## Higher order finite-difference frequency domain analysis of 2-D photonic crystals with curved dielectric interfaces

Optics Express, Vol. 17, Issue 5, pp. 3305-3315 (2009)

http://dx.doi.org/10.1364/OE.17.003305

Acrobat PDF (255 KB)

### Abstract

A high-order finite-difference frequency domain method is proposed for the analysis of the band diagrams of two-dimensional photonic crystals. This improved formulation is based on Taylor series expansion, local coordinate transformation, boundary conditions matching, and the generalized Douglas scheme. The nine-point second-order formulas are extended to fourth-order accuracy. This proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces.

© 2009 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

3. 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]

6. 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]

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

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

12. W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightw. Technol. **10**, 295–305 (1992). [CrossRef]

13. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express **10**, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853. [PubMed]

17. J. Xia and J. Yu, “New finite-difference scheme for simulations of step-index waveguides with tilt interfaces,” IEEE Photon. Technol. Lett. **15**, 1237–1239 (2003). [CrossRef]

18. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightw. Technol. **20**, 1609–1618 (2002). [CrossRef]

19. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces,” J. Lightw. Technol. **26**, 971–976 (2008). [CrossRef]

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

20. 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), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

13. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express **10**, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853. [PubMed]

20. 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), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

21. 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]

20. 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), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

16. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightw. Technol. **18**, 243–251 (2000). [CrossRef]

22. I. Harari and E. Turkel, “Accurate finite difference methods for time-harmonic wave propagation,” J. Comput. Phys. **119**, 252–270 (1995). [CrossRef]

19. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces,” J. Lightw. Technol. **26**, 971–976 (2008). [CrossRef]

## 2. The high-order FDFD scheme

*a*and

*r*denote the lattice distance and the radius of element rods, respectively. The element rods can be either dielectric rods or air columns. As indicated in [9

**44**, 2688–2695 (1996). [CrossRef]

*z*direction and periodic in the transverse plane, we will only consider the in-plane propagation for calculating the band structures and let the propagation constant in the

*z*direction, that is,

*β*be zero.

_{z}### 2.1. High-order relations between grid points

*ε*and

_{L}*ε*, respectively. The quantity

_{R}*ϕ*denotes

*E*for TM modes and

_{z}*H*for TE modes at grid points. A local cylindrical coordinate system with origin

_{z}*O*′ can be defined in terms of the normal vector

*r*̂, the tangential vector

*θ*̂, and the effective radius

*R*. To derive a correct high-order finite-difference formula, the general relation between the field at a sampled point, such as

*ϕ*|

_{(m,n)}in Fig. 2, and the fields at nearby points, must first be found. The basic procedure for deriving this relation is similar to those provided in the previous work [19

19. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces,” J. Lightw. Technol. **26**, 971–976 (2008). [CrossRef]

*ϕ*|

_{(m+1,n+1)}and

*ϕ*|

_{(m,n)}as an example, the steps that link the fields at two grid points can be illustrated as

*ϕ*|

_{R;r-θ}, which is the field just to the right of the interface, and its derivatives must be linked with those on the other side of the interface. The required equations with derivative terms up to second order can also be found explicitly in [19

**26**, 971–976 (2008). [CrossRef]

*θ*, then its derivatives with respect to

*θ*must also be continuous in this tangential direction. Therefore, if a low-order boundary matching equation of the form

*θ*will also apply.

**26**, 971–976 (2008). [CrossRef]

*k*

_{0}is the free-space wavenumber. Besides of the above relations, high-order derivative equations with respect to

*r*only, that is, ∂

*/∂*

^{p}ϕ*r*must still be obtained. They can be found by combining the Helmholtz equation with Maxwell’s equations or the Helmholtz equation itself. For example, combining the Helmholtz equation on

^{p}*E*with the curl equation ∇ ×

_{z}**E**= -

*jωμ*

**H**yields

*H*with another curl equation ∇ ×

_{z}**H**=

*jωε*

**E**yields

### 2.2. High-order formulation with the GD scheme

*m*,

*n*),

*ϕ*|

_{(m,n)}, and its derivatives. Similar linear equations at

*N*grid points, including the central sampled point (

*m*,

*n*) itself, can be expressed in matrix form as,

**Φ**is the vector that contains the fields at

*N*grid points;

**M**is an

*N*-by-

*N*matrix that contains coefficients obtained using the steps described in the preceding subsection, and

**D**

_{(m,n)}is the vector that contains the field quantity at point (

*m*,

*n*) and its derivatives with respect to

*x*or

*y*, respectively, up to

*N*required terms. Unlike in Eq. (28) in [19

**26**, 971–976 (2008). [CrossRef]

**M**is 9-by-9, then the nine-point second-order formulation as provided in [19

**26**, 971–976 (2008). [CrossRef]

**26**, 971–976 (2008). [CrossRef]

*D*′ and

_{x}*D*″ denote the first and the second-order finite-difference formulas for the differentiation with respect to

_{xx}*x*and are given by Eq. (13). This form implies that differentiations with respect to

*x*and

*y*can be decoupled. However, this implication does not apply when a slanted or curved step-index interface lies between grid points. To derive a correct formula, both second-order terms differentiated with respect to

*x*and

*y*must be taken into account simultaneously. Use the field quantity

*E*as an example and assume that the formula obtained from Eq. (13) can be expressed as following with the residual third-order terms

_{z}*g*

_{1}and

*g*

_{2}may come from both

*D*″ and

_{xx}*D*″. Although ∂

_{yy}^{3}

*E*/∂

_{z}*x*

^{2}∂

*y*and ∂

^{3}

*E*/∂

_{z}*x*∂

*y*

^{2}seem to be third-order terms, it should be included in the nine-point second-order formulas using Eq. (13). Thus

*D*‴ and

_{xxy}*D*‴ are elements of

_{xyy}**D**

_{(m,n)}and their second-order formulas can be obtained from Eq. (13). We then add proper third-order terms at both sides of Eq. (15) as

### 2.3. Implementation of the high-order formulation

*k*and

_{x}*k*are the wavenumbers in the

_{y}*x*and

*y*directions, respectively.

**12**, 1397–1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

## 3. Numerical results

*ε*

_{A1}= 8.9 and that of the air is

*ε*

_{air}= 1. The radius of the alumina rod is

*r*= 0.2

*a*, where

*a*is the lattice constant as shown in Fig. 1(a). Each point along the boundary of the first Brillouin zone shown as the middle inset in Fig. 4(a) determines the values of

*k*and

_{x}*k*that we use in Eqs. (18a), (18b) and thus in Eq. (20). The calculated band diagrams of the TE and TM modes are plotted in Figs. 4(a) and (b), respectively. The band structures marked with red circles are the results obtained by using our nine-point high-order finite-difference scheme with 30×30 grid points, and the solid blue lines are the results obtained using the MIT Photonic-Bands package [23

_{y}23. S. G. Johnson and J. D. Joannopoulos, “The MIT Photonic-Bands Package home page [on line],” http://ab-initio.mit.edu/mpb/.

**26**, 971–976 (2008). [CrossRef]

**k**at the

**M**point in the first Brillouin zone as the middle inset in Fig. 4(a), i.e.,

**k**= (

*π*/

*a*,

*π*/

*a*, 0), and search for the eigen frequencies using different numbers of grid points. We calculated the results of the four most fundamental bands using the nine-point second-order scheme of [19

**26**, 971–976 (2008). [CrossRef]

**12**, 1397–1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

13. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express **10**, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853. [PubMed]

**12**, 1397–1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

**26**, 971–976 (2008). [CrossRef]

**26**, 971–976 (2008). [CrossRef]

**12**, 1397–1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

**10**, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853. [PubMed]

**26**, 971–976 (2008). [CrossRef]

**12**, 1397–1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]

**26**, 971–976 (2008). [CrossRef]

*ε*= 11.4 and radius

*r*= 0.2

*a*in the air, where

*a*is the lattice constant as shown in Fig. 1(b). Each point along the boundary of the first Brillouin zone shown as the middle inset in Fig. 6(a) determines the values of

*k*and

_{x}*k*that we use in Eqs. (19a)–(19c) and thus in Eq. (20). The calculated band diagrams of the TE and TM modes are plotted in Figs. 6(a) and (b). The band structures marked with red circles are the results obtained using our nine-point high-order finite-difference scheme with 30 × 30 grid points and adopting the modified unit cell as shown in Fig. 3(c). The solid blue lines are the results obtained using the MIT Photonic-Bands package [23

_{y}23. S. G. Johnson and J. D. Joannopoulos, “The MIT Photonic-Bands Package home page [on line],” http://ab-initio.mit.edu/mpb/.

**k**at the K point in the first Brillouin zone as the middle inset in Fig. 6(a), i.e.,

**k**= [2

*π*/(√3

*a*),2

*π*/(3

*a*),0], and use similar processes to calculate the band structure as in the square lattice case. The convergence behaviors of the four most fundamental bands are illustrated in Figs. 7(a)–(d). The lines notations used here are the same as those in Figs. 5(a)–(d). It shows again that our method provides better numerical convergence compared with other methods.

## 4. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

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

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

5. | R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. |

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

7. | M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” Appl. Phys. |

8. | A. Taflove and S. C. Hagness, |

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

10. | K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. |

11. | G. R. Hadley and R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightw. Technol. |

12. | W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightw. Technol. |

13. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express |

14. | M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” Proc. Inst. Elect. Eng. J. |

15. | C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” Proc. Inst. Elect. Eng. J. |

16. | Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightw. Technol. |

17. | J. Xia and J. Yu, “New finite-difference scheme for simulations of step-index waveguides with tilt interfaces,” IEEE Photon. Technol. Lett. |

18. | Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightw. Technol. |

19. | Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces,” J. Lightw. Technol. |

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

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

22. | I. Harari and E. Turkel, “Accurate finite difference methods for time-harmonic wave propagation,” J. Comput. Phys. |

23. | S. G. Johnson and J. D. Joannopoulos, “The MIT Photonic-Bands Package home page [on line],” http://ab-initio.mit.edu/mpb/. |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 2, 2009

Revised Manuscript: February 11, 2009

Manuscript Accepted: February 13, 2009

Published: February 17, 2009

**Citation**

Yen-Chung Chiang, "Higher order finite-difference frequency domain analysis of 2-D photonic crystals with curved dielectric interfaces," Opt. Express **17**, 3305-3315 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3305

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### References

- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- 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]
- M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
- R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett. 61, 495-497 (1992). [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]
- M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," Appl. Phys. 87, 8268-8275 (2000). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition, (Artech House, MA USA, 2005).
- H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996). [CrossRef]
- K. Bierwirth, N. Schulz, and F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microwave Theory Tech. 34, 1104-1114 (1986). [CrossRef]
- G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol. 13, 465-469 (1995). [CrossRef]
- W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992). [CrossRef]
- Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10,853-864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=OE-10-17-853. [PubMed]
- M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 135, 56-63 (1988).
- C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).
- Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000). [CrossRef]
- J. Xia and J. Yu, "New finite-difference scheme for simulations of step-index waveguides with tilt interfaces," IEEE Photon. Technol. Lett. 15, 1237-1239 (2003). [CrossRef]
- Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002). [CrossRef]
- Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008). [CrossRef]
- C.-P. Yu and H.-C. Chang, "Compact finite-difference frequency-domain method for the analysis of twodimensional photonic crystals," Opt. Express 12,1397-1408 (2004), http://www.opticsexpress.org/abstract.cfm?uri=OE-12-7-1397. [CrossRef] [PubMed]
- 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]
- I. Harari and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," J. Comput. Phys. 119, 252-270 (1995). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "The MIT Photonic-Bands Package home page [on line]," http://ab-initio.mit.edu/mpb/.

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