Abstract

A new method for solving the wave equation is presented that is nonparaxial and can be applied to wide-angle beam propagation. It shows very good stability characteristics in the sense that relatively larger step sizes can be taken. An implementation by use of the collocation method is presented in which only simple matrix multiplications are involved and no numerical matrix diagonalization or inversion is needed. The method is hence faster and is also highly accurate.

© 2004 Optical Society of America

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  1. D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
    [CrossRef] [PubMed]
  2. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743–1745 (1992).
    [CrossRef] [PubMed]
  3. Y. Chung, N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photon. Technol. Lett. 6, 540–542 (1994).
    [CrossRef]
  4. W. P. Huang, C. L. Xu, “A wide-angle vector beam propagation method,” IEEE Photon. Technol. Lett. 4, 1118–1120 (1992).
    [CrossRef]
  5. J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
    [CrossRef]
  6. J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
    [CrossRef]
  7. Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
    [CrossRef]
  8. H. Rao, M. J. Steel, R. Scarmozzino, R. Osgood, “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
    [CrossRef]
  9. H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
    [CrossRef]
  10. Y. Y. Lu, S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002).
    [CrossRef]
  11. C. Vassallo, “Limitations of the wide-angle beam propagation method in non-uniform systems,” J. Opt. Soc. Am. A 13, 761–770 (1996).
    [CrossRef]
  12. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate solution of the Helmholtz equation by Lanczos orthogonalization for media with loss or gain,” Opt. Lett. 17, 10–12 (1992).
    [CrossRef] [PubMed]
  13. Q. Luo, C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
    [CrossRef]
  14. A. Sharma, A. Agrawal, “Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation,” Opt. Commun. 216, 41–45 (2003).
    [CrossRef]
  15. A. Sharma, S. Banerjee, “Method for propagation of total fields or beams through optical waveguides,” Opt. Lett. 14, 96 (1989).
    [CrossRef] [PubMed]
  16. A. Taneja, A. Sharma, “Propagation of beams through optical waveguiding structures: comparison of the beam propagation method (BPM) and the collocation method,” J. Opt. Soc. Am. A 10, 1739–1745 (1993).
    [CrossRef]
  17. A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed. (EMW, Cambridge, Mass., 1995), pp. 143–198.
  18. I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
    [CrossRef]
  19. H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
    [CrossRef]

2003 (1)

A. Sharma, A. Agrawal, “Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation,” Opt. Commun. 216, 41–45 (2003).
[CrossRef]

2002 (2)

Y. Y. Lu, S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002).
[CrossRef]

Q. Luo, C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[CrossRef]

2000 (2)

H. Rao, M. J. Steel, R. Scarmozzino, R. Osgood, “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

1999 (1)

1997 (1)

Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
[CrossRef]

1996 (3)

C. Vassallo, “Limitations of the wide-angle beam propagation method in non-uniform systems,” J. Opt. Soc. Am. A 13, 761–770 (1996).
[CrossRef]

I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

1995 (1)

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

1994 (1)

Y. Chung, N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photon. Technol. Lett. 6, 540–542 (1994).
[CrossRef]

1993 (1)

1992 (3)

1990 (1)

1989 (1)

Agrawal, A.

A. Sharma, A. Agrawal, “Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation,” Opt. Commun. 216, 41–45 (2003).
[CrossRef]

Banerjee, S.

Betty, I.

H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

Chung, Y.

Y. Chung, N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photon. Technol. Lett. 6, 540–542 (1994).
[CrossRef]

Dagli, N.

Y. Chung, N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photon. Technol. Lett. 6, 540–542 (1994).
[CrossRef]

El-Refaei, H.

H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Glasner, M.

Hadley, G. R.

Huang, W. P.

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

Ilic, I.

I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Koshiba, M.

Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
[CrossRef]

Law, C. T.

Q. Luo, C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[CrossRef]

Lu, Y. Y.

Y. Y. Lu, S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002).
[CrossRef]

Luo, Q.

Q. Luo, C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[CrossRef]

März, R.

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

Matsubara, K.

Nakano, H.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

Nolting, H.-P.

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

Osgood, R.

H. Rao, M. J. Steel, R. Scarmozzino, R. Osgood, “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Rao, H.

Ratowsky, R. P.

Scarmozzino, R.

H. Rao, M. J. Steel, R. Scarmozzino, R. Osgood, “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Sekiguchi, M.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

Sharma, A.

A. Sharma, A. Agrawal, “Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation,” Opt. Commun. 216, 41–45 (2003).
[CrossRef]

A. Taneja, A. Sharma, “Propagation of beams through optical waveguiding structures: comparison of the beam propagation method (BPM) and the collocation method,” J. Opt. Soc. Am. A 10, 1739–1745 (1993).
[CrossRef]

A. Sharma, S. Banerjee, “Method for propagation of total fields or beams through optical waveguides,” Opt. Lett. 14, 96 (1989).
[CrossRef] [PubMed]

A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed. (EMW, Cambridge, Mass., 1995), pp. 143–198.

Shibayama, J.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

Steel, M. J.

Tanabe, T.

Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
[CrossRef]

Taneja, A.

Tsuji, Y.

Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
[CrossRef]

Vassallo, C.

Wei, S. H.

Y. Y. Lu, S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002).
[CrossRef]

Xu, C. L.

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

Yamauchi, J.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

Yevick, D.

H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett. (6)

Y. Chung, N. Dagli, “A wide-angle propagation technique using an explicit finite-difference scheme,” IEEE Photon. Technol. Lett. 6, 540–542 (1994).
[CrossRef]

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

H. El-Refaei, I. Betty, D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

Y. Y. Lu, S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 8, 1361–1363 (1996).
[CrossRef]

Q. Luo, C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[CrossRef]

IEEE Trans. Magn. (1)

Y. Tsuji, M. Koshiba, T. Tanabe, “A wide-angle beam propagation method based on a finite element scheme,” IEEE Trans. Magn. 33, 1544–1547 (1997).
[CrossRef]

J. Lightwave Technol. (4)

H. Rao, M. J. Steel, R. Scarmozzino, R. Osgood, “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

I. Ilić, R. Scarmozzino, R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

A. Sharma, A. Agrawal, “Wide angle and bi-directional beam propagation using the collocation method for the non-paraxial wave equation,” Opt. Commun. 216, 41–45 (2003).
[CrossRef]

Opt. Lett. (4)

Other (1)

A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed. (EMW, Cambridge, Mass., 1995), pp. 143–198.

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Figures (7)

Fig. 1
Fig. 1

Error in propagation (ERR) as a function of the number of propagation steps with Δz for the graded-index waveguide.5

Fig. 2
Fig. 2

ERR with the tilt angle of the graded-index waveguide5 for propagation up to 100 μm.

Fig. 3
Fig. 3

ERR as a function of the number of propagation steps with Δz for the step-index waveguide.6

Fig. 4
Fig. 4

ERR with the tilt angle of the step-index waveguide6 for propagation up to 100 μm.

Fig. 5
Fig. 5

ERR as a function of the number of propagation steps with Δz for the benchmark step-index waveguide.19

Fig. 6
Fig. 6

ERR with the tilt angle of the benchmark step-index waveguide19 for propagation up to 100 μm.

Fig. 7
Fig. 7

ERR with the reference refractive index for the benchmark step-index waveguide19 for propagation up to 100 μm with step size 0.1 μm at 40°.

Tables (1)

Tables Icon

Table 1 Comparison of Error–Power Loss in Propagation to 100 μm in the Benchmark a Step-Index Waveguide for TE10 Modes with Different Methods

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

2ψx2+2ψz2+k02n2(x, z)ψ(x, z)=0,
Φ/z=H(z)Φ(z),
Φ(z)=ψψz,H(z)=01-t2-k02n20.
H(z)=H1+H2(z)=01-t2-k02nr20+00k02(nr2-n2)0.
Φ(z+Δz)=PQ(z)PΦ(z)+O((Δz)3),
P=exp(12H1Δz),Q(z)=exp(H2Δz).
Q(z)=10k02(nr2-n2)Δz1,
(H2)m=0,m2,
ψ(x, z)=n=1Ncn(z)ϕn(x),
d2Ψ/dz2+[S0+k02nr2I+R(z)]Ψ(z)=0,
Ψ(z)=ψ(x1, z)ψ(x2, z)ψ(xN, z),
R(z)=k02Δn2(x1, z)000Δn2(x2, z)000Δn2(xN, z),
Φ(z)=ΨdΨ/dz,
P=expΔz20I-(S0+k02nr2I)0,
Q(z)=I0-R(z)I,
ERR=1-ψexact*ψcalcdx2|ψinp|2dx|ψexact|2dx,
d2Ψ/dz2+SΨ(z)=0,
Φ/z=H1Φ(z),
H1=0I-S0
Φ(z+Δz)=exp(H1Δz)Φ(z).
Ψ(z+Δz)=cos(SΔz)Ψ(z)+1Ssin(SΔz)Ψ(z),
Ψ(z+Δz)=-Ssin(SΔz)Ψ(z)+cos(SΔz)Ψ(z).
exp(H1Δz)=cos(SΔz)1Ssin(SΔz)-Ssin(SΔz)cos(SΔz).
SΔz=V(ΛΔz)V-1,Λ=diag.(Λi)
cos(SΔz)=Vcos(ΛΔz)V-1,
sin(SΔz)=Vsin(ΛΔz)V-1.
exp(H1Δz)=V00V×cos(ΛΔz)1Λsin(ΛΔz)-Λsin(ΛΔz)cos(ΛΔz)×V-00V-1.
ϕn(x)=cos(vnx)forn=1, 3, 5,N-1,
=sin(vnx)forn=2, 4, 6,N,
xj=2jN+1-1L,j=1, 2, 3,N.
S=AGA-1+k02nr2I=A(G+k02nr2I)A-1,
G=diag.(-v12-v22-v32-vN2).
V=A,Λi=k02nr2-vi2.
V-1=A-1=2N+1AT.
exp(H1Δz)=2N+1A00A×c100s1000c200s2000cN00sNs˜100c1000s˜200c2000s˜N00cN×AT00AT,
ci=cos(ΛiΔz),si=1/Λisin(ΛiΔz),
s˜i=-Λisin(ΛiΔz).

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