Advanced Studies in Theoretical Physics

Vol. 9, 2015, no. 15, 723 - 735

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.511106

Investigation of the Transverse Electromagnetic

Modes in an Embedded Optical Waveguide

Ali Çetin and M. Selami Kiliçkaya

Department of Physics

Eskişehir Osmangazi University, 26480

Meşelik Campus, Eskişehir, Turkey Copyright © 2015 Ali Çetin and M. Selami Kiliçkaya. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The eigenvalue equations are obtained for TE and TM modes in an embedded

dielectric slab waveguide. The normalized propagation constant is determined by

using eigenvalue equations. The effective waveguide structure is determined by

refractive indexes obtained by means of an effective index method. The spatial

graphs are plotted for the least low order modes E(x) and E(y) in the waveguide.

Keywords: Optical waveguide, effective index, waveguide modes, integrated

optics

1 Introduction

Use of optical signals as a means of carrier in optical communications is evident

since the invention of lasers in 1960 [1]. Optical waveguides are the key

components of modern integrated optics. It was 1969, Miller [2] introduced the term

“Integrated Optics” referring to the fabrication and integration of several photonic

components on a common planar substrate. These components include beam splitter,

gratings, couplers, polarizers, interferometers, sources and detectors. In the last

three decades, a considerable amount of work had been reported on the study of

propagation characteristics of optical waveguides used in integrated optics [3-15].

The investigation of mode fields in dielectric waveguide is fundamental to the

analysis of integrated optical devices. Mode propagation is an important

requirement for optical waveguides.

724 Ali Çetin and M. Selami Kiliçkaya

Many methods have been developed for the modal analysis of these optical

waveguides. In addition, various methods were proposed to determine the refractive

index profile of optical waveguides to predict the waveguide characteristics and

thus to control waveguide production process [16-21].

Rectangular waveguide is a basic optical waveguide structure. To analyze the

normalized modes of rectangular structure, we followed an analytical approach

based on approximate results that developed by Marcatili [22, 23].

The cutoff frequency is the lowest frequency for which a mode will propagate in it.

The cutoff frequency is found using the characteristic equation of the Helmholtz

equation for electromagnetic waves. Marcatili’s method of the calculation for

rectangular dielectric rod waveguides has been modified for uniaxial anisotropic

dielectric materials [24] and is applied to modes starting from cutoff.

2 Theory

There are many types of rectangular waveguides which can be used in an integrated

optical circuit. One of these waveguide structures is an embedded waveguide

illustrated in Figure 1. For the structure to guide light the refractive index of the

structure must be slightly larger than the surrounding media [25]. We will determine

the mode structure of an embedded waveguide.

Figure 1. An embedded waveguide.

The cross-section of an embedded waveguide is shown in Figure 2(a). There are

nine different dielectric regions in the structure. The mode analyzing is very

difficult since it is impossible to satisfy all of the boundary conditions in this

structure simultaneously.

A difficulty occurs in shaded four regions that encountered in mode analyzing.

These four regions act as the coupling regions for the x and y solutions of the field.

Above cutoff, mode is confined in the core while in corner regions the energy is

negligible. In the case close to cutoff, mode has the most amount of energy outside

the core. The x and y dependent solutions are related strongly to boundary

conditions in the corner regions.

Investigation of the transverse electromagnetic modes 725

(a) (b)

Figure 2. a) Schematic representation of a dielectric waveguide.

b) Illustration of effective index method.

Note that small difference between refractive indexes of core and cladding is a

primary condition in such modeling [25]. The Ey field amplitude just out of the core

equals to the value in the core and it decreases exponentially away from the core.

Therefore, if the field in the core is no close to the mode cutoff and thus it is stronger

compared to out of the core counterpart. Similar properties are valid for TM modes.

Two types of modes must be distinguished. First type is polarized in x-direction

( x

nmE ), second type is polarized in y-direction ( y

nmE ). The longitudinal components

Ez and Hz must satisfy the reduced wave equation

2 2

2j2 2

0Kx y

(1)

where parameter Kj is defined as

1 2

2 2 2

j jK n k (2)

Refractive indices for fields in these five regions are given by 1

n , 2

n , 3

n , 4

n

and 5

n respectively.

We can define x

nmE mode in the first region (core region of the waveguide) [26]

2 2 21x x1 x x y y

x

A( j ) n k sin x cos yE

(3)

y

1y x x y y

Aκ( j ) cos κ x δ sin κ y δE

β (4)

1z x x y yAcos κ x δ cos κ y δE (5)

1x 0H (6)

726 Ali Çetin and M. Selami Kiliçkaya

1 2

20

1y 1 x x y y

0 x

kH jA n sin x cos y

(7)

1 2

y20

1z 1 x x y y

0 x

kH A n sin x sin y

(8)

where 2

0 0 0k ,

xδ and

yδ are the phase parameters. The separation constants

x y and must satisfy

2 2 2 2 2 2

1 1 x yK n k (9)

The sine and cosine functions were selected such that the contributions of Ez and

Hz result in the same functional dependence of the transverse components [27, 28].

The phase parameters x and y which are added to the arguments of sine and cosine

functions are necessary for the general solution of the modes.

All field components in regions 2-5 vanish since fields decay at the infinite

distances out of the core. Decaying exponential functions are used away from the

core. Thus, following equations are written in region 2 [26]

2 2 2

2 2

2x x x 2 y y 2 2

2

n kE jA cos h cos y exp x h

(10)

2y 0E (11)

2z x x 2 y y 2 2Acos h cos y exp x hE (12)

2xH 0 (13)

1 2

20

2y 2 x x 2 y y 2 2

0 2

kH jA n cos[ h ] cos y exp x h

(14)

1 2

y20

2z 2 x x 2 y y 2 2

0 2

κε kH A n cos κ δ h sin κ y δ exp γ x h

μ γ β

(15)

with

2 2 2 2 2 2

2 2 y 2K n k (16)

Similarly, we can find amplitude values of the fields relative to transverse field

components at core-cladding boundary regions 3, 4 and 5 respectively. A transverse

electric field component is calculated by appropriate amplitude selection. Hz

component must be continuous at core boundary 0x and 2

x h in regions

2 and 3. Our analysis is completed by calculating modes of dielectric waveguide at

appropriate boundary conditions. Ey component vanishes since it is very small

compared to all other components. Thus, two following equations are obtained [26]

Investigation of the transverse electromagnetic modes 727

2 2

1 2

x x 2 x x 2

x 2

n nsin κ δ h cos κ δ h 0

κ γ (17)

and

22

31

x x x x

x 3

nnsin cos 0

(18)

By using above two equations an eigenvalue equation is established for x

eigenvalue as

2 2

3 2 2 32

x 2 1 x 42 2 2

3 2 x 1 2 3

n γ n γtan κ h n κ

n n κ n γ γ

(19)

where 1 2

2 2 2 2

2 1 xγ n n k κ and

1 22 2 2 2

3 1 3 xγ n n k κ .

2γ and

3γ are respectively

exponential decay constants in regions 2 and 3.

By means of two equations obtained from the requirement of continuity of Hz, we

get for y eigenvalue

2

y 1 y 4 5 y 4 5tan( h ) (20)

where 1 2

22 2 2

4 1 4 yγ n n k κ

and

1 222 2 2

5 1 5 yγ n n k κ

.

4γ and

5γ are

respectively exponential decay constants in regions 4 and 5.

Oncex

and y

are determined, the propagation constant is obtained from Eq. (9)

1 2

2 2 2 2

1 x yn k (21)

Our investigation is not valid close to the cutoff since the fields detach themselves

from the core and reach strongly to the shaded region in Figure 2. This breakdown

of theory near cutoff is apparent from the propagation constant . We write Eq. (21)

in the form

1 2

2 2 2 2 2 2 2

2 2 2 x y 2βh n k h V κ κ h (22)

If 2

n is larger than any of the other refractive indices (except 1n ), total internal

reflection will first break down at the core boundary 2

x h . V is defined as the

frequency coefficient that combines the difference of the square of the refractive

indices of core and medium 2 whereby giving information about the operating

wavelength and the width of the core [27, 28]

1 2

2 2

1 2 2V n n kh (23)

Effective index method, firstly proposed by Knox and Toulios for fundamental

mode such that Marcatili’s results are more accurate [22, 29]. Effective index

method is a widely adopted approximation technique to determine the propagation

constant and field distribution of a rectangular optical waveguide [30-32], i.e.

generalised waveguide dispersion characteristics is able to analyse for the optical

channel waveguides using the EIM [33].

As shown in Figure 2(b), two-dimensional problem is converted to one dimensional

728 Ali Çetin and M. Selami Kiliçkaya

problem by effective index method [34, 35]. To use this method, waveguide is

stretched out along its axis. Thus, a planar slab waveguide is formed along y axis.

The thin waveguide is analyzed in terms of indices to form the structure whereas

thick waveguide is analyzed using the effective index obtained from first waveguide

analysis. We can analyze thin one-dimensional waveguide in terms of TE modes to

find the allowed value of using wavelength and mode.

When is found, the effective index of slab is determined by

eff

0

nk

(24)

where k0 is wave number in free space. After effective index is determined we

return to actual structure and stretch it vertically forming a slab waveguide in x-

direction. The modes for this waveguide can be determined by using effective index

instead of original index value. However, to obtain more accurate results by

effective index method waveguide modes must be far away from cutoff [37, 38]. In

EI method, the eigenvalue of the equivalent slab waveguide is an approximate index

value of the original waveguide [39].

3 Results and Discussions

To use effective index method in horizontal structure, at first the effective indices

of three regions are found. Since field components are polarized in vertical direction

a TE mode must exist in x-direction. In symmetric waveguides for TE modes by

using eigenvalue we obtain

x

x 2

x

tan h 2

(25)

A structure is designed with n1=1.5, n1=n2=n3=n4=n5=1.499 provided that

1 2

2 2 2 2

x 1 2 0 xγ n n k κ .

The most important guiding condition for waveguide is that 2 eff 1

n n n ,

remembering that n2 is the outer region index where n1 is the guiding structure

known as core index and neff is the effective index [36]. If all modes of some slab

region are below cut-off, heuristics have to be applied to provide the necessary

effective indices whenever mode profiles are not defined [40].

We assume that waveguide operates at 2h 20 μm and λ 1.3 μm . Root of Eq.

(25) is found by using a numerical analysis method due to Brent [41]. By means of

this root, we obtain

3 1

xκ 1.1298 10 cm

3 1

xγ 2.3936 10 cm .

The graphical representation of eigenvalue equation Eq.(19) for x

nmE modes is

shown in Figure 3. Numerical computation of the intersection points of the

corresponding function reduces to the search of zeros which can be performed

effectively.

Investigation of the transverse electromagnetic modes 729

Since the propagation constant β depends on the waveguide properties and the

wavelength, it is convenient to describe light propagation in terms of a normalized

propagation constant. Normalized propagation constant b is related to β by the

definition

2

eff 2

2 2

1 2

β / n nb

n n

(26)

Symmetric waveguide is formed by applying the same procedure for two sided

regions. In this structure x-polarized field exists as TE wave. In similar way, an

eigenvalue is obtained as

2

y1

y 1 2

2 y

γntan(κ h 2)

n κ (27)

by taking 2

h 20 μm as in the symmetric structure.

Figure 3. Graphical representation of Eq. (19).

The function xf κ , λ is the right hand side of Eq. (19).

From the numerical analysis solution of Eq. (27) we obtain

3 1

yκ 1.1302 10 cm

3 1

yγ 2.3934 10 cm

Writing xκ and yκ values in Eq. (21), we obtain an effective index value for

designed waveguide structure:

effn 1.4995

A plot of normalized propagation constant b as a function of the frequency

parameter V for the rectangular dielectric waveguide is shown in Figure 4. Since

the effective waveguide is symmetric, we can use the Eqs. (28) and (29) [42, 43].

x 2

x2

x 2

x h 22

cos xA , x h 2

cos h 2E(x)

Ae , x h 2

(28)

730 Ali Çetin and M. Selami Kiliçkaya

y 1

y1

y 1

y h 21

cos yA , y h 2

cos h 2E(y)

Ae , y h 2

(29)

Figure 4. Graphical representation of the propagation

constant b as a function of frequency coefficient V.

Using these equations, we plotted graphs of mode fields for the least low order

spatial modes. These graphs are shown in Figure 5.

(a)

(b)

Figure 5. Calculated mode intensities for Eqs. (28) and (29).

Satisfactory results are calculated for normalized propagation constant β in far from

cutoff by using the analytical approximation method developed by Marcatili.

4 Concluding Remarks

An approximation method to compute propagation modes of embedded waveguides

having rectangular cross section was presented. Only Ex and Ey modes are

supported by the waveguides due to weakly guidance. Eigenvalue equations can be

solved numerically very fast. Guided modes are determined as a function of

Investigation of the transverse electromagnetic modes 731

wavelength in a dielectric embedded waveguide from Figure 4 whose graphical

representation of Eq.(19) which is derived with respect to effective index method.

It is shown that the guided mode number changes relative to height h2 and

wavelength. As a result, the optimum β propagation constants are fastly calculated

which required the field amplitudes.

References

[1] G. Lifante, Integrated Photonics: Fundamentals, John Wiley Sons, West

Sussex, 2003. http://dx.doi.org/10.1002/0470861401

[2] S. E. Miller, A survey of integrated optics, IEEE Journal of Quantum

Electronics, 8 (1972), 199-205. http://dx.doi.org/10.1109/jqe.1972.1076910

[3] P. K. Mishra, A. Sharma, Analysis of single mode inhomogeneous planar

waveguides, Journal of Lightwave Technology, 4 (1986), 204-212.

http://dx.doi.org/10.1109/jlt.1986.1074699

[4] A. N. Kaul, S. I. Hoasin, K. Thyagarajan, A simple numerical method for

studying the propagation characteristics of single mode graded index planar

optical waveguides, Transactions on Microwave Theory and Techniques, 34

(1986), 288-292. http://dx.doi.org/10.1109/tmtt.1986.1133325

[5] A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, Numerical analysis of planar

optical waveguides using matrix approach, Journal of Lightwave Technology, 5

(1987), 660-667. http://dx.doi.org/10.1109/jlt.1987.1075553

[6] D. J. Vezzetti, M. Munowitz, Analysis of finite rib waveguides by matrix

methods, Journal of Lightwave Technology, 8 (1990), 1228-1234.

http://dx.doi.org/10.1109/50.57845

[7] W. Huang, H. A. Haus, A simple variational approach to optical rib waveguides,

Journal of Lightwave Technology, 9 (1991), 56-61.

http://dx.doi.org/10.1109/50.64923

[8] A. Chen, P. Berini, D. Feng, S. Tanev, V. P. Tzolov, Efficient and accurate

numerical analysis of multilayer planar optical waveguides in lossy anisotropic

media, Optics Express, 7 (2000), 260-272.

http://dx.doi.org/10.1364/oe.7.000260

[9] P. R. Chaudhuri, S. Roy, Analysis of arbitrary index profile planar optical

waveguides and multilayer nonlinear structures: A simple finite difference

algorithm, Optical and Quantum Electronics, 39 (2007), 221-237.

http://dx.doi.org/10.1007/s11082-007-9076-6

732 Ali Çetin and M. Selami Kiliçkaya

[10] A. Kumar, T. Srivastava, Performance of the effective index method in the

modeling of nanoscale rectangular apertures in a real metal, Optics

Communications, 281 (2008), 4526-4529.

http://dx.doi.org/10.1016/j.optcom.2008.04.040

[11] X. Qian, A. C. Boucouvalas, Synthesis of symmetric and asymmetric planar

optical waveguides, IET Optoelectronics, 1 (2007), 185-190.

http://dx.doi.org/10.1049/iet-opt:20060090

[12] H. Ding, K. T. Chan, Solving planar dielectric waveguide equations by

counting the number of guided modes, IEEE Photonics Technology Letters,

9 (1997), 215-217. http://dx.doi.org/10.1109/68.553096

[13] A. Seida, M. Osama, Propagation of electromagnetic waves in a rectangular

tunnel, Applied Mathematics and Computation, 136 (2003) 405-413.

http://dx.doi.org/10.1016/s0096-3003(02)00052-8

[14] X. Li, Z. Wang, H. Liu, Optimizing initial chirp for efficient femtosecond

wavelength conversion in silicon waveguide by split-step Fourier method,

Applied Mathematics and Computation, 218 (2012), no. 24, 11970-11975.

http://dx.doi.org/10.1016/j.amc.2012.05.065

[15] S. Kim, Analysis of the convected Helmholtz equation with a uniform mean

flow in a waveguide with complete radiation boundary conditions, Journal of

Mathematical Analysis and Applications, 410 (2014), no. 1, 275-291.

http://dx.doi.org/10.1016/j.jmaa.2013.08.018

[16] H. Zhu, Z. Cao, Q. Shen, Construction of the refractive index profiles for few-

mode planar optical waveguides, Optics Communications, 260 (2006), 542-

547. http://dx.doi.org/10.1016/j.optcom.2005.11.011

[17] A. K. Thander and S. Bhattacharyya, Study of Optical Wave Guide Using HOC

Scheme, Applied Mathematical Sciences, 8 (2014), no. 79, 3931-3938.

http://dx.doi.org/10.12988/ams.2014.44255

[18] Y. Ding, Z. Q. Cao, Q. S. Shen, Determination of optical waveguide refractive

index profiles with the inverse analytic transfer matrix method, Optical and

Quantum Electronics, 36 (2004), 489-497.

http://dx.doi.org/10.1023/b:oqel.0000025769.11690.2d

[19] K. S. Chiang, Q. Liu, K. P. Lor, Refractive index profiling of buried planar

waveguides by an inverse Wentzel-Kramer-Brillouin method, Journal of

Lightwave Technology, 26 (2008), 1367-1373.

http://dx.doi.org/10.1109/jlt.2008.923637

Investigation of the transverse electromagnetic modes 733

[20] Z. H. Dong, D. Yuan, C. Z. Qi, S. Q. Shun, Determination of field

configuration and eigenvalues of planar optical waveguides with arbitrary

index profiles, Chin. Phys. Lett., 22 (2005), 1580-1583.

http://dx.doi.org/10.1088/0256-307x/22/7/006

[21] A. Cetin, E. Ucgun, M. S. Kilickaya, Determining the Effective Refractive

Index of AlGaAs-GaAs Slab Waveguide Based on Analytical and Finite

Difference Method, Journal of Physical Science and Application 2 (2012),

no. 9, 381-385.

[22] E. A. J. Marcatali, Dielectric rectangular waveguide and directional coupler

for integrated optics, Bell Syst. Tech. J., 48 (1969), 2071-2102.

http://dx.doi.org/10.1002/j.1538-7305.1969.tb01166.x

[23] K. Kawano, T. Kitoh, Introduction to Optical Waveguide Analysis: Solving

Maxwells Equation and the Schrödinger Equation, John Wiley Sons, 2001.

http://dx.doi.org/10.1002/0471221600.ch7

[24] S. N. Dudorov, D. V. Lioubtchenko, A. V. Raisanen, Modification of

Marcatili’s Method for the calculation of anisotropic rectangular dielectric

waveguides, IEEE Transactions on Microwave Theory Techniques, 50 (2002),

1640-1642. http://dx.doi.org/10.1109/tmtt.2002.1006427

[25] D. Gloge, Weakly guiding fibers, Applied Optics, 10 (1971), 2252-2258.

http://dx.doi.org/10.1364/ao.10.002252

[26] D. Marcuse, Theory of Optical Waveguides, Academic Press, New York, 1974.

[27] Z.Wang and J. F. Dong, Analysis of guided modes in asymmetric left-handed

slab waveguides, Progress in Electromagnetics Research, 62 (2006), 203-

215. http://dx.doi.org/10.2528/pier06021802

[28] K. Okamoto, Fundamentals of Optical Waveguides, Elsevier, Amsterdam,

2006.

[29] C. Chien Huang, C. Chih Huang, An efficient and accurate semivectorial

spectral collocation method for analyzing polarized modes of rib waveguides,

Journal of Lightwave Technology, 23 (2005), 2309-2317.

http://dx.doi.org/10.1109/jlt.2005.850041

[30] K. S. Chiang, K. M. Lo, K. S. Kwok, Effective index method with built-in

perturbation correction for integrated optical waveguides, Journal of

Lightwave Technology, 14 (1996), 223-228.

http://dx.doi.org/10.1109/50.482267

734 Ali Çetin and M. Selami Kiliçkaya

[31] K. S. Chiang, Analysis of rectangular dielectric waveguides: Effective-index

method with built-in perturbation correction, Electronics Letters, 28 (1992),

388-390. http://dx.doi.org/10.1049/el:19920243

[32] K. S. Chiang, Analysis of the effective-index method for the vector modes of

rectangular-core dielectric waveguides, IEEE Transactions on Microwave

Theory and Techniques, 44 (1996), 692-700.

http://dx.doi.org/10.1109/22.493922

[33] T. K. Lim, H. Melchior, Effective index method for generalised waveguide

dispersion characteristics analysis of optical channel waveguides, Electronics

Letters, 27 (1991), 917-918. http://dx.doi.org/10.1049/el:19910574

[34] Y. Chung, N. Dağlı, An assessment of finite difference beam propagation

method, IEEE Journal of Quantum Electronics, 26 (1990), 1335-1339.

http://dx.doi.org/10.1109/3.59679

[35] C. M. Kim, B. G. Jung, C. W. Lee, Analysis of dielectric rectangular

waveguide by modified effective-index method, Electronics Letters, 22

(1986), 296-298. http://dx.doi.org/10.1049/el:19860202

[36] M. Anani, C. Mathieu, H. Abid, S. Lebid, Y. Amar, Calculation of the

effective indices of a GaN/InxGa1-xN optical guiding structure,

Microelectronics Journal, 38 (2007), 505–508.

http://dx.doi.org/10.1016/j.mejo.2007.03.016

[37] K. S. Chiang, C. H. Kwan, K. M. Lo, Effective-index method with built-in

perturbation correction for the vector modes of rectangular-core optical

waveguides, Journal of Lightwave Technology, 17 (1999), 716-722.

http://dx.doi.org/10.1109/50.754804

[38] T. M. Benson, P. C. Kendall, Variational techniques including effective and

weighted index methods, Progress in Electromagnetics Research, 10 (1995),

1-40.

[39] Y. T. Han, J. U. Shin, D. J. Kim, S. H. Park, Y. J. Park, H. K. Sung, A rigorous

2D approximate tecnique for 3D waveguide structures for BPM calculations,

ETRI Journal, 25 (2003), 535-537.

http://dx.doi.org/10.4218/etrij.03.0203.0020

[40] O. V. Ivanova, M. Hammer, R. Stoffer, E. Van Groesen, A variational mode

expansion mode solver, Optical and Quantum Electronics, 39 (2007), 849-

864. http://dx.doi.org/10.1007/s11082-007-9124-2

[41] J. Kiusalaas, Numerical Methods in Engineering with MATLAB, Cambridge

Investigation of the transverse electromagnetic modes 735

University Press, New York, 150-155, 2005.

http://dx.doi.org/10.1017/cbo9780511614682

[42] C. R. Pollock, Fundamentals of Optoelectronics, Irwin, Chicago, 1995.

[43] A. S. Supa’at, A. B. Mohammad, N. M. Kassım, Modelling techniques for

rectangular dielectric waveguides–ribs waveguides, Jurnal Teknologi, 36

(2002), 129-143. http://dx.doi.org/10.11113/jt.v36.571

Received: November 17, 2015; Published: November 24, 2015