Abstract

Nonlinear optical effects can be enhanced by photonic crystal microcavities and be used to develop practical ultra-compact optical devices with low power requirements. The finite-difference time-domain method is the standard numerical method for simulating nonlinear optical devices, but it has limitations in terms of accuracy and efficiency. In this paper, a rigorous and efficient frequency-domain numerical method is developed for analyzing nonlinear optical devices where the nonlinear effect is concentrated in the microcavities. The method replaces the linear problem outside the microcavities by a rigorous and numerically computed boundary condition, then solves the nonlinear problem iteratively in a small region around the microcavities. Convergence of the iterative method is much easier to achieve since the size of the problem is significantly reduced. The method is presented for a specific two-dimensional photonic crystal waveguide-cavity system with a Kerr nonlinearity, using numerical methods that can take advantage of the geometric features of the structure. The method is able to calculate multiple solutions exhibiting the optical bistability phenomenon in the strongly nonlinear regime.

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

2010 (1)

2009 (3)

2008 (1)

2007 (3)

2006 (2)

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol.24,3448–3453 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A23,3217–3222 (2006).
[CrossRef]

2004 (2)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004).
[CrossRef] [PubMed]

P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun.238,169–175 (2004).
[CrossRef]

2003 (3)

K. J. Vahala, “Optical microcavities,” Nature424,839–846 (2003).
[CrossRef] [PubMed]

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B62,R7683–R7686 (2000).
[CrossRef]

1991 (1)

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

1987 (1)

Agarwal, G. S.

Bao, G.

Baruch, G.

G. Baruch, G. Fibich, and S. Tsynkov, “A high-order numerical method for the nonlinenar Helmholtz equation in multi-dimensional layered media,” J. Comput. Phys.228,3789–3815 (2009).
[CrossRef]

Bermel, P.

Bowden, C. M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Bravo-Abad, J.

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express15,16161–16176 (2007).
[CrossRef] [PubMed]

Centeno, E.

E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B62,R7683–R7686 (2000).
[CrossRef]

Danckaert, J.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

Eggleton, B. J.

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).

Fan, S.

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

Felbacq, D.

E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B62,R7683–R7686 (2000).
[CrossRef]

Fibich, G.

G. Baruch, G. Fibich, and S. Tsynkov, “A high-order numerical method for the nonlinenar Helmholtz equation in multi-dimensional layered media,” J. Comput. Phys.228,3789–3815 (2009).
[CrossRef]

Fobelets, K.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

Gibbs, H.

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

Gupta, S. D.

Hagness, S. C.

A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

Hammer, M.

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003).
[CrossRef]

Hoekstra, H. J. W. M.

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

Hu, Z.

Huang, Y.

Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B24,2860–2867 (2007).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol.24,3448–3453 (2006).
[CrossRef]

Joannopoulos, J. D.

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express15,16161–16176 (2007).
[CrossRef] [PubMed]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004).
[CrossRef] [PubMed]

Johnson, S. G.

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express15,16161–16176 (2007).
[CrossRef] [PubMed]

Kivshar, Y. S.

Kwan, P. K.

P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun.238,169–175 (2004).
[CrossRef]

Li, S.

Lu, Y. Y.

Midrio, M.

Mingaleev, S. F.

Reinisch, R.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

Rodriguez, A.

Slusher, R. E.

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).

Soljacic, M.

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express15,16161–16176 (2007).
[CrossRef] [PubMed]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004).
[CrossRef] [PubMed]

Suryanto, A.

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003).
[CrossRef]

Talflove, A.

A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
[CrossRef]

Tsynkov, S.

G. Baruch, G. Fibich, and S. Tsynkov, “A high-order numerical method for the nonlinenar Helmholtz equation in multi-dimensional layered media,” J. Comput. Phys.228,3789–3815 (2009).
[CrossRef]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature424,839–846 (2003).
[CrossRef] [PubMed]

van Groesen, E.

A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003).
[CrossRef]

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

Veretennicoff, I.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

Vitrant, G.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

Xu, Z.

Yuan, J.

Yuan, L.

Zheltikov, A. M.

J. Comput. Phys. (1)

G. Baruch, G. Fibich, and S. Tsynkov, “A high-order numerical method for the nonlinenar Helmholtz equation in multi-dimensional layered media,” J. Comput. Phys.228,3789–3815 (2009).
[CrossRef]

J. Lightw. Technol. (2)

J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol.24,3448–3453 (2006).
[CrossRef]

J. Nonlinear Opt. Phy. Mater. (1)

A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (7)

Nat. Mater. (1)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004).
[CrossRef] [PubMed]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature424,839–846 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun.238,169–175 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quant. Electron. (1)

A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003).
[CrossRef]

Phys. Rev. B (2)

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991).
[CrossRef]

E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B62,R7683–R7686 (2000).
[CrossRef]

Other (5)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
[CrossRef]

A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

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Figures (6)

Fig. 1
Fig. 1

A PhC waveguide-cavity system with a microcavity at the center and two semi-infinite waveguides. The truncated domain S (for m = 5) is the rectangle enclosed by the red lines.

Fig. 2
Fig. 2

Transmission spectrum of the PhC waveguide-cavity system.

Fig. 3
Fig. 3

Magnitude of the wave field excited by an incident Bloch mode with amplitude A0 = 1 at frequency ωL/(2πc) = 0.28815.

Fig. 4
Fig. 4

(a): Magnitude of the leaky cavity mode at complex frequency ωcL/(2πc) = 0.28815−0.00058i. (b): Magnitude of the wave field excited by an incident Bloch mode at frequency ωL/(2πc) = 0.28815. The fields are normalized for comparison.

Fig. 5
Fig. 5

Normalized output power as a function of the nomalized incident power at frequency ωL/(2πc) = 0.286397.

Fig. 6
Fig. 6

Wave field patterns (magnitude of u) corresponding to the three solutions marked as A, B and C in Fig. 5.

Equations (32)

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

ρ x ( 1 ρ u x ) + ρ y ( 1 ρ u y ) + k 0 2 [ ε ( x , y ) + 3 4 χ ( 3 ) ( x , y ) | u | 2 ] u = 0 ,
u ( x , y ) = ϕ ( x , y ) e i β x ,
r + u = 𝒟 u + A 0 f on Ω c ,
S = { ( x , y ) | 0 < x < 7 L , ( m + 0.5 ) L < y < ( m + 0.5 ) L } ,
u = 0 , y = ± ( m + 0.5 ) L .
u x = u + A 0 ( + ) ϕ 0 , x = 0 ,
u x = + u , x = 7 L ,
Ω j k = { ( x , y ) | x j 1 < x < x j , y k 1 < y < y k }
[ x v j 1 , k x v j k y h j , k 1 y h j k ] = Λ [ v j 1 , k v j k h j , k 1 h j k ] ,
u ( x , y ) = n = c n ψ n ( r ) e i n θ ,
[ x v 3 , m + 1 x v 4 , m + 1 y h 4 m y h 4 , m + 1 r u | Ω c ] = Λ ˜ [ v 3 , m + 1 v 4 , m + 1 h 4 m h 4 , m + 1 u | Ω c ] ,
Λ = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] , Λ ˜ = [ Λ ˜ 11 Λ ˜ 12 Λ ˜ 13 Λ ˜ 14 Λ ˜ 15 Λ ˜ 21 Λ ˜ 22 Λ ˜ 23 Λ ˜ 24 Λ ˜ 25 Λ ˜ 31 Λ ˜ 32 Λ ˜ 33 Λ ˜ 34 Λ ˜ 35 Λ ˜ 41 Λ ˜ 42 Λ ˜ 43 Λ ˜ 44 Λ ˜ 45 Λ ˜ 51 Λ ˜ 52 Λ ˜ 53 Λ ˜ 54 Λ ˜ 55 ] ,
Λ 21 v j 1 , k + ( Λ 22 Λ 11 ) v j k = Λ 23 h j , k 1 + Λ 24 h j k Λ 12 v j + 1 , k Λ 13 h j + 1 , k 1 Λ 14 h j + 1 , k = 0.
n = 1 , n k 2 m + 1 k n v 0 n + ( Λ 11 k k ) v 0 k Λ 12 v 1 k Λ 13 h 1 , k 1 Λ 14 h 1 k = A 0 g n ,
Λ 21 v 2 , m + 1 + ( Λ 22 Λ ˜ 11 ) v 3 , m + 1 + Λ 23 h 3 m + Λ 24 h 3 , m + 1 Λ ˜ 12 v 4 , m + 1 Λ ˜ 13 h 4 m Λ ˜ 14 h 4 , m + 1 = Λ ˜ 15 u | Ω c ,
𝒜 U = u | Ω c + A 0 G ,
[ v 3 , m + 1 v 4 , m + 1 h 4 m h 4 , m + 1 ] = 𝒞 u | Ω c + A 0 f 0 ,
𝒟 = [ Λ ˜ 51 Λ ˜ 52 Λ ˜ 53 Λ ˜ 54 ] 𝒞 + Λ ˜ 55 , f = [ Λ ˜ 51 Λ ˜ 52 Λ ˜ 53 Λ ˜ 54 ] f 0 .
2 u r 2 = 1 r u r + 1 r 2 2 u θ 2 + k 0 2 ( ε 1 + 3 4 χ ( 3 ) | u | 2 ) u = 0
( 2 r 2 + 1 r r + 1 r 2 2 θ 2 + k 0 2 ε 1 ) u ( n ) + 3 4 k 0 2 χ ( 3 ) [ 2 | u ( n 1 ) | 2 u ( n ) + ( u ( n 1 ) ) 2 u ¯ ( n ) ] = 3 3 k 0 2 χ ( 3 ) | u ( n 1 ) | 2 u ( n 1 ) ,
u ( r , θ ) = u ( r , θ ˜ ) for a 2 < r < 0 and θ ˜ = ( θ + π ) mod ( 2 π ) ,
r j = a 2 cos ( j π / q ) for 0 j q , θ k = ( 2 k 1 ) π / M for 1 k M ,
u j k = { u q j , k + M / 2 , for 1 k M / 2 , u q j , k M / 2 , for 1 + M / 2 k M ,
r [ u 0 k u 1 k u q k ] 𝒲 [ u 0 k u 1 k u q k ] = [ w 00 w ˜ 0 w 0 q w ^ 0 𝒲 ^ w ^ q w q 0 w ˜ q w q q ] [ u 0 k u 1 k u q k ] ,
2 θ 2 [ u j 1 u j 2 u j M ] [ u j 1 u j 2 u j M ] ,
1 u ( n ) + 2 γ diag { | u ( n 1 ) | 2 } u ( n ) + γ diag { [ u ( n 1 ) ] 2 } u ¯ ( n ) + 2 u 0 ( n ) = γ diag { | u ( n 1 ) | 2 } u ( n ) ,
u ( n ) = [ u 11 ( n ) , u 12 ( n ) , , u 1 M ( n ) , u 21 ( n ) , u 22 ( n ) , , u 2 M ( 2 ) , , u ( q 1 ) / 2 , 1 ( n ) , , u ( q 1 ) / 2 , M ( n ) ] T
1 = ^ I + ( 𝒲 ^ ) I + 2 + k 0 2 ε 1 , 2 = w ^ 0 I + w ^ q I ^ .
I ^ = [ 0 I 2 I 2 0 ] ,
r u 0 ( n ) = w ˜ 0 I u ( n ) + ( w 00 + w 0 q I ^ ) u 0 ( n ) ,
r + u 0 ( n ) = 𝒟 u 0 ( n ) + A 0 f ,
w ˜ 0 × I u ( n ) + ( w 00 + w 0 q I ^ σ 𝒟 ) u 0 ( n ) = A 0 σ f ,

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