.
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A Novel Approach to Study Wave Characteristics of Multimode Interference Couple: Polarized TE Modes
Abstract
In this paper, we propose a new method to study wave propagation in Longitudinally invariant waveguides with arbitrary index profile. In our method, we keep, in the wave equation, the second order derivative of transverse wave field with respect to the propagation direction, which is usually neglected in paraxial approximation utilized in many cases. An explicit expression for the wave field at any longitudinal position along an optical waveguide is derived, thus excluding the use of beam propagation algorithm for computation. This study has demonstrated that our approach yields the same results as those by using a commercial software in which a beam propagation method with Pade' approximation is used.
Keywords:Fourier cosine series, matrix method, index profile, optical waveguides.
Chin-Sung Hsiao : Instructor, Department of Electrical Engineering, HIT
I.Introduction
Beam- propagation methods (BPM) have been frequently used to compute the wave field along an optical waveguide.
With these methods, the field distribution at any longitudinal position of a waveguide can be found by either finite- difference [1- 4] or finite-element [5-8] discretization in the transverse domain. These conventional methods, especially the finite-difference BPM algorithm, are nowadays highly referenced and used in commercial softwares [e.g. BPM-CAD and BeamPROP].
In dealing with the wave propagation with the paraxial approximation valid in use, the second order derivative with respect to propagation distance in the wave equation is simply neglected. This then results in a set of first-order linear differential equations, which can be easily solved. If the paraxial approximation is invalid, as is true in many cases, the aforementioned second order derivative should remain in order to obtain exact and
accurate solutions in solving the corresponding wave propagation problems.
A recurrence formula for the BPM scheme has been widely employed to consider the effect of the aforementioned second-second derivative in the so-called wide-angle approximation [9,10] . The Pade' approximant is commonly used as one kind of such approximation. The numerical results obtained by using the Pade' approximation in a BPM method are more accurate and closer to exact ones when a higher-order Pade' approximant operator is used [11, 12].
II. The Proposed method
We assume that the transverse domain of the wave field is one-dimensional. That is, we deal with the following wave equation in our problem
The resultant wave equation in the example of slab waveguides with longitudinally invariant structures
corresponds to the following matrix equation:
Here we have expressed the electric field as
where N is a large enough number, T the window (or the period) for Fourier cosine series, and the propagation constant. In deriving Eq. (2), we have assumed the wave propagates in the z direction and x is the one-dimensional transverse coordinate.
The matrix B in Eq. (2) is a constant full matrix, and E is defined as
( ), where t represents
transpose. In the conventional BPM ( beam propagation method ) method for solving Eq.(1) with Pade' approximation used, a recurrence formula is employed to obtain approximate results. Basically Eq.
(2) can be solved with similar approximation used.
However, such Pade' approximation in dealing with matrix arithmetic could result
in quite complicated computing algorithm and yield computational inefficiency. Here we propose a novel method to solve the second -order matrix equation ( i.e., Eq.
(2)). Since the matrix B in Eq. (2) is not diagonal, it is difficult to solve it directly.
In this new method, Eq. (2) is first transformed into a matrix equation such as
Here the vector F is defined as E=Y F, where Y is a matrix containing all the eigenvectors of B; is the diagonal matrix with its diagonal elements being the eigenvalues of B (see Eq. (5) below).
Once it is solved, the vector E can be obtained. Eq. (3) can now be readily solved because is diagonal. To show how to solve it, we first note that the matrix equation in (3) corresponds to a set of second-order ordinary differential equations:
where (i=0,1,2,....N) are
( )
elements of F and Λ ,respectively. That is, F and Λ are defined here as
and
The solution of Eq.(4) is simply exp exp
depend on the initial conditions. This solution is explicit in expression and henceforth the vector E can be explicitly obtained as
where and are constants and are the eigenvector of B.
As we can see above, the conventional beam propagation method would not be used to solve the wave propagation problem. In the method proposed here,
simply a set of ordinary differential equations are to be solved. The whole method is quite efficient in computation.
We have used the method to solve an MMIC (multimode interference coupler) problem as shown in Fig.1, where the input waveguide is single-mode with the core index 1.8 and the cladding index 1.446; and the multimode waveguide has the same refractive indices for the core and the cladding as the input waveguide. In the study here, we have assumed a TE wave is launched at the input end of the multimode waveguide. The electric field distribution at the output end of the multimode waveguide calculated by use of the proposed method with N=500 is shown in Fig.2 (a). To compare our method with others, we have also used a commercial software. Fig.2 (b) shows the corresponding electric field distribution obtained by BeamPROP with the approximation of Pade' order (4,4) (the output end of the multimode waveguide is at z=60un). It can be seen from Figs.2 (a) and (b) that the discrepancy between both
results (i.e., in Figs. (a) and (b)) is quite negligible. Fig. 2(c) shows the corresponding result as Pade' (1,0) approximation is used by the commercial software. This is equivalent to the result obtained with paraxial approximation used, i.e., the result obtained with the term
in Eq. (2) neglected.
III. Numerical results
Fig.1 MMIC slab waveguide under study
Fig.2(a)
.
Fig. 2(b)
Fig. 2(c)
Fig,2 Electric field distributions of the MMIC shown in Fig. 2. (a) and (b) are the results obtained by use of the proposed method and BeamPROP, respectively. (c) is the result obtained with paraxial approximation used. The result of 2(a) is in good agreement with that of 2(b) at z=60um.
V. References:
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IV. Conclusion
In conclusion, we have proposed a new method for investigating the wave propagation in an optical waveguide with a longitudinally invariant structure. An explicit expression for the field at any longitudinal position along the waveguide can be obtained in this method, thus excluding the use of beam propagation algorithm for computation. The computational results are quite correct as they compared with a conventional BPM method. We strongly believe that the proposed method is quite efficient in computation.
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