## Semi-analytical approach for guided mode resonance in high-index-contrast photonic crystal slab: TE polarization |

Optics Express, Vol. 21, Issue 18, pp. 20588-20600 (2013)

http://dx.doi.org/10.1364/OE.21.020588

Acrobat PDF (1963 KB)

### Abstract

In high-contrast (HC) photonic crystals (PC) slabs, the high-order coupling is so intense that it is indispensable for analyzing the guided mode resonance (GMR) effect. In this paper, a semi-analytical approach is proposed for analyzing GMR in HC PC slabs with TE-like polarization. The intense high-order coupling is included by using a convergent recursive procedure. The reflection of radiative waves at high-index-contrast interfaces is also considered by adopting a strict Green’s function for multi-layer structures. Modal properties of interest like band structure, radiation constant, field profile are calculated, agreeing well with numerical finite-difference time-domain simulations. This analysis is promising for the design and optimization of various HC PC devices.

© 2013 Optical Society of America

## 1. Introduction

1. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**, 5751–5758 (1999). [CrossRef]

7. V. Pacradouni, W. J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, and S. R. Johnson, “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B **62**, 4204–4207 (2000). [CrossRef]

8. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. **74**, 7–9 (1999). [CrossRef]

16. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nature Photon. **4**, 447–450 (2010). [CrossRef]

17. A. Mekis, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Two-dimensional photonic crystal couplers for unidirectional light output,” Opt. Lett. **25**, 942–944 (2000). [CrossRef]

18. S. Peng and G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A **13**, 993–1005 (1996). [CrossRef]

3. D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**, 2038–2059 (1997). [CrossRef]

9. M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. **88**, 191105 (2006). [CrossRef]

10. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasersbased on two-dimensional photoniccrystal resonators,” Opt. Express **16**, 5206–5217 (2008). [CrossRef] [PubMed]

11. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature (London) **457**, 174–178 (2009). [CrossRef]

12. T. Lu, S. Chen, L. Lin, T. Kao, C. Kao, P. Yu, H. Kuo, S. Wang, and S. Fan, “GaN-based two-dimensional surface-emitting photonic crystal lasers with AlN/GaN distributed bragg reflector,” Appl. Phys. Lett. **92**, 011129 (2008). [CrossRef]

13. H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science **319**, 445–447 (2008). [CrossRef]

14. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science **293**, 1123–1125 (2001). [CrossRef] [PubMed]

15. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: Lasers producing tailored beams,” Nature (London) **441**, 946 (2006). [CrossRef]

16. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nature Photon. **4**, 447–450 (2010). [CrossRef]

19. Y. Zhou, M. Huang, C. Chase, V. Karagodsky, M. Moewe, B. Pesala, F. Sedgwick, and C. Chang-Hasnain, “High-index-contrast grating (HCG) and its applications in optoelectronic devices,” IEEE J. Sel. Top. Quantum Electron. **15**, 1485–1499 (2009). [CrossRef]

22. Y. Zhou, M. C. Y. Huang, and C. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high-contrast grating,” IEEE Photon. Technol. Lett. **20**, 434–436 (2008). [CrossRef]

20. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics **1**, 119–122 (2007). [CrossRef]

21. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “Single mode high-contrast subwavelength grating vertical cavity surface emitting lasers,” Appl. Phys. Lett. **92**, 171108 (2008). [CrossRef]

22. Y. Zhou, M. C. Y. Huang, and C. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high-contrast grating,” IEEE Photon. Technol. Lett. **20**, 434–436 (2008). [CrossRef]

23. J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. **109**, 067401 (2012). [CrossRef] [PubMed]

24. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature (London) **499**, 188–191 (2013). [CrossRef]

25. N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. a. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nature Nano. **2**, 515–520 (2007). [CrossRef]

26. B. Zhen, S. L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. USA (2013). In press. [CrossRef] [PubMed]

27. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. **181**, 687–702 (2010). [CrossRef]

28. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

29. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B **84**, 195119 (2011). [CrossRef]

33. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express **19**, 24672–24686 (2011). [CrossRef] [PubMed]

30. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express **20**, 15945–15961 (2012). [CrossRef] [PubMed]

31. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

32. Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express **21**, 565–580 (2013). [CrossRef] [PubMed]

33. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express **19**, 24672–24686 (2011). [CrossRef] [PubMed]

## 2. High-index-contrast coupled-wave theory

29. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B **84**, 195119 (2011). [CrossRef]

33. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express **19**, 24672–24686 (2011). [CrossRef] [PubMed]

### 2.1. High-index-contrast PC slabs vs. Low-index-contrast PC slabs

*As*

_{x}_{1−}

*, Al*

_{x}*Ga*

_{x}_{1−x}As, etc. are all around 11∼12). In the LC PC slabs, Bloch waves are weakly guided, the electric field decays slowly in the cladding, and the vertical energy distribution is not restricted within the PC layer (the green curve in Fig. 1(a)). Hence, under the weakly guided condition, both the in-plane wave interactions and the vertical radiations in the PC layer are rather mild. Therefore, in the reciprocal space, treating the basic waves (blue arrows in Fig. 1(e)) as sources to excite high-order waves and radiative waves, and the excited waves coupling back to the basic waves is a good approximation, which we refer to as the LC approximation. Besides, also because of the low index contrast, the reflection between interfaces is negligibly small. Thus, a Green’s function for infinite homogeneous materials is able to provide accurate analyses on the band-edge modes and their radiation properties for LC PC slabs [29

29. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B **84**, 195119 (2011). [CrossRef]

31. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

### 2.2. Structure description

*a*= 410 nm is designed to match the wavelength of light in the waveguide such that the photonic lattices serve to provide a 2D distributed feedback effect. Then, the reciprocal base vector is given by

*β*

_{0}= 2

*π/a*. The average permittivity of the PC layer is given by

*ε*=

_{PC}*fε*+ (1 −

_{a}*f*)

*ε*, where

_{b}*ε*= 1 is the permittivity of air,

_{a}*ε*= 11.97 is the permittivity of the photonic crystal (Si), and

_{b}*f*is the filling factor given by

*f*=

*S*

_{air}_{−}

_{hole}/a^{2}(i.e., the fraction of the area of a unit cell occupied by air holes). Let the PC lies in a square lattice with arbitrarily shaped holes. Figure 1(d) depicts examples of air-hole shapes that are considered later in this paper: (i) circles and (ii) equilateral triangles. For the PC slab where the side length is sufficiently large (exceeding 300

*a*or 100

*μ*m for practical devices), the structure can be reasonably regarded as infinite in the XY plane and the in-plane loss can be neglected [35

35. H. Ryu, M. Notomi, and Y. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B **68**, 045209 (2003). [CrossRef]

### 2.3. High-order wave coupling

*E*,

_{x}*E*, 0). Thus, using the Bloch theorem and combining the master equation of electric components with the Fourier expansion of material permittivity, we obtain the GMR coupling equations in the reciprocal space [29

_{y}**84**, 195119 (2011). [CrossRef]

31. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

*n*

_{0}is the refractive index,

*k*

_{0}=

*ω/c*(

*ω*is the angular frequency and

*c*is the light velocity in vacuum),

*m*and

_{x}*n*are the wave orders in the xy plane, and

_{y}*ξ*is the component of the Fourier expansion of permittivity with an index of (

_{mn}*m*,

*n*). The above wave equations include all possible waves whose amplitudes are explicitly dependent on the vertical position

*z*. As we defined previously [29

**84**, 195119 (2011). [CrossRef]

*m*

^{2}+

*n*

^{2}| = 1), high-order waves (|

*m*

^{2}+

*n*

^{2}

*| >*1), and radiative wave (|

*m*

^{2}+

*n*

^{2}| = 0).

*m*,

*n*)) is determined by two factors: the intensity of source waves (denoted by (

*m′*,

*n′*)) and the Fourier component of the permittivity:

*ξ*

_{m−m′,n−n′}, which defines the coupling route of corresponding waves. Usually, the nearer the Bloch waves in the reciprocal space, the larger

*ξ*

_{m−m′,n−n′}is, and vice versa. For HC PC slabs with a considerable filling factor, the intense energy confinement renders high-order waves significant and

*ξ*

_{m−m′,n−n′}can be still large even if Bloch waves are far away from each other. So, the decisive factors of the high-order coupling, which are negligible in LC PC slabs, become significant in HC PC slabs, making the high-order coupling an important contribution in the whole wave interaction mechanism. That is the reason why the LC approximation no longer applies to the analysis of GMR in HC PC slabs.

36. W. Streifer, D. R. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. **12**, 422–428 (1976). [CrossRef]

**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

37. M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. **84**, 1196–1203 (1998). [CrossRef]

### 2.4. Green’s function for vertical structure with considerable reflection

*G*(

*z,z′*) = −

*i/*2

*β*·

*e*

^{−}

^{iβ|z}^{−}

*for an infinite homogenous space gives accurate results for the analysis of GMR in LC PC slabs [29*

^{z′|}**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

**19**, 24672–24686 (2011). [CrossRef] [PubMed]

*h*is the thickness of the PC slab, and Δ is the normalized deviation from the Γ point in the x or y direction [31

_{PC}**86**, 035108 (2012). [CrossRef]

*κ*can be very small. Using the Silicon-GaAs interface for example, the corresponding amplitude of

*κ*at Γ point is about 0.012, which leads to a 0.014% reflection. On the contrary,

*κ*becomes much larger in the HC case. At the air-silicon interface, the corresponding amplitude of

*κ*is about 0.5, which leads to a considerable 20% reflection. This validates the necessity of adopting a strict reflection-included Green’s function for HC PC slabs.

### 2.5. Coupled wave equations

**C**also becomes self-consistent and the eigenvalue equation is given by [31

**86**, 035108 (2012). [CrossRef]

**V**= (

*R*,

_{x}*S*,

_{x}*R*,

_{y}*S*)

_{y}^{T}. The complex frequencies

*ω*can be obtained from the eigenvalues by solving Eq. (4) as

*ω*=

*ck*. The Q factors of the band-edge modes can be determined from the real and imaginary parts of

*ω*, as

*Re*(

*ω*)/|2

*Im*(

*ω*)|, without using any ambiguous definition of the effective refractive index. The radiation constant

*α*can be obtained from the Q factor [3

_{r}3. D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**, 2038–2059 (1997). [CrossRef]

**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

*α*=

_{r}*β*

_{0}/

*Q*= (2

*π/a*)/

*Q*.

## 3. Results and discussion

*ω*and the radiation constant

*α*: the modal power loss due to the surface emission. A truncation of the summation terms with an appropriate order of

_{r}*m*and

*n*is required to obtain numerical results. Here, we define a quantity

*D*such that |

*m*,

*n*| ≤

*D*. As discussed in our previous study [29

**84**, 195119 (2011). [CrossRef]

*D*= 10 is a large enough truncation number for HC-CWT. Hence, we use

*D*= 10 in all the following calculations.

27. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. **181**, 687–702 (2010). [CrossRef]

38. M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express **13**, 2869–2880 (2005). [CrossRef] [PubMed]

*z*direction and periodic boundary conditions in the

*x*and

*y*directions. The Q factors and radiation constants were obtained from the numerical simulation, and compared directly with CWT analysis results.

### 3.1. In-plane and vertical field profiles

34. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. **89**, 021101 (2006). [CrossRef]

39. K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express **16**, 6033–6040 (2008). [CrossRef] [PubMed]

**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

**19**, 24672–24686 (2011). [CrossRef] [PubMed]

### 3.2. Frequency and radiation analysis

**84**, 195119 (2011). [CrossRef]

*ξ*is very small for large (

_{mn}*m, n*)). A small

*ξ*for high-order (

_{mn}*m, n*) restricts the amplitude of high-order coupling (Eq. (1) and (2)). Thus, the LC approximation applies again to small filling factors. Nevertheless, with large filling factors, high-order coupling is no longer negligible. The necessity of including high-order coupling is validated by the significant errors of LC-CWT in Figs. 7 and 8.

### 3.3. Band structure

**86**, 035108 (2012). [CrossRef]

2. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**, 235112 (2002). [CrossRef]

**84**, 195119 (2011). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

2. | S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

3. | D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. |

4. | M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. Young, S. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. |

5. | M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and purcell enhancement from thin-film 2-d photonic crystals,” J. Lightwave Technol. |

6. | P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B |

7. | V. Pacradouni, W. J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, and S. R. Johnson, “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B |

8. | M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. |

9. | M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. |

10. | L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasersbased on two-dimensional photoniccrystal resonators,” Opt. Express |

11. | Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature (London) |

12. | T. Lu, S. Chen, L. Lin, T. Kao, C. Kao, P. Yu, H. Kuo, S. Wang, and S. Fan, “GaN-based two-dimensional surface-emitting photonic crystal lasers with AlN/GaN distributed bragg reflector,” Appl. Phys. Lett. |

13. | H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science |

14. | S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science |

15. | E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: Lasers producing tailored beams,” Nature (London) |

16. | Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nature Photon. |

17. | A. Mekis, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Two-dimensional photonic crystal couplers for unidirectional light output,” Opt. Lett. |

18. | S. Peng and G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A |

19. | Y. Zhou, M. Huang, C. Chase, V. Karagodsky, M. Moewe, B. Pesala, F. Sedgwick, and C. Chang-Hasnain, “High-index-contrast grating (HCG) and its applications in optoelectronic devices,” IEEE J. Sel. Top. Quantum Electron. |

20. | M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics |

21. | M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “Single mode high-contrast subwavelength grating vertical cavity surface emitting lasers,” Appl. Phys. Lett. |

22. | Y. Zhou, M. C. Y. Huang, and C. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high-contrast grating,” IEEE Photon. Technol. Lett. |

23. | J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. |

24. | C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature (London) |

25. | N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. a. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nature Nano. |

26. | B. Zhen, S. L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. USA (2013). In press. [CrossRef] [PubMed] |

27. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. |

28. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

29. | Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B |

30. | Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express |

31. | C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B |

32. | Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express |

33. | C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express |

34. | K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. |

35. | H. Ryu, M. Notomi, and Y. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B |

36. | W. Streifer, D. R. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. |

37. | M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. |

38. | M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express |

39. | K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(160.5293) Materials : Photonic bandgap materials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: July 18, 2013

Revised Manuscript: August 14, 2013

Manuscript Accepted: August 15, 2013

Published: August 26, 2013

**Citation**

Yi Yang, Chao Peng, and Zhengbin Li, "Semi-analytical approach for guided mode resonance in high-index-contrast photonic crystal slab: TE polarization," Opt. Express **21**, 20588-20600 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20588

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### References

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