International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

“Modal analysis of an optical wave-guide having curvilinear square- shape cross-section.”

*Sandeep K. Nigam, *A.K.Srivastava, *S.P. Singh.

Abstract: In this contribution the modal analysis of a new type of non-conventional optical waveguide having curvilinear square shape

cross-section is carried out. Dispersion curves of the waveguide are also obtained. The characteristic equations have been derived by using Goell’s point matching method (GPMM) under weak-guidance approximation. The dispersion curves are also interpreted in two different cases. It has been observed that dielectric waveguide has more number of modes in comparison to metallic waveguide.

Key words: Optical waveguide, Dispersion curve, Modal analysis, curvilinear square shape cross-section.

Introduction

Much research work has done during the last forty years in the field of optical fiber technology. This leads to high capacity and high transmission rate system [1-8].Wave guide of unusual structure have generated great interest in comparison of conventional structure of optical wave-guides[9-14]. The materials used for production of optical waveguides like dielectrics, metals chiral, liquid crystal, polymers etc., have also played great role in revolutionized the communication technology[15-18].
In this paper an optical wave-guide having curvilinear square shape cross-section is proposed. This structure is generated by embedding two symmetrical inverted cardioids. This proposed wave guide is analyzed for two different cases. In one case, all boundaries of proposed wave-guide are taken as surrounded by dielectric material, while in other case they are taken as surrounded by conducting material. For both cases modal characteristic equations and corresponding dispersion curves are
obtained by using Goell’s point matching method
(GPMM) [19].
When two symmetrical inverted cardioids are
embedded in a common cladding, the shape may appear as shown in Figure (1). Waves are guided

Figure.1: Two symmetrical inverted cardioids embedded

in a common cladding to generate proposed structure.

Theory

For a wave guide with core and cladding refractive
inside cardioids core and proposed structure is a
indices

n1 and n2 .

n1  n2

n1 Can take as smaller

part of cladding. The theoretical study in this region
is carried out by us and it may be compared by
experimental findings. The dielectric wave-guides are fundamental building block of integrated optics. They are not only used as transmission medium but also as components: filters, and directional couplers.
than one under weak guidance approximation.
To study the proposed optical wave guide Goell’s point matching method (GPMM) is employed. For this scalar wave equation in cylindrical polar co-ordinate(r,, z) system may be written as
Therefore modal analysis of such system is very important.

 2  n

 2

 0

(1)

*Department of physics, K.N.I.P.S.S. sultanpur-

2281189, (U.P) India. nigam.kni@gmail.com,spsingh.kni@gmail.com

c  t

Here n stands for refractive index of the core or cladding region as the case may be and the function stands for

International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

the z-component of the electric field (EZ ) or magnetic field
(HZ ). Considering harmonic variation of  with t and z, it
can be written as



cladding region,  2



 2 

   0

r 2 

and in guiding

   0 exp

j( t   z) . Now equation (1)

region,  2

  - 

2   0 The solution can be

takes the form

2

2  2 2 

taken as the sum of a series where each term is the
product of Bessel’s function of the first kind

   1   1

    n 

  2    0

(2)

J (x) and trigonometric function of same order

 r 2

r r

r 2  2  2 



where as in the outer non-guiding region, the

Here, ,  and c

are Optical angular frequencies
solution can be taken as linear combination of the
z- Component of the propagation vector, and velocity of light in free space respectively. If  and
product of modified Bessel’s functions of second

kind K ( x) and the trigonometric functions.

 are permittivity of and permeability of the
If  1

and  2

represents the solution in the
medium respectively then equation (2) can be
core and cladding regions respectively, then we have
written as

2 2

 1  A0 J 0

(u r)  A1 J1

(u r) cos   B1 J1

(u r) sin 

   1   1

    2  0

 A2 J 2 (u r) cos 2  B2 J 2 (u r) sin 2  .............(7)

 r 2

2

r  r

2

r 2  2

2

(3)
and

 2  C0 K 0 (wr)  C1 K1 (wr) cos  D1 K1 (wr) sin 

Such that 

    

 C2 K 2

(wr) cos 2  D2 K 2

(wr) sin 2  ..... 8

To use separation of variables technique function

 can be given as

The parameters ‘r’ and ‘  ’ in the above equations
(7) and (8) represents the polar-coordinates of the

 (r, , z, t) 

f1 (r) f 2 ( ) exp j( t   z)

(4)
various points on the boundary of optical wave guide. The parameters ‘u’ and ‘w’ is defined as
With this expression equation (3), can be written as

 2 n 

u 2   1 

  2 , core - region p arameter.

( 9)

 2 f r

f r 

 2 

  

1 ( ) 1

2

1 ( )  2 

2  1

(r)  0

2 2  2 n 2 

r r

2

r  r 

(5)

w  

   , cladding - region p arameter. ( 10)

  

And

 f1 ( )  2 f

 2 2

( )  0

(6)
In Goell’s point-matching method, the fields in core region and cladding region are matched at selective
here  is a non-negative integer.
The solution of equation (5) in terms of Bessel function and the solution of equation (6) in term of
points on boundary of the wave guide. In this case
eighty points are taken on the core-cladding boundary to obtain reasonable results. Matching the
trigonometric functions like

cos( ) or

field along with their derivatives at chosen points on
the boundary of the wave-guide in each case

sin() can be found. The solution of the core as

well as the cladding region can be taken as the linear combination of product of Bessel’s functions and trigonometric functions of various orders. In
following equations are found.

International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

39 39

 Aq J q (urk ) cos(q k )   Bq J q (urk ) sin(q k )

q 0

39

q 0

  Cq K q (wrk ) cos(q k )   Dq K q (wrk ) sin q k  0

q 0

For k=1, 2, 3……..80.
(11)

 39

39 

u Aq J 'q (urk ) cos q k   Bq J 'q (urk ) sin q k 

q 0

 39

q 0 

39 

 wCq K 'q (wrk ) cos q k   Dq K 'q (wrk ) sin q k 

q 0

q 0 

For k= 1, 2, 3……..80. (12)
The prime terms in the above equation represents differentiation with respect to arguments and
Figure.2:Cross-sectional view of dielectric curvilinear square core optical wave guide.

quantities rk and k

are the polar co-ordinates of the
core-cladding boundary. After taking the summation explicitly, a set of 160 simultaneous linear equations
involving the constants

Aq , Bq , Cq and Dq are

obtained .These coefficients form a 160  160 determinant  I .A non–trivial solution may exist for these set of equations

 I  0

(13)
Equation (13) is the characteristic equation, which
contains all the information about the modal
properties of proposed waveguides. The solution of equation (13) gives propagation constants  for
sustained modes in core region. In both cases, as shown in figure (2) and in figure (3), dispersion curve are obtained and interpreted. The equation (13) gives the normalized propagation constant.

Figure.3:Cross-sectional view of metal loaded

  2



k 2



 n2 

(14)

curvilinear square optical wave guide.

b   

(n 2  n 2 )

For first few guided modes for proposed waveguide. The wave-guide parameter V can be written as,

1 2

International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

 2  2 2

composed of dielectric material i.e. a dielectric

V    d

 0 

n1  n2

(15) .l
optical wave-guide. While, figure-5 shows the dispersion curves for that case, when the boundaries
Similar analysis can be carried out for the case, when
a conducting material has surrounded the entire
boundary in place of dielectric. In this case, all boundaries are metallic; fields on all four sides must vanish. Therefore derivative part has no significance, which results in 80 simultaneous equations involving 80 unknown constants. These equation are converted to an 8080 determinant
of waveguide is composed of conducting material
i.e. a metallic optical wave-guide. These obtained curves have the standard shape. However, there are many possible modes, but we have shown only six modes for dielectric optical wave-guide (DOWG) and first two modes for metallic optical wave-guide (MOWG).
=\  II
consisting of constants

Aq , Bq , Cq and Dq .For existence of non- trivial solution

 II  0

(16)
The solution of equation (16) gives normalized propagation constants for a few guided modes. Results and Discussion: To understand physical consequences of characteristic equations (13) and (16), the dispersion curves of sustained modes
have to obtain. For refractive index of core (n 1 ) and cladding (n 2 ) are taken as 1·48 and 146 respectively.
The wave length  of light in free space is taken as

0

155 m
.For each case, the left hand side of
corresponding characteristic equation is plotted against the admissible values of 

( k0 n1    k0 n2 ) for fixed value of d, and zero crossings are noted. The zero crossing value corresponds to a particular sustained mode. Several such curves are plotted for different values of d for a given mode. The quantity d is related to wave-guide parameter and we have,

Figuire.4.Dispersion curves for DOWG.

V  2 d

n 2  n 2 .

Here, we have obtained some important and
interesting preliminary insights, which are as

1 2

0 follows:

With the help of V , we can calculate the normalized propagation constants ‘b’ using equation (14). In this way, b-V curves (dispersion curves) can be plotted for each mode and for each case. The same procedure is used for characteristics equation (16) for obtaining the b-V curves.
Figure-4 shows dispersion curves (b-V) for that case
when a boundary of proposed waveguide is

1. The dielectric wave guide (DOWG) and the metallic optical waveguide (MOWG) shows the mode bunching properties near the cut-off value. In the case of standard square dielectric optical waveguide normally all modes lie between V= 0  0 and V= 170 .Whereas in our proposed wave-guide all the modes lie between V=3.12 and V=4.80.

2. It has been found that first cut-off value shifts from

International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

V=0.0 to V=3.12 and the sixth cut-off value shifts from
V=1.70 to V= 4.80 approximately. The distortion of standard square shape into curvilinear square shape is also responsible for an increase in the number of sustained modes for V=4.80(say). This means that more power will be transmitted through proposed optical wave-guide than a standard square wave-guide with common value of d.

Figure.5: Dispersion curves for MOWG.

3. Near the cut-off value for the lowest order modes for standard square wave-guide, the effect of curvilinear square distortion is to shifts to the cut- off value slights towards a larger value of V.

4.In case of dielectric optical wave guide (DOWG), the cut-

off values V are crowded in the region=3.12-4.80, where as in the case of metallic wave guide it is separated and reduced which is an interesting feature and is also expected.

Conclusion: It can be concluded that proposed DOWG

are useful for transmitting more power. The DOWG are more sensitive to small distortion in shape. The metallic waveguide can be used as mode filter while DOWG can be used for switching action in optical system.
The authors hope that the predicted results will be of
sufficient interest to induce researchers worldwide to take up the experimental verification of the results in the present communication

Acknowledgement: Authors are thankful to Prof.S.P.Ojha and Dr. Vivek Singh for their academic help and encouragement throughout this work.

References:

1. A.Ghatak and K.Thyagarajan, Introduction to fiber optics, Cambridge University press, Cambridge, England, 1999.
2. N.S.Kapany, fiber optics principles and applications, Acadmemic,New York, 1967.
3. A.K.Ghatak and B.Kulshaw, V.Nagrajan, and
B.D.Kurana, Trend in fiber and optical communications and Viva Book Pvt. Ltd. New Delhi, India, 1996.
4. KeiserG: Optical Fiber Communication. McGrew- Hill, Singapore 1986.
5. Tamir T: Integrated optics. Springer Verlagm New
York 1979.
6. Marcuse D: Light Transmission Optics . Van
Nostrand, New York 1972.
7. Dyott RB, Day CR, Brain MC: Glass Fiber
waveguide with Triangular core. Electron Lett. 9 (1973) 228-290.
8. Kawakami S, Nishidha S: Characteristics of a doubly clad optical fiber with low index inner cladding. IEEE J. Quantum Electron. QE-10 (1974)
879.
9. A Kumar, V.Thyagaranjan and A.K.Ghatak:,
''Analysis of rectangular core dielectric waveguides: An accurate perturbation approach'', Opt. Lett. 8, (1983).63-65,
10. U.K. Singh, P. Khastgir, O.N. Singh: Modal analysis of annular optical waveguides bounded by non-coaxial dielectric cylindrical boundaries by Goell’s point matching method. Optik 109 (1998)
150-154.
11. V. Mishra, P.K choudhury,P.Khastgir and S.P.Ojha., An Modal Propgation of a wavguide with a regular pentagonal cross-section with conducting and non-conducting boundaries, Micro wave Opt. Technol Lett 8 (1995). 280-282.
12. V.Singh, S.P. Ojha and L.K.singh, Modal behavior cut-off condition and dispersion

International journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012

ISSN 2229-5518

characteristics of an optical waveguide with core
cross-section bounded by two spirals ,Microwave . and optical Technology Letters, 21(1999) 121-124.
13. H.P. Singh, Vivek Singh and V.P. Arora , Modal
analysis and dispersion curves of curvilinear triangular cored light guide ,using Goell’s point matching method, optik119,1(2008)29-33.
14. C.B.P. Verma ,U.K. Singh ,V.B. Singh, O.N. Singh-II . Confocal and concentric annular elliptic light guide and their modal characteristics. Optik
112 .3 (2001),105-108.
15. A.K.Singh, P.Kastgir, O.N.Singh and S.P.Ojha, Local field configuration modes in weakly guiding fiber with a canonically annular core:A comparatively study, Japan J App Phy 32 (1993), 171.
16. M.P.S. Rao, V. Singh, B.Prasad, and S.P.Ojha, An analytical study of the dispersion curves of an annular wave guide made of liquid crystal,Photon Optoelectron, 5 (1998), 73-88.
17. S.Ganguly, Kh.S. Singh, S.P. Ojha, and P.Khastgir, Modal cutoff conditions for a chirowaveguide with a chiral core of large annular cross-section bounded by a dielectric media, Microwave Opt Techno Lett 8 (1995), 18-21.
18. N.Kumar, S.K.Srivastava, and S.P.Ojha, A
comparative study of modal dispersion characteristics of different types concave shaped dielectrics waveguide, Microwave Opt. Technol Lett
,4(2002),337-342.
19. J.E. Goell, A circular- harmonic computer
analysis of rectangular dielectric waveguides, Bell
Syst Tech J 48 (1969), 2133-2160.
20. J.E. Goell, Rib waveguide for integrated optical circuits, App. Opt. 12 (1973) 2797-2798.
21 H. Furuta, H. Noda, A. Ihaya, Novel optical
waveguide for integrated optics, Appl.Opt. 13 (1974)
322-326.
22. A.A. Oliner, Acoustic surface waveguide and comparisons with optical waveguides, IEEE Trans. Microwave Theory Tech. MTT-24 (1976) 914-920.
23. Kamiya, H.Yanai, Propagation characteristics of a partially metal clad optical waveguide: metal clad optical strip line, App. Opt. 14 (1975) 322-326.
24 S.E. Miller, Integrated optics: an introduction, Bell
Syst. Tech. J. 48 (1969), 2059-2069.
25. M.J. Adams, An Introduction to Optical waveguides,Willey, Chichester, 1981 (182pp).
.