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JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 6, NO. 6, JUNE 1988
Single-Mode Optical Waveguides and Directional
Couplers with Rectangular Cross Section: A
Simple and Accurate Method of Analysis
ANURAG SHARMA, PRASANNA K. MISHRA, AND AJOY K. GHATAK
single-mode waveguides, in which the propagating mode
has a substantial fraction of its power outside the channel
region. Hence, Marcatili's analysis leads to considerable
error for such practical waveguides. This approximation
was improved upon by Knox et al. [10] by introducing
the concept of "effective index." Although the effective
index method provides a better approximation and has
generally been used [10]-[12], the accuracy that can be
obtained by this method for single-mode waveguides is
only marginally better than that of Marcatili's method.
Perturbation [13], [14] or variational [15] methods can be
used to improve the accuracy of the propagation constant
evaluation to some extent; however, these do not improve
the accuracy of the modal field calculation.
I. INTRODUCTION
We propose here a simple approximation for the propINGLE-MODE optical waveguides with rectangular
cross sections are the building blocks of most inte- agation characteristics of single-mode rectangular wavegrated optical devices; hence, a knowledge of their prop- guides and show that it is extremely accurate in the pracagation characteristics is important for the design of such tical region of single-mode operation. The method is based
devices. The evaluation of the propagation characteristics on the scalar variational analysis with a new and more
involves the solution of the electromagnetic boundary accurate trial field [16]. This trial field enables us to devalue problem for the dielectric guiding structure. How- fine equivalent waveguiding structures consisting of slab
ever, it is not possible to obtain analytical solutions of the waveguides. Using the equivalent guiding structures, we
boundary value problem for rectangular waveguides; one have obtained,x in a simple way, the two fundamental vechas to either use numerical techniques [l]-[8] or develop tor modes E u and £{, [9] of the original rectangular
approximate solutions [9]-[15]. The numerical tech- waveguide [17]. A comparison with available exact nuniques, such as circular harmonic analysis [1], direct nu- merical results shows that the results obtained using the
merical integration of the field equations [2]-[3], finite present method are much more accurate than those obelement anlaysis [4], field expansion in orthogonal func- tained using other approximations; hence, the present aptions [5]—[8], etc., require extensive computations and do proximation is more suitable for evaluating characteristics
not lead to simple analytical forms for the modal fields. of single mode waveguides. Furthermore, the equivalent
This has led to the development of various approximate guiding structures obtained in our method also greatly
simplify the analysis of composite guiding structures such
methods.
as directional couplers [17].
The earliest approximate method is due to Marcatili [9],
and it gives good accuracy for a well-confined mode.
However, most integrated optical devices are based on
II. THE SCALAR MODE
Abstract—A new method for obtaining the propagation characteristics of single-mode optical waveguides with rectangular cross section
is presented. The method is based on the scalar variational principle
using the cosine-exponential trial field. This form of trial field leads to
the definition of equivalent guiding structures consisting of homogeneous slab waveguides, which are then used to obtain, in a very simple
way, the vector modes of a rectangular waveguide and the coupling
characteristics of directional couplers consisting of two rectangular
waveguides. We have also included the results of some representative
calculations along with the results obtained using other approximate
methods and the exact numerical method. A comparison shows that
our method is much more accurate than the existing approximate
methods, particularly in the region of single-mode operation.
S
Manuscript received October 9. 1987; revised December 21, 1987. This
work was partially supported by the Department of Science and Technology (India) and The National Bureau of Standards (U.S.A.).
A. Sharma and A. K. Ghatak are with the Physics Department. Indian
Institute of Technology Delhi, New Delhi-110016. India.
P. K. Mishra was with the Physics Department, Indian Institute of Technology Delhi, New Delhi-110016, India, on leave from G. M. College,
Sambalpur-768004, Orissa. India.
IEEE Log Number 8820204.
As mentioned above, our method consists of first obtaining a simple and accurate model for the modal field in
the scalar approximation and then using this model to define equivalent guiding structures, which allow accurate
approximation of the vector modes, Exu and Eyu. In this
section we discuss the model for the scalar mode.
We consider a waveguide with a refractive index dis-
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JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 6. NO. 6, JUNE 1988
tribution given by
y
n-(x, y) = nc,
all JC
y > b,
n
x < a
< b,
= K,
n
2
= n p,
y
x > a
< b,
t
%
p2t>
<
1
(a)
y < -b,
all JC
(1
= n
where all dimensions are defined in Fig. l(a). Such a
structure reduces to a rectangular waveguide when np =
nc = ns < n0, to a channel waveguide when nc < np =
ns < no, and to an embossed waveguide when nc = np <
ns < no.
A suitable model for the scalar mode can be obtained
by assuming a modal field distribution of appropriate form
involving adjustable parameters and using this field in the
variational analysis for the propagation constant (3. The
values of the parameters that maximize (3 define a field
which closely approximates the actual modal field, and
the value of (3 thus obtained is also very close to the actual
propagation constant [18]. The accuracy of the modal field
and the propagation constant (3 thus obtained depends on
the choice of the trial field. The trial field can be either a
single function or a series of orthogonal functions. The
latter class of trial fields require extensive computations
and are more suitable for use with multimode waveguides
[5]-[8]. A variety of single function trial fields have been
suggested and used [16], [19]—[21], but of these the
cosine-exponential trial field gives the best results in the
single mode region as we have shown in an earlier paper
[16]. The field is assumed to be separable in its x- and ydependences:
* ( x , y) = * v ( x ) * v ( y )
n
n
y2
n
yi
n
y3
c
n
-2a
T
t
-x
1
*•
(c)
s
n
x2
nx2
x1
2a
< —
2b
*•
(b)
Fig. 1. Schematic of (a) a waveguide with rectangular cross section, (b)
the equivalent .r-slab, and (c) the equivalent j'-slab waveguides.
where k0 = co/c is the free-space wavenumber. Substitution for "^ (x, >•) from (2)-(4) in (5) and straightforward
integration result in a closed form expression for /3~ in
terms of/?, q, and a [16]. The value of (32 is then maximized with respect to p, q, and a using a standard optimization routine. The main advantage of this form of the
trial field lies in the fact that both Vx(x) and ¥ v ( v) correspond to the exact modal field of some slab waveguide
which can be defined once /?, q, and a are known. This
allows us to define an equivalent guiding structure for the
rectangular waveguide, as discussed in Section III.
In order to illustrate the accuracy of the cosine-exponential trial field, we have included here a typical example. We consider a rectangular waveguide (nc = np = ns)
with no — ns « no with an aspect ratio of b/a = 0.5.
The variation of the normalized propagation constant
(2)
and °fyx and ¥,. are given by
D
^ v ( x ) = A\ cos (px/a),
(6)
\x\ < a
= Ax cos (p) exp [-/?{|x/a| - 1} tan (p)],
as a function of the normalized frequency
V = koa(n;, - ni)
(3)
x\ > a
and
^ v ( y ) = A2 cos [(qy/b) - a],
is shown in Fig. 2 along with the results obtained by using
other approximate methods and the exact numerical results [1]. More examples to illustrate the accuracy of using
the cosine-exponential trial field are given in [16].
\y\ < b
= A2 cos (q - a) exp \-q{{y/b) - 1 }
• tan (q — a ) ] ,
(7)
y > b
III.
= A2 cos (q + a) exp [q{{y/b) + l}
• tan (q + a)],
y < -b
J
r\
OO
(EGS)
As mentioned in Section II, the forms of ¥x(x) and
^ v ( >') correspond to the exact scalar modes of step-index
slab waveguides with index variation along the jc-axis and
the j-axis, respectively. Thus, one can define a slab
(4)
where p, q, and a are the three adjustable parameters,
suitable values of which are obtained by maximizing the
stationary expression for /3 (see, e.g., [18]):
OO
THE EQUIVALENT GUIDING STRUCTURES
r*
OO
rt
dx d\
— oo J — oo
n\x,y)
OO
J
| * ( x , y)f d x d y ft
CO
—
oo
, y)
— OO t) —OO
dx dy
J
(5)
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SHARMA ei al.: WAVEGUIDES AND COUPLERS WITH RECTANGULAR CROSS SECTION
0.5
The determination of HVI and nvl poses some problem,
which can be solved in the following way.
Since p and q represent the transverse phase constants
along the x- and the y-directions, respectively, and 0 is
the phase constant along the z-direction, these must satisfy the relation
0.4
p2 + q2a2/b2 + (32a2 = (konoa)
0.3
(12)
within the accuracy of the variational method. By analogy, we can define nxX such that
(konxlaf
0-2
0-1
(1969)
Kumar etal.(1963)
Marcatili (1969)
Eft. index
0-0
0.6
V/TT
0.4
0.8
1.0
Fig. 2. Normalized propagation constant B as a function of normalized
frequency V/TT for the fundamental scalar mode of a rectangular waveguide.
waveguide given by
n2{x) = n 2 ,,
x/a\ < 1
x/a\ > 1
(8)
such that ^. V (JC) is the solution of the scalar wave equation
for this index distribution. The difference nx] — nx2 can
be completely deduced from the variation of the field itself; thus, using the scalar wave equation, we obtain
=p2 + (32a2
- n
x2)
=
(13)
It may be noted that as the dimension of the waveguide
along the y-axis increases, qa/b decreases, approaching
zero as the dimension tends to infinity (i.e., when the
rectangular waveguide reduces to a slab waveguide), and
nx\ approaches no, which should indeed be the case. Thus,
nxX from (13) and nx2 from (9) define a slab waveguide
with dielectric interface in the y-z plane; we shall refer to
this as the jc-slab. This x-slab is such that the propagation
constant (3 and the phase constant along the x-axis p, for
its mode are the same as those for the mode of the actual
waveguide (within the accuracy of the variational
method). Therefore, this x-slab is equivalent to the given
waveguide as far as the characteristics along the x-direction, i.e., those characteristics which have to take into
account the y-z interface, are concerned. For example,
for obtaining the coupling characteristics of a directional
coupler with its constituent waveguides separated along
the x-axis (see Section V) or for obtaining the vector mode
E\\, which behaves like a TM0 mode in the x-slab, it
would be sufficient to consider the x-slab (see Section IV).
In terms of the normalized parameters, the x-slab is defined by
n\x)
2
= (k0noaf - q2a2/b2.
=
n2x,
=nl-(U2
-P2)/(koa)\
x/a\ < 1
dx2
= n\2 = n2x] - p2 sec2 (p)/(koa)',
dx1
^x
x/a
> 1
(14)
\x\>a
where
= (p/af + (p/af tan2 (p)
- = a\k-on-o - (3 )
which gives
15)
and the corresponding wave equation would be
n
2
2
«.vi ~ x2 = P sec (p)/{koa)~.
(9)
Similarly, one can define an equivalent _y-slab waveguide
n\y) = n2,,
\y/b\ < 1
= n;2,
2
= « 3,
d2^x/dx2 + [k;,n2(x) - (32} * , ( * ) = 0.
In an exactly similar manner, one can define the y-slab in
terms of the normalized parameters as:
n\y) = n2,
y/b > 1
y/b < - 1
(16)
= n20 - (U2 - q2a2/b2)/(koa)\
y/b\ < 1
(10)
= n\.2 = n2y - q2 sec2 (q - o)/(kob)~,
with
y/a > 1
= «
n2, - «2.3 =
2
?
sec2 (9 + a)/{k,,bf-
2
3
= n]\ - q2 sec2 (q + o)/{k0b)~,
y/a < - 1
(17)
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JOURNAL OF LIGHTWAVE TF.CHNOLOGY. VOL. 6. NO. 6. JUNE 1988
and the corresponding wave equation would be
2
2
d *y/dy
2
+ [kW(y) - (3 ] %{y) = 0.
tained from
We shall make use of these equivalent guiding structures (EGS) to obtain the vector modes of a rectangular
waveguide in Section IV and to obtain the coupling characteristics of directional couplers consisting of two parallel rectangular waveguides in Section V.
IV.
THE VECTOR MODES
In general, a rectangular waveguide which supports a
single scalar mode would allow two orthogonally polarized vector modes, namely E\x and E\x [1], [9], which
have slightly different propagation properties. These two
modes, if coupled by unavoidable irregularities, may
cause undesirable effects when a polarization sensitive device is used at the end of the waveguide. Thus, in order
to design efficient integrated optical devices, such as directional couplers, filters, mode converters, etc., it is of
interest to study the vector modal properties of such rectangular-core waveguides. There are few methods available for this purpose. Of these the circular harmonic analysis by Goell [1] requires extensive numerical
computations, and the closed form solution given by Marcatili [9] essentially neglects the modal field in the corner
regions, thereby resulting in poor accuracy. We make use
of the EGS discussed in Section III to obtain the vector
modes of a rectangular core waveguide.
The two fundamental vector modes E\x and E\x are hybrid in nature and all the components of the electric and
the magnetic fields are nonzero. However, the E\x mode
has strong Ex and Hy components, with its other components described by
Ex, Hy
:,
H.
E,, Hx
Hence, it behaves as a TM0 mode in the x-slab (of EGS)
and as a TE0 mode in the y-slab. Thus one can approximate Ex by
(19)
Ex = Fx{x) Fy(y)
(p2
(18)
where Fx(x) corresponds to the TM0 mode of the x-slab
and Fv(y) corresponds to the TE0 mode of the y-slab.
Hence
V.
DIRECTIONAL COUPLERS
A directional coupler consists of a pair of waveguides
placed parallel to each other so that the evanescent field
of one waveguide penetrates into the other waveguide and
couples into the latter waveguide's propagating mode. If
one of the two waveguides is excited, then after a certain
length, most of its power is transferred to the other waveguide. The length of the coupler at which maximum power
transfer occurs is referred to as the coupling length L,..
The coupling length depends upon the structure and refractive indices of the constituent waveguides and the intervening isolation region. These directional couplers are
used in a variety of integrated optical devices such as
power dividers, input-output couplers, modulators, filters, switches, etc.
In this section we again use EGS and obtain the coupling characteristics of a directional coupler consisting of
two identical rectangular-core waveguides (see Fig. 4).
Since the two waveguides are separated along the x-axis,
we use the equivalent x-slab waveguides. The refractive
index distribution of the equivalent slab coupling structure is given by (see Fig. 4)
< 2a + d
d <
(20)
< d
and
Fx(x)
(23)
where (3 is the scalar propagation constant. For the £",,
mode, one can carry out a similar analysis by finding the
eigenvalue corresponding to TM0 mode of the y-slab.
To illustrate the accuracy of the method, we consider a
rectangular waveguide with n0 - 1.5, ns = nc = np =
1.0 and aspect ratio b/a = 0.5, for which the exact results are available [1]. The variation of the normalized
propagation constant B as a function of K/TT for the two
fundamental vector modes is shown in Fig. 3, along with
the results obtained by using Marcatili's method [9] and
the exact numerical results of Goell [1]. As can be seen
from the figure, our approximation gives an extremely accurate description of the fundamental vector modes in the
region of interest for the operation of integrated optical
devices, i.e., in the wavelength region just below the cutoff of the next higher mode, which is about V = 0.8TT [1].
n2(x) = n
FAy) = %(y)
~p2)/a2
=
(\/n2x) c o s (px/a),
\x\
<
> 2a + d
a
= (l/«.v2) cos (p)
• e x p [ - a { ( \ x \ / a ) - []],
\x\ > a
(21)
where a~ = p~ sec" (p) — p~ andp is obtained by solving
the eigenvalue equation corresponding to the TM0 mode
of the x-slab; that is
p tan (p) = a(nxX/nx2)~.
(22)
The propagation constant, (ix, of the E]x mode is thus ob-
(24)
where 2d is the separation between the two waveguides
and nxi, nx2 are defined in Section III. Thus, the problem
of coupling between two rectangular waveguides reduces
to a much simpler problem of coupling between two slab
waveguides, and one obtains an analytical expression for
the coupling length L( (see, e.g., [23])
U_
(3a2
=
TT[1 +/?tan
2p2 sin2 (p)
exp [p tan (p){2d/a}]
(25)
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SHARMA el al.: WAVEGUIDES AND COUPLERS WITH RECTANGULAR CROSS SECTION
05,
0.4
n =1.0
0.3
x X x x x GOELL
PRESENT
MARCATILI
0.2
0.1
0.0
1.0
Fig. 3. Normalized propagation constant fi as a function of normalized
frequency V/ -K for the fundamental vector modes of a rectangular waveguide.
n
n t
P
n
2b
_L
(a) <
*
n
x?
c
n
P
o
n
n
P
o
2b
n
y2
n
yi
\
2
a
2a
n
x1
<
* n
»«-2d—> <
n
x2
2a
2a
n
x1
»
(O
n
y3
^
n
x2
(b)
Fig. 4. Schematic of (a) the directional coupler consisting of two identical
rectangular core waveguides, and (b and c) the equivalent slab coupling
structure.
where /3 and p correspond to each of the two constituent
waveguides and are obtained from the variational analysis
as discussed in Section II. In Fig. 5, we have shown the
variation of the normalized coupling length Lc/(la2 as a
function of the normalized separation d/a for a typical
directional coupler consisting of two identical rectangular
waveguides. The corresponding calculations using Marcatili's method [9] lead to considerably different results
for the coupling length, as shown in Fig. 5. The differ-
ence is expected due to the inherent error in Marcatili's
method, as shown in Fig. 2.
VI. SUMMARY AND CONCLUSION
We have developed a new method to obtain the characteristics of homogeneous waveguides in rectangular
cross sections. The method, which is based on the scalar
variational calculations, defines for the given waveguide
equivalent guiding structures which can then be used to
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JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 6. NO. 6. JUNE 1988
|11]
|12]
113]
114]
115]
|16]
|17]
|181
119]
|20]
|21]
Fig. 5. Variation of the normalized coupling length Lc as a function of the
normalized separation d / a for a rectangular waveguide directional coupler with /!,, = 1.5, n, = n% = np = 1.0 and bja = 0.5.
[22]
[23]
obtain the vector modes of the waveguide and the coupling characteristics of a directional coupler made up of
two such waveguides. The accuracy of the method and of
the calculation of vector modes is illustrated with examples, and it is shown that this approach gives much more
accurate results than those obtained using other approximate methods.
REFERENCES
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Anurag Sharma was born in Bareilly. India, on
May 7, 1955. He received the B.Sc. degree from
Agra University, Agra. India, in 1972, and the
M.Sc. degree in physics, the M.Tech. degree in
applied optics, and the Ph.D. degree in optical
waveguides from Indian Institute of Technology,
Delhi. India, in 1974. 1976. and 1979. respectively.
In 1978 he joined the Department of Physics,
Indian Institute of Technology, Delhi, where he
JSi,
is currently an Assistant Professor of Physics.
During 1982-1983. he visited the Institut fur Hochfrequenztechnik and
Quantenelektronik, Universitiit Karlsruhe, Karlsruhe, West Germany, as
an Alexander von Humboldt Fellow.
Dr. Sharma is a recipient of the Indian National Science Academy's
Medal for Young Scientists (1986) for his contributions in the gradient
index optics. He is also an Associate Member of the International Center
for Theoretical Physics, Trieste (Italy). His current research interests are
in fiber and integrated optics, gradient-index optics, and optics of visual
photoreceptors.
Prasanna K. Mishra was born in Sambalpur,
Orissa, India, on September 1, 1953. He received
the B.Sc. and M.Sc. degrees from Sambalpur
University, Orissa, in 1973 and 1975, respectively.
Since 1976 he has been a Lecturer of Physics
in the Education and Youth Services Department
of Government of Orissa, currently serving at
G.M. College, Sambalpur, Orissa. During 19821985 he was on study leave from G.M. College,
to work for the Ph.D. degree at the Indian Insti-
SHARMA et al.: WAVEGUIDES AND COUPLERS WITH RECTANGULAR CROSS SECTION
tute of Technology, Delhi, India, which he received in 1986 in the field of
fiber and integrated optics.
Dr. Mishra is a fellow of the Optical Society of India. His currrent interests are in single-mode fibers and integrated optical waveguides.
Ajoy K. Ghatak was born in Lucknow, India, on
November 9, 1939. He received the B.Sc. degree
from Agra University, Agra, India, the M.Sc. degree in physics from Delhi University, Delhi, India, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1957, 1959, and 1963,
respectively.
In 1962, he worked for five months in the Reactor Physics Group at General Atomics, San
Diego, CA, and from 1963 to 1964, he worked as
Research Associate in the Department of Nuclear
1125
Engineering at Brookhaven National Laboratory, Upton, NY. From 1964
to 1966, he was in the Department of Physics, Delhi University, and since
1966, he has been at the Indian Institute of Technology, New Delhi, where
he is currently Professor of Physics. From 1973 to 1974, he was Visiting
Scientist in the Department of Electrical Engineering, Drexel University.
Philadelphia, PA, and from 1977 to 1978, he was Visiting Fellow in the
Departments of Applied Mathematics and Engineering Physics, Institute of
Advanced Studies, Australian National University, Canberra. He was Visiting Professor at the University of New South Wales, Sydney, Australia,
and at the National University of Singapore, during 1983-1984 and 19841985, respectively. During the summer of 1987, he was Visiting Professor
at the Institut fur Hochfrequenztechnik und Quantenelectronik, Universitiit
Karlsruhe, West Germany.
Dr. Ghatak is a recipient of the 1979 Bhatnagar Award (Instituted by
CSIR, India). His current interests are in the theory of optical waveguides
and propagation of electromagnetic waves. He has published over 130 research papers and several books.