Abstract

We present a Wannier-function-based time-domain method for photonic-crystal integrated optical circuits. In contrast to other approaches, this method allows one to trade CPU time against memory consumption and therefore is particularly well suited for the treatment of large-scale systems. As an illustration, we apply the method to the design of a photonic-crystal-based sensor, which utilizes a dual Mach–Zehnder–Fano interferometer.

© 2011 Optical Society of America

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  1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
    [CrossRef]
  2. K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
    [CrossRef]
  3. Y. Jiao, S. Fan, and D. A. B. Miller, Opt. Lett. 30, 141 (2005).
    [CrossRef] [PubMed]
  4. Y. Jiao, S. Fan, and D. A. B. Miller, Opt. Lett. 30, 302 (2005).
    [CrossRef] [PubMed]
  5. D. Hermann, M. Schillinger, S. F. Mingaleev, and K. Busch, J. Opt. Soc. Am. B 25, 202 (2008).
    [CrossRef]
  6. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
    [CrossRef]
  7. A. Quarteroni, Numerical Mathematics (Springer, 2009).
  8. O. Schenk and K. Gärtner, Electron. Trans. Numer. Anal. 23, 158 (2006).
  9. G. L. G. Sleijpen and D. Fokkema, Electron. Trans. Numer. Anal. 1, 11 (1993).
  10. K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.
  11. A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, Opt. Express 16, 11376 (2008).
    [CrossRef] [PubMed]
  12. A. E. Miroshnichenko and Yu. S. Kivshar, Appl. Phys. Lett. 95, 121109 (2009).
    [CrossRef]
  13. M. S. Muradoglu, A. R. Baghai-Wadji, and T. W. Ng, J. Opt. Soc. Am. A 27, 757 (2010).
    [CrossRef]

2010 (2)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

M. S. Muradoglu, A. R. Baghai-Wadji, and T. W. Ng, J. Opt. Soc. Am. A 27, 757 (2010).
[CrossRef]

2009 (1)

A. E. Miroshnichenko and Yu. S. Kivshar, Appl. Phys. Lett. 95, 121109 (2009).
[CrossRef]

2008 (2)

2007 (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

2006 (1)

O. Schenk and K. Gärtner, Electron. Trans. Numer. Anal. 23, 158 (2006).

2005 (2)

2003 (1)

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

1993 (1)

G. L. G. Sleijpen and D. Fokkema, Electron. Trans. Numer. Anal. 1, 11 (1993).

Avniel, Y.

Baghai-Wadji, A. R.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Blum, C.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Busch, K.

D. Hermann, M. Schillinger, S. F. Mingaleev, and K. Busch, J. Opt. Soc. Am. B 25, 202 (2008).
[CrossRef]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Fan, S.

Fokkema, D.

G. L. G. Sleijpen and D. Fokkema, Electron. Trans. Numer. Anal. 1, 11 (1993).

Gärtner, K.

O. Schenk and K. Gärtner, Electron. Trans. Numer. Anal. 23, 158 (2006).

Graham, A. M.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Hermann, D.

D. Hermann, M. Schillinger, S. F. Mingaleev, and K. Busch, J. Opt. Soc. Am. B 25, 202 (2008).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Jiao, Y.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, Opt. Express 16, 11376 (2008).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. E. Miroshnichenko and Yu. S. Kivshar, Appl. Phys. Lett. 95, 121109 (2009).
[CrossRef]

Köhl, M.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Mack, P.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Miller, D. A. B.

Mingaleev, S. F.

D. Hermann, M. Schillinger, S. F. Mingaleev, and K. Busch, J. Opt. Soc. Am. B 25, 202 (2008).
[CrossRef]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko and Yu. S. Kivshar, Appl. Phys. Lett. 95, 121109 (2009).
[CrossRef]

Muradoglu, M. S.

Ng, T. W.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, Opt. Express 16, 11376 (2008).
[CrossRef] [PubMed]

Quarteroni, A.

A. Quarteroni, Numerical Mathematics (Springer, 2009).

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Schenk, O.

O. Schenk and K. Gärtner, Electron. Trans. Numer. Anal. 23, 158 (2006).

Schillinger, M.

D. Hermann, M. Schillinger, S. F. Mingaleev, and K. Busch, J. Opt. Soc. Am. B 25, 202 (2008).
[CrossRef]

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Sleijpen, G. L. G.

G. L. G. Sleijpen and D. Fokkema, Electron. Trans. Numer. Anal. 1, 11 (1993).

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Wolff, C.

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

Zhang, L.

Appl. Phys. Lett. (1)

A. E. Miroshnichenko and Yu. S. Kivshar, Appl. Phys. Lett. 95, 121109 (2009).
[CrossRef]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Electron. Trans. Numer. Anal. (2)

O. Schenk and K. Gärtner, Electron. Trans. Numer. Anal. 23, 158 (2006).

G. L. G. Sleijpen and D. Fokkema, Electron. Trans. Numer. Anal. 1, 11 (1993).

J. Mod. Opt. (1)

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, J. Mod. Opt. , to be published.

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

K. Busch, S. F. Mingaleev, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Other (1)

A. Quarteroni, Numerical Mathematics (Springer, 2009).

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Figures (3)

Fig. 1
Fig. 1

Performance characteristics regarding memory consumption (left panel) and CPU time (right panel) of our WFTD approach using different solvers: PARDISO (crosses), BiCGstab(l) with a sparse matrix scheme (triangles), and BiCGstab(l) with a low-memory implicit matrix–vector- product scheme (squares). Also shown are MEEP computations (circles) with a grid spacing that gives comparable accuracy.

Fig. 2
Fig. 2

Layout of a dual Mach–Zehnder–Fano interferometer with add–drop filter. The underlying PhC consists of a square array of silicon posts ( ϵ Si = 12 , r / a = 0.18 ). The add–drop filter is realized via low-index posts ( ϵ = 1.74 ; gray circles in the shaded region on the right-hand side). An ultra-low-index post with variable refractive index ( ϵ = 1.00 1.05 ; open circle in the shaded region on the left-hand side) implements a tunable Fano-type defect within the Mach–Zehnder interferometer. See the text for details.

Fig. 3
Fig. 3

Transmittance spectra of the dual MZFI for different detunings ( ϵ = 1.00 , solid curve; ϵ = 1.01 , short-dashed curve; ϵ = 1.05 , long-dashed curve) of the ultra-low- index post. The left panel displays the spectrum at the position of probe 1, which exhibits a steep slope. The add–drop filter has been designed to operate at the center of this slope for zero detuning ( ϵ = 1.00 ). Even small values of the detuning lead to a significantly reduced transmittance into the cross port (probe 2; right panel).

Equations (7)

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2 E ( r , t ) = 1 c 2 t 2 ( ϵ p ( r ) + δ ϵ ( r ) ) E ( r , t ) .
F α = 1 c t E α , β A α β E β = 1 c t β ( C α β + D α β ) F β .
A α β = R 2 d 2 r W α * ( r ) 2 W β ( r ) , C α β = R 2 d 2 r W α * ( r ) ϵ p ( r ) W β ( r ) , D α β = R 2 d 2 r W α * ( r ) δ ϵ ( r ) W β ( r ) .
t E α ( m ) 1 Δ t ( a 0 E α ( m ) + a 1 E α ( m 1 ) + a 2 E α ( m 2 ) ) .
a 0 E α ( m ) = c Δ t F α ( m ) ( a 1 E α ( m 1 ) + a 2 E α ( m 2 ) ) ,
β ( c 2 Δ t 2 A α β + a 0 2 ( C α β + D α β ) ) F β ( m ) = c Δ t β A α β ( a 1 E β ( m 1 ) + a 2 E β ( m 2 ) ) a 0 β ( C α β + D α β ) ( a 1 F β ( m 1 ) + a 2 F β ( m 2 ) ) .
D α β γ ( NL ) = R 2 d 2 r W α * ( r ) χ ( 2 ) ( r ) W β ( r ) W γ ( r ) ,

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