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Surface Science
0 North-Holland
96 (1980) 41-53
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OPTICS OF ANISOTROPIC
ALGEBRA
LAYERED MEDIA: A NEW 4 X 4 MATRIX
Pochi YEH
Rockwell International
Received
20 August
Science Center, Thousand Oaks, California 91360,
USA
1979
A new 4 X4 matrix algebra, which combines and generalizes
Abel&s’ 2 X 2 matrix method
and Jones’ 2 X 2 matrix method,
is introduced
to investigate
plane-wave
propagation
in an
arbitrarily
anisotropic
medium. In this new method, each layer of finite thickness is represented
by a propagation
matrix which is diagonal and consists of the phase excursions
of the four
partial plane waves. Each side of an interface is represented
by a dynamical
matrix that depends
on the direction of the eigenpolarizations
in the anisotropic
medium.
1. Introduction
Anisotropic
thin films have become increasingly important in a number of
modern optical systems such as guided-wave propagation in integrated optics, narrow-band birefringent
filters and, many semiconductor
devices. Many of these
active and passive optical devices require the epitaxial growth of anisotropic thin
films. The design characteristics of these devices are strongly dependent on the
understanding
of electromagnetic
propagation in anisotropic layered media. In
addition, ellipsometry
of anisotropic
multilayer systems also requires a precise
understanding
of electromagnetic
propagation
in these media. Ellipsometry
of
anisotropic multilayer systems may be an ultimate optical technique for the characterization of the anisotropic thin film properties, which include the orientation of
the optical axes, the refractive indices, and the film thicknesses.
A general theory of electromagnetic
propagation in isotropic layered media and
the 2 X 2 matrix method are described in the pioneering analysis of Abel&s [l]. A
similar theory on the electromagnetic
propagation in periodic layered media was
published by Yeh et al. [2]. Several interesting new phenomena are predicted and
observed experimentally
in these media; these include Bragg waveguiding [3,4],
optical surface waves [5,6] and injection lasers [7,8]. A general theory on the
propagation of electromagnetic
radiation in birefringent layered media, especially
the Sole layered media [9], was recently published by Yeh [lo]; new phenomena
such as the exchange Bragg scattering and the oscillatory evanescent waves are
found in these media [lO,ll].
A 4 X4 differential matrix method has been devel41
42
P. Yeh / Optics
of anisotropic layered media
oped by Berreman [12,13] to study the reflection and the transmission of an arbitrarily polarized light by stratified anisotropic planar structures.
In the case of isotropic layered media, the electromagnetic
radiation can be
divided into two independent
(uncoupled)
modes: s-modes (with electric field
vector E perpendicular to the plane of incidence) and p-modes (with electric field
vector E parallel to the plane of incidence). Since they are uncoupled, the matrix
method involves the manipulation of 2 X 2 matrices only. In the case of birefringent
layered media, the electromagnetic
radiation consists of four partial waves, Mode
coupling takes place at the interface where an incident plane wave produces waves
with different polarization states due to the anisotropy of the layers. As a result,
4 X4 matrices are needed in the tnatrix method.
Before proceeding with the many applications envisaged for birefringent layered
media, it is necessary to understand precisely and in detail the nature of electromagnetic wave propagation in these media. Although a number of special cases have
been analyzed, a simple and general theory is not available.
The case of plane wave incidence at a plane interface between two biaxial media
has been solved by many workers [14-l 61; however, these results are not in a
useful form for treating electromagnetic
propagation in birefringent
multilayer
media. The case of plane wave propagation in a three-layer structure consisting of
birefringment
media has also been solved [17-201; however, in each case, other
than in ref. /20], the structure involves at least one isotropic layer. Even the results
obtained in ref. [ZO] are not systematic enough for treating the general tnultilayer
birefringent media. Holmes and Feucht [Zl] treated the reflection and transmission
properties of a stack of birefringent
crystals theoretically.
Their analysis is
restricted to the case where one principal axis is normal to the plane of incidence
and the incident waves have their electric fields polarized parallel to the same plane.
Recently, Stamnes and Sherman [22] obtained exact solutions for the reflected and
transmitted fields which result when an arbitrary electromagnetic wave is incident
on a plane interface separating two uniaxial media, Their results, however, are not
in a useful form for treating birefringent layered media.
This paper describes a general theory of electromagnetic plane wave propagation
in birefringent layered media. The theoretical approach is general, so that many
situations considered previously will be shown to be special cases of the formalism.
New concepts of dynamical matrix as well as propagation matrix are introduced:
this makes it possible to write out the transfer matrix in terms of diagram representation.
2. Propagation
of plane waves in homogeneous
anisotropic
media
First a brief review of the propagation of plane waves in homogeneous anisotropic media is in order. The approach and formulation are different from the traditional one [23] because of the demands of their appli~tion
in this problem. The
43
P. Yeh / Optics of anisotropic layered media
Cartesian coordinate system is chosen such that the z axis is normal to the interfaces. Since the medium is not isotropic, the propagation characteristics depend on
the direction of propagation, The orientations of the crystal axes are described by
the Euler’s angles 8, #, J/ with respect to a fixed xyz coordinate system. The
dielectric tensor in the xyz coord~ate system is given by
where el, ez e3 are the principal dielectric
tion matrix, which is given by f24] :
J, cos d, - cos 0 sin @sin $
constants
and A is the coordinate
rota-
-sin$cosQ,-cosOsin@cosJI
sin 0 sin ct,
-sin*
-sin B cos # . (2)
sin#+cosB
sin B cos I$
cos~cosJ,
cos e
:
Since A is orthogonal, the dielectric tensor E in xyz coordinate must be symmetric,
i.e., eff = eii. The electric field can be assumed to have exp[i(m t (3~ + ~2 - wt)]
dependence in each crystal layer, which is assumed to be homogeneous. Since the
whole birefringent layered medium is homogeneous in the xy plane, LYand @remain
the same t~oughout
the layered medium. Therefore, these two com~nents
(QI,8)
of the propagation vector are chosen as the dynamical variables to characterize the
electromagnetic
waves propagating in layered media. Given a: and /I, the z component y is determined directly from the wave equation in momentum space:
kX(kX~~~2~~~=0,
(3)
or equivalently
In order to have nontrivial plane-wave solutions, the determinant of the matrix in
eq. (4) must vanish. This gives a quartic equation in y which yields four roots yo,
u = 1, 2, 3, 4. These roots may be either real or complex. Since all the coefficients
of this quartic equation are real, complex roots are always in conjugate pairs. These
four roots can also be obtained graphically from fig. 1, if they are real. The plane of
incidence [25] is defined as the plane formed by CY&
+ #I$ and i. The intersection of
this plane with the normal surface yields two closed curves which are symmetric
44
Fig. 1. Graphic
method
P. Yeh /Optics
of‘anisotropic
to determine
the propagation
layered
media
constants
from the normal
surface
with respect to the origin of the axes. Drawing a line from the tip of the vector c& +
0-9 parallel to the z^direction yields, in general, four points of intersection. These
four wave vectors k, = ok + /3j + y$ all lie in the plane of incidence, which also
remains the same throughout the layered medium because o( and fl are constant.
However, the four group velocities associated with these partial waves are, in
general, not lying in the plane of incidence. If all the four wave vectors k, are real,
two of them have group velocities with positive z component and the other two
have group velocities with negative z component. The z component of the group
velocity vanishes when Y,, becomes complex. The polarization of these waves is
given by
where u = 1, 2, 3, 4 and NO’s are the normalization
constants such that fi. p = 1.
The electric field of the plane electromagnetic waves can thus be written
4
E=~A,ri,exp[i(ax+&+y,z-wt)].
(6)
0=1
Partial
waves with
complex
propagation
vectors
cannot
exist in an infinite
P. Yeh / Optics of‘anisotropic
layered media
45
homogeneous birefringent
medium. If the medium is semi-infinite, the exponentially damped partial waves are legitimate solutions near the interface, and the field
envelope decays exponentially
as a function of z, where z is the distance from the
interface. These exponentially
damped partial waves are called evanescent waves.
The evanescent waves in birefringent
media in general have complex y’s, i.e., y =
yK t iyt. In a uniaxially birefringent medium, the ordinary evanescent wave has a
purely imaginary y. If the three principal dielectric constants are all real, then these
partial waves with complex y’s can be shown to have their Poynting vectors parallel
to the interface. In other words, the energy is flowing parallel to the interface and
the propagation is lossless as it should be. A mathematical proof is given in appendix A of ref. [lo] for the special case of extraordinary
evanescent waves in a
uniaxially birefringent medium.
3. Matrix method
The 4 X4 matrix algebra, which analyzes the propagation of monochromatic
plane waves in birefringent layered media, can now be introduced. The approach is
general so that the results will be used later on for many special cases of propagation in anisotropic layered media. The materials are assumed to be nonmagnetic so
that 1_1=constant throughout the whole layered medium. The dielectric permittivity
tensor e in xyz coordinates is given by
40)
3
z<z,,
41),
z.
<z <z,,
4%
z1
<z <z,,
e=..
(7)
ZN-1
.?N
<i! <zN,
<z.
The electric field distribution
within each homogeneous anisotropic layer can be
expressed as a sum of those four partial waves. The complex amplitudes of these
four partial waves constitute the components of a column vector. The electramagnetic field in the nth layer of the anisotropic layered medium can thus be represented by a column vector A Jn), u = 1,2, 3,4.
As a result, the electric field distribution in the same nth layer can be written
4
E=
oGl
hAn)Mn)
ew{i[m
+Pv
+r&)(z
-
z,)
-
atI 1.
(8)
46
i? Yeh / Optics of‘anisotropie
layered media
The cofumn vectors are not independent of each other. They are related through
the continuity conditions at the interfaces. As a matter of fact, only one vector (or
any four components
of four different vectors) can be arbitrarily chosen. The
magnetic field distribution is obtained by using Maxwell’s equations and is given by
4
A.(n)(i,(12)expCi[axfpy+y,(n)(z-z,)-wtl},
ff=g,
B*(n) = (QWYW)
Q,(n)
(9)
2
(10)
k,(n) = a.2 + pj + y*(n) 3 I
(1x1
Note that the ijO(n
are no longer unit vectors.
Imposing the continuity of EX, E.V,H,, and H.,, at the interface z = z,,_t Ieads to
5 A&r - l)ri,(n
u=r
- 1) *g= 5 A.(n)~~i,(n).~exp[-iy,(n)t,],
0=1
(12)
$r A,@ - l)&(fi
- 1) *P= kr
(13)
@$ A&
- I)?&
- 1) *.%= 5 A,(n)
0=1
Q$r A&
- l)c?,(n
- 1)-P = ~~n,(i,)g~~)-pexpi-iroDI)f,,i,
where f, = z, -- z,
~~(~z)~~(~)
-.PexpI-ly&)f,l,
ci,(n)* .? exp[-kyo(n)r,]
,
(14)
(151
, (12= 1,2, ... . N.
These four equations
=fl-‘(n
can be rewritten
as a matrix equation
- 1) D(n) P(n)
where
(17)
41
P. Yeh / Optics of anisotropic layered media
rev[-irl(fl)f,l
0
0
P(n)= i
0
0
0
0
0
0
exp[-ir&)bJ
0
0
0
0
exp[-ir,(~~M1.
(18)
exp[-ir,(n>t,l
The matrices D(n) are called dynamical matrices because they depend only on
the direction of polarization of those four partial waves. The dynamical matrices
are defined in a way such that they are block-diagonalized
when the mode coupling
disappears. This requires that Ar and A2 are the amplitudes of the plane waves of
the same mode (polarization)
such that the plane wave with amplitude Ar propagates to the right, whereas the plane wave with amplitude Aa propagates to the left.
Likewise A3 and A4 are the amplitudes of the plane waves of the same mode, propagating, respectively to the right or left. The matrices, P(n),
are called propagation
matrices, and depend only on the phase excursion of these four partial waves. The
transfer matrix is defined as
Tn- 1 ,n =ryn
- l)D(n)P(n).
09)
Eq. (16) can thus be written
= Tn-r,n
The matrix equation
which relates A(0)and A(s)is therefore given by
wheres=N+
1 and tN+l20.
Eqs.(16) and (21) show how systematic the matrix method is for treating electromagnetic
propagation in anisotropic layered media. If eq. (21) is represented
graphically in fig. 2, two dynamical matrices can be seen to be associated with each
layer. The overall transfer matrix is the product of all these matrices from left to
right. This completes the theoretical formulation of the 4 X 4 matrix method.
The matrix method just described is an exact approach to the propagation of
48
P. Yelr / Optics of anisotropic
0
laJ,ered media
1
D-‘(0
) Il(N)
(1) P(1) D-l{1
Fig. 2. Diagram
S
N
representation
P(N)
of ‘tl he matrix
D-l(N
(S)
method.
electromagnetic
radiation in anisotropic layered media. Both birefringent phase
retardation and thin film interference are considered. This differs from the traditional 2 X 2 Jones calculus [26] which neglects the reflection from each interface.
Therefore, in calculating the transmission and reflection properties of some birefringent filters, the results obtained from these methods are expected to be different. In fact, they are only very different in the fine structures of the spectral
responses [27,28] when the birefringence of the material is small.
In addition, there are several interesting optical phenomena in periodic birefringent layered media which have been analyzed by using this 4 X4 matrix algebra.
These include the indirect optical bandgap, exchange Bragg reflection and exchange
Sole-Bragg transmission. Details can be found in ref. [IO].
4. Reflection
and transmission
The matrix method just discussed is very useful in the calculation of the reflectance and transmittance
of an anisotropic layered medium. Because of the anisotropy of the medium, mode coupling appears at the interfaces. Therefore, there are
four complex amplitudes associated with the reflection and another four associated
with the transmission. These eight complex amplitudes can be expressed in terms of
the matrix elements of the overall transfer matrix. To illustrate this, one considers,
without loss of generality, the case of an anisotropic layered medium sandwiched
between two isotropic ambient and substrate media. Assume that the light is
incident from the left side of the structure, and let A,,A,, B,, B, and C,, C, be the
incident, reflected, and transmitted electric field amplitudes, respectively. By employing the matrix method described in section 3, a transfer matrix can be found
for any given anisotropic layered structure such that
As
4
=
A,
M,,
MS, M33 M34
B p,
_M 4’
A442
M43
M44
!
(22)
CP
IJ0
P. Yeh /Optics of anisotropic layered media
The reflection and transmission
the matrix elements as follows:
coefficients
are defined and expressed in terms of
(23)
rss
M,,M33 -M&f,,
YSP
A~=O =%M,,
Ml,43
-M,,M,,
A~=O =M,,M,,
(25)
9
(26)
,
(27)
-M,,M,,
-4
=
3
-M,df31
MS3
t ss =
(24)
-M,,M,,
=%M,,
t sp
49
/ip=o =%M,,
1
- M,,M,,
’
-Ml,
=%M,,
c
(1
t,, = $
p A,=0
(28)
(29)
-M&f,,
Ml1
(30)
=M,,M,,-M,,M,,’
These reflection and transmission formulas are extremely useful in the calculation of the spectral response of an anisotropic layered structure. The matrix elements are obtained by carrying out the matrix multiplication
in eq. (21). The
general explicit forms are normally not available. For fast results, a computer
program is in general required. Even for the special case of periodic layered medium, closed forms for the reflectance and transmittance
are too complicated to
derive. These eight complex amplitudes are spectrally correlated (see ref. [lo]).
5. Ellipsometry
of anisotropic
layered structures
Ellipsometry has long been recognized as one of the most accurate techniques
for determining the optical properties of materials. The basic mathematical problem
involved for the case of isotropic films has been discussed by VaCic”ek [29] and
many other workers [30.31]. In recent years, many mathematical techniques have
been developed for the ellipsometric study of anisotropic layered media; many of
these are special cases of the uniaxial system, with the optic axis either perpendi-
50
P. Yeh / Optics of anisotropic layered media
cular or parallel to the surface [32-381. The differential 4 X4 matrix method
described by Berreman 1121 provides a much more general approach to the problem of stratified anistropic media, including continuous variation of the refractive
indices in the media. De Smet [39] has recently presented a paper at the Third
Conference on Ellipsometry in which he discussed a 4 X4 matrix formalism. This
4 X 4 matrix formalism also used the amplitudes of the electric fields and magnetic
fields as the elements of a column vector which is function of position. In the
matrix method discussed in secion 3, a constant column vector is associated with
each layer,
The new 4 X4 matrix algebra just discussed is very useful in calculating the
reflectance and transmittance
amplitudes which are externally measurable via the
ellipsometric techniques. For example, the ellipse of polari~tion
for the reflected
light can be expressed in terms of the matrix elements as
(31)
where xi represents the ellipse of polarization for the incident light. In the ellipsometric determination
of the refractive indices, crysta1 axes orientations
and
thicknesses of the films several independent measurements obviously must be made.
Since there are so many variables involved, a computer program is generally
required to determine these unknowns efficiently.
6. Guided waves
Birefringent multilayer waveguides, especially titanium diffused lithium niobate
waveguides [40,41], are becoming increasingly important in integrated optics. The
waveguiding principle is similar to that of the isotropic case. Waves are to be
evanescent in the regions outside the guiding layer. The propagation characteristics,
however, depend on the direction of propagation. The analytic treatment for the
general multilayer birefringent
waveguide suffers from the serious difficulty of
solving an eigenvalue problem involving a large number of simultaneous linear equations. A systematic approach is to use the matrix method described in section 3
which involves the ~nipulation
of 4 X 4 matrices.
As a result of successive matrix multip~~tions,
a linear relation between the
fields on both sides of a finite birefringent layered medium is obtained. The reflectance and transmittance
coefficients have been shown to be expressible in terms of
the elements of the overall transfer matrix. It will be shown in the following that
the poles of the reflectance coefficients play an important role in the guided-mode
theory of birefringent layered media.
A basic problem in particle physics is that the poles in the scattering amplitude,
which are assumed to dominate the scene, correspond to exchange of particles
carrying definite angular momentum
[42]. In other words, a resonance scattering
P. Yeh /Optics of anisotropic layered media
51
corresponds to an eigenstate of the composite system. It was suggested by Regge
[43] in 1959 that the angular momentum be treated as a complex continuous
variable. In particle scattering, the angular momentum corresponds to the impact
parameter, while in the optics of birefringent layered media the direction of incidence (or equivalently, o. and /3) is the corresponding variable. The a! and /I variables
can now be extended into complex variables, and the poles of the reflectance amplitudes, which correspond to the scattering amplitudes, can be sought. In general,
the poles occur at complex values of 01and f3, and each of these poles corresponds
either to a guided mode or to a leaky mode.
From eqs. (23) through (30) it was concluded that the poles of the reflectance
amplitudes occur at
MllM,, -~13~31
=o.
(32)
It is important to notice that at the poles of the reflectance amplitudes, the reflectivities are infinite. In order to fulfill the finiteness of the electromagnetic field, the
solution of the Maxwell equations consists of outgoing waves only. Eq. (32) is
actually the mode dispersion relation
(33)
for a given birefringent
w = w3,@p>
7
p,
=
layered structure.
Eq. (33) can also be written
(34)
(a't /3”)‘“,
0 n = tan-’
(fl/o)
(36)
The subscript en in eq. (34) indicates that the dispersion relation between o and &,
depends on the direction of propagation defined by 0, in the xy plane. In order to
be a confined mode, the field amplitude must decay to zero at infinity (z = +m).
Therefore, the propagation constant, /In, must be big enough so that the z components of the propagation vectors (i.e., 7,) are complex. Outgoing waves with
complex propagation constant are evanescent waves. Therefore, the optical energy
is guided by the structure and propagates parallel to the layers.
Because of mode coupling, pure TE or TM waves, in general, do not exist.
Except for some cases with special crystal orientations,
most of the guided modes
are a mixture of ordinary waves and extraordinary
waves. Another distinct feature
of the guided waves in birefringent layered structure is the evanescent waves in a
birefringent substrate. In the case of isotropic media, the evanescent waves have a
pure imaginary propagation constant. This however, is no longer true in birefringent
layered media. A guided wave in birefringent layered waveguide has, in general, two
complex y’s in the birefringent substrates. This makes the evanescent wave decay
exponentially with an oscillatory intensity distribution.
52
P. Yeh /Optics
of‘anisotropic
layered
media
7. Conclusion
A new 4 X 4 matrix algebra has been developed for investigation of the propagation of electromagnetic
radiation in anisotropic layered media. The concepts of
dynamical matrix and propagation matrix have been introduced
to clarify this
method and to make it systematic. Diagram representation is also made possible via
the use of these two matrices. Applications of this new matrix method in the ellipsometry of anisotropic layered media, as well as the guided wave in these media,
have been illustrated.
References
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[lo]
[ll]
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[13]
[14]
[15]
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[ 17 j
[18]
[lP]
[20]
[21]
[22]
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[24]
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D.W. Berreman, J. Opt. Sot. Am. 62 (1972) 502.
See, for example,
R.M.A. Azzam and N.M. Bashara, Ellipsometry
and Polarized
Light
(North-Holland,
Amsterdam,
1977) section 4.7, p. 340.
A. Wunsche, Ann. Physik(Leipzig)
25 (1970) 201.
J. Schesser and G. Eichmannb,
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C.B. Curry, ElectromagneticTheory
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Ii. Schopper, 2. Physik 132 (1952) 146.
A.B. Winterbottom,
Kgl. Norske Videnskab.
Selskab, Skrifter 1 (1955) 17.
A.M. Goncharenko
and F.J. Federov, Opt. Spektrosk.
14 (1962) 94; Opt. Spectrosc.
14
(1963) 48.
J. Schesser and G. Eichmann,
J. Opt. Sot. Am. 62 (1972) 786.
D.A. Holmes and D.L. Feucht, J. Opt. Sot. Am. 56 (1966) 1763.
J.J. Stamesand
G.C. Sherman, J. Opt. Sot. Am. 67 (1977) 683.
M. Born and E. Wolf, Principles of Optics (Pergamon,
Oxford, 1964).
See, for example,
H. Goldstein,
Classical
Mechanics
(Addison-Wesley,
Reading,
MA,
1965).
[25] See, for example,
[26]
[27]
[28]
[29]
[3O]
J.M. Stone,
tion 17-5.
R.C. Jones, J. Opt. Sot.
Ref. [lO],p.
752.
P. Yeh, Opt. Commun.
A. Vasicek, J. Opt. Sot.
See, for example, O.S.
1965).
Radiation
Am. 31 (1941)
and Optics
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New York,
1963)
sec-
488.
29 (1979) 1.
Am. 37 (1947) 145.
Heavens, Optical Properties
of Thin Solid Films (Dover,
New York,
P. Yeh /Optics ~~arlisoiro~ic layered media
53
[ 311 See, for example, ref. [ 131, ch. 3.
[32] R&W. Graves, J. Opt. Sot. Am. 59 (1969) 1225.
[33] D.J. D&yam and M. Moskovits, Appl. Opt. 9 (1970) 1868.
[34] D. den Engelsen, J. Opt. Sot. Am. 61 (1971) 1460.
[35] D.J. De Smet, .I. Opt. Sot. Am. 63 (1973) 958.
[36] D.J. De Smet, J. Opt. Sot. Am. 64 (1974) 631.
1371 R.M.A. Azzam and N.M. Basbara, J. Opt. Sot. Am. 64 (1974) 128.
[38] M. E~hazIy-ZaghIou~ R.M.A. Azzam and N.M. Bashara, Surface Sci. 56 (1976) 293.
[39] D.J. De Smet, Surface Sci. 56 (1976) 293.
[40] I.P. Kaminow and J.R. Carruthers, AppL Phys. Letters 22 (1973) 326.
[41] R.V. Schmidt and I.P. Kaminow, Appi. Phys. Letters 25 (1974) 458.
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[43] T. Regge, Nuovo Cimento 14 (1959) 9.51.
Discussion
J.B. Theeten (Laboratories D’Electronique et de Physique Appliquee): In your calculation
on exchange Bragg scattering in those layered structures, what do you expect the effect of a
nonabrupt transition between two successive layers to be on the “quality” of the guided wave?
P. Yeh: In the event when there is a non-abrupt transition between two successive layers,
the phe~amena of guided waves and exchange Bragg reflection still exist. Wowever, the matrix
method developed here becomes only an approximation to the exact solution provided the
transition region near the interface is much smaller than the wavelength. Exchange Bragg reflection and guided waves normally happen under different conditions, i.e., different LYand 0.