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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 2 — Jan. 22, 2007
  • pp: 402–407
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A three-dimensional wide-angle BPM for optical waveguide structures

Changbao Ma and Edward Van Keuren  »View Author Affiliations


Optics Express, Vol. 15, Issue 2, pp. 402-407 (2007)
http://dx.doi.org/10.1364/OE.15.000402


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Abstract

Algorithms for effective modeling of optical propagation in three- dimensional waveguide structures are critical for the design of photonic devices. We present a three-dimensional (3-D) wide-angle beam propagation method (WA-BPM) using Hoekstra’s scheme. A sparse matrix algebraic equation is formed and solved using iterative methods. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation, along with a technique for shifting the simulation window to reduce the dimension of the numerical equation and a threshold technique to further ensure its convergence. These techniques can ensure the implementation of iterative methods for waveguide structures by relaxing the convergence problem, which will further enable us to develop higher-order 3-D WA-BPMs based on Padé approximant operators.

© 2007 Optical Society of America

1. Introduction

The classical finite-difference beam propagation method (FD-BPM) [1

01. M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17,3990–3998 (1978). [CrossRef] [PubMed]

-5

05. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation,” J. Quantum. Electron. 26,1335–1339 (1990). [CrossRef]

] is a technique for calculating the electromagnetic field transmitted through materials with inhomogeneous refractive index profiles. The electric field is discretized on a planar grid perpendicular to the direction of propagation. The field on the following plane is calculated based on a numerical solution of the scalar Helmholtz equation using the Crank-Nicholson scheme [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

], and solved using the highly efficient Thomas algorithm [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

] along with transparent boundary conditions (TBC) [7

07. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16,624–626 (1991). [CrossRef] [PubMed]

]. BPM simulations have been used extensively to simulate propagation in 2-D, but 3-D methods are more difficult to implement. 3-D algorithms for the paraxial case have been simplified using the alternating-direction implicit (ADI) method [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

] by splitting the 3-D equation into two 2-D steps. Besides the limitation to paraxial beams, these methods also restrict the simulations to a low refractive index contrast ratio between the core and cladding of the waveguide [4

04. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6,150–161 (2000). [CrossRef]

-5

05. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation,” J. Quantum. Electron. 26,1335–1339 (1990). [CrossRef]

, 8

08. M. D. Feit and J. A. Fleck Jr., “Analysis of rib waveguides and couplers by the propagation method,” J. Opt. Soc. Am A 7,73–79 (1990). [CrossRef]

-9

09. C. Vassallo, “Reformulation for the beam-propagation method,” J.Opt. Soc. Am. A 10,2208–2216 (1993). [CrossRef]

]. Efforts have been taken to relax these limitations for wide-angle (WA) simulations [10

10. J. Gerdes and R. Pregla, “Beam-propagation algorithm based on the method of lines,” J. Opt. Soc. Amer. A 8,389–394 (1991). [CrossRef]

-19

19. Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. 14,345–347 (1997). [CrossRef]

], of which the 2-D wide-angle BPM based on Hoekstra’s scheme [18

18. H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, “On the accuracy of the finite difference method for applications in beam propagating techniques,” Opt. Commun. 94,506–508 (1992). [CrossRef]

, 19

19. Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. 14,345–347 (1997). [CrossRef]

] is simple and practical, because the wide-angle term is included in the formulation by simply replacing the second-order derivative using the derivative of the first-order derivatives. Hadley has also developed WA-BPMs using the Padé approximant operators and the multistep method, which are widely used for 2-D structures, which, however, the author has also claimed to be restricted for 3-D structures due to the lack of efficient solvers [16

16. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17,1426–1428 (1992). [CrossRef] [PubMed]

, 17

17. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17,1743–1745 (1992). [CrossRef] [PubMed]

]. A 3-D wide-angle BPM was recently developed using the efficient Thomas algorithm, Hoekstra’s scheme and the splitting of the 3-D Fresnel wave equation into three 2-D wave equations [20

20. C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14,4668–4674 (2006). [CrossRef] [PubMed]

]. However, both this splitting method and the ADI method may cause splitting errors. In this work, we demonstrate a new 3-D wide-angle BPM based on Hoekstra’s scheme that does not require the splitting of the Fresnel wave equation or the use of the ADI method. As a result, the 3-D equation is approximated more strictly by only one numerical equation. The coefficients matrix of the algebraic equation of this method is not tridiagonal but large and sparse, and its dimension (number of elements) is usually on the order of >108. In this case, use of direct matrix equation solvers is prohibitively expensive and in some cases impossible even with the best available computing power. This large sparse matrix equation, however, can be solved using iterative methods, such as the bi-conjugate gradients method (BICG) and the bi-conjugate gradients stabilized method (BICGSTAB) [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

].

While a different iterative method, the Orthogonal Minimum Residual (ORTHOMIN), has been proposed for 3-D structures [21

21. W. P. Huang and C. L. Xu, “A wide-angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4,1118–1120 (1992). [CrossRef]

], there are no published efforts demonstrating how the well-known convergence problem can be relaxed for the simulations of 3-D optical waveguide structures. When solving a large sparse matrix equation Ax = b , iterative methods have convergence problems if either the coefficient matrix A is ill-conditioned or the elements of the vector b have a wide range of values [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

]. A large simulation window is needed for wide-angle simulations, which may result in an ill-conditioned large size algebraic equation with a corresponding convergence problem. In order to overcome this problem, we introduce a simulation window shifting scheme to decrease the dimension of the equation and a threshold technique to reduce the volatility of field distribution b.

2. Formulation

The 3-D scalar Helmholtz equation is given by

2Ex2+2Ey2+2Ez2+k02n2xyzE=0
(1)

where k0 denotes the free-space wavenumber, E(x,y,z)exp(iωt) is the electric field component with angular frequency ω, and n(x, y, z) is the refractive index profile. If we assume the slowly varying envelope approximation (SVEA), then E(x, y, z) can be separated into two parts: the complex field amplitude ψ(x,y,z) (the axially slowly varying envelope term) and a propagation factor exp(-ik0n0z) (the rapidly varying phase term). The field is then expressed as

Exyz=ψxyzexp(ik0n0z)
(2)

where n0 is the refractive index in the cladding. Substituting Eq. (2) into Eq. (1) yields

iaψ2ψz2=2ψx2+2ψy2+bψ
(3)

with a = 2k 0 n 0 and b = k 2 0(n 2 - n 2). In the usual paraxial formulation, the second-order derivative ∣∂2 ψ/∂z 2∣ in Eq. (3) is dropped, giving

iaψz=2ψx2+2ψy2+bψ
(4)

Rather than dropping this second order derivative, we can discretize Eq. (3) in the z direction into the following

iaψl+1ψlΔz1Δz[ψzl+1ψzl]
=2x2(ψl+1+ψl2)+2y2(ψl+1+ψl2)+(bψ)l+1+(bψ)l2
(5)

and use Eq. (4) at the different z positions l and l+1 to find expressions for the first derivative with respect to z

ψzl=1ia[2ψlx2+2ψly2+(bψ)l]
(6a)
ψzl+1=1ia[2ψl+1x2+2ψl+1y2+(bψ)l+1]
(6b)

Substituting Eq. (6a) and Eq. (6b) into Eq. (5), and discretizing the equation, we find

Aψm1,jl+1+Bψm,j1l+1+Cψm,jl+1+Dψm,j+1l+1+Eψm+1,jl+1
=Rψm1,jl+Sψm,j1l+Tψm,jl+Uψm,j+1l+Vψm+1+jl
(7)

where

{A=E=1Δx21+2iaΔzB=D=1Δy21+2iaΔzC=1Δx2+1Δy224iaΔz+2iaΔz12iaΔzbm,jl+1R=V=1Δx2+1+2iaΔzS=U=1Δy2+1+2iaΔzT=1Δx2+1Δy224iaΔz+2iaΔz+1+2iaΔzbm,ji
(8)

Interestingly, Eq. (7) with its coefficients in Eq. (8) is same as the result when applying Padé (1, 1) approximant operator to the 3-D wave equation [16

16. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17,1426–1428 (1992). [CrossRef] [PubMed]

]. The second-order truncation error of the algorithm is maintained as in tridiagonal BPMs. However, the numerical algebraic equation (7) is not tridiagonal and cannot be solved using the Thomas method. Equation (7) is an M 2 by M 2 matrix equation for an M by M mesh grid, which requires a large amount of computer resources, further resulting in the inapplicability of direct matrix equation solvers, if the matrix is fully stored. For example, a relatively small simulation window with 200 by 200 mesh grids will have a 40,000 by 40,000 matrix equation, which requires 25.6 GB dynamic storage space only for the coefficient matrix of Eq. (7). Additional matrices of the same size are typically needed in direct matrix equation solvers. As a result, direct methods would be prohibitively expensive and in some cases impossible even with the best available computing power. Some direct solvers can store the matrix sparsely, i.e. only the non-zero elements. However, Gaussian elimination method or LU decomposition method in these direct solvers might generate non-sparse triangular matrices, which will require a large amount of computer memory as the matrix is fully stored. Whether the generated triangular matrices are sparse or not is quite problem dependent. Fortunately, each row of the coefficient matrix of Eq. (7) has no more than five non-zero values. As a result, this sparse matrix equation can be efficiently solved using iterative methods [6

06. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

].

Wide-angle simulations, such as those for a tilted waveguide or a branch of a Y-switch, need a large simulation window, which can also be computationally demanding. To ease this problem, we introduce a simulation window shifting technique, in which the window is shifted in the transverse direction every few steps, maintaining the main part of the pulse around the center of the window and avoiding computations involving large numbers of edge cells. In other words, the window is numerically shifted along the flow of the light energy. This is performed by dropping cells at the two edges closer to the origin in transverse directions x and y and generating the same number of cells at the two opposite edges. These new cells are initialized with zero field values. The number of steps along the direction of propagation before the window is shifted depends on the waveguide geometry, but is such that the peak position is kept within several grid points of the origin. This technique can be directly applied to a tilted waveguide structure. It can also be applied to a Y-branch by splitting the simulation window into two after the two branches are decoupled.

Iterative methods may have convergence problems when the coefficient matrix of the equation A is ill-conditioned or the vector b is widely distributed. In our simulations, all nonzero elements of the coefficient matrix are in a narrow range, indicating that the matrix is not ill-conditioned. However, the values of the elements of the vector b are widely distributed due to the wide range of the field [Ψ] in the equation. For example, a normalized Gaussian beam with a waist radius of 3.0 μm drops exponentially to the order of 10-16 roughly 18 μm transversely away from the center of the pulse. In order to implement the simulations using iterative methods, we narrow the field distribution by setting the elements of the field profile to zero if they are below some threshold. This reduces the volatility of the field profile to a large extent. Simulations performed without using the threshold technique confirm that iterative solvers will not converge if the threshold technique is not used.

3. Simulations and discussions

Table 1. The relative L2 norm errors and relative position shift

table-icon
View This Table

a Δy is set equal to Δx, and Δz = 0.2 μm for all simulations. ball errors and shifts are in percentage.

The incorporation of the wide-angle effects enables the more accurate simulation for devices, while the use of the window shift significantly reduces the computational time required. For example, a simulation of the propagation in the tilted channel waveguide described above, with a step size of 0.1 μm, required ∼380 seconds with our new method, compared to ∼1000 seconds with a traditional BPM, and ∼950 seconds with our non-iterative 3D wide angle method, described earlier [20

20. C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14,4668–4674 (2006). [CrossRef] [PubMed]

]. These simulations were all run on an Intel P4 based PC with a 3.0 GHz CPU and 2 GB DDR400 RAM. In addition, our simulations do not seem to show any problems of convergence at longer path lengths. Changing the total path length from 60 to 300 μm, while keeping the step size constant at 0.2 μm, had no effect on the number of iterations needed at each step (between 10 and 20) to reach a fixed tolerance of 10-6.

In summary, we derived a new form of 3-D wide-angle BPM based on Hoekstra’s scheme and an iterative matrix solution, and demonstrated its utility for two cases: the wide-angle propagation of a Gaussian beam with a 30 degree tilt in free space, and the wide-angle propagation in a single-mode waveguide channel 30 degree tilted with respect to z axis. This method yields good convergence and low errors. The simulation window shifting technique is useful for wide-angle simulations of large photonic structures. The threshold technique should be adopted to ensure convergence for any relatively large simulation windows which may result in widely distributed field profile. The threshold technique, however, cannot be applied if there is a considerable amount of radiation leaving the simulation window, from which no widely distributed field profile results. We are currently trying to adapt the new algorithm to a semi-vectorial formulation and investigating other large sparse matrix equation solving methods. This method also enables us to develop higher order 3-D wide-angle algorithms using Padé approximant operators and multistep method as was developed by Hadley for 2-D simulations [16

16. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17,1426–1428 (1992). [CrossRef] [PubMed]

, 17

17. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17,1743–1745 (1992). [CrossRef] [PubMed]

].

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 0348955. The authors also express gratitude to Professor Amy Liu in the Department of Physics at Georgetown University, for her helpful discussion.

References and links

01.

M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17,3990–3998 (1978). [CrossRef] [PubMed]

02.

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quantum. Electron. 26,S185–S197 (1994). [CrossRef]

03.

J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71,803–810 (1981). [CrossRef]

04.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6,150–161 (2000). [CrossRef]

05.

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation,” J. Quantum. Electron. 26,1335–1339 (1990). [CrossRef]

06.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

07.

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16,624–626 (1991). [CrossRef] [PubMed]

08.

M. D. Feit and J. A. Fleck Jr., “Analysis of rib waveguides and couplers by the propagation method,” J. Opt. Soc. Am A 7,73–79 (1990). [CrossRef]

09.

C. Vassallo, “Reformulation for the beam-propagation method,” J.Opt. Soc. Am. A 10,2208–2216 (1993). [CrossRef]

10.

J. Gerdes and R. Pregla, “Beam-propagation algorithm based on the method of lines,” J. Opt. Soc. Amer. A 8,389–394 (1991). [CrossRef]

11.

R. P. Ratowsky and J. A. Fleck Jr., “Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction,” Opt. Lett. 16,787–789 (1991). [CrossRef] [PubMed]

12.

P. -C. Lee, D. Schulz, and E. Voges, “Three-dimensional finite difference beam propagation algorithms for photonic devices,” J. Lightwave Technol. 10,1832–1838 (1992). [CrossRef]

13.

P. -C. Lee and E. Voges, “Three-dimensional semi-vectorial wave-angle beam propagation method,” J. Lightwave Technol. 12,215–225 (1994). [CrossRef]

14.

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21,1082–1087 (2004). [CrossRef]

15.

S. L. Chui and Y. Y. Lu, “NWide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner,” J. Opt. Soc. Am. A 21,420–425 (2004). [CrossRef]

16.

G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17,1426–1428 (1992). [CrossRef] [PubMed]

17.

G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17,1743–1745 (1992). [CrossRef] [PubMed]

18.

H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, “On the accuracy of the finite difference method for applications in beam propagating techniques,” Opt. Commun. 94,506–508 (1992). [CrossRef]

19.

Z. Ju, J. Fu, and E. Feng, “A simple wide-angle beam-propagation method for integrated optics,” Microwave Opt. Technol. Lett. 14,345–347 (1997). [CrossRef]

20.

C. Ma and E. V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14,4668–4674 (2006). [CrossRef] [PubMed]

21.

W. P. Huang and C. L. Xu, “A wide-angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4,1118–1120 (1992). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(130.2790) Integrated optics : Guided waves

ToC Category:
General

History
Original Manuscript: December 8, 2006
Revised Manuscript: January 11, 2007
Manuscript Accepted: January 16, 2007
Published: January 22, 2007

Citation
Changbao Ma and Edward Van Keuren, "A three-dimensional wide-angle BPM for optical waveguide structures," Opt. Express 15, 402-407 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-402


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References

  1. M. D. Feit and J. A. FleckJr., "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978). [CrossRef] [PubMed]
  2. D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994). [CrossRef]
  3. J. Van Roey, J. van der Donk, and P. E. Lagasse, "Beam-propagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803-810 (1981). [CrossRef]
  4. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000). [CrossRef]
  5. Y. Chung and N. Dagli, "An assessment of finite difference beam propagation," J. Quantum. Electron. 26, 1335-1339 (1990). [CrossRef]
  6. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).
  7. G. R. Hadley, "Transparent boundary condition for beam propagation," Opt. Lett. 16, 624-626 (1991). [CrossRef] [PubMed]
  8. M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990). [CrossRef]
  9. C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993). [CrossRef]
  10. J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991). [CrossRef]
  11. R. P. Ratowsky and J. A. FleckJr., "Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction," Opt. Lett. 16, 787-789 (1991). [CrossRef] [PubMed]
  12. P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992). [CrossRef]
  13. P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994). [CrossRef]
  14. A. Sharma and A. Agrawal, "New method for nonparaxial beam propagation," J. Opt. Soc. Am. A 21, 1082-1087 (2004). [CrossRef]
  15. S. L. Chui and Y. Y. Lu, "Wide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner," J. Opt. Soc. Am. A 21, 420-425 (2004). [CrossRef]
  16. G. R. Hadley, "Wide-angle beam propagation using Padé approximant operators," Opt. Lett. 17, 1426-1428 (1992). [CrossRef] [PubMed]
  17. G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992). [CrossRef] [PubMed]
  18. H. J. W. M. Hoekstra, G. J. M. Krijnen and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992). [CrossRef]
  19. Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997). [CrossRef]
  20. C. Ma and E. V. Keuren, "A simple three dimensional wide-angle beam propagation method," Opt. Express 14, 4668-4674 (2006). [CrossRef] [PubMed]
  21. W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992). [CrossRef]

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