AN INVESTIGATION OF THE LINEAR ELECTROOPTIC 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 tripleexposure photograph showing a 10X microscope
objective (to the left), a gallium arsenide rectangular substrate with a
semicircular 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 ZnSGaAs sample as well as from smoke used to help indicate the
path of the beam. This beam (slightly offaxis 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 semicircle, 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 cubicstructure zinc sulfide films on gallium arsenide
substrates. As a result of that meeting, it was agreed that ZnSon
GaAs samples would be made available for electrooptical 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 electrooptic 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
OuasiGeometrical RayOptics Approach.........,........ 12
III. BASIC PHYSICS OF INHOMOGENEOUS ASYMMETRIC PLANAR OPTICAL
WAVEGUIDES........................................... 17
Introduction .. . . . . . . . . . .17
OuasiGeometrical RayOptics 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
QuasiGeometrical RayOptics 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 ELECTROOPTIC EFFECT OF EPITAXTAL ZINC
SULFIDE FILMS ........................................... 60
The Princival Axes..................................... 60
The Transverse ElectroOptic 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 Setup.................................. 77
viii
Page
VIII. EXPERIMENTAL RESULTS AND OBSERVATIONS......,............. 81
Light Guiding..,................................. 81
ElectroOptic 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 ELECTROOPTIC 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 quasigeometrical
rayoptics 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 generalindexgradient
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 electrooptic
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,
lowindex inhomogeneous layer of zinc sulfide near the boundary, it is
concluded that 1) 9095% 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,
electrooptic 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 rayoptics approach, and a more rigorous approach using
electric and magnetic vector potentials to solve Maxwell's equations.
Having done this, the theoretical and physicalinsight 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 generalindexgradient
planar optical waveguides.
After homogeneous and inhomogeneous waveguides are discussed, the
results of waveguiding and electrooptic 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 II1. 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,
xO
~Te:
w n4
xw I~
/ i / nS/
u79 > ~15 ~ Y1,
t~ h, ng s~e 8
Figure IIi. Crosssectional 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 freespace 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
zcomponent 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 zaxis, 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~) ews
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,
Mik o9 8 II(51)
Hence, as the TE + TM wave propagates down the guide according to
ei 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
OuasiGeometrical RayOptics 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 II2, the plane wave propagating down the guide may be thought
of as a zigzagging within the guide, plus an evanescent wave
Figure II2. Geometry of a homogeneous optical waveguide.
X
propagating in the superstrate and in the substrate at the same
velocity in the zdirection as in the guide. This is consistent with
the statement made earlier in this chapter that the wavevector component
along the zaxis, 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 II2 represent normals to the pro
pagating, zigzagging 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 g9aQ, 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 quasigeometrical rayoptics 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 homogeneouswaveguide 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 xdirection
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 III1, the refractive index of the guide has
its maximum at the n n boundary and then continuously decreases
a g
in the negative xdirection. The index can be written as
where
nS (X)n9 o JIII(2)
OuasiGeometrical RayOptics Approach
It is convenient to begin with a quasigeometrical approach.
The justification of the results will be discussed afterwards.
Letting B still be the wavevector component along the zaxis, then 8
can be written as
X=~
n,
fw ~J,~
x=0
Figure III1. Crosssectional 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~acilcu~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 III1, 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)
a3 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 righthand 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
xcomponent of the optical ray path along QR. [The product k r is
positive.] The third integral is due to the "zcomponent 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 quasigeometrical rayoptics 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 (3n11O4
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 quasigeometric rayoptics 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 quasigeometrical 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 refractionandreflection in general
indexgradient planar optical waveguides.
OuasiGeometrical RayOptics 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 IV1. 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 IV1.
Crosssectional 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 IV1, 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 IV2a, 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 IV3. 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 IV2a
Figure IV2b
Figure IV2. Refraction of light at a boundary with an abrupt change
of refractive index (Figure IV2a) and in a medium with
a continuously varying index (Figure IV2b).
x~O
Figure IV3. 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 secondorder 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 secondorder 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 Evector 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 kW/
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) = 0.1pm1, 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~iYJ)
IV(32)
Thus, from Eq. (30),
Sa
on n lu
IV(33)
hence [from Eq. (10)]
" = an, o~ b'lw)lxcw)
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 IV4. 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=HIb
SP
_83~
4eq;~n
x= +A
c fX
3
e +$
r~3
1= 0
f=
Figure IV4.
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 secondorder
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 IV5 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 IV5. 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 IV4 and IV5), 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. D20. 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 quasigeometrical 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 xMj1""IV(58)
[where optical waveguide notation has been used for the righthand 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 steepindexprofile
values used with Eq. (27) are chosen, and the e1 point of the index
profile occurs when x = D/2 = 0.5pm, then d = 2pm1. 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, nonlinear effects such as secondharmonic 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 quasigeometric rayoptics 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 8ZnS, 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 ZnSGaAs 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 ZnSfilm/GaAssubstrate 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 106/0C: for gallium arsenide the value is 6.5 x 106/0C (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 Itekgrown 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 105, 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 II1 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
zaxis
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 Itekgrown 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 II1 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 ELECTROOPTIC 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 electroonptic 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 VI1. 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 VI2. 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 VI1. 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 ElectroOptic 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 electrooptic effect.
[This is not to be confused with the transverse electric (TE) mode of
propagation.] The light also has a kvector component k no cosO which
is parallel to e, resulting in the longitudinal electrooptic 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 electrooptic 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 electrooptic effect in a thinfilm waveguide is orders of
magnitude greater than the voltage necessary to produce a transverse
effect. For a pure longitudinal effect, the halfwave 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 halfwave voltage is26
Using ho = 0.6328um, no = nZ = 2.35, and r41 = 2.07 x 101 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 electrooptic 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
20ipmthick zinc sulfide film would break down long before any significant
longitudinal effect occurred.
Thus it can be concluded that the transverse electrooptic effect
is the primary mechanism for electrooptic 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 IIIV), the transverse component of the kvector
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 Efield 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 Evector 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)
5r 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., timeaveraged) 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
/aa ) D 3 r Y/ oW VI(41)
This is the same result obtained for the bulk transverse electrooptic
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
parallelplate capacitors in series. The result is
\J V 0EE 7 VI(42)
where Ed and Wd are the lowfrequency dielectric constant and thickness,
respectively,of the layer: EZ is the lowfrequency 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 zpropagation
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 Evector 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 Itekgrown 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 VI1], it is necessary
to cleave each sample twice (indicated in Figure VII1), 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 onesided razor blade in the proper orientation on the
GaAs side of the ZnSonGaAs 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 ZnSonGaAs 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 glassplate 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 VII2.
Cleave Edges
Input
Light
Figure VII1. Top view of cleavededge sample for light propagation
in the z' direction [the (011) direction].
Sliced Edges
Input
Light
* CI
X, x
~Y*' (O11)
~ Ip'O)
Figure VII2. Top view of slicededge 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 VII3. 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 ZnSon
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 electrooptic measurements.
Experimental Setup
A heliumneon 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 halfwave 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 electrooptic investigation of zinc
sulfide thinfilm samples.
Figure VII3.
At the output end, a microscope, consisting of a 20X or 44X objective
and a 10X eyepiece, is used. This experimental setup is shown
schematically in Figure VII4. The input objective, the ZnSonGaAs
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.
Halfwave
Plate
Microscope
rf Analyzer
20X Objective
Imaged
Output
GaAs
Substrate
Al jMS
Figure VII4.
Side view of experimental setup 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 VITI2: Figure VIII1 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 VIII3 and VIII4),
and to the bottom of the guide (Figure VIIT5), 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 VIII2). 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 VIII2.
Endview photograph of the output intensity distribution
of the cleavededge 18pmthick ZnS guide in Figure VIII1,
but with a higher magnification (780X); wavelength
Ao = 0.6328pm. Figure VIII2 is a mirror image of Figure
VIII1, as may be seen by comparing the asymmetry of the
crystal defect in the center of Figure VIII1 with the
(same) defect in the upper right of Figure VIII1. 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 VIII3. Same as in Figure VIII2, 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 VIII3
showing the symmetry.
Figure VIII5. Same as in Figure VIII2, 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.nSonGaAs 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 VIII6 and VIII7 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
"'builtin" part of the ZnSGaAs 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, cubicstructured ZnS
film.) If the line were G~aAs, then light would strongly reflect from
Figure VIII6. Endview photograph of the cleaved edge of an 11pmthick
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 3pmthick layer of
dielectric parylene used to isolate the metal electrode
from the zinc sulfide. Vertical microcrystalline
cracks in the sample are also evident.
Figure VIII7. Same as Figure VIII6 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 VIII6 and VIII7). This is due
to the high index nG = 3.87, resulting in a high normalincidence 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 VIII6, the thicker black line
at the top of the ZnS film is due to a 3pmthick 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.
