## Efficient numerical method for analyzing optical bistability in photonic crystal microcavities |

Optics Express, Vol. 21, Issue 10, pp. 11952-11964 (2013)

http://dx.doi.org/10.1364/OE.21.011952

Acrobat PDF (4381 KB)

### 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

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

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] .

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

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] .

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. 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] .

## 2. Problem formulation and linear results

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. 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] .

*L*and

*a*, respectively. The dielectric constants of the cylinders and the surrounding medium are

*ε*

_{1}and

*ε*

_{2}, respectively. A waveguide is formed by reducing the radii of one array of cylinders to

*a*

_{1}(

*a*

_{1}<

*a*) in the PhC. A microcavity is created by increasing the radius of one cylinder to

*a*

_{2}(

*a*

_{2}>

*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] 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,

*k*

_{0}=

*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.

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] .

*ω*/(2

_{c}L*πc*) = 0.28815 − 0.00058

*i*(the time dependence is assumed to be

*e*

^{−iωt}). 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*/(2

*πc*) = 0.28815 in Fig. 4(b).

## 3. Domain reduction process

*. Our boundary condition has the following form where*

_{c}*∂*Ω

*denotes the boundary of Ω*

_{c}*,*

_{c}*r*is the radial coordinate with the center of Ω

*as the origin,*

_{c}*) partial derivative with respect to*

_{c}*r*,

*f*is a function defined on

*∂*Ω

*,*

_{c}*𝒟*is an operator that acts on functions defined on

*∂*Ω

*, and*

_{c}*A*

_{0}is the amplitude of the incident Bloch mode. In a discrete approximation,

*∂*Ω

*is sampled by*

_{c}*M*points,

*f*is approximated by a column vector of length

*M*, and

*𝒟*is approximated by an

*M*×

*M*matrix. In the following, we present a numerical method for computing

*f*and

*𝒟*by solving the linear problem outside Ω

*.*

_{c}*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 2

*m*+ 1 unit cells in the vertical direction. That is 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. On the two vertical edges of

*S*, the boundary conditions are where

*ϕ*

_{0}is the fundamental propagating Bloch mode of the PhC waveguide,

*ℒ*

^{−}and

*ℒ*

^{+}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*> 7

*L*) 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] .

## 4. Nonlinear solver

*. For that purpose, it is natural to use the polar coordinate system. Equation (1) becomes for*

_{c}*r*∈ (0,

*a*

_{2}) and

*θ*∈ [0, 2

*π*]. Newton’s method is widely used to solve nonlinear problems. For Eq. (17), it gives

*u*

^{(n−1)}is a previous iteration of the solution,

*u*

^{(n)}is the current iteration to be determined, and

*ū*

^{(n)}is the complex conjugate of

*u*

^{(n)}. To start the iterative process, we need an initial guess

*u*

^{(0)}.

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] .

*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] .

*r*to (−

*a*

_{2},

*a*

_{2}) by so that Eq. (18) is valid for

*r*∈ (−

*a*

_{2},

*a*

_{2}) and

*θ*∈ [0, 2

*π*]. We discretize

*r*and

*θ*by where

*q*is an odd integer and

*M*is an even integer (the number of discretization points on

*∂*Ω

*). Notice that*

_{c}*r*

_{0}=

*a*

_{2},

*r*= −

_{q}*a*

_{2}, and

*r*= 0 is avoided since

*q*is an odd integer. Denoting

*u*(

*r*,

_{j}*θ*) by

_{k}*u*, then Eq. (19) implies that for (

_{jk}*q*+ 1)/2 ≤

*j*≤

*q*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 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*,

*w̃*_{0}and

*w̃**are row vectors of length*

_{q}*q*− 1, and

*ŵ*_{0}and

*ŵ**are column vectors of length*

_{q}*q*− 1. The second order partial derivative with respect to

*r*can be approximated by the matrix

*𝒯*=

*𝒲*

^{2}. Similarly, the second order partial derivative with respect to

*θ*is approximated as where

*ℱ*is the

*M*×

*M*second order Fourier differentiation matrix [26

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

*∂*,

_{r}*r*,

_{j}*θ*) for 1 ≤

_{k}*j*≤

*q*− 1 and 1 ≤

*k*≤

*M*. With a further application of the symmetry relation (20), Eq. (18) is discretized as where is a column vector for

*u*

^{(n)}at all interior points, and 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,

*ℛ*= diag{1/

*r*

_{1}, 1/

*r*

_{2},..., 1/

*r*

_{(q−1)/2}},

*u*

^{(n)}on

*∂*Ω

*,*

_{c}*Î*is the

*M*×

*M*matrix given by where

*I*

_{2}is the (

*M*/2) × (

*M*/2) identity matrix. On the boundary of Ω

*, i.e. at*

_{c}*r*=

*a*

_{2}, the partial derivative of

*u*with respect to

*r*can be evaluated using the first row of Eq. (21): where the superscript “−” indicates the one-side derivative taken from the interior of Ω

*. Meanwhile, boundary condition (3) gives where the one-side derivative is taken from the exterior of Ω*

_{c}*. The continuity condition of*

_{c}*ρ*

^{−1}

*∂*gives us another relation between

_{r}u**and**

*u*

*u*_{0}: where

*σ*= 1 or

*σ*=

*ε*

_{1}/

*ε*

_{2}for TE or TM polarization, respectively. If

*u*^{(n−1)}is given, we can solve Eqs. (23) and (26) to find

*u*^{(n)}and

*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

*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 Ω

*, and*

_{c}*q*is the number of point for discretizing the radial variable

*r*in Ω

*. On a personal computer with a 2.4GHz CPU, the boundary condition (3) can be constructed within 8 seconds. For the nonlinear problem in Ω*

_{c}*, the linear system for each iteration involves a 2222 × 2222 coefficient matrix, and it can be solved in less than 1 second.*

_{c}

*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

*P*is used as the initial guess for a slightly larger

_{in}*P*. For sufficiently large

_{in}*P*, 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

_{in}*P*in each step. Since the middle branch is unstable, it cannot be calculated by time-domain methods, such as the nonlinear FDTD [6

_{in}**15**,16161–16176 (2007) [CrossRef] [PubMed] .

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] .

*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

*P*is decreased and increased in each step, respectively.

_{in}## 6. Conclusions

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] .

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] .

## Acknowledgments

## References and links

1. | H. Gibbs, |

2. | C. M. Bowden and A. M. Zheltikov, “Nonlinear optics of photonic crystals,” J. Opt. Soc. Am. B |

3. | R. E. Slusher and B. J. Eggleton, |

4. | K. J. Vahala, “Optical microcavities,” Nature |

5. | M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

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 |

7. | G. S. Agarwal and S. D. Gupta, “Effect of nonlinear boundary conditions on nonlinear phenomena in optical resonators,” Opt. Lett. |

8. | J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B |

9. | M. Midrio, “Shooting technique for the computation of the plane-wave reflection and transmission through one-dimensional nonlinear inhomogeneous dielectric structres,” J. Opt. Soc. Am. B |

10. | 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. |

11. | 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. | P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun. |

13. | A. Talflove and S. C. Hagness, |

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. |

15. | Z. Xu and G. Bao, “A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects,” J. Opt. Soc. Am. A |

16. | E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B |

17. | S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B |

18. | J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol. |

19. | Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express |

20. | R. W. Boyd, |

21. | Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B |

22. | S. Li and Y. Y. Lu, “Efficient method for analyzing leaky cavities in two-dimensional photonic crystals,” J. Opt. Soc. Am. B |

23. | Z. Hu and Y. Y. Lu, “A simple boundary condition for terminating photonic crystal waveguides,” J. Opt. Soc. Am. B |

24. | Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. |

25. | J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A |

26. | L. N. Trefethen, |

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 |

**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

- H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).
- C. M. Bowden and A. M. Zheltikov, “Nonlinear optics of photonic crystals,” J. Opt. Soc. Am. B19,2046–2048 (2002). [CrossRef]
- R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).
- K. J. Vahala, “Optical microcavities,” Nature424,839–846 (2003). [CrossRef] [PubMed]
- M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991). [CrossRef]
- M. Midrio, “Shooting technique for the computation of the plane-wave reflection and transmission through one-dimensional nonlinear inhomogeneous dielectric structres,” J. Opt. Soc. Am. B18,1866–1871 (2001). [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]
- 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]
- P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun.238,169–175 (2004). [CrossRef]
- A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
- 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]
- Z. Xu and G. Bao, “A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects,” J. Opt. Soc. Am. A27,2347–2353 (2010). [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]
- S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B19,2241–2249 (2002). [CrossRef]
- 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]
- Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express16,17383–17399 (2008). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics (Academic, 1992).
- 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]
- 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]
- Z. Hu and Y. Y. Lu, “A simple boundary condition for terminating photonic crystal waveguides,” J. Opt. Soc. Am. B29,1356–1360 (2012). [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. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A23,3217–3222 (2006). [CrossRef]
- L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000). [CrossRef]
- 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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