Abstract

We present an accurate and efficient method of modeling second-harmonic generation in two-dimensional structures by use of eigenmode expansion. By using the undepleted-pump approximation we uncouple the calculations for the fundamental and second harmonic. Expansion of the field in eigenmodes gives rise to a linear matrix formalism. The method includes reflections and is especially suited for periodic structures. Several examples, including a two-dimensional photonic-crystal-cavity device, are studied.

© 2005 Optical Society of America

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  1. M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
    [CrossRef]
  2. C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. G. Nalesso, "Conversion efficiency for second-harmonic generation in photonic crystals," J. Opt. Soc. Am. B 18, 348-351 (2001).
    [CrossRef]
  3. Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
    [CrossRef]
  4. W. Nakagawa, R. C. Tyan, and Y. Fainman, "Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation," J. Opt. Soc. Am. A 19, 1919-1928 (2002).
    [CrossRef]
  5. F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
    [CrossRef]
  6. A. Locatelli, D. Modotto, C. De Angelis, F. M. Pigozzo, and A.-D. Capobianco, "Nonlinear bidirectional beam propagation method based on scattering operators for periodic microstructured waveguides," J. Opt. Soc. Am. B 20, 1724-1731 (2001).
    [CrossRef]
  7. B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
    [CrossRef]
  8. P. Bienstman, and R. Baets, "Optical modeling of photonic crystals and VC-SELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
    [CrossRef]
  9. B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
    [CrossRef]
  10. Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
    [CrossRef]
  11. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).
  12. Camfr simulation software is freely available from http://camfr.sourceforge.net/.
  13. C. De Angelis, G. Nalesso, D. Modotto, M. Midrio, A. Locatelli, and J. S. Aitchison, "Multiple-scale coupled-mode theory for second-harmonic generation in one-dimensional periodic structures," J. Opt. Soc. Am. B 20, 1853-1865 (2003).
    [CrossRef]
  14. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
    [CrossRef]
  15. 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]

2004 (1)

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
[CrossRef]

2003 (2)

2002 (4)

W. Nakagawa, R. C. Tyan, and Y. Fainman, "Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation," J. Opt. Soc. Am. A 19, 1919-1928 (2002).
[CrossRef]

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

2001 (4)

1999 (2)

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

Aitchison, J. S.

Baets, R.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
[CrossRef]

P. Bienstman, and R. Baets, "Optical modeling of photonic crystals and VC-SELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Bertolotti, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Bienstman, P.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
[CrossRef]

P. Bienstman, and R. Baets, "Optical modeling of photonic crystals and VC-SELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Bloemer, M. J.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Bowden, C. M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Capobianco, A.-D.

Centini, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

D'Aguanno, G.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

De Angelis, C.

Dumeige, Y.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Fainman, Y.

Fink, Y.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Gringoli, F.

Ibanescu, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Jeong, Y.

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

Jiang, Z. M.

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

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]

Johnson, S. G.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

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]

Lee, B.

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

Letartre, X.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Levenson, A.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Locatelli, A.

Maes, B.

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
[CrossRef]

Midrio, M.

Modotto, D.

Nakagawa, W.

Nalesso, G.

Nalesso, G. G.

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Pigozzo, F. M.

Raineri, F.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Scalora, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Shi, B.

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

Sibilia, C.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Soljacic, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Tyan, R. C.

Wang, X.

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

Zhou, X. F.

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

Electron. Lett. (1)

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, "Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal," Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

J. Appl. Phys. (1)

B. Shi, Z. M. Jiang, X. F. Zhou, and X. Wang, "A two-dimensional nonlinear photonic crystal for strong second harmonic generation," J. Appl. Phys. 91, 6769-6771 (2002).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Quantum Electron. (2)

P. Bienstman, and R. Baets, "Optical modeling of photonic crystals and VC-SELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

B. Maes, P. Bienstman, and R. Baets, "Modeling of Kerr nonlinear photonic components with mode expansion," Opt. Quantum Electron. 36, 15-24 (2004).
[CrossRef]

Phys. Rev. E (3)

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Other (2)

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

Camfr simulation software is freely available from http://camfr.sourceforge.net/.

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Figures (8)

Fig. 1
Fig. 1

Structure with three invariant sections along z. Mode amplitude vectors at input and output are indicated.

Fig. 2
Fig. 2

Interface followed by an invariant section.

Fig. 3
Fig. 3

Geometry of the waveguide example.

Fig. 4
Fig. 4

Generated electric field in the center at the end of the waveguide section, versus half the waveguide width. The light curve is the result with one mode, the heavy curve, with multiple modes.

Fig. 5
Fig. 5

Transverse grating structure. Only the section in dotted line is needed in the simulation because of symmetry.

Fig. 6
Fig. 6

Grating SHG intensity in the forward direction versus length d, normalized with the fundamental wavelength λ.

Fig. 7
Fig. 7

Photonic crystal device geometry. The upper (lower) half shows the electric field of the fundamental (second harmonic) at resonance. Dipole and quadrupole modes are clear. Because of high Q the field in the lower waveguide is barely visible.

Fig. 8
Fig. 8

Photonic crystal device results. The transmission of fundamental T ω and second harmonic T 2 ω are shown, together with the generated second-harmonic power P 2 ω , by curves. P 2 ω is normalized and all curves are plotted versus the fundamental ω. Values for T ω 2 T 2 ω are indicated by circles.

Equations (18)

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E ( r ) = i = 1 N F i exp ( j β i z ) E i ( r t ) = i F i E i ( r t ) ,
H ( r ) = i F i exp ( j β i z ) H i ( r t ) = i F i H i ( r t ) ,
t 2 E + k 2 E = β 2 E ,
t ( E i × H j ) u z d l = δ i j ,
[ F 2 B 1 ] = S [ F 1 B 2 ] = [ T 12 R 21 R 12 T 21 ] [ F 1 B 2 ] ,
2 E ω + k ω 2 E ω = ω 2 μ 0 ϵ 0 2 d nl E 2 ω E ω * ,
2 E 2 ω + k 2 ω 2 E 2 ω = ( 2 ω ) 2 μ 0 ϵ 0 d nl E ω 2 ,
2 E 2 ω + k 2 ω 2 E 2 ω = ( 2 ω ) 2 μ 0 P nl ,
E ω = i A i exp ( j β ω , i z ) E ω , i ,
E 2 ω = i C i ( z ) exp ( j β 2 ω , i z ) E 2 ω , i = i C i ( z ) E 2 ω , i .
d C i d z = 2 j k 0 2 k A k 2 exp ( j Δ β k i z ) β 2 ω , i d nl E ω , k 2 H 2 ω , i 4 j k 0 2 l , m A l A m exp ( j Δ β l m i z ) β 2 ω , i d nl E ω , l E ω , m H 2 ω , i ,
O k i = d n l E ω , k 2 H 2 ω , i ,
O l m i = d n l E ω , l E ω , m H 2 ω , i .
C i ( z ) = C i ( 0 ) 2 k 0 2 k A k 2 exp ( j Δ β k i z ) 1 β 2 ω , i Δ β k i O k i 4 k 0 2 l , m A l A m exp ( j Δ B l m i z ) 1 β 2 ω , i Δ β l m i O l m i .
[ F 2 B 1 ] = [ diag [ exp ( j β 2 ω , i L ) ] 0 0 diag [ exp ( j β 2 ω , i L ) ] ] [ F 1 B 2 ] + [ N fw N bw ] ,
[ F 3 B 1 ] = [ P ( L ) T 12 P ( L ) R 21 P ( L ) R 12 T 21 P ( L ) ] [ F 1 B 3 ] + [ P ( L ) R 21 N bw + N fw T 21 N bw ] .
[ F out B out ] = S [ F in B in ] + [ N tot , fw N tot , bw ] .
T ( ω ) = γ 2 ( ω ω res ) 2 + γ 2 .

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