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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11952–11964
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Efficient numerical method for analyzing optical bistability in photonic crystal microcavities

Lijun Yuan and Ya Yan Lu  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11952-11964 (2013)
http://dx.doi.org/10.1364/OE.21.011952


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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.

© 2013 OSA

1. Introduction

Optical bistability in dielectric structures with the Kerr nonlinearity has been extensively studied since 1980s [1

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

]. It has applications in all-optical devices such as optical switches, but these applications are often limited by the weak nonlinear response of conventional dielectric materials, thus require high operating power. The development of photonic crystals (PhCs) has made the applications of optical bistability feasible, because nonlinear effects can be dramatically enhanced by PhC microcavities [2

2. C. M. Bowden and A. M. Zheltikov, “Nonlinear optics of photonic crystals,” J. Opt. Soc. Am. B 19,2046–2048 (2002) [CrossRef] .

6

6. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express 15,16161–16176 (2007) [CrossRef] [PubMed] .

].

Except for simple one-dimensional problems [7

7. G. S. Agarwal and S. D. Gupta, “Effect of nonlinear boundary conditions on nonlinear phenomena in optical resonators,” Opt. Lett. 12,829–831 (1987) [CrossRef] [PubMed] .

12

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

], not many numerical methods are available for simulating nonlinear optical phenomena, including optical bistability, in two- or three-dimensional structures. Most researchers use a time domain method, such as the finite-difference time-domain (FDTD) method [13

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

], since the nonlinear terms are relatively easy to handle in the time domain. However, FDTD requires a small grid size to resolve large index-contrast and a small time step to ensure numerical stability. Furthermore, the implementation of realistic material dispersion and the truncation of unbounded domains (especially when the structure is inhomogeneous at infinity) can be rather complicated. For two-dimensional (2D) structures, the nonlinear interactions between light and Kerr media can be modeled by a nonlinear Helmholtz equation. In general, the nonlinear Helmholtz equation can be discretized by a standard numerical method such as the finite difference or finite element method, and solved by an iterative method, such as Newton’s method or the nested iteration scheme [14

14. 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] .

, 15

15. Z. Xu and G. Bao, “A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects,” J. Opt. Soc. Am. A 27,2347–2353 (2010) [CrossRef] .

]. However, this approach is not very popular, since the iterative methods often converge very slowly or fail to converge for interesting problems with strong nonlinear effects and unstable solutions. Meanwhile, more efficient but approximate frequency-domain methods have been developed for problems with special geometric properties. For example, the multi-scattering theory was used to analyze optical bistability in finite-size 2D photonic crystals with microcavities [16

16. E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62,R7683–R7686 (2000) [CrossRef] .

]. Effective discrete equations were derived by expanding the wave field in localized modes to analyze 2D PhC waveguides with nonlinear defects [17

17. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19,2241–2249 (2002) [CrossRef] .

]. A perturbation theory and a coupled-mode theory have also been developed to analyze nonlinear PhC devices [6

6. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express 15,16161–16176 (2007) [CrossRef] [PubMed] .

, 18

18. 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] .

]. However, all these methods are based on some analytic assumptions. For example, in the multi-scattering theory, the wave field in the nonlinear cylinders is assumed to be a constant. This assumption becomes invalid when the wavelength is comparable to the radii of the cylinders.

In this paper, we develop a rigorous and efficient numerical method for analyzing nonlinear optical phenomena enhanced by microcavities. The method is presented for a PhC waveguide-cavity structure with the Kerr nonlinearity, where the nonlinear effect is concentrated in a microcavity. Our approach is to reduce the nonlinear problem to a small region around the microcavity. The linear problem in the exterior domain is solved first (and only once) to establish a boundary condition around the nonlinear region. The small nonlinear problem is then solved by an iterative method. To take advantage of the many identical unit cells in the PhC waveguide-cavity structure, we use the Dirichlet-to-Neumann (DtN) map method [19

19. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16,17383–17399 (2008) [CrossRef] [PubMed] .

] to solve the linear problem. For a linear unit cell, the DtN map is an operator that maps the wave field to its normal derivative on the cell boundary, and it is approximated by a small matrix. Only a few different DtN maps are needed, since most of the unit cells are identical. This paper is organized as follows. In section 2, the problem is formulated and linear results are presented. In section 3, the procedure for establishing the boundary condition for the small nonlinear problem is described. In section 4, we present a numerical method for solving the nonlinear Helmholtz equation based on Newton’s iterative scheme and a pseudospectral method. Numerical results for the nonlinear problem are given in section 5.

2. Problem formulation and linear results

We consider a 2D PhC waveguide-cavity structure as previously studied in [6

6. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express 15,16161–16176 (2007) [CrossRef] [PubMed] .

, 18

18. 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] .

] and shown in Fig. 1. It is formed by introducing a defect cylinder and two semi-infinite waveguides in an otherwise perfectly periodic and infinite PhC. The background PhC consists of a square lattice of parallel and infinitely long circular dielectric cylinders surrounded by a homogenous medium with a lower refractive index. The lattice constant and the radii of the cylinders are L and a, respectively. The dielectric constants of the cylinders and the surrounding medium are ε1 and ε2, respectively. A waveguide is formed by reducing the radii of one array of cylinders to a1 (a1 < a) in the PhC. A microcavity is created by increasing the radius of one cylinder to a2 (a2 > a), it is placed in the waveguide core with three regular cylinders on each side. In a Cartesian coordinate system where the waveguide axis and the cylinder axes are parallel to the x- and z-axis, respectively, the governing equation for polarized electromagnetic waves in a Kerr medium is [20

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

]
ρx(1ρux)+ρy(1ρuy)+k02[ε(x,y)+34χ(3)(x,y)|u|2]u=0,
(1)
where u is the z-component of the electric or magnetic field for the transverse electric (TE) or transverse magnetic (TM) polarization, respectively, ε(x, y) is the dielectric function, k0 = ω/c is the free space wavenumber, ω is the angular frequency, c is the speed of light in vacuum, χ(3)(x, y) is an element of the third order nonlinear susceptibility and it is only non-zero in nonlinear media, and ρ = 1 or ρ = ε for the TE or TM polarization, respectively.

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 Transmission spectrum of the PhC waveguide-cavity system.
Fig. 3 Magnitude of the wave field excited by an incident Bloch mode with amplitude A0 = 1 at frequency ωL/(2πc) = 0.28815.

As a linear structure, the PhC waveguide-cavity system shown in Fig. 1 has a leaky cavity mode. It is a special solution of the linear Helmholtz equation satisfying outgoing radiation conditions in the waveguides, but it only exists for a complex frequency. The real part of the complex frequency corresponds to the peak frequency in the transmission spectrum shown in Fig. 2, and the imaginary part gives the damping rate of the field amplitude. The wave field of the leaky cavity mode looks like the wave field at the peak frequency, but its magnitude actually increases as the horizontal distance to the microcavity increases and blows up at infinity. We calculate the leaky cavity mode by the DtN-map method presented in [22

22. S. Li and Y. Y. Lu, “Efficient method for analyzing leaky cavities in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 26,2427–2433 (2009) [CrossRef] .

], and obtain the complex frequency ωcL/(2πc) = 0.28815 − 0.00058i (the time dependence is assumed to be eiωt). Figure 4(a) shows the wave field of the leaky cavity mode to the left of the cavity center. Notice that the wave field increases in the waveguide core as the distance to the microcavity is increased. As a comparison, we show the wave field excited by an incident Bloch mode at the peak frequency ωL/(2πc) = 0.28815 in Fig. 4(b).

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.

3. Domain reduction process

To solve the linear problem, the infinite PhC waveguide-cavity structure has to be truncated. For frequencies in the bandgap of the background PhC, the wave field decays exponentially to zero as |y| → ∞. Therefore, the wave field sufficiently far away form the waveguide core in the y direction can be accurately approximated by zero. The computation domain S is the rectangular region enclosed by the red lines in Fig. 1. It covers seven unit cells in the horizontal direction and 2m + 1 unit cells in the vertical direction. That is
S={(x,y)|0<x<7L,(m+0.5)L<y<(m+0.5)L},
where m is a positive integer (m = 5 in Fig. 1). Zero Dirichlet boundary conditions are imposed on the top and bottom edges of S, i.e.
u=0,y=±(m+0.5)L.
(4)
On the two vertical edges of S, the boundary conditions are
ux=u+A0(+)ϕ0,x=0,
(5)
ux=+u,x=7L,
(6)
where ϕ0 is the fundamental propagating Bloch mode of the PhC waveguide, and + are operators acting on functions of y. These operators are approximated by matrices when y is discretized. Equations (5) and (6) are derived by expanding the field in the waveguides (x < 0 or x > 7L) in a complete set of Bloch modes. A detailed derivation of these boundary conditions are given in [19

19. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16,17383–17399 (2008) [CrossRef] [PubMed] .

].

With the DtN maps Λ and Λ̃, an equation can be set up for each edge of the unit cells. First, we write Λ and Λ̃ in block forms as
Λ=[Λ11Λ12Λ13Λ14Λ21Λ22Λ23Λ24Λ31Λ32Λ33Λ34Λ41Λ42Λ43Λ44],Λ˜=[Λ˜11Λ˜12Λ˜13Λ˜14Λ˜15Λ˜21Λ˜22Λ˜23Λ˜24Λ˜25Λ˜31Λ˜32Λ˜33Λ˜34Λ˜35Λ˜41Λ˜42Λ˜43Λ˜44Λ˜45Λ˜51Λ˜52Λ˜53Λ˜54Λ˜55],
(10)
where Λjk and Λ̃jk for j, k = 1, 2, 3, 4 are N × N matrices, Λ̃j5 for j = 1, 2, 3, 4 are N × M matrices, Λ̃5k for k = 1, 2, 3, 4 are M × N matrices, and Λ̃55 is an M × M matrix. The equation for an edge is established by matching the normal derivative of u using the DtN maps of two neighboring unit cells or boundary conditions (5) and (6). For example, if a vertical edge (related to field vjk) is not on the boundary of S or the defect unit cell, the x-derivative xvjk can be evaluated from the DtN maps of the two unit cells Ωjk and Ωj+1,k. This gives rises to
Λ21vj1,k+(Λ22Λ11)vjk=Λ23hj,k1+Λ24hjkΛ12vj+1,kΛ13hj+1,k1Λ14hj+1,k=0.
(11)
If the edge is on the left boundary of S (related to field v0k), we have
n=1,nk2m+1knv0n+(Λ11kk)v0kΛ12v1kΛ13h1,k1Λ14h1k=A0gn,
(12)
where kn are blocks of the matrix approximation of , and g = [g1, g2,..., g2m+1]T is the vector approximation of (+)ϕ0(0, y). For an edge on the boundary of the defect unit cell, we establish the equation using Λ and Λ̃. For example, for the edge with the corresponding field v3,m+1, the equation is
Λ21v2,m+1+(Λ22Λ˜11)v3,m+1+Λ23h3m+Λ24h3,m+1Λ˜12v4,m+1Λ˜13h4mΛ˜14h4,m+1=Λ˜15u|Ωc,
(13)
where u|Ωc is assumed to be given.

4. Nonlinear solver

To solve the inhomogeneous Helmholtz equation (18), we use a mixed Fourier-Chebyshev pseudospectral method [26

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

, 27

27. L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 26,587–594 (2009) [CrossRef] .

]. More precisely, the Chebyshev collocation method is used to discretize the radial direction and the Fourier pseudospectral method is used to discretize the angular direction. To avoid the singularity at r = 0, we follow the approach presented in [26

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

], and extend the radial variable r to (−a2, a2) by
u(r,θ)=u(r,θ˜)fora2<r<0andθ˜=(θ+π)mod(2π),
(19)
so that Eq. (18) is valid for r ∈ (−a2, a2) and θ ∈ [0, 2π]. We discretize r and θ by
rj=a2cos(jπ/q)for0jq,θk=(2k1)π/Mfor1kM,
where q is an odd integer and M is an even integer (the number of discretization points on Ωc). Notice that r0 = a2, rq = −a2, and r = 0 is avoided since q is an odd integer. Denoting u(rj, θk) by ujk, then Eq. (19) implies that for (q + 1)/2 ≤ jq
ujk={uqj,k+M/2,for1kM/2,uqj,kM/2,for1+M/2kM,
(20)
Therefore, although the computation domain is doubled by the extension of r to negative values, the total number of unknowns still corresponds to the original domain. In the pseudospectral method, the partial derivative with respect to r is approximated as
r[u0ku1kuqk]𝒲[u0ku1kuqk]=[w00w˜0w0qw^0𝒲^w^qwq0w˜qwqq][u0ku1kuqk],
(21)
where 𝒲 is the Chebyshev differentiation matrix [26

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

], 𝒲̂ is the middle (q − 1) × (q − 1) block of 𝒲, 0 and q are row vectors of length q − 1, and ŵ0 and ŵq are column vectors of length q − 1. The second order partial derivative with respect to r can be approximated by the matrix 𝒯 = 𝒲2. Similarly, the second order partial derivative with respect to θ is approximated as
2θ2[uj1uj2ujM][uj1uj2ujM],
(22)
where is the M × M second order Fourier differentiation matrix [26

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

].

The pseudospectral method applies the matrix approximations for r, r2 and θ2 to Eq. (18), and assumes that the approximate equation remains valid at all interior points (rj, θk) for 1 ≤ jq − 1 and 1 ≤ kM. With a further application of the symmetry relation (20), Eq. (18) is discretized as
1u(n)+2γdiag{|u(n1)|2}u(n)+γdiag{[u(n1)]2}u¯(n)+2u0(n)=γdiag{|u(n1)|2}u(n),
(23)
where
u(n)=[u11(n),u12(n),,u1M(n),u21(n),u22(n),,u2M(2),,u(q1)/2,1(n),,u(q1)/2,M(n)]T
is a column vector for u(n) at all interior points, and
1=^I+(𝒲^)I+2+k02ε1,2=w^0I+w^qI^.
In the above, ⊗ is the Kronecker product, 𝒯̂ is the middle (q−1)×(q−1) block of matrix 𝒯, I is the M × M identity matrix, = diag{1/r1, 1/r2,..., 1/r(q−1)/2}, u0(n)=[u01(n),u02(n),,u0M(n)]T is the vector for u(n) on Ωc, Î is the M × M matrix given by
I^=[0I2I20],
where I2 is the (M/2) × (M/2) identity matrix. On the boundary of Ωc, i.e. at r = a2, the partial derivative of u with respect to r can be evaluated using the first row of Eq. (21):
ru0(n)=w˜0Iu(n)+(w00+w0qI^)u0(n),
(24)
where the superscript “−” indicates the one-side derivative taken from the interior of Ωc. Meanwhile, boundary condition (3) gives
r+u0(n)=𝒟u0(n)+A0f,
(25)
where the one-side derivative is taken from the exterior of Ωc. The continuity condition of ρ−1ru gives us another relation between u and u0 :
w˜0×Iu(n)+(w00+w0qI^σ𝒟)u0(n)=A0σf,
(26)
where σ = 1 or σ = ε1/ε2 for TE or TM polarization, respectively. If u(n−1) is given, we can solve Eqs. (23) and (26) to find u(n) and u0(n). Since the complex conjugate of u(n) appears in Eq. (23), it is necessary to work with the real and imaginary parts of u(n). On the other hand, since the structure is symmetric about the x-axis, we can reduce the size of the linear system by one half. As a result, we only need to solve a (qM/2) × (qM/2) linear system in each iteration.

5. Numerical results: nonlinear case

Fig. 5 Normalized output power as a function of the nomalized incident power at frequency ωL/(2πc) = 0.286397.
Fig. 6 Wave field patterns (magnitude of u) corresponding to the three solutions marked as A, B and C in Fig. 5.

The results in Fig. 5 and Fig. 6 are obtained with m = 10, N = 11, M = 44, and q = 101, where m is the number of remaining unit cells in each side of the waveguide in the truncated domain, N is the number of points for discretizing each edge of the square unit cells, M is the number of points for discretizing the boundary of the nonlinear cylinder Ωc, and q is the number of point for discretizing the radial variable r in Ωc. On a personal computer with a 2.4GHz CPU, the boundary condition (3) can be constructed within 8 seconds. For the nonlinear problem in Ωc, the linear system for each iteration involves a 2222 × 2222 coefficient matrix, and it can be solved in less than 1 second.

The lower branch of the solution curve in Fig. 5 can be easily obtained with a zero initial guess u(0) = 0. Typically, Newton’s method converges in less than 20 iterations. It can be calculated more efficiently by a continuation scheme, where the solution obtained for one value of Pin is used as the initial guess for a slightly larger Pin. For sufficiently large Pin, the non-linear problem has only one solution. In that case, Newton’s method again converges for the zero initial guess. The upper branch of the solution curve is then calculated by a continuation scheme that slightly decreases Pin in each step. Since the middle branch is unstable, it cannot be calculated by time-domain methods, such as the nonlinear FDTD [6

6. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express 15,16161–16176 (2007) [CrossRef] [PubMed] .

, 18

18. 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] .

], but it can still be calculated in the frequency-domain. To find the solution marked as B in Fig. 5, we start Newton’s iteration where u(0) is the product of solution C with a number s ∈ (0, 1). For a properly chosen s, the iterations converge to a solution (i.e. solution B) which is different from A or C. Once the solution B is obtained, we use the continuation scheme to calculate the left and right parts of the middle branch where Pin is decreased and increased in each step, respectively.

6. Conclusions

For the PhC structure considered in this paper, we make use of the so-called Dirichlet-to-Neumann (DtN) maps of the unit cells to calculate Bloch modes of the PhC waveguides as in [21

21. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24,2860–2867 (2007) [CrossRef] .

], truncate domains with PhC waveguides extending to infinity as in [19

19. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16,17383–17399 (2008) [CrossRef] [PubMed] .

], set up linear system of equations on edges of the unit cells only, and finally replace the linear part by a boundary condition. The nonlinear problem is solved iteratively by Newton’s method and the discretization is based on a combined Fourier-Chebyshev pseudospectral method. The optical bistability of the structure appears in a strong nonlinear regime where multiple solutions exist. To find the relation between the input and output power, we have also used a continuation scheme to improve the efficiency. The DtN map and the pseudospectral methods are used to take advantage of the the special geometry of the concerned structure. For a general problem with a microcavity enhancing the nonlinearity, a more general numerical method, such as the finite element method, may be more appropriate.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (project No. 11201508), the Scientific Research Foundation of Chongqing Technology and Business University (project No. 20125605), and by City University of Hong Kong (project No. 7002864).

References and links

1.

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

2.

C. M. Bowden and A. M. Zheltikov, “Nonlinear optics of photonic crystals,” J. Opt. Soc. Am. B 19,2046–2048 (2002) [CrossRef] .

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

6.

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express 15,16161–16176 (2007) [CrossRef] [PubMed] .

7.

G. S. Agarwal and S. D. Gupta, “Effect of nonlinear boundary conditions on nonlinear phenomena in optical resonators,” Opt. Lett. 12,829–831 (1987) [CrossRef] [PubMed] .

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J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44,8214–8225 (1991) [CrossRef] .

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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] .

12.

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

13.

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

14.

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] .

15.

Z. Xu and G. Bao, “A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects,” J. Opt. Soc. Am. A 27,2347–2353 (2010) [CrossRef] .

16.

E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62,R7683–R7686 (2000) [CrossRef] .

17.

S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19,2241–2249 (2002) [CrossRef] .

18.

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] .

19.

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16,17383–17399 (2008) [CrossRef] [PubMed] .

20.

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

21.

Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24,2860–2867 (2007) [CrossRef] .

22.

S. Li and Y. Y. Lu, “Efficient method for analyzing leaky cavities in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 26,2427–2433 (2009) [CrossRef] .

23.

Z. Hu and Y. Y. Lu, “A simple boundary condition for terminating photonic crystal waveguides,” J. Opt. Soc. Am. B 29,1356–1360 (2012) [CrossRef] .

24.

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] .

25.

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23,3217–3222 (2006) [CrossRef] .

26.

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

27.

L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 26,587–594 (2009) [CrossRef] .

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(190.1450) Nonlinear optics : Bistability
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: February 15, 2013
Revised Manuscript: April 12, 2013
Manuscript Accepted: April 24, 2013
Published: May 8, 2013

Citation
Lijun Yuan and Ya Yan Lu, "Efficient numerical method for analyzing optical bistability in photonic crystal microcavities," Opt. Express 21, 11952-11964 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11952


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References

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  18. 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]
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  21. 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]
  22. S. Li and Y. Y. Lu, “Efficient method for analyzing leaky cavities in two-dimensional photonic crystals,” J. Opt. Soc. Am. B26,2427–2433 (2009). [CrossRef]
  23. Z. Hu and Y. Y. Lu, “A simple boundary condition for terminating photonic crystal waveguides,” J. Opt. Soc. Am. B29,1356–1360 (2012). [CrossRef]
  24. 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]
  25. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A23,3217–3222 (2006). [CrossRef]
  26. L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000). [CrossRef]
  27. L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B26,587–594 (2009). [CrossRef]

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