## Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls |

Optics Express, Vol. 19, Issue 24, pp. 24672-24686 (2011)

http://dx.doi.org/10.1364/OE.19.024672

Acrobat PDF (1087 KB)

### Abstract

The coupled-wave theory (CWT) is extended to a photonic crystal structure with arbitrary sidewalls, and a simple, fast, and effective model for the quantitatively analysis of the radiative characteristics of two-dimensional (2D) photonic-crystal surface-emitting lasers (PC-SELs) has been developed. For illustrating complicated coupling effects accurately, sufficient numbers of waves are included in the formulation, by considering their vertical field profiles. The radiation of band-edge modes is analyzed for two in-plane air-hole geometries, in the case of two types of sidewalls: i.e. “tapered case” and “tilted case.” The results of CWT analysis agree well with the results of finite-difference time-domain (FDTD) numerical simulation. From the analytical solutions of the CWT, the symmetry properties of the band-edge modes are investigated. In-plane asymmetry of the air holes is crucial for achieving high output power because it causes partial constructive interference. Asymmetric air holes and tilted sidewalls help in inducing in-plane asymmetries. By breaking the symmetries with respect to the two orthogonal symmetric axes of the band-edge modes, the two factors can be tuned independently, so that the radiation power is enhanced while preserving the mode selectivity performance. Finally, top-down reactive ion etching (RIE) approach is suggested for the fabrication of such a structure.

© 2011 OSA

## 1. Introduction

1. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. **75**, 316–318 (1999). [CrossRef]

1. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. **75**, 316–318 (1999). [CrossRef]

7. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Optics Express. **12**, 1562–1568 (2004). [CrossRef] [PubMed]

10. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, and J. Faist, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Optics 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 , **457**, 174–178 (2009). [CrossRef] [PubMed]

12. 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]

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

3. 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]

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

14. 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]

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

6. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. **39**, 689–700 (2003). [CrossRef]

20. M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. **28**, 1653–1662, (1992). [CrossRef]

17. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. **43**, 2327–2335 (1972). [CrossRef]

19. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. **21**, 144–150 (1985). [CrossRef]

7. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Optics Express. **12**, 1562–1568 (2004). [CrossRef] [PubMed]

21. 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]

22. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **45**, 788–795 (2010). [CrossRef]

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with TE polarization — a general approach,” Phy. Rev. B. (submitted), http://arxiv.org/abs/1107.1772.

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with TE polarization — a general approach,” Phy. Rev. B. (submitted), http://arxiv.org/abs/1107.1772.

24. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Air-hole design in a vertical direction for high-power two-dimensional photonic-crystal surface-emitting lasers,” JOSA B . **27**, 1204–1207 (2010). [CrossRef]

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with TE polarization — a general approach,” Phy. Rev. B. (submitted), http://arxiv.org/abs/1107.1772.

## 2. Coupled-wave theory for inhomogeneous vertical structure

*ɛ*=

_{PC}*fɛ*+ (1 –

_{a}*f*)

*ɛ*, where

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

_{a}*ɛ*= 12 is the permittivity of the dielectric material (GaAs), and

_{b}*f*= 0.16 is the filling factor. The slab structure supports only a single waveguide mode propagating in the

*XY*plane; and the wavevector is denoted by

*β*. Let the PC lie in a square lattice (lattice constant

*a*) with arbitrarily shaped holes; then, the reciprocal base vector is given by

*β*

_{0}= 2

*π*/

*a*. For the PC-SELs where the side length is sufficiently large (exceeding 300

*a*or 100

*μm*for practical devices), the structure can be reasonably regarded as infinite in the

*x*–

*y*plane and the in-plane loss can be neglected [25

25. H. Y. Ryu, M. Notomi, and Y. H. 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]

*k*

_{0}is the wavenumber in vacuum. For the transverse-electric (TE) mode, the field should be

**= (**

*E**E*,

_{x}*E*, 0), and it can be expanded using the Bloch theorem, as

_{y}*E*(

_{x}*z*) = Σ

*E*(

_{x,mn}*z*)

*e*

^{–imβ0x–inβ0y},

*E*(

_{y}*z*) = Σ

*E*(

_{y,mn}*z*)

*e*

^{–imβ0x–inβ0y}. Besides, the permittivity can similarly be expanded as Here,

*ɛ*

_{0}(

*z*) is the average permittivity and

*ξ*(

_{mn}*z*) = 0 for the no-periodicity case. In the case of a general inhomogeneous vertical structure,

*ɛ*

_{0}(

*z*) and

*ξ*(

_{mn}*z*) are not necessarily constant in the

*z*direction. Then, we get the wave equation for all

**components,**

*E**z*; solving such equation is difficult because all the waves are coupled to one another. As we defined previously [23

*m*

^{2}+

*n*

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

*m*

^{2}+

*n*

^{2}| > 1), and radiative wave (|

*m*

^{2}+

*n*

^{2}| = 0). Most of energy is concentrated in the basic waves for the resonant case, and therefore, treating the four basic waves as sources to excite high-order waves and radiative waves, and the excited waves coupling back to the basic waves would be a good approximation. Coupled equations describing the interactions between the four basic waves can be obtained, and the four band-edge modes at the second-order Γ-point can be obtained as shown in Fig. 1(b).

### 2.1. Basic waves

*R*,

_{x}*S*,

_{x}*R*,

_{y}*S*with identical vertical profiles Θ

_{y}_{0}(

*z*). For a infinite periodical PC structure, the in-plane amplitudes are independent of

*x,y*. Without loss of generality, we focus on the case

*m*= 1,

*n*= 0: Eqs. 4 and 5 lead to

*β*=

*β*

_{0}, and hence, the profile Θ

_{0}(

*z*) for basic waves can be determined by the waveguide mode profile; the basic wave profile satisfies following equation Θ

_{0}(

*z*) can be solved by using transfer matrix method (TMM) [23

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

*z*when the air hole possesses an arbitrary sidewall, the PC layer must be divided into multiple layers and constant permittivity must be assumed within every single layer, if necessary [26

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

*k*

_{0}is generalized by including a small deviation from the desired frequency

*k*, as

*k*

_{0}=

*k*+

*δk*. The radiative waves are denoted as

*E*

_{x,00}= Δ

*E*(

_{x}*z*),

*E*

_{y,00}= Δ

*E*(

_{y}*z*); hence, the coupled equation of the basic wave is obtained from Eq. 9 and Eq. 10, as Multiplication of the complex conjugate field

*E*(

_{y}*z*) and high-order profile

*E*(

_{y,m′n′}*z*) are still unknown, and they will be determined in the subsequent sections.

*δk*(implying temporally decaying modes) are used here. Simultaneous Bragg coupling and energy leakage perturb the bands corresponding to complex propagation constant, however, the two mechanisms are decoupled in the case of complex frequency [27

27. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Optics Express **15**, 680–694 (2007). [CrossRef] [PubMed]

### 2.2. Radiative waves

*E*(

_{y}*z*), the terms with (

*m, n*) = (0, 0) are grouped from Eq. 4: Then, the approximation that only basic waves are important for exciting radiative waves is applied:

*G*(

*z,z*′) = −

*i*/2

*β*·

*e*

^{−iβ|z–z′|}for infinite homogenous space gives accurate results [23

*E*(

_{x}*z*).

### 2.3. High-order waves

*E*(

_{x,mn}*z*) and

*E*(

_{y,mn}*z*); then we get

*G*(

_{mn}*z, z*′). In addition, only basic waves are assumed to excite high-order waves and the generalized Green function is used for better accuracy (see Appendix): where

*E*component) and under the transversity condition ∇ · [

_{z}*ɛ*(

*r*)

**(**

*E**r*)] = 0, Eq 16 gives where

### 2.4. Coupled wave equations

**V**= (

*R*,

_{x}*S*,

_{x}*R*,

_{y}*S*) with the coupling matrix

_{y}**C**: The complex frequencies

*ω*can be obtained from the eigen values by solving Eq. 20 as

*ω*= (

*k*+

*δk*)/

*c*. In this case, 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*)|, and directly compared with the FDTD simulation results without using any ambiguous definition of the effective refractive index. Nevertheless, for the infinite PC structure, only the radiative wave causes radiation loss in the spatial domain and afford a corresponding finite Q factor in the time domain. The Q factor describes the same radiative characteristics of PC-SELs as do the radiative constant derived from the complex propagating constant, but from another point of view. Thus, the radiation constant can be obtained from the Q factor using the following equation [23

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

## 3. Radiative characteristics of air holes with arbitrary sidewalls

### 3.1. Air-hole shape and calculation parameters

*θ*, and

*θ*= 0° indicates an ideal vertical sidewall. For the “tilted case,” the direction of shift of the center position is defined as the tilt direction

*ϕ; ϕ*= 0° is in the

*X*= 0 direction, which matches with the square lattice of the PC. Two air-hole shapes, CC and RIT, will be discussed.

*z*-dependent permittivity, i.e.,

*ɛ*

_{0}(

*z*) and

*ξ*(

_{mn}*z*). For the “tapered case” shown in Fig. 2(a–b), the sizes of the air holes vary along the PC layer, and hence the sidewall can be represented by the linear change in the filling factor. Because the air-hole shape and center positions are unchanged in

*z*-direction,

*ξ*(

_{mn}*z*) only changes with the filling factor. The average permittivity

*ɛ*

_{0}(

*z*) also varies in the

*z*-direction, and hence the PC layer must be divided into multiple layers when applying the TMM to basic wave profile calculation [26

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

*r*induces an extra phase shift

*e*

^{−iβ0·Δr}to

*ξ*(

_{mn}*z*). As a result,

*ɛ*

_{0}(

*z*) is constant within the PC layer, whereas

*ξ*(

_{mn}*z*) varies with the linear change in the phase shift in the

*z*-direction. In this case, the PC slab can be treated as a single homogeneous layer in the TMM calculation.

*x*×

*y*×

*z*of 40 × 40 × 640 pixels, corresponding to 1 × 1 × 16 lattice periods, with the PML in the

*z*direction and the periodic boundary in the

*xy*plane [29

29. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications **181**, 687C702 (2010) [CrossRef]

### 3.2. Calculation results and discussion

*m*

^{2}+

*n*

^{2}| = 1) to radiative waves (|

*m*

^{2}+

*n*

^{2}| = 0). The main contribution to the radiation power is from the basic waves (|

*m*

^{2}+

*n*

^{2}| = 1), where most of the energy is concentrated. As shown in Eq. 14, the radiation can be approximated as the superposition of these four basic waves. On the basis of the eigenvectors solved by the CWT, we first explain the in-plane symmetries of resonant modes A and B, assuming ideal vertical sidewalls, referred to “uniform case,” and then analyze the “tapered case” and “tilted case” using the insights from the symmetry.

#### Uniform case

*Rx*,

*Ry*and

*Sx*,

*Sy*, as shown in Fig. 3.

**E**is a vector while

**H**is a pseudovector, mode A with CC air holes is antisymmetric with respect to the

*Y*=

*X*and

*Y*= −

*X*axes, but mode B is symmetric with respect to the same axes. As a result of these symmetries and the equal amplitudes of the eigenvectors, the combination of these four basic waves according to Eq. 14 leads to complete destructive interference, which minimizes the radiative power and yields a large Q factor for modes A and B, as confirmed by the FDTD simulation [23

*Y*= −

*X*direction for mode A but in the

*Y*=

*X*direction for mode B.

#### Tapered case

*z*and −

*z*directions are not identical and can be calculated from the generalized Green function (see Appendix). To improve the laser efficiency, the radiative power can be maximized in one direction by imposing a vertically asymmetric design. Detail discussion on this topic is beyond the scope of this paper, but it indicates that the arbitrary sidewall provides more freedom to optimize the performance of PC-SELs.

#### Tilt case

*ϕ*= 0° (Figs. 5(a) and 6(a)), the radiation constant of mode B increases more rapidly than that for mode A, and the radiation discrimination significantly changes with the tilt angle. As we have explained, the symmetric axes of modes A and B are

*Y*=

*X*and

*Y*= −

*X*and the tilt

*ϕ*= 0° breaks the in-plane symmetry for both the symmetric axes. The symmetry breaking contributes radiations to modes A and B simultaneously, but the contribution differs with the tilt angle. As a result, the radiation constant curves diverge strongly for the CC case shown in Fig. 5(a) where the radiation is contributed entirely by the tilted-sidewall-induced asymmetries. One the other hand, in the RIT case in Fig. 6(a), the in-plane asymmetries induced by the asymmetric air holes shape and the tilted sidewall in the

*ϕ*= 0° direction cannot be combined linearly, and hence the radiation discrimination changes significantly with the tilt angles

*ϕ*= 135°, i.e., the

*Y*= −

*X*direction. Fig. 5(b) shows that the curves are degenerate, and Fig. 6(b) shows that, the gap between two curves is maintained at all tilt angles, which indicates that radiation discrimination is determined only by the in-plane air-hole geometry and not by the tilted sidewall. This can be explained as follows. When asymmetric air holes and tilted sidewall break the symmetries along the two orthogonal symmetric axes of modes A and B, radiative waves with two normal polarizations are generated. Hence, radiative waves with normal polarization contribute to the total radiation power orthogonally and independently, and can be linearly combined. As a result, the use of in-plane asymmetric air holes and vertically tilted sidewalls can help enhance the radiation power without affecting the mode selectivity performance. As shown in Fig. 6(b), the radiation at a moderate tilt angle can be much higher than that in the ideal vertical case, which is a very promising feature for high-power PC-SELs devices.

#### Discussions

*xy*plane due to the boundary conditions of finite PC cavity. As a result, the finite structure gives rise to more complicate near-filed distributions, and hence, leads to a variety of far-field patterns through Fraunhofer diffraction [8

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

30. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for square-lattice photonic-crystal lasers with TE polarizaiton — finite-size effects,” Optics Express. (in preparation). [PubMed]

### 3.3. Fabrication method of tilted air-hole

31. S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, “Direct creation of three-dimensional photonic crystals by a top-down approach,” Nature Materials **8**, 721–725 (2009). [CrossRef] [PubMed]

## 4. Conclusion

22. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **45**, 788–795 (2010). [CrossRef]

30. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for square-lattice photonic-crystal lasers with TE polarizaiton — finite-size effects,” Optics Express. (in preparation). [PubMed]

## Appendix: Analytical solution for the Green function

*h*

_{−N},

*h*

_{−N+1},...,

*h*

_{−1},

*h*

_{1},

*h*

_{2},

*h*, where

_{N}*h*is the interface between the layers, and the PC layer ∈ [

_{i}*h*

_{1},

*h*

_{−1}]. Assume that

*z*′ ∈ [

*h*

_{1},

*h*

_{−1}] lies in the PC layer; for

*z*>

*z*′, the Green function

*G*(

*z,z*′) can be written as where

*G*(

*z,z*′)

*∂G*(

*z, z*′)/

*∂z*should be continuous across the interface, so that

*A*= 0. This would lead to For

_{N}*z*<

*z*′, we notate the amplitude as

*C*,

*D*instead of

*A,B*, and the Green function

*G*(

*z,z*′) becomes Similarly, we have Consider

*D*

_{−N}= 0, and thus,

*G*(

*z,z*′) should ensure continuity at

*z*=

*z*′ and the “jump condition”

*G*′(

*z*′

_{+0},

*z*′) −

*G*′(

*z*′

_{−0},

*z*′) = 1 Finally, and Here

*κ*

_{p1}and

*κ*

_{p2}present the reflection induced by the refractive index contrast between individual layers. When the reflections are negligible, i.e.,

*κ*

_{p}_{1},

*κ*

_{p}_{2}→ 0, above Green function reduces to the infinite homogeneous space form

## Acknowledgments

## References and links

1. | M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. |

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

3. | 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 |

4. | M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B. |

5. | G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel, “Operating characteristics of a semiconducting polymer laser pumped by a microchip laser,” Appl. Phys. Lett. |

6. | Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. |

7. | D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Optics Express. |

8. | E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature , |

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, and J. Faist, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Optics 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 , |

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

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

14. | 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. |

15. | M. Kamp, “Photonic crystal lasers: On-chip beam steering,” Nature Photonics, News and Views , |

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

17. | H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. |

18. | W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers” IEEE J. Quantum Electron. |

19. | R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. |

20. | M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. |

21. | 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. |

22. | K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. |

23. | Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with TE polarization — a general approach,” Phy. Rev. B. (submitted), http://arxiv.org/abs/1107.1772. |

24. | S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Air-hole design in a vertical direction for high-power two-dimensional photonic-crystal surface-emitting lasers,” JOSA B . |

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

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

27. | Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Optics Express |

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

29. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications |

30. | Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for square-lattice photonic-crystal lasers with TE polarizaiton — finite-size effects,” Optics Express. (in preparation). [PubMed] |

31. | S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, “Direct creation of three-dimensional photonic crystals by a top-down approach,” Nature Materials |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(160.5298) Materials : Photonic crystals

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 12, 2011

Revised Manuscript: September 8, 2011

Manuscript Accepted: September 13, 2011

Published: November 17, 2011

**Citation**

Chao Peng, Yong Liang, Kyosuke Sakai, Seita Iwahashi, and Susumu Noda, "Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls," Opt. Express **19**, 24672-24686 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24672

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