## Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation

Optics Express, Vol. 15, Issue 1, pp. 207-218 (2007)

http://dx.doi.org/10.1364/OE.15.000207

Acrobat PDF (296 KB)

### Abstract

In-plane light propagation in two-dimensional (2D) photonic crystals (PCs) has been investigated by using the finite element method (FEM) in frequency domain. Conventionally, the band structures of 2D PCs were calculated by either the plane-wave expansion method (PWEM) or the finite difference time domain method. Here, we solve the eigenvalue equations for the band structures of the 2D PCs using the adaptive FEM in real space. We have carefully examined the convergence of this approach for the desired accuracy and efficiency. The calculated results show some discrepancies when compared to the results calculated by the PWEM. This may be due to the accuracy of the PWEM limited by the discontinuous nature of the dielectric functions. After acquiring the whole information of the dispersion relations within the irreducible Brillouin zone of the 2D PCs, the in-plane photon density of states for both the transverse electric (TE) and transverse magnetic (TM) modes can be calculated, accurately. For the case, the width of the complete band gap predicted by the FEM is much smaller, only about 65 % of that calculated by the PWEM. Therefore, the discrepancy in the prediction of complete band gaps between these two methods can be quite large, although the difference in band structure calculations is only a few percent. These results are relevant to the spontaneous emission by an atom, or to dipole radiation in two-dimensional periodic structures.

© 2007 Optical Society of America

## 1. Introduction

16. K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B **51**,4672–4675 (1995). [CrossRef]

23. A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B **58**,7252–7259 (1998). [CrossRef]

26. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A **14**,3323–3332 (1997). [CrossRef]

27. W. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express **8**,203–208 (2001). [CrossRef] [PubMed]

28. J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, “Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition,” Opt. Soc. Am. B **15**,2316–2324 (1998). [CrossRef]

36. A. Figotin and Y. A. Godin, “The Computation of Spectra of Some 2D Photonic Crystals,” J. Comput. Phys. **136**,585–598, 1997. [CrossRef]

22. H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B **45**,13962–13972 (1992). [CrossRef]

21. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E **58**,3896–3908 (1998). [CrossRef]

## 2. Formulation

### 2.1. In-plane band structure

*ε*(

*r*) and

*μ*(

*r*) are the permittivity and permeability functions of the PCs, respectively, and

*ω*is the angular eigen-frequency. In a 2D periodic system, the dielectric function is a periodic function of

*x*and

*y*. We assume that the materials are linear, homogeneous, isotropic, lossless, and nonmagnetic. We have

*ε*(

_{r}*x*,

*y*) is the dielectric function profile, and

*ε*and

_{a}*ε*are the dielectric constants of the air and dielectric regions, respectively. The two master equations are reduced to two homogeneous Helmholtz’s equations for the air (dielectric) region

_{d}*x*,

*y*) and homogeneous in the third (

*z*). For light propagating in the

*xy*-plane, we can separate the modes into two independent polarizations, TM and TE modes, and consider the band structure and photon density of states of each. The propagation properties of TM and TE modes can be characterized by the field components parallel to the rods,

*E*(

_{z}*x*,

*y*) and

*H*(

_{z}*x*,

*y*), respectively. For the in-plane propagation (

*k*= 0), the Helmholtz’s equations, Eqs. (4) and (5), for the air (dielectric) region can be rearranged as:

_{z}*xy*-plane, we have

*ε*(

**r**

_{∥}) =

*ε*(

**r**

_{∥}+

**R**), where

**r**

_{∥}is the in-plane position vector and

**R**is the linear combination of the primitive lattice vectors. By applying Bloch theorem, we can focus our attention on the values of the in-plane wave vector,

**k**

_{∥}=

*k*̂ +

_{x}x*k*, that are in the first Brillouin zone. We can calculate the in-plane band structures of 2D PCs by solving Eqs. (6) and (7) with the following boundary conditions,

_{y}ŷ*k*and

_{x}*k*in the irreducible Brillouin zone.

_{y}*r*= 0.38

*a*and dielectric constant

*ε*= 9, and the corresponding reciprocal lattice are shown in Fig. 1(a) and Fig. 1(b), respectively, where

_{d}*a*is the lattice constant and Γ: (0,0),

*X*: (

*π*/

*a*(1,0)), and

*M*: [

*π*/

*a*(1, 1)) are the high-symmetry points at the corners of the irreducible Brillouin zone (1/8 of the first Brillouin zone, in shaded region). The cross-sectional views of a triangular array of air columns with a radius

*r*= 0.4297

*a*drilled in a dielectric substrate of dielectric constant

*ε*= 11.9, and the corresponding reciprocal lattice are shown in Fig. 1(c) and Fig. 1(d), respectively. The corresponding high-symmetry points at the corners of the irreducible Brillouin zone (1/12 of the first Brillouin zone, in shaded region) are Γ : (0,0),

_{d}*M*: (

*π*/

*a*(1, -1/ √3)), and

*K*:(

*π*/

*a*(4/3,0)).

*ωa*/2

*πc*) is calculated and plotted as a function of Bloch’s vector (

**k**

_{∥}) as the boundary Γ

*XM*Γ of the irreducible Brillouin zone. For that of the triangular lattice, the corresponding path is Γ

*MK*Γ. Table 1 shows the reciprocal lattice paths and the corresponding ranges of

*k*

_{∥}and phase changes,

**k**

_{∥}∙

**R**

_{1}and

**k**

_{∥}∙

**R**

_{2}, for both the square and triangular lattices. Note that the direct lattice basis vectors

**R**

_{1}and

**R**

_{2}are (

*a*, 0) and (0,

*a*), respectively, for the square lattice and those of the triangular lattice are (

*a*,0) and (

*a*/2, √3

*a*/2).

### 2.2. Finite element method

### 2.3. In-plane photon density of states

*XM*of the irreducible Brillouin zone. For that of the triangular lattice, the corresponding

*k*-points are distributed inside the triangle Γ

*MK*. Then, the dispersion relations can be plotted as 3D (

*k*-

_{x}*k*-

_{y}*ω*) diagrams and it will be a surface for each band. To perform the 2D PDOS calculation, we construct two equifrequecy lines

*ω*(

*k*,

_{x}*k*) =

_{y}*ω*and

*ω*(

*k*,

_{x}*k*) =

_{y}*ω*+ Δ

*ω*), where

*ω*) is an arbitrary value of the frequency and Δ

*ω*is an infinitesimal increment. The differential surface element in

*k*-space within the lines is

*ω*,

*ω*+

*dω*), by definition, is

*ω*,

*ω*+

*dω*) is obtained from dividing the surface calculated in Eq. (11) by the surface corresponding to one mode, (2

*π*)

^{2}/

*S*, where

*S*is the surface of the system. By definition, we have

*dN*(

*ω*) ≡

*D*(

*ω*)

*dω*. Finally, we arrive at the expression for the PDOS as

## 3. Results and discussion

*M*point for the TE and TM modes of a triangular array. As one can see in the figure, the fractional errors (

*δω*/

*ω*) of all eigenvalues are smaller than 10

^{-4}when the mesh number increases to 16,340. The accuracy and convergence of FEM have been demonstrated. The dispersion relations of 2D photonic crystals are basically controlled by the dielectric contrast, the lattice type, and the filling ratio. For illustrations, two cases have been calculated by this finite element based approach. One is an example in the textbook [13], a square array of dielectric columns with a radius

*r*= 0.38

*a*and dielectric constant

*ε*= 9. The band structures for the TE and TM modes of the square lattice are plotted in Fig. 4(a). The in-plane wave vectors go along the edges of the irreducible Brillouin zone, from Γ to

_{d}*X*,

*X*to

*M*, and

*M*to Γ, as shown in the inset of Fig. 4 (a). Clearly, the lattice of dielectric cylinders has a gap for the TM modes but not for the TE modes. Our calculated results are similar to those in Ref [13]. The other case is a triangular array of air columns with a radius

*r*= 0.4297

*a*drilled in a silicon substrate (

*ε*≈ 11.9) in Ref. [21

_{d}21. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E **58**,3896–3908 (1998). [CrossRef]

*M*,

*M*to

*K*, and

*K*to Γ, as shown in the inset of Fig. 4 (b). A complete photonic band gap opens at

*ωa*/2

*πc*~ 0.4. The calculated results are quite similar to those in Ref [21

21. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E **58**,3896–3908 (1998). [CrossRef]

**58**,3896–3908 (1998). [CrossRef]

*k*-points (Γ,

*M*,

*K*) for both the TE and TM modes is plotted in Figure 6. As one can see in the figure, the frequency shift approximately increases as the frequency increases. For the case, the discrepancy in band structure calculations is up to about 4 %. This may be due to the accuracy of the PWEM limited by the discontinuous nature of the dielectric functions [22

22. H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B **45**,13962–13972 (1992). [CrossRef]

*k*-points uniformly distributed in the irreducible Brillouin zone have been calculated for the two cases via the FEM approach. It means that, due to the symmetry consideration, we effectively discretize the first Brillouin zone into 6,241 and 9,852 grid points for the square array and the triangular array, respectively. As Wang

*et al*. found that a grid number of 256,000 is enough for the 3D case [47

47. X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps,” Phys. Rev. Lett. **88**,093902 (2002) [CrossRef] [PubMed]

**58**,3896–3908 (1998). [CrossRef]

## 4. Conclusion

## Acknowledgement

## References and links

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

2. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) |

3. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

4. | D. Kleppner, “Inhibited Spontaneous Emission,” Phys. Rev. Lett. |

5. | A. O. Barut and J. P. Dowling, “Quantum electrodynamics based on self-energy: Spontaneous emission in cavities,” Phys. Rev. A |

6. | H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A |

7. | J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A |

8. | T. Suzuki and P. K. L. Yu, “Emission power of an electric dipole in the photonic band structure of the fcc lattice,” J. Opt. Soc. Am. B |

9. | A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, “Dipole relaxation in dispersive photonic band-gap structures,” Phys. Rev. A |

10. | A. S. Sánchez and P. Halevi, “Spontaneous emission in one-dimensional photonic crystals,” Phys. Rev. E |

11. | P. Halevi and A. S. Sánchez, “Spontaneous emission in a high-contrast one-dimensional photonic crystal,” Opt. Commun. |

12. | M. C. Lin and R. F. Jao, “Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation,” Phys. Rev. E |

13. | K. Sakoda, |

14. | M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B |

15. | Z. Y. Li and Y. Xia, “Omnidirectional absolute band gaps in two-dimensional photonic crystals,” Phys. Rev. B |

16. | K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B |

17. | K. Sakoda, “Transmittance and Bragg reflectivity of two-dimensional photonic lattices,” Phys. Rev. B |

18. | R. Hillebrand, W. Hergert, and W. Harms, “Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness,” Phys. stat. sol. (b) |

19. | M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B |

20. | O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, “Generalized field propagator for arbitrary finite-size photonic band gap structures,” Phys. Rev. Lett. |

21. | K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E |

22. | H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B |

23. | A. J. Ward and J. B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B |

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

25. | C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B |

26. | G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A |

27. | W. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express |

28. | J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, “Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition,” Opt. Soc. Am. B |

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

30. | M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Tech. |

31. | G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. |

32. | D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. |

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

34. | B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modeling,” IEE Proc. -Sci. Meas. Technal. |

35. | W. J. Kim and J. D. O’Brien, “Optimization of a two-dimensional photonic crystal waveguide branch by simulated annealing and the finite-element method,” J. Opt. Soc. Am. B |

36. | A. Figotin and Y. A. Godin, “The Computation of Spectra of Some 2D Photonic Crystals,” J. Comput. Phys. |

37. | R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E |

38. | E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B |

39. | J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B |

40. | L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E |

41. | D. Hermann, M. Frank, K. Busch, and P. wölfle, “Photonic band structure computations,” Opt. Express |

42. | J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. |

43. | A. Taflove, |

44. | J. Jin, |

45. | A. Yariv and P. Yeh, |

46. | C. Kittel, |

47. | X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(160.4760) Materials : Optical properties

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 14, 2006

Revised Manuscript: December 23, 2006

Manuscript Accepted: December 31, 2006

Published: January 8, 2007

**Citation**

Ming-Chieh Lin and Ruei-Fu Jao, "Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation," Opt. Express **15**, 207-218 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-207

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey 1995).
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997). [CrossRef]
- E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
- D. Kleppner, "Inhibited Spontaneous Emission," Phys. Rev. Lett. 47, 233-236 (1981). [CrossRef]
- A. O. Barut and J. P. Dowling, "Quantum electrodynamics based on self-energy: Spontaneous emission in cavities," Phys. Rev. A 36, 649-654 (1987). [CrossRef] [PubMed]
- H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996). [CrossRef] [PubMed]
- J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992). [CrossRef] [PubMed]
- T. Suzuki and P. K. L. Yu, "Emission power of an electric dipole in the photonic band structure of the fcc lattice," J. Opt. Soc. Am. B 12, 570-582 (1995). [CrossRef]
- A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997). [CrossRef]
- A. S. Sánchez and P. Halevi, "Spontaneous emission in one-dimensional photonic crystals," Phys. Rev. E 72, 056609 (2005). [CrossRef]
- P. Halevi and A. S. Sánchez, "Spontaneous emission in a high-contrast one-dimensional photonic crystal," Opt. Commun. 251, 109-114 (2005). [CrossRef]
- M. C. Lin and R. F. Jao, "Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation," Phys. Rev. E 74, 046613 (2006). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin 2001).
- M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998). [CrossRef]
- Z. Y. Li and Y. Xia, "Omnidirectional absolute band gaps in two-dimensional photonic crystals," Phys. Rev. B 64, 153108 (2001). [CrossRef]
- K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995). [CrossRef]
- K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995). [CrossRef]
- R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000). [CrossRef]
- M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
- O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999). [CrossRef]
- K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998). [CrossRef]
- H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992). [CrossRef]
- A. J. Ward and J. B. Pendry, "Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B 58, 7252-7259 (1998). [CrossRef]
- H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996). [CrossRef]
- C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995). [CrossRef]
- G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 14, 3323-3332 (1997). [CrossRef]
- W. Zhang, C. T. Chan, and P. Sheng, "Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres," Opt. Express 8, 203-208 (2001). [CrossRef] [PubMed]
- J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, "Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition," J. Opt. Soc. Am. B 15, 2316-2324 (1998). [CrossRef]
- W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials," J. Comput. Phys. 150, 468-481 (1999). [CrossRef]
- M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000). [CrossRef]
- G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000). [CrossRef]
- D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000). [CrossRef]
- D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376, 1999. [CrossRef]
- B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002). [CrossRef]
- W. J. Kim and J. D. O’Brien, "Optimization of a two-dimensional photonic crystal waveguide branch by simulated annealing and the finite-element method," J. Opt. Soc. Am. B 21, 289-295 (2004). [CrossRef]
- A. Figotin and Y. A. Godin, "The Computation of Spectra of Some 2D Photonic Crystals," J. Comput. Phys. 136, 585-598, 1997. [CrossRef]
- R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609, 2004. [CrossRef]
- E. Moreno, D. Erni and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120, 2002. [CrossRef]
- J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715, 1996. [CrossRef]
- L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001). [CrossRef]
- D. Hermann, M. Frank, K. Busch, and P. w¨olfle, "Photonic band structure computations," Opt. Express 8, 167-172 (2001). [CrossRef] [PubMed]
- J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992). [CrossRef] [PubMed]
- A. Taflove, Computational Electrodynamics: The Finite- Difference Time-Domain Method, Artech, Boston, Mass. (1995).
- J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-Interscience, New York 2002).
- A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York 1984).
- C. Kittel, Introduction to Solid State Physics (Wiley, New York 1976).
- X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002) [CrossRef] [PubMed]

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