Abstract

We present a powerful sensitivity analysis method for devices in a photonic crystal. The method is based on a Wannier basis field expansion and efficient matrix analysis techniques for finding eigenvalue and transmission gradients with respect to the perturbation. The method permits fast analysis of a large number of dielectric perturbation situations for multiple devices in a photonic crystal. We verify the method with finite-difference time-domain and plane-wave expansion calculations.

© 2005 Optical Society of America

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References

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics:?the Finite-Difference Time-Domain Method, 2nd ed., Artech House Antennas and Propagation Library (Artech House, Norwood, Mass., 2000).
  2. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  3. G. Veronis, R. W. Dutton, and S. Fan, Opt. Lett. 29, 2288 (2004).
    [CrossRef] [PubMed]
  4. J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
    [CrossRef]
  5. D. M. Whittaker and M. P. Croucher, Phys. Rev. B 67, 085204 (2003).
    [CrossRef]
  6. K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, J. Phys. Condens. Matter 15, R1233 (2003).
    [CrossRef]
  7. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  8. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, England, 1990), p. 561.

2004

2003

D. M. Whittaker and M. P. Croucher, Phys. Rev. B 67, 085204 (2003).
[CrossRef]

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

2002

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

2001

2000

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

Albert, J. P.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

Bertho, D.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

Busch, K.

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

Cassagne, D.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

Croucher, M. P.

D. M. Whittaker and M. P. Croucher, Phys. Rev. B 67, 085204 (2003).
[CrossRef]

Dutton, R. W.

Fan, S.

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Garcia-Martin, A.

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

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics:?the Finite-Difference Time-Domain Method, 2nd ed., Artech House Antennas and Propagation Library (Artech House, Norwood, Mass., 2000).

Hermann, D.

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

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, England, 1990), p. 561.

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, England, 1990), p. 561.

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

Jouanin, C.

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

Mingaleev, S. F.

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

Schillinger, M.

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

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics:?the Finite-Difference Time-Domain Method, 2nd ed., Artech House Antennas and Propagation Library (Artech House, Norwood, Mass., 2000).

Veronis, G.

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and M. P. Croucher, Phys. Rev. B 67, 085204 (2003).
[CrossRef]

J. Phys. Condens. Matter

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

Opt. Express

Opt. Lett.

Phys. Rev. B

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, Phys. Rev. B 61, 4381 (2000).
[CrossRef]

D. M. Whittaker and M. P. Croucher, Phys. Rev. B 67, 085204 (2003).
[CrossRef]

Phys. Rev. E

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Other

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, England, 1990), p. 561.

A. Taflove and S. C. Hagness, Computational Electrodynamics:?the Finite-Difference Time-Domain Method, 2nd ed., Artech House Antennas and Propagation Library (Artech House, Norwood, Mass., 2000).

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Figures (2)

Fig. 1
Fig. 1

Reflection spectra for a PC waveguide bend [see inset: high-index rods n=3.4 in air, rod radius 0.18a, where a is the lattice constant]. The two highlighted rods (open circles) in the inset are perturbed by a 10% radius increase. Solid and dashed curves, spectrum from the FDTD method, before and after perturbation, respectively; filled circles, original spectrum found by use of Wannier expansion; circles, perturbed spectrum found by use of Eq. (9).

Fig. 2
Fig. 2

(a) Resonator formed by three missing rods. The computational size is 9 by 9 lattice constants, larger than the area shown. The PC parameters are the same as in Fig. 1. (b)–(d) Sensitivity maps for the three eigenfrequencies of the resonator. The shading above each rod shows the changes to ω in units of 2πc/a×10-4 when the particular rod’s radius increases by 10%. Using WBG sensitivity analysis, it took 10 s to calculate all three maps on a Pentium 3 computer.

Equations (9)

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I+iDia=1ω2Aa,    aan,R,
DinR,mS=R2Wn,R*rΔirWm,Srd2r.
Bx=y,
B+δDx=y,
E+A-1Dδ0a=βa,    β=c/ω2,
Dδ0nR,mS=rhγ=0,rWn,R*sWm,Ssds,
dβγdγγ=0=wA-1Dδ0vwv.
dβγdγγ=0=wjA-1Dδjvwv,
dxγdγ=-B-1Dδ0B-1y.

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