Finite Element Beam Propagation Method for Nonlinear

Optical Waveguides

Takashi Yasui, Masanori Koshiba, Akira Niiyama, and Yasuhide Tsuji

Graduate School of Engineering, Hokkaido University, Sapporo, Japan 060-8628

SUMMARY

A new beam propagation method based on the finiteelement method (FE-BPM) has been developed for theanalysis of nonlinear optical waveguides. A formulation forthe FE-BPM that is applicable not only to the TE mode butalso to the TM mode is presented. Various techniques forenhancing the performance of the FE-BPM are introduced,including the Padé equation of wide-angle beam propaga-tion, the transparent boundary condition, the perfectmatched layer condition for prevention of spurious reflec-tion from the edge of the analysis region, and an algorithmfor adaptive updating of the reference index of refractionand the finite-element grids. Beam propagation analysis ofspatial soliton emission in a nonlinear optical waveguide isperformed in order to investigate the performance of thenew FE-BPM. © 1999 Scripta Technica, Electron CommJpn Pt 2, 82(4): 47�53, 1999

Key words: Beam propagation method; finite ele-ment method; nonlinear optical waveguide; Padé equation;transparent boundary condition; perfect matched layer con-dition.

1. Introduction

Beam propagation methods (BPM), which are widelyused for the beam propagation analysis of opticalwaveguides, include FFT-BPM based on the fast Fouriertransform (FFT), FD-BPM based on the finite difference

method (FDM), and FE-BPM based on the finite elementmethod (FEM) [1]. In particular, the FE-BPM is superiorto the FFT-BPM because it is applicable to a waveguidewith a large index difference or with a strong polarizationdependence. In the FE-BPM, it is natural to select theelement order and the number of elements with referenceto the required computational accuracy and to use nonuni-form grids. Further, the method has additional propertiesnot available in the FFT-BPM and the FD-BPM, such as theability to improve computational efficiency without de-grading numerical accuracy by adaptive updating in thedirection of propagation.

The FE-BPM with the characteristics describedabove has been used for beam propagation analysis in anonlinear optical waveguide [2�6]. Its application has beenlimited to the TE mode, since the nonlinear effect is sepa-rated by means of the split step method [2, 4�6]. Althoughapplication of the Runge�Kutta method has been attempted[3], the analysis has still been limited to the TE mode. Sincethe paraxial equation (Fresnel equation) continues to beused in these conventional FE-BPM approaches [2�6], it isdifficult to apply the method to wide-angle beam propaga-tion analysis. Further, there is no provision for preventionof spurious reflection from the edge of the analysis region.Hence, research on the FE-BPM for the analysis of non-linear optical waveguides lags substantially behind that ofthe linear case.

In this paper, a new formulation of the FE-BPM forthe analysis of a nonlinear optical waveguide that can beapplied to the TM mode as well as the TE mode is presented.Several procedures for improving the performance of theFE-BPM are introduced: the use of the Padé equation to

CCC8756-663X/99/040047-07© 1999 Scripta Technica

Electronics and Communications in Japan, Part 2, Vol. 82, No. 4, 1999Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-C-I, No. 7, July 1998, pp. 417�422

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deal with wide-angle beam propagation [7�10], the appli-cation of the transparent boundary condition (TBC) [9�12]and the perfect matched layer condition (PML) [13, 14] toprevent spurious reflection from the edge of the analysisregion, and the use of an algorithm to adaptively change thereference index of refraction and the finite-element grids[10, 15]. By means of the FE-BPM developed here, beampropagation analysis is carried out for spatial soliton emis-sion in a nonlinear optical waveguide [4]. In addition to theTE mode, the behavior of TM mode propagation is pre-sented for the first time. It is pointed out that the step widthin the propagation direction must be set to a much smallervalue in FE-BPM analysis of a nonlinear optical waveguidethan in that of a linear waveguide.

2. Fundamental Equations

Let us consider a planar (two-dimensional) opticalwaveguide. Let the propagation direction be z, the lamina-tion direction be y, and the direction normal to these be x.It is assumed that there is no structural variation in the xdirection �w /wx 0�. Also, the optical Kerr effect is as -sumed, so that the nonlinear index of refraction is

where nL is the linear index of refraction, c the speed oflight, nc the nonlinearity coefficient, and E the electric fieldvector.

From Maxwell�s equations, the wave equation belowis obtained:

where k0 is the free-space wave number, and ), p, and q aregiven in terms of the x components of the electric andmagnetic fields, Ex and Hx, as

Also, s is a parameter related to the PML, whose selectionis described later.

Let the appropriate reference index of refraction ben0. Let

Substituting Eq. (5) into Eq. (2) and assuming wp /wz | 0for the TM mode, for a slowly varying I�y, z�, the funda-mental equation for wide-angle beam propagation analysisis obtained:

Naturally, s = 1 in the region outside the PML.

3. Discretization by Finite Element Method

The waveguide cross section (analysis region)y1 d y d yM is divided by second-order line elements and theFEM based on Galerkin�s method is applied to Eq. (6). Thenthe following equation is obtained:

where {I} is the nodal electric field or magnetic fieldvector, and {0} is the null vector. The subscript M corre-sponds to the total number of nodes, and the superscript Tindicates the transpose. Further, the matrices [K], [M], and[K]* are given by

where {N} is the shape function vector and 6e indicatessummation over all elements. The method for setting theparameters k1 and kM in the matrix [K]* with regard to theTBC is described later. Naturally, k1 kM 0 if the TBC isnot used.

Equation (7) can be modified by the Padé approxima-tion [7] as

where the matrices [K~] and [M~ ] are given by

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

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4. Crank�Nicolson Method

When the Crank�Nicolson method is applied to Eq.(12) in the z direction,

In the above, 'z is the step width in the propagation direc-tion, and the subscripts i and i + 1 indicate quantities relatedto the i-th and (i + 1)-th propagation steps. The matrix[A]i is LU-decomposed and the forward and backwardsubstitutions are repeated, with the bandedness and sparse-ness of the matrices [A]i and [B]i taken into account. Then,without computing the inverse of the matrix [A]i, the valueof {I}i�1 can be derived from the value of {I}i.

The finite-element grids are adaptively updated ateach propagation step [10, 15]. Also, the reference index ofrefraction n0 is updated at each calculation step [10, 15] as

where � indicates the conjugate transpose.If the distribution of the hypothetical electric and

magnetic conductivities in the PML in the thickness direc-tion has a square profile, the parameter s for the PML isgiven by the following [13]:

where O is the free-space wavelength, d is the thickness ofthe PML, y0 is the y coordinate of the surface of the PML,and R is the theoretical reflection coefficient.

On the other hand, the parameters k1 and kM relatedto the TBC are automatically determined [12] from

The real parts of k1,i and kM,i must be positive so that thecondition of radiation to the external region is satisfied.

The electric field needed for the evaluation of thenonlinear index of refraction is given for each element as

for the TE mode and as

for the TM mode. Here, {I}e is the electric field or magneticfield at the element node and Z0 is the free-space impedance.

5. Numerical Results and Discussion

First, let us consider a nonlinear optical waveguidewith a linear cladding and a nonlinear cladding on bothsides of the linear core, as shown in Fig. 1. The wavelengthof the input light is O = 1.3 Pm and the waveguide parame-ters are [4]

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

Fig. 1. Nonlinear optical waveguide with a linear corebounded by linear and nonlinear claddings.

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When the fundamental TE and TM modes are incident onsuch a nonlinear optical waveguide, the power Pcore in thecore after propagation of 300 Pm is plotted as a function ofthe input power Pin in Fig. 2. The step width in the propa-gation direction is 'z 0.03 Pm. When the input powerexceeds a certain threshold value, the optical power in thecore decreases suddenly and a spatial soliton is emitted.After spatial soliton emission, the power in the core in-creases again, then decreases. At this point, a second spatialsoliton is emitted. There is little difference in the thresholdvalue of spatial soliton emission between the TE and TMmodes. Figures 3 and 4 show the spatial soliton emission atinput powers Pin 280 and Pin = 360 W/m, respectively.

In order to study the effect of the step width 'z in thepropagation direction on the solution, the calculations are

carried out again with 'z = 0.3 and 0.003 Pm; the results

are shown in Figs. 5 and 6. In the case of a linear waveguide,

the convergence is usually reached within a wavelength. Itis insufficient in the case of a nonlinear waveguide. Thespatial soliton is radiated obliquely to the predetermined

propagation direction (z). Hence, the refractive index pro-

file varies significantly in the z direction. Figure 7 shows

the position ypeak of the peak of the spatial soliton afterpropagation of 300 Pm as a function of the step width. In

order to obtain a converged solution, it is necessary to make

the value of 'z several hundredths to several thousandths.

Care is required in setting the step width along the propa-gation direction. Although the PML or TBC is installed at

Fig. 2. Normalized power trapped in a linear core as afunction of input power.

Fig. 3. One-solition emission (Pin = 280 W/m). Fig. 5. Influence of propagation step size ('z 0.3 Pm).

Fig. 4. Two-solition emission (Pin 360 W/m).

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the analysis region edges, no substantial difference is ob-served in the results.

Finally, we consider the soliton coupler shown in Fig.8 in which the spatial soliton emission phenomenon is used.The wavelength of the input light is O = 1.3 Pm and thewaveguide parameters are [9]

Fig. 7. Position of soliton peak as a function of 'z /O.

Fig. 8. Nonlinear optical waveguide coupler.

Fig. 6. Influence of propagation step size ('z = 0.003Pm).

Fig. 9. Spatial soliton emission and trapping (T = 0).

Fig. 10. Spatial soliton emission and trapping (T = S).

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Based on the results shown in Fig. 7, the step width in thepropagation direction is chosen as 'z 10�3 Pm �'z / O |8 u 10�4�. The light input to the central linear core is asuperposition of the signal light with power P0 and thecontrol light with power P1. The signal light is the funda-mental mode of a linear symmetric three-layeredwaveguide with indices of refraction of 1.57 and 1.55 in thecore and the clad and a core thickness of 5 Pm. The controllight is the first higher-order mode in the same waveguide.The phase difference of the control light with respect to thesignal light is T. Figures 9 and 10 show the emission andcapture of a spatial soliton when T = 0 and S. Here, P0 =232 W/m and P1 = 15 W/m. It is found that the nonlinearcore for capture of the spatial soliton can be actively se-lected by means of the phase difference T. There is verylittle difference between the TE and TM modes in theemission angle and coupling length of the spatial soliton,and hence the polarization dependence is very small.

6. Conclusions

The FE-BPM is formulated for nonlinear opticalwaveguide analysis. As the fundamental equation, the Padéequation is used to deal with wide-angle beam propagation.The TBC and PML are introduced to prevent spuriousreflection from the edges of the analysis region. Othertechniques for improving the performance of the FE-BPM,such as an adaptive reference index of refraction and adap-tive grids, are taken into account. Specifically, beam propa-gation analysis is carried out for spatial soliton emission ina nonlinear optical waveguide. It is shown in particular thatthe step width in the propagation direction must be madesufficiently small in nonlinear waveguide analysis.

Although the TBC and the PML are found to beeffective as absorbing boundary conditions, no sufficientstudy has been carried out for determination of the optimumparameters of the PML for a nonlinear optical waveguide.This is a topic of future investigation.

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AUTHORS (from left to right)

Takashi Yasui (student member) graduated from Fukui University in 1997 and entered the M.S. course at HokkaidoUniversity. He has been engaged in research on optical and wave electronics.

Masanori Koshiba (member) graduated from Hokkaido University in 1971 and completed the M.S. course in 1973. Aftercompleting the doctoral course in 1976, he joined Kitami Institute of Technology as a lecturer, and later became an associateprofessor there. In 1979, he moved to Hokkaido University, and has been a professor since 1987. He has been engaged in researchon optical and wave electronics. He holds a D.Eng. degree. He received a Best Paper Award from IEICE in 1987 and 1997. Heis the author of Fundamentals of Finite Element Method for Optics and Waves (Morikita), Optical Waveguide Analysis (Asakura),Optical Waveguide Analysis (McGraw�Hill), and Optical Waveguide Theory by the Finite Element Method (KTK ScientificPublishers/Kluwer Academic Publishers). He is also a coauthor or chapter contributor of 10 books. He is a member of theInstitute of Image Information and Television Engineers of Japan, the Institute of Electrical Engineers of Japan, the Japan Societyof Simulation Technology, the Japan Society for Computational Methods in Engineering, the Japan Society of AppliedElectromagnetics and Mechanics, and the Japan Society for Computational Engineering and Science, and is a senior memberof IEEE.

Akira Niiyama (student member) graduated from Hokkaido University in 1994 and completed the M.S. course in 1996.He then entered the doctoral course. He has been engaged in research on optical and wave electronics.

Yasuhide Tsuji (member) graduated from Hokkaido University in 1991 and completed the M.S. course in 1993. Hecompleted the doctoral course in 1996 and joined Hokkaido Institute of Technology as a research associate, and was subsequentlyappointed a lecturer in there. In 1997, he became an associate professor in the Department of Electronic and InformationEngineering, Graduate School, Hokkaido University. He has been engaged in research on optical and wave electronics. He holdsa D.Eng. degree. He received a Best Paper Award from IEICE in 1997. He is a member of the Japan Society of Applied Physicsand of IEEE.

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