Abstract

A simple strategy for accurately recovering discontinuous functions from their Fourier series coefficients is presented. The aim of the proposed approach, named spectrum splitting (SS), is to remove the Gibbs phenomenon by making use of signal-filtering-based concepts and some properties of the Fourier series. While the technique can be used in a vast range of situations, it is particularly suitable for being incorporated into fast-Fourier-transform-based electromagnetic mode solvers (FFT-MSs), which are known to suffer from very poor convergence rates when applied to situations where the field distributions are highly discontinuous (e.g., silicon-on-insulator photonic wires). The resultant method, SS-FFT-MS, is exhaustively tested under the assumption of a simplified one-dimensional model, clearly showing a dramatic improvement of the convergence rates with respect to the original FFT-based methods.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
    [Crossref]
  2. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
    [Crossref]
  3. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [Crossref] [PubMed]
  4. E. Silvestre, T. Pinheiro-Ortega, P. Andrés, J. J. Miret, and A. Ortigosa-Blanch, "Analytical evaluation of chromatic dispersion in photonic crystal fibers," Opt. Lett. 30, 453-455 (2005).
    [Crossref] [PubMed]
  5. J. M. López-Doña, J. G. Wangüemert Pérez, and I. Molina-Fernández, "Fast-Fourier-based three-dimensional full-vectorial beam propagation method," IEEE Photon. Technol. Lett. 17, 2319-2321 (2005).
    [Crossref]
  6. A. Ortega-Moñux, J. G. Wangüemert-Pérez, I. Molina-Fernández, E. Silvestre, and P. Andrés, "Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides," IEEE Photonics Technol. Lett. 18, 1128-1130 (2006).
    [Crossref]
  7. A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
    [Crossref]
  8. D. Marcuse, "Solution of the vector wave equation for general dielectric waveguides by the Galerkin method," IEEE J. Quantum Electron. 28, 459-465 (1992).
    [Crossref]
  9. G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [Crossref]
  10. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [Crossref]
  11. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [Crossref]
  12. Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E 67, 046607 (2003).
    [Crossref]
  13. C. H. Sauvan, P. Lalanne, and J. P. Hugonin, "Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory," Opt. Quantum Electron. 36, 271-284 (2004).
    [Crossref]
  14. A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
    [Crossref]
  15. D. Gottlieb and C. W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Rev. 39, 644-668 (2005).
    [Crossref]
  16. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, (SIAM Publications, 1998).

2007 (1)

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[Crossref]

2006 (2)

A. Ortega-Moñux, J. G. Wangüemert-Pérez, I. Molina-Fernández, E. Silvestre, and P. Andrés, "Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides," IEEE Photonics Technol. Lett. 18, 1128-1130 (2006).
[Crossref]

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[Crossref]

2005 (3)

D. Gottlieb and C. W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Rev. 39, 644-668 (2005).
[Crossref]

E. Silvestre, T. Pinheiro-Ortega, P. Andrés, J. J. Miret, and A. Ortigosa-Blanch, "Analytical evaluation of chromatic dispersion in photonic crystal fibers," Opt. Lett. 30, 453-455 (2005).
[Crossref] [PubMed]

J. M. López-Doña, J. G. Wangüemert Pérez, and I. Molina-Fernández, "Fast-Fourier-based three-dimensional full-vectorial beam propagation method," IEEE Photon. Technol. Lett. 17, 2319-2321 (2005).
[Crossref]

2004 (1)

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, "Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory," Opt. Quantum Electron. 36, 271-284 (2004).
[Crossref]

2003 (1)

Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E 67, 046607 (2003).
[Crossref]

2001 (2)

1996 (3)

1992 (1)

D. Marcuse, "Solution of the vector wave equation for general dielectric waveguides by the Galerkin method," IEEE J. Quantum Electron. 28, 459-465 (1992).
[Crossref]

1985 (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Marcuse, "Solution of the vector wave equation for general dielectric waveguides by the Galerkin method," IEEE J. Quantum Electron. 28, 459-465 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (2)

J. M. López-Doña, J. G. Wangüemert Pérez, and I. Molina-Fernández, "Fast-Fourier-based three-dimensional full-vectorial beam propagation method," IEEE Photon. Technol. Lett. 17, 2319-2321 (2005).
[Crossref]

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate analysis of photonic crystal fibers by means of the fast-Fourier-based mode solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. Ortega-Moñux, J. G. Wangüemert-Pérez, I. Molina-Fernández, E. Silvestre, and P. Andrés, "Enhanced accuracy in fast-Fourier-based methods for full-vector modal analysis of dielectric waveguides," IEEE Photonics Technol. Lett. 18, 1128-1130 (2006).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C. H. Sauvan, P. Lalanne, and J. P. Hugonin, "Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory," Opt. Quantum Electron. 36, 271-284 (2004).
[Crossref]

Phys. Rev. B (1)

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[Crossref]

Phys. Rev. E (1)

Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E 67, 046607 (2003).
[Crossref]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[Crossref]

SIAM Rev. (1)

D. Gottlieb and C. W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Rev. 39, 644-668 (2005).
[Crossref]

Other (1)

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, (SIAM Publications, 1998).

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 (10)

Fig. 1
Fig. 1

Qualitative representation of discontinuous function spectrums. From N 0 , coefficients of g ( x ) and s ( x ) are the same.

Fig. 2
Fig. 2

Original (solid curve) and modified (dashed curve) step function s ( x ) .

Fig. 3
Fig. 3

(a) Reconstruction of the function g ( x ) (SS method indistinguishable from the analytical result). (b) Absolute error of the recovered function g ̃ ( x ) in the vicinity of x = π 2 (right discontinuity). Dashed curve, Fourier sum; bold solid curve, SS method. Number of harmonics N = 64 .

Fig. 4
Fig. 4

Relative error of high-frequency band versus the number of coefficients N.

Fig. 5
Fig. 5

Product by a function operation in (a) Fourier coefficients domain, (b) space domain, and (c) space domain combined with the SS technique.

Fig. 6
Fig. 6

Step-index slab. Relative error of the effective index ( N eff ) . Analytical value, N eff = 3.10305 . X 0 = 1.5 μ m . Dashed curve, FFT-MS; bold solid curve SS-FFT-MS.

Fig. 7
Fig. 7

Step-index slab. Electric field profile of the fundamental mode. Dotted curve, analytical result; dashed curve, FFT-MS; bold solid curve, SS-FFT-MS (indistinguishable from the analytical result). Number of harmonics N = 64 .

Fig. 8
Fig. 8

Directional coupler. (a) Relative error of the calculated N eff for the two lowest-order (even and odd) TM supermodes. Analytical values, N eff even = 2.7215 and N eff odd = 2.6138 . (b) Relative error of the coupling length L c . Analytical value, L c = 7.1913 . X 0 = 4 μ m .

Fig. 9
Fig. 9

Directional coupler. (a) Electric field distribution of the even supermode. (b) Electric field distribution of the odd supermode. Number of harmonics N = 100 .

Fig. 10
Fig. 10

Directional coupler. Coupling length L c as a function of the core separation s. Inset, relative error of the coupling length. Number of harmonics N = 100 . X 0 = 4 μ m .

Equations (18)

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

t ̃ ( x ) = k = N N T k exp ( j k 2 π X 0 x ) ,
T k = 1 X 0 X 0 t ( x ) exp ( j k 2 π X 0 x ) d x .
g ( x ) = s ( x ) + g c ( x ) ,
G ¯ = S ¯ + G c ¯ ,
s j ( x ) = { g ( x j ) , x < x j g ( x j + ) , x x j } ,
s ( x ) = j s j ( x ) .
g ( x j + ) g ( x j ) = C j .
g ̃ ( x j ) = k = N N G k exp ( j k 2 π X 0 x j ) = g ( x j ) + g ( x j + ) 2 .
g ( x ) = { 6 ( 1 + x π ) , x < π 2 3 ( 1 x π ) , x π 2 12 ( 1 x π ) , x > π 2 } .
g ( x 1 + ) g ( x 1 ) = 1 2 ,
g ̃ ( x 1 ) = g ( x 1 ) + g ( x 1 + ) 2 = 2.25 ,
g ( x 2 + ) g ( x 2 ) = 4 ,
g ̃ ( x 2 ) = g ( x 2 ) + g ( x 2 + ) 2 = 3.75 ,
ξ HF ( N ) = 10 log ( k > N G k S k 2 k G k S k 2 ) .
x { 1 ϵ ( x ) x [ ϵ ( x ) e x ( x ) ] } + k 0 2 ϵ ( x ) e x ( x ) = β 2 e x ( x ) ,
[ M ] ̿ E x ¯ = β 2 E x ¯ ,
[ M ] ̿ = [ [ D ] ̿ [ P ( ϵ 1 ) ] ̿ [ D ] ̿ [ P ( ϵ ) ] ̿ + k 0 2 [ P ( ϵ ) ] ̿ ] ,
e x ( x j + ) e x ( x j ) = ϵ ( x j ) ϵ ( x j + ) .

Metrics