

search and for their editing of this manuscript. An expression of

gratitude goes to Dr. Julius Feinleib who suggested the idea for this

research project. The continued encouragement and advice of Dr. Robert

Hills, Jr., and Dr. F. Dow Smith are gratefully acknowledged. The

technical discussions with Dr. Stephen G. Lipson, Mr. Donald W. Wilmot,

and Mr. Paul Vohl, the assistance of Mr. James B. McCallum with crystal

preparation, the skillful typing of Mrs. Kathleen Gerow, and the

excellent reproduction of this manuscript by Mr. Lawrence W. Gay and

his staff have all contributed immensely to the success of this research

effort. The support of Mr. Richard T. Turpin, Mr. Robert S. Hilbert,

Mr. Walter S. Crowell, Mr. William J. Davis, and Mr. Richard J. Wollensak

is greatly appreciated. Finally, I wish to acknowledge my former under-

graduate optics and physics professor, Dr. Roy C. Gunter, Jr., and my

former high school physics teacher, Mr. Harold F. Desmond; their

inspiration has in a very large way contributed to the achievement of

my scientific and educational goals.





PREFACE


On June 26, 1972 Dr. Stanley S. Ballard, Professor of Physics,

University of Florida, visited the Optical Systems Division of Itek

Corporation and met with this author and with Dr. Julius Feinleib,

Manager of the Physics Scientific Staff (part of Itek's Central

Research Laboratories). Dr. Feinleib described Itek's capability to

fabricate cubic-structure zinc sulfide films on gallium arsenide

substrates. As a result of that meeting, it was agreed that ZnS-on-

GaAs samples would be made available for electro-optical waveguide

experiments.

Subsequently, preliminary experiments and analyses were performed,

and a proposed line of research was submitted to the author's Ph.D.

supervisory committee. That proposal suggested both a theoretical

and an experimental research effort. The theoretical part was to

concentrate on the propagation of light in inhomogeneous waveguides.

The experimental part was to examine the electro-optic effect of zinc

sulfide waveguides. This dissertation presents the results of research

encompassing both aspects of the proposed research.





TABLE OF CONTENTS








Page


FRONTISPIECE .............................................. iii


ACKNOWL~EDGMENTS.................................... v

PREFACE............................................. vi

ABSTRACT ................................................ x


1. INTRODUCTION...................................... 1


II. BASIC PHYSICS OF HOMOGENEOUS ASYMMETRIC PLANAR OPTICAL

WAVEGUIDES . .. . . .. . .. . ., 2

Introduction .. . . . . . . . . . . 2

Vector Potentials .................. 2

TE Mdes............................................


TM Modes .. . . . . . . * * * - 9

TE + TM Modes . . . .. . . . . .. ..11


Ouasi-Geometrical Ray-Optics Approach.........,........ 12

III. BASIC PHYSICS OF INHOMOGENEOUS ASYMMETRIC PLANAR OPTICAL

WAVEGUIDES........................................... 17

Introduction .. . . . . . . . . . .17


Ouasi-Geometrical Ray-Optics Approach ..,,............. 17

Solutions for E and Hi....,......,,......~.........~. 26


IV. BASIC PHYSICS OF TOTAL INTERNAL REFRACTION IN INHOMOGENEOUS

PLANAR OPTICAL WAVEGUIDES... .............................. 29

Introduction............................... ........ 29


vii





Page

Quasi-Geometrical Ray-Optics Approach.................. 29


Solutions for E and H.................................. 40


1. The Linear Approximation...................... 40


2. TE Modes..................................... 42


3. TM Modes ................... ................... 48


V. GUIDING O~F LIGHT IN EPITAXIAL ZINC SULFIDE FILMS............ 54


Introduction.................................... 54


Total Internal Refraction.............................. 54


Partial Internal Reflection............................ 56


VI. THE LINEAR ELECTRO-OPTIC EFFECT OF EPITAXTAL ZINC


SULFIDE FILMS ........................................... 60


The Princival Axes..................................... 60


The Transverse Electro-Optic Effect.................... 63


Intensity Modulation for Propagation in the


(011) Direction ....................................... 65


1. TE Modes..................................... 65


2. TM Modes..................................... 66


3. TE + TM Modes................................. 66


Intensity Modulation for Propagation in the


(001) Direction.................................... 71


1. TE Modes..................................... 71


2. TM Modes..................................... 71


3. TE + TM Modes................................. 72


VII. EXPERIMENTAL PROCEDURE.................................. 74


Sample Preparation................................ 74

Experimental Set-up.................................. 77


viii





Page

VIII. EXPERIMENTAL RESULTS AND OBSERVATIONS......,............. 81


Light Guiding..,................................. 81

Electro-Optic Modulation.......................... 92

IX. SUMMARY.......................................... 108

APPENDIX A. APPLICATION OF MAXWELL'S EQUATIONS AND VECTOR

POTENTIALS TO HOMOGENEOUS AND INHOMOGENEOUS

PLANAR OPTICAL WJAVEGUIDES ................... ...... 110

Maxwell's Equations......................... 110

Vector Potentials........................... 113


Homogeneous Waveguides ................... ... 116

Inhomogeneous Waveguides ................... .122

APPENDIX B. HOMOGENEOUS ASYMMETRIC WAVEGUIDES................. 127

TE Modes.................. ................. 127

TM Modes ................... ................. 143

APPENDIX C. INHOMOGENEOUS ASYMETRIC WAVEGUIDES............... 157

TE Modes ................................... 157

TM Modes ................................... 179

APPENDIX D. TOTAL INTERNAL REFRACTION IN INHOMOGENEOUS

rJA.VEGUIDES................................. 193

Introduction.............................. 193

TE Mlodes................... ..........:...... 195

TMI Modes.................................... 210

APPENDIX E. THE PRINCIPAL AXES AND INDEXES OF CUBIC ZINC

SULFID)E ................ .................. .....,,220

LIST OF REFERENCES,.....,......,....~................... 225

BIOGRAPHICAL SKETCH...................... 227





Abstract of Dissertation Presented to the Graduate Council
of the Univiversity of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


AN INVESTIGATION OF THE LINEAR ELECTRO-OPTIC EFFECT OF EPITAXIAL
ZINC SULFIDE FILMS FOR INTEGRATED OPTICS

By

JOHN FR~ANKLIN EBERSOLE

March, 1974

Chairman: Stanley S. Ballard
Major Department: Physics


The basic physics of homogeneous asymmetric waveguides is reviewed,

allowing ready comparison of similar concepts between homogeneous and

inhomogeneous optical waveguides. It is shown that a quasi-geometrical

ray-optics approach leads to the same results obtained from more rigorous

mathematics, permitting a practical understanding of the phenomenon

of total internal refraction and/or reflection in general-index-gradient

planar optical waveguides. The chase change of w/2 (not 0 or n) occurring

upon total internal refraction is explained as being due to a cylindrical

focussing of the light within the guide. Wraveguiding and electro-optic

modulation of light in epitaxial zinc sulfide films on gallium arsenide

substrates are also discussed. Two mechanisms for light guiding at the

ZnS/GaAs boundary are described: total internal refraction or partial

internal reflection. From the discovery of a very thin, transparent,

low-index inhomogeneous layer of zinc sulfide near the boundary, it is

concluded that 1) 90-95% of the zinc sulfide thin film is homogeneous,

and 2) due to this thin inhomogenous laver, the zinc sulfide guide is

isolated from the gallium arsenide substrate: therefore the guide is

not a leaky one but instead is a true dielectric waveguide. Finally,

electro-optic measurements are presented which agree with those nre-

dicted from theory.





CHAPTER I

INTRODUCTION




This dissertation begins with a discussion of homogeneous waveguides

(even though they have been treated by other authors) and develops two

systematic means for understanding the phenomenon of total internal

refraction and/or reflection in inhomogeneous waveguides; a quasi-

geometrical ray-optics approach, and a more rigorous approach using

electric and magnetic vector potentials to solve Maxwell's equations.

Having done this, the theoretical and physical-insight discussions of the

inhomogeneous case should present no problems, for although the mathe-

matics are more complicated and more tedious, the approach is basically

the same as in the homogeneous case.

Because of the complexity of the mathematics in the theoretical

part, the long derivations have been relegated to several appendices.

Then, in the main body, summaries of the important results are

presented with physically based explanations, permitting a practical

understanding of the propagation of light in general-index-gradient

planar optical waveguides.

After homogeneous and inhomogeneous waveguides are discussed, the

results of waveguiding and electro-optic light modulation experiments

with epitaxial zinc sulfide films are presented.





CHAPTER II

BASIC PHYSICS OF HOMOGFNEOUS ASYMMETRIC PLANAR OPTICAL WAVEGUIDES



Introduction

A planar or slab (as opposed to a circular fiber) optical wave-

guide is one that confines light in one dimension only, as shown in

Figure II-1. Light propagating in the + 2 direction is guided or

confined by the two surfaces at x = 0, -W by total internal reflec-

tion. There is no confinement in the v direction. A homogeneous

optical waveguide is one that has an index of refraction n_ which is

constant throughout the guide. An asymmetric guide has a substrate

refractive index ns which is not equal to the superstrate index n ,

1.e., nS / n With no loss of generality, ns is assumed to be

greater than n i.e., ns > n A large number of optical waveguides

can be classified as homogeneous asymmetric planar waveguides.

Vector Potentials

In order to describe the propagation of light in optical

waveguides, the method of dual vector potentials, is reviewed in

Appendix A. Beginning, with Maxwell's equations, the wave equations

for E and F are obtained and the form of solution is show~n. Then,

for the case where no charge density or currents are present, it is

shown that both E and H can be expressed in terms of a magnetic vector

potential A and an electric vector potential F. The wave equations

for A and F for both the homogeneous and inhomogeneous cases are





I
i n,
x-O
~-Te:
w n4

x-w I~

/ i / nS/


u79 > ~15 ~ Y1,

t~ --h, ng s~e 8-









Figure II-i. Cross-sectional view of light propagating in a
homogeneous asymmetric planar optical wavegui~e.





~i~(~ Fl )i~-)+R~~ 1I? A O


~s a ~---5
V' F~ -~ /Ih nFO


II-(2)


Of course, for the homogeneous case, Vn2 = 0 in Eq. (1). Here

ko = 2w/X = m/c, ho is the free-space wavelength, w is the

angular frequency, and c is the speed of light in free space. The

following expressions result for E and H~:


E=~ ~TI-(3)


-J -_r -r
1~ ~KA `iL/9~o) V (~~


II-(4)


For the choice of potential


x


II-(5)


II-(6)


Ind z are Cartesian unit normals), it is shown that



x I


A A
(where x, v, a


I-(7)


Ax 3


II-(8)


f`_- iL~ai~o~_U -x,
'bzax



~ 23


II-(9>


II-(10)









a II (11)



t,-;/~ aa x II-(12)


For the choice



II-(13)



F'~ Fx II-(14)



hnEx = 0, Ez = 0, and Ry = 0), in which case, because there is no
A
z-component of E, a pure transverse electric (TE) mode of propagation

exists. For the choice



II-(15)


F =0
II-(16)



then Ey = 0, Ex = 0, Hz = 0, and a pure transverse magnetic (TM)

mode of propagation exists. It may be seen that a field of any

polarization in an asymmetric planar optical waveguide can be expressed

as the sum (i.e., the superposition) of a TE field and a TM field.

TE Modes

In Appendix B, homogeneous asymmetric waveguides are discussed in

in detail for both TE and TM modes. After an analysis involving boun-

dary conditions, simultaneous equations, and complex algebra, the

















































II- (25)


following solutions for E and H are obtained for TE modes:


11- (17)


C = x HY +z 2

:.t);ij;(t?


II- (18)


II- (19)


iRox (Xe wt 2)


(K,(x 2, t)


II- (20)


II-(21)


Here B is the wavevector component along the z-axis, i.e.,


,,,~ngSV1.B1 Cionstuct


II- (22)


and has a constant value in all three of the na, n and ns regions.
Eq (22) implies that the wave travels at the same velocity vz in all
three regions since vz = cB. Finally,


sFae


aF, e


Q9


CQO~~~ 9r ea


-Csr y20


II- (23)


t i~
"Y


c~oo Jgx + e6_~~) fWWL


Cba~(FqT~~m~e'(nr~~


II- (24)


-;9d
s


01


3








and, from Eqs. (8), (10), and (12), H (- i/k )aE /az and
x o y
Hz = (i/k BE /ax). The exact form of Ho and Ho is given in

Aptpendix B, Eqs. B-(83) through B-(86). These values for Eoy, Hox'
and H show that the wave is confined to the n region (the wave-
oz a

guide) and that the fields go to zero as x +t + m. The coefficient F2

is arbitrary, being determined, for instance, by how much light is

coupled into the guide. The remaining parameters as determined in

Appendix B are given below:



II-(26)



II-(27)



3 II-(28)



"=t~~' (/g)II-(29)



~;=t,' (sS) I-(30)



In the course of obtaining the solutions for ETE and HTE, two

useful TE equations were obtained:





~9'-n")''aII-(31)





where (W i)TF is the minimum waveguide thickness required in order to

propagate a mode of order m. Eq. (31) is called the characteristic

waveguide equation.

Eqs. (31) and (32) show that, depending on h0, na, n and us'

only certain modes can propagate in a film of thickness W. Further,

for ns # n there is a cutoff thickness, W ,' below which no guided

wave can be propagated, even for m = 0. It may be noted that E4. (31)

is a transcendental equation in O.

Fq. (32) is derived from E~q. (31) using the following argument.

In order for the waves to leak out or escape (i.e., for propagation

to cease), the waves must escape either into the na region or into

the ns region. For such a situation, at least one of the functions
-ax > sx <
e [for x = ] or e [for x =- W] in Eqs. (23) and (25) must no

longer be decaying exponentials (which are real functions) but must

instead be purely imaginary. From Eqs. (26) and (28), it may be seen
-ax sx
that when n si~ne < n or when n sine < n ,then e or e
g a g s)
respectively is imaginary. Thus when either



5~n~ ;n~nqII-(33)



or








then waves will begin to leak out. When sin0 = ns/n then 8 is at

the critical angle for total internal reflection at the n -n
g s

boundary. Since n > n total reflection still occurs at the n n

boundary. When sine < n /n plane waves escape into the ns medium,





and propagation ceases, i.e., the mode is cut off. Using Eq. (34)

in Eq. (31) results in Eq. (32).

Referring to Eq. (31), it may be seen that O can never reach the

value of n/2 (propagation straight down the guide), since in the

limit as 6 + r/2, W + m, which is physically unrealistic. Put

another way, no longer is there really any waveguiding as 6 +t w/2.

Combining this fact with Eq. (34) results in the relationship







as the range of values 6 for which waveguiding (without cutoff) can

occur, as determined by Eq. (31).

TM Modes

Appendix B3 also contains a derivation of E and H for TM modes

propagating in a homogeneous asymmetric waveguide. The solutions are


E = x E + z E,
11-(36)



II-(37)


Ex ix, z t) = Eo x (x~) ew-s
II-(38)



11-(39)



Hy 'i Z'ijr~Y 1) eII-(40)





Here B has the value given in Eq. (22) and is a constant. The x

dependence is given by





dA, e 3 -FwK~O II-(42)





795 ~w =-~ II-(43)



From Eqs. (7), (9), and (11), Ex = (i/k n )3H /9z and E =

(-i/k n )aH /ax. The exact form of Eo and Eo is given in Eqs. B-(157)

through B-(162). The coefficient A2 is arbitrary, as was the

coefficient F2 in the TE case. The parameters a, g, and s are given
TM TM
by Eqs. (26) (28). The quantities # and (p are given by
ag gs








II-(46)



-t-~ [~~i~s 1"4)]II-(47)



I n thy e oure of obtinn the solutions for ETM and HTM in






AppndxB the characterotistic wavegouid n uofeutions for the n H~Mi





TM case were obtained:



II-(48)







2 2
Eq. (49) is similar to Eq. (32) except for the factor n /n in the
g a
argument of the arctangent. From Eqs. (48) and (49), it may be seen

that the value of W for the TM case must be larger than W in the TE
TM TE
case from Eqs. (31) and (32), owing to the fact that 4 > 4 and
ag ag
TM TE
> 4 Finally, Eq. (35) applies equally well to TM modes for
gs gs
the allowable values of O for waveguiding without cutoff, as

determined by Eq. (48).

TE + TM Modes

As mentioned above and shown in more detail in Appendix A, a

field of any polarization in a homogeneous asymmetric planar guide can

be expressed as the sum (i.e., the superposition) of a TE field and a

TM field. Further, the components obtained from the TE derivation

[E Hx, and Hz] obey the waveguide and cutoff equations given by

Eqs. (31) and (32), whereas the components obtained from the TM

derivation [Ex, Ez' and H ] obey Eqs. (48) and (49). For propagation

of a TE + TM wave in a homogeneous guide of thickness W, the angle

O = OTE in Eq. (31) is not equal to the angle B = OTE in Eq. (48) due
TE TM TE TM
to the different phases and 4 As a consequence,
ag' ag gs' gs
the propagation constant a must be different for the TE and TM cases,





Mi-k o9 -8 II-(51)


Hence, as the TE + TM wave propagates down the guide according to

e-i the phase difference aB between the TE and the TM parts of the

TE + TM wave is



TE TN II- (52)







where L is the length of the guide. Thus a TE + TM wave, initially

linearly polarized, will become more or less elliptically polarized,

depending on OT and OTM [which are determined by Eqs. (31) and (48)]

and L, for a given ko and n .

Concerning the minimum thickness (W )necessary to
min TE + TM

propagate a given mode, the larger of (W l)T or (W )T [given by

Eqs. (32) and (49)] must be the minimum thickness. Owing to the
2 2
factor n /n in the argument of the arctangent of Eq. (49), (W)
g a min TM

is the larger (since ng > n ). Hence the cutoff thickness for TE + TN
modes is







Ouasi-Geometrical Ray-Optics Approach

It is possible to derive the waveguide equations using a quasi-

geometrical approach similar to one given by Tien and Ulrich.3 Referring

to Figure II-2, the plane wave propagating down the guide may be thought

of as a zig-zagging within the guide, plus an evanescent wave





Figure II-2. Geometry of a homogeneous optical waveguide.


X





propagating in the superstrate and in the substrate at the same

velocity in the z-direction as in the guide. This is consistent with

the statement made earlier in this chapter that the wavevector component

along the z-axis, 8, has a constant value k n sin0 in all three
o g
regions (guide, substrate and superstrate) for a TE or for a TM wave.

Physically, if this were not so, eventually a point would be reached

along the guide for which a wave within the guide would be

unaccompanied by an evanescent wave outside the guide -- clearly an

impossibility for a totally internally reflected guided wave.

The optical rays in Figure II-2 represent normals to the pro-

pagating, zig-zagging plane wavefronts within the guide. It may be

seen that the time it takes for the light to travel along the path

PQR must be equal to the time it takes for the light (the evanescent

wave) in the na region (or, equivalently, in the ns region) to

travel along the path PR. Another way of expressing this is to state

that the phase delay experienced by the light wave in the guide must

be equal (to within an even multiple of w) to the phase delay

experienced by the evanescent wave, i.e.,








the evanescent wave's phase delay is:








The phase delay in the guide has three contributions, one due

to the optical path along PQR, another due to the phase change





-20, resulting from total internal reflection from the ng n
boundary, and the third change -24gs due to total internal reflection
from the ng ns boundary:


Ci>QR) Ptny dCaq -~99J


j~h~e d~4),


II-(57)


Substituting Eqs. (56) and (58) into Eq. (55) produces:


II-(58)


~PP, ng W/cboB- a ~P~ ag4,_ amir


rarit~B=


II-(59)


Since 8 = k n sin0, Eq. (59) can be written as
o g








( al o s




or A ,co8 g9-aQ, d~-


II-(62)


II-(63)


Eq. (63) is identical with the characteristic waveguide equations
(31) and (48), providing that the correct values of (ag and Ogs are used.





According to Born and Wolf,4 the phase changes ~ and Qg for the TE

and TM cases of total internal reflection are identical to the values

given by Eqs. (29), (30), (44) and (45). Further, once the waveguide

equations are known, it is then possible to obtain the waveguide

cutoff equations for (W i)T and (W i) .' Finally, the extension to

TE + TM waves is the same as given above -- by the superposition of

TE and TM waves.

Physically, the waveguide equation (63) [i.e., Eqs. (31) or (48)]

represents the condition that, after two reflections (one at each

boundary), the wavefront is in phase with the original wavefront. If

this were not so, many wavefronts, each differing in phase by some amount,

might eventually add up somewhere in the guide such that they would

destructively interfere. By conservation of energy, the light would

then have to pass out into the na or ns regions, i.e., the light

could no longer propagate in the n_ medium.

It may be seen that a quasi-geometrical ray-optics approach can

be a useful way of treating wave propagation in an optical waveguide.

It provides a means for understanding the physics of the problem at

hand; at the same time it results in an accurate representation of

the homogeneous-waveguide equation.





CHAPTER III

BASIC PHYSICS OF INHOMOGENEOUS ASYMMETRIC PLANAR OPTICAL WrAVEGUIDES



Introduction

An inhomogeneous optical waveguide is one that has a variable

index of refraction n within the guide. The refractive index profile

treated in this dissertation is assumed to lie in the x-direction

only, i.e., transverse to the net direction of propagation. Further,

the index profile is assumed to be a monotonic function of x, such

that the value of the index is higher near the superstrate. That

is, as indicated in Figure III-1, the refractive index of the guide has

its maximum at the n n boundary and then continuously decreases
a g
in the negative x-direction. The index can be written as







where



nS (X)n9 o JIII-(2)



Ouasi-Geometrical Ray-Optics Approach

It is convenient to begin with a quasi-geometrical approach.

The justification of the results will be discussed afterwards.

Letting B still be the wavevector component along the z-axis, then 8

can be written as





X=-~


n,
-fw ~J,~


x=0


Figure III-1. Cross-sectional view of light propagating in an

inhomogeneous asymmetric planar optical waveguide.


A


A_
A


r=-w





o-~ ,x III)rr-(3)

In Appendix C, solutions were obtained to the wave equation assuming

that 8 was a constant independent of x for both TE and TM waves, i.e.,

as in the homogeneous case



e=""t"~III-(4)



oP III-(5)



where



1, ~G~a-cilcu~xIII- (6)



Eq. (4) implies that the wave travels at the same velocity vz in the

Direction in all three regions: guide, superstrate, and substrate.

Eq. (6) is simply another way of writing Snell's Law.

Referring to Figure III-1, it may be seen that as the ray travels

from point P to point Q, the ray is constantly bent away from the

normal to the n n boundary, just as predicted by Eq. (6). Upon
a g
reflection at the ne ns boundary, the ray travels from point q to

point R, this time being constantly refracted toward the normal. At

the n n boundary it is totally internally reflected once again,
a g
In order to obtain the characteristic waveguide equation for the

inhomogeneous case, the same argument given in the homogeneous case

is useful. That is, the time it takes for the light to travel along

the path PqR must be equal to the time it takes for the light







































































I


(an evanescent wave) in the na region to travel along P As

before, this can also be expressed in terms of phase delays:


jla e ~ )+ 9m ~ en i = os delaj III- (7)



The evanescent wave's phase delay is







The phase delay in the guide has three contributions: one due to the

changing optical path caused by the index gradient, Ik dr, where
E A A A A
r = xx + yy + zz (x, y, and z are unit normals): the other two are

due to the reflections at the n n and n ns boundaries: -24a
a g g a
and -20gs. The values of #ag and #gs can be obtained using Eqs.

II-(29) and II-(30) for the TE case and Eqs. II-(44) and II-(45) for

the TM case. Defining






then



III-(10)





a-3 III-(12)



a)hgsni /,l~i~ III-(13)





where



III-(14)



III-(15)



The phase delay in the guide can thus be written as







Further,
O





oa I (.x)- sc a III- (17)


The first integral on the right-hand side of Eq. (17) is due to the x-

component of the optical ray path along PQ. [The product k is

negative, hence the factor (-dx).] The second integral is due to the

x-component of the optical ray path along QR. [The product k r is

positive.] The third integral is due to the "z-component of the path along

PR. [The product~ k is positive also.] Since, by Eq. (6),

n (x)sin9(x) is a constant, Eq. (17) becomes







Thus, with the aid of Eqs. (8), (16), and (18), Eq. (7) becomes





_p -ao 5 dn97s III-(19)






o a III-(20)



Eq. (20) is thus the waveguide equation for asymmetric inhomogeneous

waveguides, as determined by the quasi-geometrical ray-optics approach.

It may be noted that when n (x) and cos0(x) are constants [as in the

homogeneous case], Eq. (20) reduces to Eqs. II-(31) or II-(48), the

waveguide equations for homogeneous waveguides.

Writing



III-(21)



then the integral in Eq. (20) can be transformed to



-YJ III-(22)


oP~ Srgi)O?]3 III-(23)



where the last step follows from Eq. (6). Thus Eq. (20) becomes





which makes it easy to directly incorporate the mathematical form of

the index gradient n (x). Sometimes the form of n (x) allows the

integral to be directly integrated analytically. Otherwise (such as

with an error function profile) it can be integrated numerically.

From Eqs. (10)-(15), it may be seen that Eq. (24) is transcendental in

both n and W. Writing


C Y) III-(25)




then, for a given value of n (0), perhaps the easiest way to solve

Eq. (24) is to pick a value for 9(0), then try values of W until one

is found which satisfies Eq. (24). By using different values of 0(0),

a set of curves can be generated for the different modes m. Also,
TM TE TM TE
since O > 4 and 4 then the value of W obtained from Eq.
ag ag gs gs)
(24) for the TM case is greater than W obtained for the TE case, as

in the homogeneous case.

For the cutoff condition, waves will begin to escape when either



S~M~j)- n/~,gIII-(26)



or



III-(27)



where



n~g' J(~)III-(28)



95 III-(29)





Using Eq. (6), Eqs. (26) and (27) are equivalent to the following

equations



IIT-(30)







respectively. It was assumed for homogenous waveguides that us > na'

Hence as 6 decreases from a maximum value of w/2 (i.e., propagation

straight down the guide), then, for a given functional form of n the

cutoff condition first to be met is given by Eq. (31). That is, waves

will start to leak out at the n ns boundary when sinO(x) = n /n (x)

[i.e., when sinO(-W) = s/n (-W), or equivalently, when sinO(0) = s/n (0)].

The waves will still be totally internally reflected at the n -n
a g
boundary. This result is identical with the result obtained for the

homogeneous case. Substituting Eq. (31) into Eqs. (10) (15) produces



III-(32)



8as III-(33)


T1 t2C / 9 ITI- (34)




Then at cutoff, Eq. (24) becomes for the TE case





and, for the TM case,



JII-(36)
s a

~~s a




n > n [since n (0) Z n (-W)]. Hence if n < n then the cutoff
s ag a
TE TM
condition is given by Eq. (30), in which case 0 4 Z 0,
gs gs
TE TM
= = 0, and Eq. (24) becomes
ag gs

~n 3r IJ 3 n ~jIII-(37)




and





It ay e eentht, ndedth susttutonof fr n and vice





[or Eq. (25)],n chan inevrxeed, the valtiue on (0) In fact n cannotc






equal n (0) either, as may be seen from the following argument. If
sin0(0) = 1, then 0(0) = w/2, and from Snell's Law (Eq. (6)]

O(x) = v/2 for all x. Hence the integrand in Eq. (20) would be zero

for all x, and in order to propagate a light wave, only an infinitely

thick (W + m) guide could satisfy Eq. (20), which is physically

unrealistic. Thus, using Fq. (25), the range of values of 9(0) for





waveguiding in the case n > n is given by the relationship
s a


5 III-(39)



and for the n > n case,
a s



,9-, r>MIII-(40)



Eqls. (39) and (40) apply to both TE and TM modes as determined by Eq.

(24). It may be seen that Eq. (39) is identical to Eq. II-(35) for

the homogeneous case.

Solutions for E and H

As in Appendix B, in Appendix C trial solutions are chosen for

the vector potentials F and A, only now a provision is made in the form

of the solution to allow for an index gradient within the guide.

Specifically, instead of the product gx in Eas. II-(24) and II-(42),

where g is a constant given by Eq. II-(27), the choice is g = g(x). The

function g(x)/k~ is often called the eikonal.5 Using the boundary
O
conditions on E and H together with the assumption that the index

in the guide is a slowly varying function of x, the following solutions

are obtained in ADoendix C for TE modes:



III-(41)



III-(42)



III-(43)





where Eqs. II-(17) through II-(21) apply equally well to the TE

inhomogeneous case. The exact form of Hox and Hoz is given in Eqs.

C-(135) through C-(138). It may be seen from E~q. (42) that both the

amplitude and phase of Eoy(x) are functions of x. The function g(x)
is given by



9(= i (3n11-O4

where q(x) is given by Eq. III-(9). The method used in Appendix C

for solving the wave equation for an inhomogeneous medium is called the

WKBJ method71 (after Wentzel, Kramers, Brillouin, and Jeffries). The

assumption that the guide has a slowly varying index profile leads to the

following requirement:






where q' and q": are the first and second derivatives of q with respect

to x. Eq. (45) is the WKBJ approximation for the TE case.10 It may

be seen that for large wavenumbers ko (i.e., small A0 the WKBJ

approximation is more easily satisfied. However, as q = n cos0
approaches zero (as n_ decreases and 6 approaches n/2), Eq. (45) cannot
be satisfied, regardless of how small q' and q" are. This situation

will be discussed in Chapter IV. When Eq. (45) is satisfied, all of

the TE results obtained from the quasi-geometric ray-optics approach

are valid, at least to the level of approximation given by Eq. (45).

In the homogeneous case q is a constant. Hence, the function g(x) in

Eq. (44) can be easily evaluated to give (koq)x = (kon cos)x, which





is the same as the product gx originally obtained in Chapter II, as

mentioned above.

For the TM case, the following solutions are obtained in Appendix

C:M


III-(46)







iThe exc form of and iE is gien inx~ Eq.n~ C-(24 thouh -(229).





contributios, n-(x) andug q -(x0), whera for the TE anhmpliteudecs


frthe eaTM cas ofel0 d is given byEs -24 truhC(2)





fist mand second deriatives ofliud n Thee siuton when Eq (49) is


antrisftied, all of the M result ob raine foro the qu aspigdeomti

merthod arIce va ida es otelvlo prxmto given byEq


(49).gI _t ~l (-1~- II(9





CHAPTER IV

BASIC PHYSICS OF TOTAL INTERNAL REFRACTION

IN INHOMOGENEOUS PLANAR OPTICAL WAVEGUIDES



Introduction

It was stated in Chapter III (and shown in Appendix C) that the

WKBJ solutions for E and H are valid as long as Eqs, III-(45) and

III-(49) for TE and TM modes resnectively are satisfied. Whnen either

Eq. III-(45) or III-(49) is not satisfied, it may be seen from Eqs.

III-(42), C-(135), III-(47), and C-(225) that, due to the factor q-/

at least two field components (Ey and Hx for the TE case, Fo and Ex

for the TM case) become infinite as q(x) = n (x) cos0(x) approaches zero

as O(x) +t w/2. Since this is physically unrealistic, some other

approach must be used to solve the wave equation near the level q = 0.

In the field of radio ionospheric physics, Buddenl1 and IWaitl2 have

developed a useful approach for describing the propagation of light

near q = 0. This method will be adapted to optical waveguides. Before

doing this, however, it is convenient to use quasi-geometrical arguments

to form a physically based explanation of the results obtained in the

more rigorous derivation. This permits a practical understanding of

the phenomenon of total internal refraction-and-reflection in general-

index-gradient planar optical waveguides.

Ouasi-Geometrical Ray-Optics Approach

First of all, it should be mentioned that -W is now chosen to be





the value of x for which O = w/2 (and q = 0) for a given mode m and

incident angle 0(0), as indicated in Figure IV-1. That is, at the

level x = -W, O(-W) = w/2, q(-W) = 0, and total internal refraction

occurs. It is convenient to let






where



Y}o) 0 o b IV- (2)



at the top of the guide, and



n (-D)= Yn Iv-(3)



where -D is the penetration depth of the index profile such that beyond

x = -D there is no longer an index gradient. For instance, no might

be the index of the film before ion implanation, and an the maximum

index change (where A~n << no) which is generally a measurable quantity.
The function b(x) thus gives the index profile, where


bio):= \ Iv-(4)



b leD) = IV an)





Figure IV-1.


Cross-sectional view of light propagating in an inhomo-
geneous planar optical waveguide by means of total internal
refraction and reflection.


k:


ng(x) > n, ~w all x
~giO),"g(-W)
si-~= R/a








As shown in Figure IV-1, after reflecting from the na n (0)

boundary, the wave is continually refracted away from the normal to

the boundary until, at the level x = -W (and O = w/2), the wave is

refracted back toward the normal as it returns toward the na n (0)

boundary. The phase change suffered by this wave has three contributions:

a) the integral of the changing optical path k dr over the path POR:

b) the phase change -24ag upon total internal reflection at the na n (0)
boundary: and c) the phase change upon total internal refraction. It is

the third contribution which is the more elusive quantity.

From one point of view, since the total internal refraction

phenomenon is purely refraction, no phase change might be expected.

On the other hand, since the light does in fact reverse its direction,

then a phase change of w might seem reasonable, since in the limit as

0 +~ -/2, the phase change 20ag due to total internal reflection approaches

n. However, neither of these points of view is correct. In fact,

the phase change is w/2, not 0 or v.

It is knownl3 that a phase change of n occurs when light passes

through the focus of a spherical lens; for a cylindrical lens, there is

a phase change of w/2. In Figure IV-2a, it may be seen that as a beam

of light is refracted at a boundary with an abrupt refractive index change,

the beam width decreases when n2 < n1. For a continuously varying

index, a similar phenomenon occurs, as shown in Figure IV-3. Here the

path of the ray is shown surrounded by a small beam width. Because the

waveguide is a planar slab (as opposed to a fiber), it may be seen that

in the region x = -W and (q = 0), the beam undergoes a cylindrical focus.

Hence, it is reasonable to expect a phase change of Ti/2 upon "reflection",

i.e., total internal refraction, at this level.





















Figure IV-2a























Figure IV-2b

















Figure IV-2. Refraction of light at a boundary with an abrupt change

of refractive index (Figure IV-2a) and in a medium with

a continuously varying index (Figure IV-2b).





x~O


Figure IV-3. Total internal refraction in a planar optical waveguide

showing the cylindrical focussing effect.





From these various phase contributions the characteristic wave-

guide equation is thus obtained:







which applies to both TE and TCM modes. [A second-order correction for

TM modes will be discussed later.] Comparison of Eq. (6) with Eq. III-

(20) shows that the only change is; to replace the term 20gs with n/2.
Eq. (6) can also be written as







similar to Eq. III-(24). Eqs. (6) and (7) are transcendental in n,

which is equivalent to saying that it is transcendental in 0(0),

according to Eq. III-(25). It may or may not be transcendental in W

too, depending on the functional form of n (x) and the integral in

Eq. (7). Eqs. (6) and (7) are useful for describing the propagation

of light in such optical waveguides and has general applicability to

any slowly varying monotonic index profile (including the error function

profile); it also allows computation of numerical answers. As before,
TM TE
since # the value of W for the TM case will exceed W for the
ag ag
TE case. It is interesting to note that, in the limit as 0(0) +t 1/2,

Eq. (7) approaches an equation for TE modes derived by Ma3rcuse.14

Following arguments given in Chapter III, the cutoff condition

occurs when either [from Eq. III-(30)]





V= n IV-(8)

or [similar to Eq. III-(31)]

V7 o IV- (9)

For >na (which is usually the case), the cutoff condition is given

by Eq. (9), i.e., when n7 = n When n < n the light can no longer
be totally internally refracted, and the light escapes below the guide.

For this cutoff condition, there is thus no w/2 phase change. For the
TE TM
exact form of the phase changes 20 and 20 q. (9) can be substituted
ag ag'
into Eqs. III-(10), III-(12), and III-(14). Further, from Eqs. III-(6)

and III-(9), q(x) can be written as



gl, I J l /1] IV-(10)



Using Eqs. (2), (9), and (10), then


0)=i tan Y~lo'/ Iv-(11)



where the (hn)2 term has been neglected compared to 2n An. Thus, at

cutoff the phases are


IV-(12)







Eq. IV-(7) then becomes an integral equation for :n


IV-(14)





where for (W )T Eq. (12) is to be used in Eq. (14), and for

(W )T Eq. (13) is to be used.

In the event that no < n a much less common situation [where

na < no + An, in order to insure total internal reflection at the

n n (0) boundary], then the cutoff condition is given when n7 = na


a g aa





which is the same for both TE and TM modes (except for the second-

order correction for the TM case, to be discussed).

As explained in Chapter HII, n can never exceed or equal the

value n (0). Thus the range of values of 0(0) for waveguiding without

cutoff in the case n > n is given by the relationship

Yl~o~ >1M> n, IV-(16)
and for n < n
o a

no tdnr r V c IV-(17)


Since 6(-W) = w/2, then from Eqs. (9) and III-(6), n = n (-W) = n.
g o

Referring to Eq. (1), this can be satisfied only when b(-W) = 0, which

from Eq. (5) shows that, at cutoff, W equals the maximum index gradient

penetration depth D.

It was mentioned above that there is a second-order correction to

the characteristic waveguide equation for TM modes. Physically, the

reason for this is as follows. In the TE case, the only component

of ETE is Ey ,which is transverse to the index gradient. Hence, as
a TE, wave propagates within the guide, the E-vector at any one point,

say xo, within the guide will "see" only one index -- n (xo) -- at a





time. For the TM case, however, since there is a component of E

along the index gradient (the Ex component), the situation becomes

more complicated. Thus when deriving the w/2 phase change for the

characteristic waveguide equation for the TM case, it was implicitly

assumed that the TM wave "sees" some sort of average index n (xo) at

at any given position xo on the ray path. As shown below, this

averaging takes place over some fraction of a wavelength of light,

say C1A(xo), where Cl is a constant to be determined and where A(x) =

Ao/n (x). Any shift 6W in the level of total internal refraction at
x =- -W can then be considered as due to the change of wavelength ~A

caused by a different refractive index above and below the ray path.

That is, the index above the ray path, say n (-W), is slightly

greater than n (-W), and the index below the path, say n (-W),

is slightly less than n (-W). Thus the wavelength hC slightly above
the ray path is smaller than X(-W), and the wavelength X slightly

below the ray path is slightly larger than X(-W). Hence the median

position of X (over which Ex is averaged) occurs at a level slightly

below x = -W. The net effect is that the Ex component sees the index

n (-W) before the ray actually reaches the level x = -W. As a result,

the TM ray is internally refracted "sooner" than in the TE case, i.e.,

at a value of Ix which is slightly smaller than in the TE case.

This has a slight tendency to offset the fact that the value of W for

the TM case (from the waveguide equation) is larger than W for the TE case.

It is possible to derive the functional form of this shift 6W.

From the above argument, it is reasonable to expect that 6W is pro-

portional to the change in wavelength 61 due to the slightly different

refractive indexes ng and n i.e.,





sw= ta~


IV-(18)


where C2 is another constant. Since X = 1/n then


-~. d"q
Y)9(-W)


Iv-(19)


Further, for an index gradient dn /dx near the level x = -W, the
change in index seen over a distance CIA is


Ax


5


IV-(20)


cLr / x=-


= s"on(-) _


IV-(21)


Combining Eqs. (18) (21) produces


\NW c c, 0~ fxl


IV-(22)


Using Eq. (1)

x="-
3


dvi b'(-w)


IV-(23)


3no dn


+an~ ~n~S (n,3


3
Y)D +


IV-(24)


Since an << no, then Eq. (22) becomes approximately


a 31Y b(W


Sw~~- cC


IV-(25)


2 2
= Av /k then


2
Writing 1
o


/* vil


h I k--W/





The product CIC2 is evaluated in the more rigorous derivation given

in Appendix D and has the value 3/8n2 [from Eq. D-(122)]. Thus



S3LY70 PoD IV-(27)



In order to get a rough estimate of how large 6W might be, if

an = 0.1, D = lym, and W = D (i~e., the wave is propagating nearly at

cutoff), then,for a linear index profile, b'(-W) = 0.1pm-1, which is

a fairly steep profile. Further, letting X0 = 0.63pm and n = 1.5,

then 6W 4.5 x 10 p1m = 0.45 A~, which is very small. For less steep

profiles, 6W will be smaller still. Thus to a very good approximation

6W can be neglected in calculations concerning the TM characteristic

waveguide equation as well as the cutoff thickness (Wmin TM'
Solutions for E and H

1. The Linear Approximation

As mentioned in the introduction of this chapter, the WKBJ

solutions for the field components E and H (for the TE case) and
oy ox
H and E (for the TM case) become infinite as q(x) approaches
oy ox
zero. In order to get around this problem, theoretical techniques

developed by Buddenl and Waitl for radio ionospheric physics can be

adapted to optical waveguides.

The assumption is now made that, for a slowly varying index pro-

file, q2 is approximately linearlyI proportional to x near the level q = 0,

i.e., near x = -W. This is nearly the same thing as saying that n (x)
is linear in this region, as may be seen from the following argument.





Expanding n (x) in a Taylor series expansion around x = -W, produces


IV-(28)
which, from Eq. (1), becomes


6, b(- w) + b I(-nw (W+ vj


Y1 X) = ,o+


IV-(29)


which is linear in x. Further, for small an


119 x) Y1


(X w)]


+- alo h7 Pb I- yJ)t +j W()


Iv-(30)


which is also linear in x. At the level q = 0 (and x = -W), 6(-W)

has the value w/2,so from Eq. III-(6)


IV-(31)


= n~i-YJ)


IV-(32)


Thus, from Eq. (30),


Sa


on n lu


IV-(33)


hence [from Eq. (10)]


" = an, o~ b'l-w)lx-c-w)


Iv-(34)


which is linear in x too. Eq. (34) is equivalent to a Taylor expansion
of q2(x) in the region near x = -W.


"(X,=





2. TE Modes

Defining the following quantity:11



/ 32) 3IV-(35)


where (q2)' is the derivative of q2 [from Eq. (34)] with respect to x,

then, as shown in Appendix D, the wave equation for Fx [Eq. A-(92), or
equivalently, Eq. D-(20)] can be written as

at F,
x~~- IV-(36)


which is called the Stokes differential equation. Its solution as

discussed by Buddenl1 is the following:






where KTE is a constant and where Ai(S) is the Airy integral function
defined as follows:


A~j)= L iea (g t L 3) Ot1 IV-(38)


Tables of Ai(5) are given by M3iller.15 A plot of Ai(S) versus 5 is

given in Figure IV-4. It may be seen that beyond the region r = 0

(q = 0, x = -W) the solution Ai(5) is an exponentially damped wave --
an evanescent wave.





Exponeht: ct
~i~e~ca~l Regi~K

r=-HI-b



SP


_83~


4eq;~n


x=- +A


c- fX
3
e- +$


r~3


1= 0
f=


Figure IV-4.


Plot of the Airy integral function Ai(S) as a function
11 12
or 5, or, equivalently, as a function of q .









There is another solution to Eq. (26) since it is a second-order

differential equation. It is another Airy integral function Bi(S).

However, beyond the level x = -W at q = 0, the function Bi(T) becomes

indefinitely largell as 5 increases, i.e., as x + -m. Since it is

physically unrealistic for Ex (and thus E) to get larger and larger at

distances farther and farther away from the waveguide, then the solution

to Eq. (26) cannot contain a multiple of Bi(r).

One of the interesting properties of Ai(S), or equivalently Ai(q2)

is that for large q (corresponding to large cos0, far away from the

level q = 0), Ai(5) asymptotically approaches the WKBJ solution for E,

as shown by Budden.11 Referring to Figure IV-5 and following the
1112 2
arguments of Buddenl and Wait, it is assumed that q is linear over

the entire region be. There is a certain range cl where lq is so

small that it violates the WKBJ approximation, so the WKBJ solutions

cannot be used. Outside this range the WKBJ solutions are good

approximations. In the regions be and de the asymptotic approximations

of Ai(S) can be used, and they must be matched to the WKBJ solutions.

In other words, outside the region near q = 0, the WKBJ solutions

should be valid; inside the region, Ai(T) must be used. At the

"boundaries" of this region, at say x = -W + D (where a is to be

determined), the two solutions must merge smoothly, thus providing

boundary conditions to be satisfied. When this is done, as shown in

Appendix D, the phase change 6R suffered by the wave upon propagating

into the linear n (i.e., linear q2) region, reflecting (i.e.,

internally refracting) at the level x = -W, and then propagating out of

the linear n_ region, is given by





be de

4% 1 1
x=I, I x- J-
11



























Figure IV-5. Plot of q2(x) as a function of x showing the linear
11,12
region be.





~~ IV-(39)



Aside from the factor i = ein/2, this phase change 6R is nothing more

than what would be predicted by the WKBJ method even though the WKBJ

solution itself breaks down in the region q = 0. When 6R is combined

with the phase change suffered by the wave in the rest of the guide,

the characteristic waveguide equation [Eq. (6)] results.

The effect of curvature of the index profile must be investigated

in order to determine the validity of Eq. (34) and the resultant

conversion of the wave equation to the Stokes equation. If an

additional term is kept in Eq. (29), then


n no+ .an~n Ib(-w) +bi( W)(w+W) Iv-(40)



and



aoIV-(41)



Then, from Eq. (10),






where



IV-(43)





When Eq. (42) is substituted into the wave equation for Ex [Eq. A-(92)],

then as shown in Appendix D the wave equation can be converted to the

Stokes equation for which Ai(5) is solution only if


IV-(45)



and this must hold for values of 5 large enough for the asymptotic

approximation of Ai(S) (and thus the WKBJ solutions) to be valid.

According to Budden,11 this will happen when 15 For jz0 = 1,

Eq. (44) requires the following linearity criterion:


)4)o .a J/3 IV-(46)



which from Eqs. (1), (43), and (44), requires that






The condition that ]r = 1 (at x = -Wi + the boundaries of the linear

ng region, as shown in Figures IV-4 and IV-5), corresponds to






It is now possible to write down the solutions for ETE and HTE'

The solutions for x = 0 are given by Eqs. III-(41), C-(135), and C-(136).

For the region (-W + a) = x = 0, the solutions are the UKBJ solutions

given by Eqa. III-(42), C-(135), and C-(137). For the region

(-W -a) = x = (-W + a), the solutions involve the Airy integral function:





where KTrE is defined in Eq. D-(58) [using Eq, D-(48)]. The exact

form of Ho and Ho is given in Eqs. D-(94) and D-(96). Finally, for

the region x = (-W a), the WKBJ exponentially decaying solution is


ECR= e Iv-(5o)



where Fq is defined in Eq. D-(62). The exact form of Ho and Ho
in this region is given in Eqs. D-(99) and D-(100).

3. TM Modes

Defining the following quantities:71


IV-(51)







(where Q is an "effective value of q"),11 then, as shown in Appendix

D, the wave equation for Ax [Eq. A-(93)] can be simplified to


aa k oi~u IV-(53)




similar to Eq. D-20. By making the assumption that n (x) varies
linearly near the level x = -W, then it is shown in Appendix D that

the difference between Q2 and q2 from Eq. (52) is proportional to

[an b'(-W)/k ] which is a very small quantity for optical waveguides.
Hence, near the level x = -W, the wave equation Eq. (53) becomes

approximately





and all the techniques described for the TE case can be used to convert

this form of the wave equation to the Stokes equation, and to then

find the Airy integral solutions.

Neglecting the difference between Q2 and q2 is equivalent to the
assumption that they have a zero at nearly the same point. As discussed

in the quasi-geometrical approach, it is possible to calculate this (very)

slight difference 6W between the TE and TM reflection points (i.e.,

the points of total internal refraction). Since 6W is so small,

Eq. (54) is valid for TM modes to a very good approximation.

The quantity Q2(x) is subject to two linearity criterions, as

compared to just one [Eq. (46)] for q2(x). Assuming Q2 obeys the

criterion given by Eq. (46) concerning the second derivative n "(-W),
then a Taylor series expansion of Q2 from Eq. (52) can still result in

a quadratic term which has the first derivative n '(-IJ) only [Eq. D-(123)].
As a result [from Eq. D-(128)],Q2 may be written as






where 6W has been neglected and where the constants Q1 and Q2 are

defined in Eqs. D-(120) and D-(126) respectively. Eq. (55) is of the

same form as Eq. (42); thus the quadratic term in Eq. (55) is also subject

to the linearity criterion given by Eq. (46), i.e.,


14 C IL a/ i IV-(56)





Using the definitions of Q1 and Q2 together with the fact that the

minimum value of nl is no before cutoff, Eq. (56) requires that






Budden1 defines the quantity B as follows:


D r" r s l~nbix 'I x-Mj1""IV-(58)



[where optical waveguide notation has been used for the right-hand side

of Eq. (58) instead of Budden's radio ionosphere notation]. Using Eq.

(58), Eq. (57) becomes


8 ~, 1~,7IV-(59)



For optical waveguides, Eq. (58) almost always satisfies Eq. (59).

That is, if the linearity criterion given by Eq. (46) is satisfied for

TE modes, then it is satisfied for TM modes also, to an extremely good

approximation. For instance, if the index profile is exponential such
that


"'"'" i~i ,"~IV-(60)



then Eq. (58) satisfies Eqs. (1) (3). If the steep-index-profile

values used with Eq. (27) are chosen, and the e-1 point of the index

profile occurs when x = -D/2 = 0.5pm, then d = 2pm-1. Also, letting
n7 = no = 1.5 and W = D, (i.e., the wave is propagating nearly at cutoff),





then b'(-W) = b'(-D) = Pnd/(1 edD) = 0.313. Thus, from Eq. (56).

B = 51, which certainly satisfies Eq. (57). For waveguide index

profiles with an = 0.01 over a distance D = 10pm, B has the value of

1089.

It might be mentioned that in the radio ionospheric case, (an)2

may not be small but may approach n In such a situation, the problem

is more complicated, and Budden discusses it in some detail.11 As

he explains it, the field component Ex imparts vertical motions

to the electrons and if Ex~ is large, then within one cycle any one

electron sees a different E~ at different: levels. In other words,

the electron does not see a constant force. Hence its motion does not

vary harmonically with time and is therefore anharmonic. Because

of this, some energy goes into harmonics, and the original wave is

attenuated. As a result, the phase factor encountered upon reflec-

tion (or total internal refraction) must have an imaginary component

to account for this loss in the TM case. Further, the phase change

is less than w/2. Maximum attenuation (i.e., conversion to harmonics)

occurs when B mL 0.45, in which case only about 72% of the incident

energy is reflected by the ionosphere.

For the optical waveguide case, B is very large; hence a

negligible loss should occur per reflection (i.e., per total internal

refraction). However, unlike radio waves which reflect once from

the ionosphere, light in a waveguide might be reflected many times

during its passage within the guide. Perhaps, for certain index

profiles, enough energy can be converted to higher harmonics to

appreciably attenuate the TM wave. And perhaps, with the proper









phase matching, non-linear effects such as second-harmonic generation

or parametric upconversion might take place. As Budden points out,11

the problem of TM attenuation is very difficult, requiring numerical

computations. Since from Eq. (16) or (17) [and Eq. III-(6)], 6(x)

is very small for small an, then, for a propagation length of a few

millimeters, the number of total internal refractions will be quite

small. Further, since B is so large (on the order of 102 or more)

for such profiles, a negligible loss per refraction should occur.11

Thus, no significant attenuation should occur for a TM wave in a slowly

varying index optical waveguide which is only a few millimeters in

length (such as the zinc sulfide films used in this research project).

It may be that: waveguides with steeper profiles and longer lengths will

produce significant attenuation of a TM wave.

It is now possible to write down the solutions for EM and HT .

For the region x~ 0, the solutions are given by Eqs. III-(46), C-(224),

and C-(227). For the region (-W + a) =x = 0, the solutions are the

WKBJ solutions given by Eqs. III-(47), C-(225), and C-(228). For the

region (-w a) x (-W + a), the solutions involve the Airy integral

function:







The exact form of E and E is given in Eqs. D-(151) and D-(152).
ox oz
Finally, for the region x = (-W a), the WKBJ exponentially decaying

solution is





where Aq is defined in Eq. D-(144). The exact form of fo and Eo in

this region is given in Eqs. D-(154) and D-(155).

From the analysis for TE and TM modes given above and in Appendix

D, all the results obtained from the quasi-geometric ray-optics approach

can be verified from considerably more rigorous mathematics. Thus it can

be concluded that a practical and accurate understanding of the phenomenon

of total internal refraction in optical waveguides can be obtained by

using physical insight arguments without the necessity of finding the

actual solutions for E and H. Of course, the conditions under which such

arguments are valid must be determined from the more rigorous approach.





CHAPTER V

GUIDING OF LIGHT IN EPITAXIAL ZINC SULFIDE FILMS



Introduction

Cubic zinc sulfide (also called 8-ZnS, zincblende, or, for naturally

occurring crystals, zincblende or sphalerite) has a bulk refractive indexl6

nZ 2.35 at 10 = 0.6328pm. The epitaxial zine sulfide films fabricated

by the Central Research Laboratories of Itek Corporation are grown on

gallium arsenide (GaAs) substrates.171 The bulk refractive index of
19
Ga~sis cmplx: n= n i<= 3.87 + 0.087i at X = 0.6328pm.

[Although the imaginary part KG of the index is only about 2%

of the real part nG, the absorption coefficient20 ar is approximately
4 -1
1.7 x 10 cm so, for A_ = 0.6328pm, no light propagation can take

place in GaAs crystals of length L greater than a few micrometers.]

Sinc K < H, ten G =nG, and the reflection coefficient R has

little dependence on KcG. Further, since nG > nZ, then light propagation

in an epitaxial zinc sulfide film on a gallium arsenide substrate cannot

take place solely by means of total internal reflection. Instead, either

partial internal reflection or, possibly (as discussed below), total

internal refraction must be the mechanism for guiding light at or near

the ZnS-GaAs boundary.

Total Internal Refraction

Cubic GaAs has a lattice spacing of 5.64 A; cubic ZnS has a natural

spacing of 5.41 A (Ref. 17). If there is a perfect lattice match at





the ZnS-film/GaAs-substrate boundary, then the spacing of the ZnS film

lattice will change from 5.64 A to a more closely packed 5.41 A,

a difference of 4.25%. It seems possible that, because of the greater

ZnS lattice spacing at the ZnS/GaAs boundary, a lower ZnS density as

well as a lower ZnS refractive index should result. Then, as the

epitaxial ZnS lattice changes from 5.64 A to 5.41 A, an index gradient

could exist within the guide, with a higher value of the index nZ at

distances increasingly farther from the ZnS/GaAs boundary. Thus, light

could propagate in the ZnS film by means of total internal refraction at

some level above the ZnS/GaAs boundary.

There is a second possible mechanism which might give rise to a

positive index gradient for nZ in addition to the lattice mismatch. The

coefficient of linear thermal expansion (at 2000C) of zinc sulfide is

6.3 x 10-6/0C: for gallium arsenide the value is 6.5 x 10-6/0C (Ref. 22).

After crystal film growth at typically 4000C, the ZnS and GaAs must cool

down simultaneously. In doing so, the smaller thermal expansion of the

ZnS film causes a small bending of the ZnS/GaAs combination, such that the

ZnS film is on the inner diameter. As a result, the density of the ZnS

film is perhaps slightly higher at the ZnS/superstrate boundary. Such

a bending was observed previously by scientists at Itek (unpublished),

and a "focal length" f of approximately one meter was observed for a

collimated beam of laser light. (To the eye, the surface appears perfectly

flat.) Thus the radius of curvature of the bending is 2f. For a ZnS

film thickness of W = 20pm or so (a typical value of the Itek-grown films

used for the light guiding experiments), then only a small change in the

radius of curvature 2f is present between the top and bottom of the film.

For a crystal length dimension LT at the top surface, then the angle





subtended by the top surface of the crystal is LT/4xif: for the bottom

surface, the same angle is subtended and can be written as LB/2ni(2f + W),

where LB is the length of the film along the bottom of the ZnS film.

Equating the above two expressions leads to LB = LT(1 + WT/2f). Since

W/2f a 10-5, it is reasonable that any index gradient due to a

differential thermal expansion should be negligible compared to an index

gradient caused by the lattice mismatch.

The question, then, is how far into the ZnS film does the lattice

mismatch have any effects. Experiments by other authors232 with zinc

sulfide films deposited on glass and fused silica substrates have

confirmed the presence of index inhomogeneities extending anywhere from

100 A to 800 A (i.e., from 20 to 150 lattice spacings) away from the

substrate/film boundary. Of course, since the substrates were amorphous,

the zinc and sulfur atoms had complete freedom to align themselves: in

fact, the ZnS films were nearly amorphous themselves, showing a

minimum of crystalline structure. On the other hand, the epitaxial

growth of zinc sulfide films on gallium arsenide substrates requires

that the ZnS molecules align themselves according to the cubic-

structured GaAs lattice. Hence, the possibility exists that any

inhomogeneity due to a lattice mismatch could extend more than a 1000 A

into the ZnS film.

Partial Internal Reflection

Partial internal reflection at the ZnS/GaAs boundary is another

possible mechanism for guiding of light in a zinc sulfide thin film.

Such a light guide is a leaky one25 since some energy is lost upon

reflection by waves at the ZnS/CaAs bonldary. Ulrich and Prettl25





discuss leaky light guides in detail and
relationships for a given mode number m:


arrive at the following


ewi; Rs, lm- ; O(m


~TE a~v


Tas 74~ YKI4)


V- (2)


=T r L)
1;~~~~ a5 I (II 7


where


CYly3_ npl) 'la

n,
a 3 3'/
ngS YI,


~TN mc~eed)


V-(5)


V-(6)


Eq. (2) can be considered as a characteristic waveguide equation for

leaky light guides. The quantities na, ns, ng and W are defined in
Figure II-1 except that, for leaky guides,


V]35 vl~m"


The quantity 6m is similar to the wavevector component along the
(from Eq. II-(22)], only now a provision is made for attenuation
light by including the term -iam'


V-(7)

A
z-axis
of the


~q 19 ~li~, W+
O g'b


~, Wg r 04,W ~





The quantity Wq is the "equivalent thickness"2 of the guide and

is the result of the fact that the totally internally reflected wave at

the n ng boundary penetrates slightly into the na region. For 1
a g

0.6328pm and W = 20pm, then k W = 200: for n = nZ = 2.35 and n =16

(the largest index of the superstrate materials used in waveguiding

experiments), then (n n 2) =/ 0.6 << k W. Ponce, for all practical
g a o
purposes, W can be used instead of W in Eqs. (1) (6) for both TE and
eq
TM modes in Itek-grown epitaxial zinc sulfide light guides.

Eqrs. (1) (4) were derived25 under the assumption that the imaginary

part KS of the refractive index ns is much less than the real part n i.e.,

K << n and ns .LT As explained in the introduction to this chapter,

the values of nG and KG satisfy this assumption. Using the value

n = nG = 3.87 together with ho = 0.6328um and W = 20pm, then, from

Eqs. (3) and (4)



V-(8)







Since the samples used for light guiding experiments had lengths of a

few millimeters, negligible attenuation should occur for the lower-

order modes (m < 5).

Ignoring the term iam in Eq. (1), then from Eq. II-(22), nm can be

written as



?m- l~ ScnlV-(10)







where em is the value of O in
Eqs. (2) and (10),


Figure II-1 for a given mode m. Combining


(W)tj)3
C~?o W)1


riTa

a"3


V-(ll)


(M~I~1
8~a ~ga
9


V- (12)


Using the values of ;\0, ng = n and W used above,


(aaxlo5~)jm+r)a


3cTn ~-, = I -


V-(13)


It may be seen that, to a very good approximation, Om is very nearly
equal to (but always slightly less than) n/2 -- propagation straight
down the guide. This is true for all but the highest order modes,
which, from Eqs. (8) and (9), are strongly attenuated anyway.





CHAPTER VI

THE LINEAR ELECTRO-OPTIC EFFECT OF EPITAXIAL ZINC SULFIDE FILMS



The Principal Axes

The optical properties of a crystal can be described in terms of

an optical indicatrix, a refractive index ellipsoidal surface. Since

zinc sulfide is isotropic, its indicatrix can be represented by the
26
following equation:



2I VI-(1)



which is the equation of a sphere. Here no = nZ, the bulk refractive

index. Upon application of an external electric field e~ (not to be confused

with the electromagnetic field E of a light wave) for the linear electro-

optic effect, then the indicatrix becomes


a VI-(2)



Here r41 is the electro-onptic coefficient of zinc sulfide.

The epitaxial films grown by Itek Corporation's Central Research

Laboratories are oriented in the (100) direction, defined as the x

direction in Figure VI-1. On top of this film an electrode is deposited.

Thus, the applied field across the film is in the x direction only, i.e.,

e = e = 0. Eq. (2) then reduces to
y z





X xri i/OD)


nlectrode I/ x / At






GaAs















Figure VI-2. Crystal orientation of epitaxial zinc sulfide films

grown by the Central Research Laboratories of Itek

Corporation. The principal axes are given by the x'

y', and z' directions which are also the normals to

the natural cleavage planes.





~3a
x +y $2
~1
3
Y)


t a r,, c, yZ =I


VI-(3)


By a proper rotation of the index ellipsoid, a new set of principal
A A
axes (x', y', z') can be obtained such that



Ixa 72 ad VI-(4)


Comparing Eqs. (3) and (4) it may

unchanged, i.e.,


A
be seen that the x axis remains


A/
) X


VI-(5)


VI-(6)


The problem thus reduces to calculating the angle R through which the

y and z axes must be rotated to y' and z'. As shown in Appendix E,

0 = w/4. The new axes x', y', and z' are shown in Figure VI-1. They

coincide with the normals to the natural cleavage planes of cubic zinc

sulfide.

In the course of obtaining the principal axes, the dependence of

the principal refractive indexes nx, n and nz on +he applied field ex

are also obtained in Appendix E. The results to first order are









o o 91 ^e~ i VI-(8)




33


where n3 r 2< I a ese ta adn have a linear
o '41 xl o< m b e ta y a z
26
dependence on the applied field e.

The Transverse Electro-Optic Effect
A, A
For propagation in either the z or z' direction [the (001) and (011)

directions, respectively], the light has a k vector component

B = k n sine which is perpendicular (i.e., transverse) to the applied
oo

electric field 4. This results in the transverse electro-optic effect.

[This is not to be confused with the transverse electric (TE) mode of

propagation.] The light also has a k-vector component k no cosO which

is parallel to e, resulting in the longitudinal electro-optic effect.

In order to get intensity modulation (as opposed to pure phase

modulation) when the ZnS guide is placed between crossed polarizers, it

is necessary that the polarization of the incident light be altered. In

order to determine if changes in polarization will occur, all incident

polarization components must first be resolved along the principal axes

(so long as the axes are perpendicular to the net direction of light

propagation). If there are two polarization components each parallel to

a different principal axis which is perpendicular to the net direction of

travel, then each polarization component will "see" a different refractive

index if an electric field is applied to the crystal. As a result, each

component will have a different phase retardation, and the light will

become elliptically polarized as it propagates down the guide. The net

effect is to rotate the plane of polarization of the incident light.





A combination transverse and longitudinal electro-optic effect can

in general be very complicated. However, for optical waveguides, there

are several factors which tend to reduce the complexity.

First of all, the angle 6 will be very nearly 900 for epitaxial zinc

sulfide films which guide light at or near the ZnS/GaAs boundary either

by partial internal reflection or by total internal refraction. In the

former case, this is obvious from Eq. V-(13). Since the longitudinal

component depends on the cosine of 0, it will be negligible for em w /2.

In the latter case, the change in index due to a lattice mismatch is

probably no more than 4.25%, i.e., an = 0.0425nZ. From Eqs. IV-(16)

and D-(73), sine(0) cannot be any less than n /(n +~ an) = 0.96, i.e.,

0(0) cannot be any less than 740. Of course, O(x) changes from 0(0) to

O(-W) = 900. Hence,very roughly, the minimum "average"! value of 6(x)

would be 820. Since sin 820= 0.99 and cos 820 = 0.14, it may be seen

that the longitudinal component (from cos 820) is, at the very most,

only 14% of the transverse component.

Even more significantly, the voltage required to produce a

longitudinal electro-optic effect in a thin-film waveguide is orders of

magnitude greater than the voltage necessary to produce a transverse

effect. For a pure longitudinal effect, the half-wave voltage (defined

as the applied voltage required to change the plane of polarization

of the incident light by 900) is given by26



'/2 Lond A f) VI- (10)



For a pure transverse effect, the half-wave voltage is26

















Using ho = 0.6328um, no = nZ = 2.35, and r41 = 2.07 x 10-1 cm/volt

(Ref. 26), then (V )= 1.2 x 10~ volts. For WJ = 20pm and L = kcm,
X/2 Long
-3
then (V /)= 10 (V ) = 47 volts. Thus, with an applied

voltage of 50 volts or so, a large rotation of the polarization will occur

as a result of the transverse electro-optic effect: a negligible effect

will be observed for the longitudinal effect.

Further, since optical waveguides are quite thin, dielectric break-

down will occur before a very large voltage can be applied to the thin-

film material. For zinr. sulfide, the breakdown voltage for practical

purposes is approximately 105 volt/cm = 10 volt/lpm (Ref. 27). Thus, a

20ipm-thick zinc sulfide film would break down long before any significant

longitudinal effect occurred.

Thus it can be concluded that the transverse electro-optic effect

is the primary mechanism for electro-optic modulation in zinc sulfide

optical waveguides. It might be pointed out that, since B = konZ sine

is a constant for homogeneous as well as inhomogeneous waveguides (as

explained in Chapters II-IV), the transverse component of the k-vector

is thus a constant for either case.

Inenity ModuEElation for Propagation in the (011) Direction

1. TE Modes

For a pure TE mode, the electric vector E is polarized parallel to

the y' axis, i.e.,





n- Y EYVI-(13)



Y VI-(14)



where E = E, (x) eiw z The wave will "see" an index n' = n = nZ
o y o Z

with zero applied field ex, and an index n = nZ n~ r41ex/2 with an

applied field. Thus, phase modulation can occur for the TE, case. However,

since the light is polarized parallel to only one principal axis, then

for any value of ex the E-field sees only one index of refraction at any

one time. Hence the polarization will not be altered, and no intensity

modulation is possible.

2. TM Modes

For a pure TM mode, the electric vector E can be written as



TY VI-(15)








Since the propagation is in the z' direction, it may be seen that there

is only one polarization component along a principal axis perpendicular

to the net direction of travel. Hence, the polarization will not be

altered, so no intensity modulation is possible; phase modulation

also is not possible.

3. TE + TM Modes

For TE + TM incident light, the E-vector can be written as



VI-(17)





67









In this situation, with an applied field ex, the speed of propagation

for E' is different from the speed of propagation for E'. Thus after
x y
travelling a length L through the crystal, unequal phase shifts 6' and

6' are introduced:


VI-(20)



F=' E~e
VI-(21)



where


S:' k~'VI-(22)



~Y- prr'VI-(23)



and


PTN=a n~3 TMVI-(24)



5-r oA TyrB VI-(25)


The values of OTE and 87~ are to be determined from the appropriate

characteristic waveguide equation, depending on the mechanism for light

guiding.









It is now assumed that the zinc sulfide light guide is placed

between crossed polarizers (where the input polarizer is oriented so as

to pass a TE + TM wave). Upon application of a field e the E' and E'
components, each at angle 450 to the analyzer axis, have components

-Ex//2 and +Ex//2~ along the analyzer axis. The amplitude EA of the
light passed by the analyzer is thus


EA J / VI-(26)


where the z' component has been neglected

above. From Eqs. (20) and (21), Eq. (26)


since OT~M /2, as explained
becomes







I transmitted by the analyzer


The mean (i.e., time-averaged) intensity

is


r=A A ~


VI-(28)


From Eq. (27) this results in






Defining


VI-(29)


I, = E~/a


VI-(30)








then


I (~, lY)I sYn ~`~,+I a 4M BTN C83 ~sx(~dyl~l


VI- (31)


Using Eqs. (22) (25) together with Eqs. (7) and (8), then the phase
shifts can be written as


x,-~n oo TN


VI- (32)


VI-(33)


~, (~- d r,,e,/a)L siL;iBTF


Thus,


o 0, l3


r,, Le, /s


x~l d


VI- (34)


where


0, oi~ L


VI- (35)


Further, since

e, = VWV



where V is the voltage across the crystal, then


VI- (36)


VI- (37)


3
+ crT n, $,


VI- (38)


VLI~,W





If the approximation is made that



7M v :er t/ VI-(39)


which is valid for epitaxial zinc sulfide light guides (as explained

above} then Eqs. (31) and (38) become


o X VI-(40)



and



/a-a ) D 3 r Y/ oW VI-(41)


This is the same result obtained for the bulk transverse electro-optic

effect,26 and intensity modulation will thus occur as the applied

voltage is varied from zero to (VA/2 Trans.
It should be mentioned that if there is a dielectric layer isolating

the metal electrode from the zinc sulfide film, then the voltage V in

Eq. (41) will not be the total voltage V but only the voltage across

the ZnS film. The relationship between V and VT can be obtained from

elementary physics since the layer and ZnS film are effectively two

parallel-plate capacitors in series. The result is


\J V 0EE 7 VI-(42)



where Ed and Wd are the low-frequency dielectric constant and thickness,

respectively,of the layer: EZ is the low-frequency dielectric constant
of zinc sulfide and has the value 8.37 + 0.8 picofarad/meter (Ref. 28).





71


Intensity Modulation for Propagation in the (001) Direction

1. TE Modes

For a pure TE mode, the electric vector 2 is polarized to the y

axis, i.e.,



VI-(43)






nr\ is;VI-(45)




The polarization components of the incident light must always be at right

angles to the direction of propagation. Since the y' and z' axes are

not perpendicular to the net direction of propagation for the z-propagation

case, then the method of resolving the incident polarization along principal

axes is not a proper way of examining the problem. Instead, the effective

contributions of the axes can be evaluated by resolving these axes along

the axis formed by the incident polarization. In this situation, any

change An' due to the y' axis is identically compensated by any change

an' = -an' [from Eqs. E-(17) and E-(18)] due to the z' axis. Fence the TE
z y
wave "sees" a constant index no = n The net effect is no change at all

(not even phase modulation when a field is applied to the crystal) In

other words, for propagation in the z direction, at least for TE modes, the

zinc sulfide appears isotropic, with or without an applied electric field.

2. TM Modes

For a pure TM mode, the electric vector E can be written as


E =x E~ + z E
VI-(46)





^I / / Al /
=X t, +t E' 42 E
3 VI-(47)



s~n~TFI y 2/ E~iC14T~IVI-(48)



Since OT~M w/2, then Eq. (48) reduces to


161 VI-(49)



which is the same as Eq. (16). Thus no intensity modulation is possible

(and, since n' = no is independent of the applied field e no phase

modulation is possible either). Even if 01TM is not quite equal to n/2,

there will still be no net phase change from the y' and z' terms for

the same reasons given for TE modes propagating in the z' direction.



3. TE + TM Modes

For TE + TM incident light, the E-vector can be written as


VI-(50)



VI-(51)





--2 t~iajl+ we,)VI- (52)




Again, since OTM 'L w/2, then


11~T Y /VI-(53)






73


Again, the same arguments used for TE and for TM modes lead to the
AtA~
conclusion that the y' and z' term will produce no net effect. Thus,

even though from the appearance of Eq. (52) it might seem that there

should be some modulation (either intensity or phase),such is not

the case. Hence the direction is an optic axis of cubic zinc sulfide

since the refractive index is the same for TE, TM, or TE + TM polar-

izations.














CHAPTER VII

EXPERIMENTAL PROCEDURE



Sample Preparation

Most of the Itek-grown samples have a circularly shaped zinc

sulfide film which does not extend all the way to the edge of the

gallium arsenide substrates. Thus, in order to propagate light in the

z' direction [the (011) direction shown in Figure VI-1], it is necessary

to cleave each sample twice (indicated in Figure VII-1), thereby

exposing one edge for input coupling of the light and another edge

at the other end for the output. This is accomplished by placing the

sharp edge of a one-sided razor blade in the proper orientation on the

GaAs side of the ZnS-on-GaAs sample and then tapping the blade slightly.

In this way a microscopically smooth edge of the ZnS film is obtained.

For propagation in the zn direction [the (001) direction], it is

necessary to slice the input and output ends of the ZnS-on-GaAs samples

(using a diamond saw) at a 450 angle to the normals to the cleavage

planes. The sample is then sandwiched between two glass plates and

subsequently fine polished. The glass-plate sandwich is used because

the first polishing attempts without the plates resulted in a bevelled

ZnS edge. As a result, light incident on the film edge was refracted

into the GaAs substrate, and propagation down the length of the thin-

film guide could not be achieved. Schematic diagrams of the resulting

samples for z" propagation are shown in Figure VII-2.


















Cleave Edges


Input
Light


Figure VII-1. Top view of cleaved-edge sample for light propagation
in the z' direction [the (011) direction].





Sliced Edges


Input
Light


* CI
X, x


~Y*' (O11)
~ Ip'O)


Figure VII-2. Top view of sliced-edge sample for light propagation

in the z direction [the (001) direction].





For electrode preparation, the top surface of the sample is

masked off with Scotch tape, leaving exposed the desired area for the

electrode. Then electrically conductive silver paint (Epoxy Products Co.,

New Haven, Conn.) is applied to this area and allowed to dry, after

which the tape is removed. The sample is then mounted as shown in

Figure VII-3. A glass substrate is used as an electrically insulating

mechanical base. Half of the top surface of the plate is covered with

electrically conducting silver paint, and the G~aAs side of the ZnS-on-

GaAs sample is placed on one portion of this paint before it dries.

(Gallium arsenide has a much lower resistivity than zinc sulfide, and

therefore it acts as the bottom electrode to the ZnS light modulator.)

To the other portion of this paint an electrical lead is affixed with

Dupont Duco cement (since the silver paint does not provide much

mechanical strength).

On the other half of the glass substrate more silver paint is

deposited (electrically isolated from the first half). Then an electrical

lead is affixed to this paint and to the top electrode on the ZnS

film using some more paint. Finally, another electrical lead is

bonded with Duco cement to the silver paint on the glass base. The

sample is thus mechanically mounted for waveguiding experiments and

electrically connected for electro-optic measurements.

Experimental Set-up

A helium-neon laser is used as the primary light source for the

experiments. The output beam of light (X0 = 0.6328pm) is linearly

polarized so that use of a half-wave plate and an angularly calibrated

linear polarizer allows easy selection of TE, TM, or TE + TM light. The

light is coupled into the guide by means of a 20X microscope objective.





Silver Paint Electrode


Electrical Lead
to Voltage Power
Supply


SSilver Paint


Electrical Lead
to Voltage Power
Supply


Silver Paint ,


XX



~'~ Y") d..
z (w3)


End view of mechanical and electrical mounting for

waveguiding and electro-optic investigation of zinc

sulfide thin-film samples.


Figure VII-3.





At the output end, a microscope, consisting of a 20X or 44X objective

and a 10X eyepiece, is used. This experimental set-up is shown

schematically in Figure VII-4. The input objective, the ZnS-on-GaAs

sample mount, and the output microscope are mounted independently on

translational stages each capable of precision motion in three dimensions.

The sample mount has an angular degree of freedom also. For the electro-

optic modulation experiments an analyzer (with transmission axis 900 to

the input polarizer axis) is placed between the ZnS sample and the output

microscope.



























Half-wave
Plate


Microscope


rf Analyzer
20X Objective


Imaged
Output


GaAs
Substrate


Al jMS


Figure VII-4.


Side view of experimental set-up for waveguiding experi-

ments (not drawn to scale).















CHAPTER VIII

EXPERIMENTAL RESULTS AND OBSERVATIONS



Light Guiding

Experiments with light guiding in epitaxial zinc sulfide films were

performed to first determine whether partial internal reflection

or total internal refraction was the primary mechanism for light guiding

at or near the ZnS/G~aAs boundary. Preliminary experiments had resulted

in an output intensity pattern with a maximum nearer the ZnS/superstrate

boundary, suggesting, perhaps, total internal refraction. This is

shown in Figure VITI-2: Figure VIII-1 is a photograph of the entire guide

illuminated in transmission by white light. Subsequent experiments with

several different samples showed that the maximum of the distribution

could be shifted to the center of the guide (Figures VIII-3 and VIII-4),

and to the bottom of the guide (Figure VIIT-5), all for the same mode.

(This is essentially an m = 0 mode, although the beginning of a conversion

to an m = 1 mode can be seen, especially in Figure VIII-2). This result

was obtained by translating the ZnS sample, located at the focal point

of the input 20X objective, in the xA direction (i.e., perpendicular to

the direction of light propagation). In this way the effective angle of

incidence of the light (defined by the central ray of the cone of light

from the objective) could be changed slightly, but not so much as to

convert the waveg~uided light to a higher order mode. Since the entire

guide (and not just the top of the guide) could be selectively





below. th ud ie beo x~ = -9) th i uesrt
















abeov the guide (i.e., beove x = 0W). h i uesrt


















































Figure VIII-2.


End-view photograph of the output intensity distribution

of the cleaved-edge 18pm-thick ZnS guide in Figure VIII-1,

but with a higher magnification (780X); wavelength

Ao = 0.6328pm. Figure VIII-2 is a mirror image of Figure

VIII-1, as may be seen by comparing the asymmetry of the

crystal defect in the center of Figure VIII-1 with the

(same) defect in the upper right of Figure VIII-1. It

may be noted that the intensity distribution has a

maximum near the top of the guide. The mode is m = 0

but approaching m = 1.





Figure VIII-3. Same as in Figure VIII-2, only now the intensity distribu-

tion has a maximum near the center of the guide.


















































































L


L.. x ^'


Intensity
(Arbitrary Units)


Figure VTIII4.


Plot of the intensity distribution in Figure VIII-3

showing the symmetry.


















































Figure VIII-5. Same as in Figure VIII-2, only now the intensity distribu-

tion has a maximum at the bottom of the guide.











"illuminated" in this manner, this suggested that the ZnS films were

fairly homogeneous throughout; if some inhomogeneity is present,

then the index profile is an extremely slowly varying one. Hence,

continual internal refraction is probably not the primary mechanism for

light guiding within a ZnS film.

The question remains: what about the effects of the lattice mis-

match between the ZnS and the GaAs? During further experiments with

various samples illuminated in transmission with white light, an

interesting phenomenon was discovered. Inadvertently, the white light

bulb used for illumination overfilled the input 20X objective with the

result that the Z.nS-on-GaAs sample was illuminated in reflection also,

i.e., light reflected off the face of the output microscope objective.

As a result, it was observed that these samples showed a thin black line

at the ZnS/GaAs boundary. This line was not visible (or, at most, was

barely discernible) when the sample was viewed in reflected light only.

This black line is shown in Figures VIII-6 and VIII-7 and is seen to be

about lum thick. The thickness of this line did not increase when a

voltage of two hundred volts or more was applied to the crystal, as might

have occurred if the line was due to some semiconductor junction effect.

Thus, it seems reasonable to conclude that this line is a permanent

"'built-in" part of the ZnS-GaAs structure.

This black line must be either zine sulfide or gallium arsenide

since no other material was present when the ZnS film was epitaxially

grown. (If there were some other material, such as oxygen, then a zinc

oxide layer would have prevented growth of a good, cubic-structured ZnS

film.) If the line were G~aAs, then light would strongly reflect from



















































Figure VIII-6. End-view photograph of the cleaved edge of an 11pm-thick

zinc sulfide guide on a gallium arsenide substrate

illuminated simultaneously in transmission and in

reflection with white light. Magnification: 440X. A

black line of about lum thickness is evident at the

ZnS/GaAs boundary. The somewhat thicker black line at

the top of the ZnS film is due to a 3pm-thick layer of

dielectric parylene used to isolate the metal electrode

from the zinc sulfide. Vertical microcrystalline

cracks in the sample are also evident.





Figure VIII-7. Same as Figure VIII-6 but with a shorter exposure to

enhance the contrast of the black line at the ZnS/GaAs

boundary.





it (as seen from the bottom of Figures VIII-6 and VIII-7). This is due

to the high index nG = 3.87, resulting in a high normal-incidence GaAs/air

reflectivity of 35% or so. Therefore,the black line is of zinc sulfide

material. Further, since this black line is prominent only when the

white light has travelled the length of the thin film crystal, it seems

plausible to also conclude that this line is a result of the lattice

mismatch. A lum layer corresponds to approximately 2000 lattice spacings.

As discussed in Chapter V, it is not unreasonable to expect that the

lattice mismatch could have effects extending beyond 1000 A. Thus, it

appears that although an inhomogeneity may exist within an epitaxial

zine sulfide film, it does not extend beyond lum from the ZnS/GaAs

boundary, a distance of only 5 10% of the total film thickness. Thus,

90 95% of the film is quite homogeneous.

As stated in the caption to Figure VIII-6, the thicker black line

at the top of the ZnS film is due to a 3pm-thick superstrate layer of

parylene (a transparent dielectric material obtainable from Union Carbide

Corp., Bound Brook, N. J.; refractive index na 1.63). Since na < nZ'

the parylene must be a leaky light guide with losses proportional to

the cube of the reciprocal of the guide thickness (cf. Chapter V).
-1
Using Eq. V-(3), the attenuation coefficient am is approximately 1.3mm

since the length of the Z.nS crystal and varylene is 5mm, no light can

propagate within the parylene but must, instead, eventually leak into the

ZnS film. Further, since the parylene is transparent and since only a

small amount of light (1 49) will reflect from the parylene/air boundary,

the illuminating light reflecting from the output microscope objective

must be transmitted in the -z direction into the parylene. This is why


the parylene appears black.




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