## Computing Photonic Crystal Defect Modes by Dirichlet-to-Neumann Maps

Optics Express, Vol. 15, Issue 22, pp. 14454-14466 (2007)

http://dx.doi.org/10.1364/OE.15.014454

Acrobat PDF (175 KB)

### Abstract

We develop an efficient numerical method for computing defect modes in two dimensional photonic crystals based on the Dirichlet-to-Neumann (DtN) maps of the defect and normal unit cells. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative. The frequencies of the defect modes are solved from a condition that a small matrix is singular. The size of the matrix is related to the number of points used to discretize the boundary of the defect cell. The matrix is obtained by solving a linear system involving only discrete points on the edges of the unit cells in a truncated domain.

© 2007 Optical Society of America

## 1. Introduction

2. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. **67**, 2017–2020 (1991). [CrossRef] [PubMed]

3. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. **67**, 3380–3383 (1991). [CrossRef] [PubMed]

4. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B **10**, 314–321 (1993). [CrossRef]

5. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E **71**, 056606 (2005). [CrossRef]

6. R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B **54**, 7837–7842 (1996). [CrossRef]

7. X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics **36**, L120–L123, (1997). [CrossRef]

8. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

9. 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]

10. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. **149**, 363–376 (1999). [CrossRef]

11. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. **150**, 468–481 (1999). [CrossRef]

12. H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. **44**, 2688–2695 (1996). [CrossRef]

13. C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express **12**, 1397–1408 (2004). [CrossRef] [PubMed]

5. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E **71**, 056606 (2005). [CrossRef]

14. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E **75**, 026703 (2007). [CrossRef]

15. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. **23**, 1514–1521 (2005). [CrossRef]

16. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B **56**, 4830–4835 (1997). [CrossRef]

17. K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics , **84**, 1210–1214 (1998). [CrossRef]

18. V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B **57**, 15242–15250 (1998). [CrossRef]

19. N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E **64**, 056614 (2001). [CrossRef]

20. S. P. Guo and S. Albin, “Numerical techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express **11**, 1080–1089 (2003). [CrossRef] [PubMed]

21. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. **16**, 816–818 (2004). [CrossRef]

15. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. **23**, 1514–1521 (2005). [CrossRef]

8. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

9. 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]

6. R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B **54**, 7837–7842 (1996). [CrossRef]

7. X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics **36**, L120–L123, (1997). [CrossRef]

10. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. **149**, 363–376 (1999). [CrossRef]

11. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. **150**, 468–481 (1999). [CrossRef]

15. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. **23**, 1514–1521 (2005). [CrossRef]

29. Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications **12**, 597–609 (1991). [CrossRef]

30. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. **21**, 1793–1807 (2003). [CrossRef]

31. L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. **16**, 2057–2059 (2004). [CrossRef]

*K*×

*K*matrices, the frequency of a defect mode is solved from the condition that a

*K*×

*K*matrix

*B*is singular. The matrix

*B*depends on the frequency and it is constructed from the DtN maps of the normal and defect cells. Although an iterative method is needed, our method is very efficient, since

*K*is typically very small. Notice that the matrix in the standard formulation corresponds to a discretization of the supercell. Here, the matrix

*B*corresponds to a discretization of the boundary of the defect cell. A large truncated domain is still used in our formulation, but only the wave field on edges of the unit cells are involved in the manipulation, because the DtN maps contain complete information about the wave field in the interior of the unit cell if the field on the edges is known. The matrix

*B*is obtained when the wave field on all edges in the truncated domain, except the edges of the defect cell, are eliminated. We demonstrate our method by numerical examples.

## 2. Eigenvalue problems for defect modes

*k*

_{0}=

*ω/c*is the free space wavenumber,

*ω*is the angular frequency,

*c*is the speed of light in vacuum,

*n*=

*n*(

**x**) is the refractive index function and

**x**= (

*x,y*). For the

*E*polarization,

*u*is the

*z*-component of the electric field and

*ρ*= 1. For the

*H*polarization,

*u*is the

*z*-component of the magnetic field and

*ρ*=

*n*

^{2}. For a 2D PhC [1], the refractive index function is periodic in two linearly independent directions. We have two primitive translation vectors

**a**

_{1}and

**a**

_{2}, such that

*l*

_{1}and

*l*

_{2}are arbitrary integers. The

*xy*plane can be divided into unit cells given periodically on the lattice defined by the two vectors

**a**

_{1}and

**a**

_{2}. The unit cell can be chosen as the parallelogram defined by

**a**

_{1}and

**a**

_{2}, but for the triangular lattice, a hexagon unit cell is often more convenient. The periodic variation of the refractive index gives rise to bandgaps which are intervals of the frequency, such that propagating Bloch waves of the form

**k**. For the defect mode problem, we assume that

*n*(

**x**) is given differently from (2) for

**x**is a small bounded region specified by one or a few unit cells. An example is shown in Fig. 1, where the PhC is composed of a triangular lattice of parallel dielectric rods and the defect cell contains a rod with a different radius and/or different refractive index. The structure is supposed to be infinite and periodic except at the defect cell, although only four rings around the defect cell are shown in Fig. 1. Our problem is to find the frequencies in the bandgaps such that the homogeneous Helmholtz equation (1) has non-zero solutions that decay to zero at infinity. This is an eigenvalue problem formulated on the entire

*xy*plane, where the eigenvalue is

*ω*

^{2}or

*k*

^{2}

_{0}.

*S*. In the supercell approach,

*S*corresponds to a unit cell with translation vectors

*m*

**a**

_{1}and

*m*

**a**

_{2}, where

*m*is a positive integer. For the triangular lattice and

*m*= 3, two different possible supercells are shown in Fig. 2. The parallelogram supercell contains

*m*

^{2}= 9 elementary parallelogram unit cells (one defect cell and eight normal unit cells). The hexagon supercell contains one hexagon defect cell, six normal hexagon cells and six corner regions. Each of these corner regions corresponds to one third of a normal hexagon unit cell. Corresponding to duplicating the supercell on the lattice specified by the vectors

*m*

**a**

_{1}and

*m*

**a**

_{2}, a periodic boundary condition is often used and it is convenient for the plane wave expansion method [3

3. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. **67**, 3380–3383 (1991). [CrossRef] [PubMed]

6. R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B **54**, 7837–7842 (1996). [CrossRef]

7. X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics **36**, L120–L123, (1997). [CrossRef]

**23**, 1514–1521 (2005). [CrossRef]

*xy*plane is truncated to a domain that contains a few rings of normal cells around the defect cell. For the triangular lattice and using hexagon unit cells, the domain truncated to one ring contains exactly one defect cell and six normal cells. This domain is in fact different from the hexagon supercell for

*m*= 3 as shown in Fig. 2, since the six corner regions are not included. The truncated domain with two rings around the defect cell is shown in Fig. 3. On the boundary of the truncated domain, we use a simple zero boundary condition.

*ω*, the general solution of the Helmholtz equation (1) in a unit cell (normal or defect) including a circular cylinder can be written down analytically in terms of the cylindrical waves. The general solution allows us to find an operator (the DtN map) that maps the wave field on the boundary of the unit cell to its normal derivative there. In section 3, we use the DtN maps of the normal and defect unit cells to establish the condition

_{d}is the boundary of the defect cell,

*B*is an operator that depends on the frequency. In practice, with the general solution in a unit cell approximated by a finite sum of

*K*special solutions, the DtN maps of the unit cells and the operator

*B*are all approximated by

*K*×

*K*matrices. For a given

*ω*, the matrix

*B*can be calculated efficiently by solving a linear system involving only the wave field on the edges of the unit cells in the truncated domain. Since we are solving the frequencies of the defect modes, we need to search

*ω*from the condition that the matrix

*B*is singular. In section 4, we develop a modified version of the secant method to solve

*ω*using the smallest singular value of

*B*. Overall, we have turned the original linear eigenvalue problem (for a non-dispersive medium) on a two-dimensional domain to a nonlinear eigenvalue problem on a curve (the boundary of the defect cell). The method is iterative, but each iteration can be evaluated efficiently.

## 3. Formulation on the boundary of the defect cell

*u*is an arbitrary solution of the Helmholtz equation (1) and

*ν*is a unit normal vector of Γ. Therefore, the DtN map Λ is the operator that maps the wave field on the boundary to its normal derivative. To find a matrix approximation to Λ, we follow the approach developed in [25

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

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

*K*, we choose

*K*points on the boundary Γ, say

**x**

_{1},

**x**

_{2}, ...,

**x**

_{K}, and also assume that the general solution of the Helmholtz equation (1) in Ω can be approximated by a sum of

*K*special solutions. That is

*ϕ*satisfies (1). Evaluating the above at the

_{j}*K*points on the boundary, we obtain

_{1}is a

*K*×

*K*matrix whose (

*i,j*) entry is

*ϕ*(

_{j}**x**

_{i}). We can also evaluate the normal derivative at these

*K*points. This leads to

*ν*=

_{i}*ν*(

**x**

_{i}) is a unit normal vector of Γ at

**x**

_{i}and Λ

_{2}is a

*K*×

*K*matrix whose (

*i,j*) entry is

*∂*(

_{νi}ϕ_{j}**x**

_{i}). Therefore, we approximate the DtN map Λ by the matrix

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

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

*ϕ*be the solution in connection with a plane incident wave. The different special solutions in (5) correspond to the same scatterer and different incident waves, and they can be solved together efficiently by a boundary integral equation method [28]. On the other hand, if the unit cell contains a few circular cylinders, we can use the multipole method [32

_{j}32. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A **11**, 2526–2538 (1994). [CrossRef]

27. S. J. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A **24**, 2438–2442 (2007). [CrossRef]

*K*can be quite small. For a hexagon unit cell, we have

*K*= 6

*N*where

*N*is the number of sampling points on each edge. Numerical examples in section 5 indicate that accurate solutions can be obtained for

*N*≤ 8.

*S*by matching the normal derivative of the wave field there. If

*N*points are used on the edge, the wave field there will be approximated by a column vector of length

*N*and the equation on that edge is in fact a system of

*N*equations. To write down these equations, we first need to specify an ordering for the matrix approximation of the DtN map Λ. Consider the hexagon unit cell shown in Fig. 4, the wave field on the boundary Γ, i.e.,

*u*|

_{Γ}, is given as the column vector [

*u*

_{1};

*u*

_{2};...;

*u*

_{6}] (following the MATLAB notation) where

*u*

_{1},

*u*

_{2}, ...,

*u*

_{6}are column vectors of length

*N*and they approximate the wave field

*u*on the six edges following a counterclockwise ordering. To ease the matching on a common edge of two neighboring unit cells, we require that the sampling points on opposite edges follow the same order. More precisely, we have ordered the points

**x**

_{i}, for 1 ≤

*i*≤ 6

*N*, such that on each edge the

*x*coordinates increase as

*i*is increased and if the

*x*coordinates are constant (on the vertical edges) the

*y*coordinates must increase. The case for

*N*=3 is shown in Fig. 4. Such an ordering may seem to be less natural as the clockwise or counter-clockwise ordering, but it allows us to avoid a permutation when writing down the equation for each edge. We also assume that the normal vectors on opposite edges point to the same direction (one into the cell Ω and the other out). With these specifications, we write down the DtN map Λ as a 6 × 6 block matrix where each block is an

*N*×

*N*matrix. We have

_{1}, Ω

_{2}and Ω

_{3}contain red, yellow and green cylinders, respectively. Let the DtN maps of the three unit cells Λ

^{(1)}, Λ

^{(2)}and Λ

^{(3)}be partitioned as 6×6 block matrices accordingly. For edge 1 in Fig. 5, we have

_{2}corresponds to the fourth edge, thus the fourth row of the DtN map Λ

^{(2)}appears in the second equation above. The above gives rise to one equation connecting the field on 11 edges of two unit cells Ω

_{1}and Ω

_{2}. Similarly, for edge 2 in Fig. 5, we have

*E*polarization. For the

*H*polarization, we modify the above equations by matching

*n*

^{-2}

*∂*on the edges, where

_{ν}u*n*is the refractive index of the background medium in each unit cell.

*S*so that the first six edges are the edges of the defect unit cell and the next 24 edges are the remaining edges of the six closest normal unit cell (i.e. the first ring). This implies that the wave field on all these edges is given as a column vector

*u*, is a column vector of length

_{i}*N*for the field on the

*i*-th edge,

*U*= [

_{d}*u*

_{1};

*u*

_{2};...;

*u*

_{6}] is a column vector of length

*K*= 6

*N*for the boundary of the defect cell,

*U*= [

_{n}*u*

_{7};

*u*

_{8};...;

*u*

_{30}] is a column vector of length 24

*N*for other edges in the first ring and

*U*is the column vector for the remaining interior edges in the truncated domain

_{f}*S*. Using the DtN maps of the normal and defect unit cells and following the procedure outlined above, we arrive at the following linear system:

*U*and

_{n}*U*are given by

_{f}*ω*, the matrix

*B*depends on

*ω*.

## 4. Searching defect mode frequencies

*B*in Eq. (10) is singular. Notice that the larger matrix

*A*in Eq. (7) is also singular at these frequencies, but it is much easier to use the smaller matrix

*B*. One way to find the defect mode is to solve

*ω*from det

*B*= 0. However, it is well known that the determinant is not a good indicator for near singularity of a matrix. Our approach is to solve the defect mode frequencies from

*s*

_{1}is the smallest singular value of the matrix

*B*, i.e., the square root of the smallest eigenvalue of

*B*B*where

*B** the transpose and complex conjugate of

*B*.

*ω*,

*s*

_{1}is non-negative and behaves like the absolute value of another function. In particular, the derivative of

*s*

_{1}with respect to

*ω*is not continuous at its zeros. However,

*s*

_{1}appears to smoothly connect to -

*s*

_{1}as

*ω*passes through a zero. This motivates us to develop a modified secant method for solving Eq. (11). In each step, we calculate two approximations. The first approximation is obtained from the standard secant method. Let

*ω*

_{0}and

*ω*

_{1}be two initial guesses satisfying

*s*

_{1}(

*ω*

_{1}) ≤

*s*

_{1}(

*ω*

_{0}), then the first approximation is

*s*

_{1}(

*ω*

_{1}) by -

*s*

_{1}(

*ω*

_{1}). Thus

*s*

_{1}at both

*ω*

^{(1)}

_{2}and

*ω*

^{(2)}

_{2}, and choose

*ω*

_{2}to be the one that gives the smaller value of

*s*

_{1}. The process is then repeated using

*ω*

_{1}and

*ω*

_{2}to find

*ω*

_{3}, etc. The iterations can be terminated at some integer

*k*, if

*s*

_{1}(

*ω*

_{k+1}) >

*s*

_{1}(

*ω*) or |

_{k}*ω*

_{k+1}-

*ω*|/|

_{k}*ω*

_{k+1}| <

*δ*, where

*δ*is a small number for error tolerance.

## 5. Numerical examples

*S*with

*p*= 6 rings around the defect cell and

*N*= 8 sampling points on each edge, we calculate the defect mode frequency with initial guesses:

*ω*

_{0}

*a*/(2

*πc*) = 0.46 and

*ω*

_{1}

*a*/(2

*πc*) = 0.47. For the error tolerance

*δ*= 10

^{-12}, the modified secant method gives us the solution

*ωa*/(2

*πc*) = 0.46798 in only four iterations. The fast convergence of the modified secant method can be explained by the graph of

*s*

_{1}as a function of the frequency shown in Fig. 6. We observe that

*s*

_{1}looks like the absolute value of a smooth function which is close to a straight line. For this problem, the defect mode belongs to the band gap given by 0.415 <

*ωa*/(2

*πc*) < 0.483. We have actually calculated the function

*s*

_{1}on the entire band gap. It turns out that

*s*

_{1}has only one zero near 0.468. Next, we check the convergence with respect to the number of sampling points on each edge, i.e,

*N*. For that purpose, we fix

*p*= 6 and

*δ*= 10

^{-12}, then calculate the defect mode frequency using different values of

*N*. Let

*ω*

^{(N)}be the frequency obtained with

*N*points on each edge, we compute a relative error using the solution obtained with

*N*= 16 as the reference solution, that is

*N*is odd. To check the convergence with respect to the number of rings

*p*, we fix

*N*= 8 and

*δ*= 10

^{-12}, and calculate the defect mode for

*p*= 3, 4,..., 12. Using the solution obtained with

*p*= 12 as the reference solution, we calculate the relative error for different values of

*p*. The results are shown in Fig. 8. As expected, the relative errors decay exponentially as

*p*is increased. To have six significant digits in the solution, the relative error should be less than 10

^{-6}. From Fig. 7 and Fig. 8, it appears that we should choose

*N*≥ 7 and

*p*≥ 9. For

*N*= 7 and

*p*= 9, we have

*ωa*/(2

*πc*) = 0.467955.

*U*from Eq. (10), where

_{d}*U*represents the wave field of the defect mode on the six edges of the defect unit cell. At the defect mode frequency, the matrix

_{d}*B*is singular, then

*U*is the eigenvector for the zero eigenvalue of

_{d}*B*. Next, we solve

*U*and

_{n}*U*from Eq. (7). If the two matrices

_{f}*D*

_{1}and

*D*

_{2}are saved, we can use Eq. (9) to evaluate

*U*and

_{n}*U*. This gives rise to the values of the eigenfunction on all edges of the unit cells. For each unit cell, we can find the coefficients {

_{f}*c*} in Eq. (5) and then evaluate the eigenfunction in its interior. For this numerical example, the eigenfunction is shown in Fig. 9.

_{j}**36**, L120–L123, (1997). [CrossRef]

17. K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics , **84**, 1210–1214 (1998). [CrossRef]

*R*= 0.2

*a*and

*ε*

_{1}=

*n*

^{2}

_{1}= 13.0, respectively. The defect cell contains a circular rod with the same refractive index and a different radius

*R*. The objective is to calculate the defect mode frequencies for various values of

_{d}*R*. The PhC without the defect has a bandgap given by 0.2644 <

_{d}/a*ωa*/(2

*πc*) < 0.4345. In our calculations, we use a truncated domain

*S*with

*p*rings around the defect cell, where 6 ≤

*p*≤ 9. When the defect mode frequency is close the band edges, the mode profile spreads out and a larger

*p*is necessary. We also use a larger value of

*N*to verify the calculations obtained with

*N*= 5. For the modified secant method, a fixed error tolerance

*δ*= 10

^{-12}is chosen. Our results are shown in Fig. 10 and they agree well with those of Sakoda [17

17. K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics , **84**, 1210–1214 (1998). [CrossRef]

33. K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express **15**, 4753–4762 (2007). [CrossRef] [PubMed]

## 6. Conclusions

*B*in Eq. (10) is singular. The size of matrix

*B*is

*K*×

*K*where

*K*is the number of sampling points used on the boundary of the defect cell. A modification of the secant method is used to search the defect mode frequencies. In each iteration, it is necessary to calculate the matrix

*B*by solving a linear system involving approximations only on the edges of unit cells in the truncated domain. Accurate results can be obtained with a small number of sampling points on each edge of the unit cells.

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. |

3. | E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. |

4. | D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B |

5. | S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E |

6. | R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B |

7. | X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics |

8. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

9. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

10. | D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. |

11. | W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. |

12. | H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. |

13. | C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express |

14. | P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E |

15. | V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. |

16. | K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B |

17. | K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics , |

18. | V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B |

19. | N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E |

20. | S. P. Guo and S. Albin, “Numerical techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express |

21. | V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. |

22. | R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter |

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

24. | J. H. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice,” Opt. Commun. |

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

26. | Y. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics |

27. | S. J. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A |

28. | J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication. |

29. | Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications |

30. | T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. |

31. | L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. |

32. | D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A |

33. | K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 4, 2007

Revised Manuscript: October 10, 2007

Manuscript Accepted: October 17, 2007

Published: October 18, 2007

**Citation**

Shaojie Li and Ya Yan Lu, "Computing Photonic Crystal Defect Modes by Dirichlet-to-Neumann Maps," Opt. Express **15**, 14454-14466 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14454

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### References

- J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, Princeton, NJ. 1995).
- S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith and S. Schultz, "Microwave Propagation in two dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991). [CrossRef] [PubMed]
- E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer and J. D. Joannopoulos, "Donor and acceptor modes in photonic band-structure," Phys. Rev. Lett. 67, 3380-3383 (1991). [CrossRef] [PubMed]
- D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall and P. M. Platzman, "Photonic band structure and defects in one and two dimensions," J. Opt. Soc. Am. B 10, 314-321 (1993). [CrossRef]
- S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005). [CrossRef]
- R. R. Villeneuve, S. H. Fan and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996). [CrossRef]
- X. P. Feng and Y. Arakawa, "Defect modes in two-dimensional triangular photonic crystals," Jpn. J. Appl. Phys. 36, L120-L123, (1997). [CrossRef]
- K. M. Ho, C. T. Chan and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- 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]
- D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376 (1999). [CrossRef]
- W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case," J. Comput. Phys. 150, 468-481 (1999). [CrossRef]
- H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996). [CrossRef]
- C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004). [CrossRef] [PubMed]
- P. J. Chiang and C. P. Yu and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007). [CrossRef]
- V. F. Rodr´ıguez-Esquerre, M. Koshiba and H. E. Hern´andez-Figueroa, "Finite-element analysis of photonic crystal cavities: Time and frequency domains," J. Lightwave Technol. 23, 1514-1521 (2005). [CrossRef]
- K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997). [CrossRef]
- K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 84, 1210-1214 (1998). [CrossRef]
- V. Kuzmiak and A. A. Maradudin, "Localized defect modes in a two-dimensional triangular photonic crystal," Phys. Rev. B 57, 15242-15250 (1998). [CrossRef]
- N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, "Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals," Phys. Rev. E 64, 056614 (2001). [CrossRef]
- S. P. Guo and S. Albin, "Numerical techniques for excitation and analysis of defect modes in photonic crystals," Opt. Express 11, 1080-1089 (2003). [CrossRef] [PubMed]
- V. F. Rodríguez-Esquerre, M. Koshiba and H. E. Hernández-Figueroa, "Finite-element time-domain analysis of 2-D photonic crystal resonant cavities," IEEE Photon. Technol. Lett. 16, 816-818 (2004). [CrossRef]
- R. Moussa, L. Salomon, F. de Fornel and H. Aourag, "Numerical study on localized defect modes in twodimensional lattices: a high Q-resonant cavity," Physica B 338, 97-102 (2003). [CrossRef]
- J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006). [CrossRef]
- J. H. Yuan and Y. Y. Lu, "Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice," Opt. Commun. 273, 114-120 (2007). [CrossRef]
- Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006). [CrossRef]
- Y. H. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 337-349 (2007).
- S. J. Li and Y. Y. Lu, "Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells," J. Opt. Soc. Am. A 24, 2438-2442 (2007). [CrossRef]
- J. H. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted for publication.
- Y. Y. Lu and S.-T. Yau, "Eigenvalues of the Laplacian through boundary integral equations," SIAM J. Matrix Anal. Appl. 12, 597-609 (1991). [CrossRef]
- T. Lu and D. Yevick, "A vectorial boundary element method analysis of integrated optical waveguides," J. Lightwave Technol. 21, 1793-1807 (2003). [CrossRef]
- L. Prkna, M. Hubalek and J. Ctyroky, "Vectorial eigenmode solver for bent waveguides based on mode matching," IEEE Photon. Technol. Lett. 16, 2057-2059 (2004). [CrossRef]
- D. Felbacq, G. Tayeb and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994). [CrossRef]
- K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan and C. M. de Sterke, "Gap-edge asymptotics of defect modes in two-dimensional photonic crystals," Opt. Express 15, 4753-4762 (2007). [CrossRef] [PubMed]

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