A Perturbation Method for Complex Root Finding of Nonlinear Electromagnetic Waves

International Journal of Infrared and Millimeter Waves, Vol. 20, No. 7, 1999

A PERTURBATION METHOD FOR COMPLEX ROOT FINDINGOF NONLINEAR ELECTROMAGNETIC WAVES

M. M. Shabat,1,* M. A. Abd-El naby,2 N. M. Barakat,3 and D. Jager1

1 Center of Semiconductor and Optoelectronics ZHO,Faculty of Electrical Engineering and Electronics

Gerhard-Mercator UniversityDuisburg, D-47057, Germany

2Department of Mathematics, Faculty of EducationAin Shams University

Cairo, Egypt3College of Education

Gaza, Gaza Strip, Palestinian Authority

Received January 12, 1999

Abstract

Aspects of the perturbation method are presented and implementedon the analytic function of the complex variable for the propagation ofnonlinear electromagnetic surface guided by a thin film. The behavior ofnonlinear TE waves guided by asymmetrical dielectric thin film, boundedby a generalizes nonlinear substrate with a permittivity of the form

e~ | E|s is analyzed. The complex roots, which correspond to the

propagation and attenuation coefficient are obtained and discussed.

Key words

Nonlinear TE waves, Lossy waveguides, Numerical and perturbationstechniques.

* Permenant address: Department of Physics, Islamic University of Gaza,P.O.Box 108 , Gaza Strip, Palestinian Authority.

1389

0195-9271/99/0700-1389$16.00/0 © 1999 Plenum Publishing Corporation

1390 Shabat, Abd-El naby, Barakat, and Jager

Introduction

In recent years, growing interest has been devoted and paid to studythe wave propagation characteristics in multilayer wave guided structures[1-8]. The analytical or numerical techniques are used to study theirpropagation characteristics. Perturbation methods [9-11] play animportant role in the solution of many theoretical problems in physics,mathematics and Engineering as numerical techniques [12-16] do not givethe desired solutions and have many difficulties especially in implementingfor the nonlinear waves.An analytical solution was developed for the electromagnetic fields in

generalized nonlinear media that has offered the opportunity to study[1-7] the dispersion relations and the field properties of nonlinear guidedwaves in lossless waveguide structure. In studying complex waveguides(lossy waveguides), it is necessary to solve analytic dispersion equationwith complex variables. In this communication, we demonstrate andimplement an efficient perturbation approach in order to solve thecomplex dispersion equation with the complex variables. Thisperturbation technique has been found to be an efficient and quickapproach and has a good agreement with numerical technique and can beused as initial starting guess solutions for the numerical technique and canbe also used to predict the behavior of non-linear surface waves and non-linear guided waves in multilayer structures.The work follows on the expansion and integrating of Ma [7] byconsidering that structure to be a complex one. The explicit analyticalexpressions for the propagation characteristics of TE-waves have beenderive and investigated.

Theory and Notations

It is worth to borrow Ma and Wolf notations [7], theory and structure.We consider a three layer structure as shown in figure 1.

Nonlinear Electromagnetic Waves 1391

Fig. 1 Waveguide structure

The structure consists of a thin optically linear film of thickness d and arefractive index nf, a linear cover with complex refractive index nc, and asemi infinite nonlinear medium forming the substrate which has a

dielectric profile

Where s0 is the free-space permittivity, ns the refractive index of the

material for | E |=0, a is an arbitrary constant , and § is an arbitrary real

number. Assuming the guide is infinite in the y-direction, then the*S

resulting fields will be independent of y i.e---=0 . The z- direction isdz

assumed to be the propagation direction, and possible waves in thisdirection are described by the complex function e -j p z. We will focus ourattention on solution of the form E~ ej(W t-Bz). For the three regions,shown in figure 1 the wave equations are


where B is the complex propagation constant (/J=#' +ij3 ) & is

the phase constant and /7" is the attenuation coefficient.

In this paper, we discuss the behavior of the guided waves. Solving theEqs. (2), (3), and (4) in the three regions, the electric field components forthe three regions can be written as:

Follows Ma and Wolf [7] the dispersion relation can be easily obtained as,

Where


X, is the parameter given by the initial exciting condition and gives themaximum of the field in the nonlinear region and E0 is the magnitudeof the electric field at X = d.

In this communication, we concentrate our efforts only on the guidedwaves as N< nf.

The linear complex refractive index and the nonlinear optical coefficient ofthe cladding layer are denoted by [18]

nc=nRc +inIC (where nRC, and nIC are the real and thenc = nRc+inIc nRc nlc

imaginary real part and the imaginary part of the complex refractive index,respectively) and n2c respectively. In such a waveguiding system, the

lossy nonlinear guided modes will be either TE or TM-polarized. For TE-polarization , the following nonlinear relative permittivity gc is

considered

where (%c = C 0 E 0 n R c n 2 c is the nonlinear coefficient of the cladding

medium, and Q0 and g0 are the light velocity and the permittivity invacuum respectively. Hence, the final complex nonlinear refractive indexin the cladding lossy nonlinear medium for TE modes is


N can be written as

A perturbation method

For the lossy waveguide (complex waveguide) as the refractive indexof the cover of the structure is complex leading the propagation constant

to be complex constant. In view of the fact that £IC/£RC is small for a

linear cover. A zero-order approximate solution of Eq. (6) can beobtained by neglecting gIC and solving the real equation which is a routine

one and can be computed by a Newton's method, and then taking theeffect of £IC into account by perturbation method [9-11].

And using Taylor expansion, then


Then we get,

or,


so the eigenvalue equation can be written as,

so Eq. (13) becomes


If we denote the right hand-side of Eq.(14) as F(N), then F( N) issmall, and by perturbation the first-order correction for N is given by

Results and discussions

A simple algorithm [12-15] for solving the analytic function [17] of thecomplex variable is developed and implemented for studying the behaviorof nonlinear electromagnetic waves in a three layered waveguidestructure. The accuracy of the present work has been checked by applyingthis method to the real structure of Ma and Wolf [7] and produced similarresults. In this communication, we concentrate on the imaginary part ofthe root of the analytic function or equations (dispersion) as the real partof the root has been found to be very close of the Ma and Wolf paper [7]taking into account their neglecting the imaginary part of the refractiveindex of the cover.


Fig.2: Computed modal index N' versus the reduced film thickness D=k0dfor various values of X0= -1.5d, 0.0d, and X0= 0.7d., ns=nc=1.55, nf=1.57(a) S=1, (b)S=2, (c)S=4

Figure 2a shows the real root of the complex variable (mode index of thedispersions equation) of the analytic function versus the reduced filmthickness D= K0d for 8= 1 for various values of x0. It shows that theincreasing of the value of X0, the value of the mode index is alsoincreasing especially for the lower value of the reduced film thickness.Figures 2b and 2c show similar characteristics for 8=2 and 8=4respectively, and the discussions of these pictures have been reportedespecially for 8=2.


Fig. 3: Computed attenuation coefficient versus the reduced film thicknessD =k0d for various values of X0= -1.5d, X0=0.0d, and x0= 0.7d,ns=nc=1.55, nf=1.57(a) 8=1, (b)S=2, (c)8=4

Each figure shows the attenuation coefficient increasing rapidly for smallvalues of the films thickness and it reaching a maximum value point andafter is decreasing.


The three figures show that the attenuation value is getting bigger as thevalue of 6 is getting smaller at the same value of X0 , so the higherattenuation is seen at S=1.Figure 3 a shows the imaginary part of the complex variable (attenuationcoefficient) of the analytic function (dispersions equation) versus thereduced film thickness D=k0d for the same above data. It shows that theattenuation can be controlled by varying the value of X0 or by the powercarried by the waves.Figures 3 a, 3b and 3c show the normalized attenuation coefficient for thesame above data for the modal index as shown in the figure 2.

ConclusionThe nonlinear characteristics for the propagation constant for a lossy

nonlinear, the waveguide is investigated and thus the attenuation of thelowest order TE (TEo) is computed. It is found that the perturbationmethod is an efficient and quick technique to find the complex root of thedispersion equation (the attenuation coefficient).

AcknowledgmentOne of the authors (M. M .S) acknowledges financial support from the

Alexander Von Humboldt-Stiftung (A.V. H), Germany and hospitality ofProf. Dr. Jager at the Gerhard Mercator University, Duisburg, Germany.

References

1. J.G. Ma, "An Exact TE-Solution in Arbitrarily Nonlinear/Anisotropic Media," in proc. MIOP '93, Sindelfingen, Germany,pp. 595-599, 1993.

2. J.P. Torres and L. Torner, " Universal Diagrams for TE WavesGuided by Thin Films Bounded by Saturable Nonlinear Media,"IEEE J. Quantum Elecrton., vol. 29, pp. 917-925, 1993.

3. G.I. Stegeman and C T. Seaton, " Nonlinear Integrated Optics," J.Appl. Phys., vol 58, no. 12, pp. R57-R78, 1985.

4. N. Saiga, "Calculation of TE Modes in Graded-Index NonlinearOptical Waveguides with Arbitrary Profile of Refractive Index," J.Opt. Soc. Amer. B, vol. 8, no.1, pp.88-94, 1991.


5. G.I. Stegeman, E.M. Wright, C.T. Seaton, J.V. Moloney, T.P.Shen, A.A. Maradudin, and R.F. Wallis, "Nonlinear Slab-GuidedWaves in Non-Kerr Like Media," IEEE J. Quantum Electron., vol.22, pp. 977-983, 1986.

6. Y. Chen, "Nonlinear Fibers with Arbitrary Nonlinearity," J. Opt.Soc. Amer. B, vol. 8, no. 11, pp. 2338-2341,1991.

7. J.G. Ma, and I. Wolff, "Propagation Characteristics of TE-WavesGuided by Thin Films Bounded by Nonlinear Media," IEEE Trans.Microwave Theory Techn., vol. 43, no.4, pp. 790-795, 1995.

8. G.I. Stegeman, J.D. Valera, C.T. Seaton, J. Sipe, and A.A.Maradudin, "Nonlinear S-Polarized surface Plasmon Polaritions,"Opt. Commun., vol. 52, no. 3 , pp. 293-296, 1984.

9. A.H. Nayfeh, 'Introduction to Perturbation Techniques," A Wiley-Interscience Publication, John Wily& Sons, N-Y,1981.

10. A. Nayfeh, "Perturbation methods" Wiley, New York, 1973.

11. S.X. She, "Metal-clad Multilayer Dielectric Waveguide: AccuratePerturbation Analysis," J. Opt. Soc. Amer. A, vol. 7, no. 9, pp.1582-1590, 1990.

12. NAG FORTRAN Library Manual Mark 6 Numerical AlgorithmGroup, Downers, Group III, 1977.

13. IMSL Library Reference Manual, 7th ed. International MathematicalLibraries, Houston, Tex, 1979.

14. W.H. Press, B.P. Flannery, S.A. Teukolskv and W.T. VetterlingNumerical Recipes, The Art of Scientific Computing, FORTRANVersion, Cambridge University Press, Cambridge, 1989.

15. MINPACK Documentation Applied Mathematical Division,Argonne National Library, Argonne, III, 1979.

16. S.D. Cont and C.D. Boor, "Elementary Numerical Analysis,"McGraw. -HILL Kogakusha, Ltd, 1972.


17. R.V. Churchill and J.W. Brown, "Complex Variable andApplications," McGraw-Hill, New York, 1984.

18. A.P. Zhao, "Numerical Modeling of TE and TM Modes in LossyNonlinear Optical Waveguides," Opt. Quantum Electron., vol. 27,pp. 735-740, 1995.

A Perturbation Method for Complex Root Finding of Nonlinear Electromagnetic Waves

Documents