Abstract

An exact finite difference (FD) representation of the second-order derivative on three nodes is presented and used to obtain an FD algorithm that allows achieving an arbitrary truncation order. The FD weights are calculated analytically using the series that expresses the field value at a given FD node in terms of the field value and its derivatives at a neighboring node, when a stepwise discontinuity in the refractive index distribution is present between the nodes. The results obtained confirm that the proposed algorithm is accurate, efficient, and achieves the predicted improved performance.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. S. Stern, IEE Proc. J. Optoelectron. 135, 56 (1988).
    [CrossRef]
  2. C. Vassallo, IEE Proc. J. Optoelectron. 139, 137 (1992).
    [CrossRef]
  3. J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
    [CrossRef]
  4. J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
    [CrossRef]
  5. Y. P. Chiou, Y. C. Chiang, and H. C. Chang, J. Lightwave Technol. 18, 243 (2000).
    [CrossRef]
  6. R. Stoffer and H. J. W. M. Hoekstra, Opt. Quantum Electron. 30, 375 (1998).
    [CrossRef]
  7. J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
    [CrossRef]
  8. B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
    [CrossRef]
  9. G. R. Hadley, J. Lightwave Technol. 16, 134 (1998).
    [CrossRef]
  10. Y. P. Chiou and C. H. Du, Opt. Express 18, 4088 (2010).
    [CrossRef] [PubMed]
  11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  12. Y. Saad, Numerical Methods for Large Eigenvalue Problems (Halsted, 1992).
  13. R. Pregla, Analysis of Electromagnetic Fields and Waves: The Method of Lines (Wiley, 2008).
    [CrossRef]

2010

2008

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

2005

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

2000

1998

R. Stoffer and H. J. W. M. Hoekstra, Opt. Quantum Electron. 30, 375 (1998).
[CrossRef]

G. R. Hadley, J. Lightwave Technol. 16, 134 (1998).
[CrossRef]

J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
[CrossRef]

1997

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

1992

C. Vassallo, IEE Proc. J. Optoelectron. 139, 137 (1992).
[CrossRef]

1988

M. S. Stern, IEE Proc. J. Optoelectron. 135, 56 (1988).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Benson, T. M.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

Chang, H. C.

Chiang, Y. C.

Chiou, Y. P.

Du, C. H.

Hadley, G. R.

Hoekstra, H. J. W. M.

R. Stoffer and H. J. W. M. Hoekstra, Opt. Quantum Electron. 30, 375 (1998).
[CrossRef]

Hu, B.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

Nakano, H.

J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
[CrossRef]

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

Pregla, R.

R. Pregla, Analysis of Electromagnetic Fields and Waves: The Method of Lines (Wiley, 2008).
[CrossRef]

Saad, Y.

Y. Saad, Numerical Methods for Large Eigenvalue Problems (Halsted, 1992).

Sekiguchi, M.

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

Sewell, P.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

Shibayama, J.

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Stern, M. S.

M. S. Stern, IEE Proc. J. Optoelectron. 135, 56 (1988).
[CrossRef]

Stoffer, R.

R. Stoffer and H. J. W. M. Hoekstra, Opt. Quantum Electron. 30, 375 (1998).
[CrossRef]

Takahashi, G.

J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
[CrossRef]

Uchiyama, O.

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

Vassallo, C.

C. Vassallo, IEE Proc. J. Optoelectron. 139, 137 (1992).
[CrossRef]

Vukovic, A.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

Wykes, J. G.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

Yamauchi, J.

J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
[CrossRef]

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

IEE Proc. J. Optoelectron.

M. S. Stern, IEE Proc. J. Optoelectron. 135, 56 (1988).
[CrossRef]

C. Vassallo, IEE Proc. J. Optoelectron. 139, 137 (1992).
[CrossRef]

IEEE Photon. Technol. Lett.

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, and H. Nakano, IEEE Photon. Technol. Lett. 9, 961 (1997).
[CrossRef]

J. Yamauchi, G. Takahashi, and H. Nakano, IEEE Photon. Technol. Lett. 10, 1127 (1998).
[CrossRef]

J. Lightwave Technol.

Microw. Opt. Technol. Lett.

J. G. Wykes, P. Sewell, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 44, 95 (2005).
[CrossRef]

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, Microw. Opt. Technol. Lett. 50, 995 (2008).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

R. Stoffer and H. J. W. M. Hoekstra, Opt. Quantum Electron. 30, 375 (1998).
[CrossRef]

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Y. Saad, Numerical Methods for Large Eigenvalue Problems (Halsted, 1992).

R. Pregla, Analysis of Electromagnetic Fields and Waves: The Method of Lines (Wiley, 2008).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the FD grid showing the relative positions of the refractive index profile discontinuities.

Fig. 2
Fig. 2

Dependence of the relative error in the propagation constant on the grid size for the seven-, nine-, and eleven-point FD stencils [11] and the algorithm presented here for k = 2 , 3, and 4 in (7) in the case of the TE mode.

Fig. 3
Fig. 3

Dependence of the relative error in the propagation constant on the grid size for the seven-, nine-, and eleven-point FD stencils [11] and the algorithm presented here for k = 2 , 3, and 4 in (7) in the case of the TM mode.

Equations (11)

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

s x ( 1 s ϕ x ) + ( n 2 k 0 2 β 2 ) ϕ = 0 ,
ϕ i + 1 = b 0 ϕ i + b 1 ϕ i + b 2 ϕ i + HOT = [ 1 q 1 2 q 2 ] × [ 1 0 0 0 θ 0 η 0 1 ] × [ e p D ϕ i e p D ϕ i e p D ϕ i ] .
b M = k = 0 M 1 k ! p k B M k ϕ i .
B 2 l = 1 η l 1 2 I ( l 1 / 2 , q η ) Γ ( l + 1 / 2 ) 2 l + 1 / 2 ( q η ) l + 1 / 2 ( 2 l ) ! ,
B 2 l + 1 = 1 η ( 2 l + 1 ) / 2 1 2 I ( l + 1 / 2 , q η ) Γ ( l + 3 / 2 ) 2 l + 1 / 2 ( q η ) l + 1 / 2 ( 2 l ) ! ,
ϕ i 1 = a 0 ϕ i + a 1 ϕ i + a 2 ϕ i + HOT ,
ϕ i = Δ 2 ϕ i j = 3 D j ϕ i ( j ) ; Δ 2 ϕ i = 1 b 1 ϕ i + 1 ( b 0 b 1 a 0 a 1 ) ϕ i 1 1 a 1 ϕ i 1 ( b 2 b 1 a 2 a 1 ) ; D j = ( b j b 1 a j a 1 ) ( b 2 b 1 a 2 a 1 ) .
ϕ = Δ ϕ i j = 3 C j ϕ i ( j ) .
ϕ i = Δ 2 ϕ i + k = 1 ( β 2 n 2 k 0 2 ) G k ϕ i ; G k ϕ i = ( D 3 k 1 Δ ϕ i D 4 k 1 Δ 2 ϕ i ) ,
D j k = ( D 3 k 1 C j D 4 k 1 C j D j + 2 k 1 ) ,
ε β = β calculated β exact β exact .

Metrics