## Analysis of two-dimensional photonic crystal with anisotropic gain |

Optics Express, Vol. 19, Issue 10, pp. 9475-9491 (2011)

http://dx.doi.org/10.1364/OE.19.009475

Acrobat PDF (1658 KB)

### Abstract

Photonic modes in a two-dimensional square-lattice photonic crystal (PC) with anisotropic gain are analyzed for the first time. A plane-wave expansion method is improved to include the gain, which depends on not only the position but also the propagation direction of each plane wave. The anisotropic gain varies the photonic band structure, the near-field distributions, and the gain dispersion curves through variation in PC symmetry. Low-threshold operation of a PC laser with anisotropic-gain material such as nonpolar InGaN requires that the direction of higher gain in the material aligns along the ΓX direction of the PC.

© 2011 OSA

## 1. Introduction

1. 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**(5862), 445–447 (2008). [CrossRef]

7. O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. **93**(17), 171112 (2008). [CrossRef]

8. E. Matioli, B. Fleury, E. Rangel, T. Melo, E. Hu, J. Speck, and C. Weisbuch, “High extraction efficiency GaN-based photonic crystal light-emitting diodes: comparison of extraction lengths between surface and embedded photonic crystals,” Appl. Phys. Express **3**(3), 032103 (2010). [CrossRef]

9. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**(23), 231114 (2007). [CrossRef]

10. S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. **37**(Part 2, No. 5B), L565–L567 (1998). [CrossRef]

11. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. **90**(2), 545–551 (2001). [CrossRef]

12. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter **49**(16), 11080–11087 (1994). [CrossRef] [PubMed]

13. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B **60**(8), 5359–5365 (1999). [CrossRef]

14. I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B **62**(23), 15299–15302 (2000). [CrossRef]

15. S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B **75**(3), 035102 (2007). [CrossRef]

16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **46**(5), 788–795 (2010). [CrossRef]

17. K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express **15**(7), 3981–3990 (2007). [CrossRef] [PubMed]

18. K. Okamoto, J. Kashiwagi, T. Tanaka, and M. Kubota, “Nonpolar m-plane InGaN multiple quantum well laser diodes with a lasing wavelength of 499.8nm,” Appl. Phys. Lett. **94**(7), 071105 (2009). [CrossRef]

19. A. Tyagi, Y.-D. Lin, D. A. Cohen, M. Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Stimulated emission at blue-green (480nm) and green (514nm) wavelengths from nonpolar (m-plane) and semipolar (11-22) InGaN multiple quantum well laser diode structures,” Appl. Phys. Express **1**, 091103 (2008). [CrossRef]

20. Y. Enya, Y. Yoshizumi, T. Kyono, K. Akita, M. Ueno, M. Adachi, T. Sumitomo, S. Tokuyama, T. Ikegami, K. Katayama, and T. Nakamura, “531 nm green lasing of InGaN based laser diodes on semipolar {20-21} free-standing GaN substrates,” Appl. Phys. Express **2**, 082101 (2009). [CrossRef]

21. P. S. Hsu, K. M. Kelchner, A. Tyagi, R. M. Farrell, D. A. Haeger, K. Fujito, H. Ohta, S. P. DenBaars, J. S. Speck, and S. Nakamura, “InGaN/GaN blue laser diode grown on semipolar (30-31) free-standing GaN substrates,” Appl. Phys. Express **3**(5), 052702 (2010). [CrossRef]

23. S.-H. Park, “Crystal orientation effects on many-body optical gain of wurtzite InGaN/GaN quantum well lasers,” Jpn. J. Appl. Phys. **42**(Part 2, No. 2B), L170–L172 (2003). [CrossRef]

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B **80**(11), 115320 (2009). [CrossRef]

*c*-axis possesses larger gain than that along the

*a*-axis [25

25. K. Okamoto, H. Ohta, S. F. Chichibu, J. Ichihara, and H. Takasu, “Continuous-wave operation of *m*-plane InGaN multiple quantum well laser diodes,” Jpn. J. Appl. Phys. **46**(9), L187–L189 (2007). [CrossRef]

26. T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in *m*-plane In_{x}Ga_{1-x}N/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. **93**(9), 091112 (2008). [CrossRef]

1. 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**(5862), 445–447 (2008). [CrossRef]

27. H. Kitagawa, T. Suto, M. Fujita, Y. Tanaka, T. Asano, and S. Noda, “Green photoluminescence from GaInN photonic crystals,” Appl. Phys. Express **1**, 032004 (2008). [CrossRef]

## 2. Calculation method

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter **44**(16), 8565–8571 (1991). [CrossRef] [PubMed]

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B **80**(11), 115320 (2009). [CrossRef]

*x*and

_{1}*x*as being in the PC plane, while coordinate

_{2}*x*is normal to the plane. The TE mode in the PC corresponds to so-called “H-polarization,” whose magnetic field is perpendicular to the PC planes. The magnetic field along the

_{3}*x*axis,

_{3}**k**is the wavevector of the EM wave,

**G**is a reciprocal vector of PC, and

*ω*is an angular frequency of the propagating EM wave. Here,

**G'**is another reciprocal vector of PC, and

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter **44**(16), 8565–8571 (1991). [CrossRef] [PubMed]

*f*,

*r*, and

*J*are the filling fraction, radius of circular region, and Bessel function, respectively. Since we here consider the anisotropic gain,

_{1}(x)**k'**-dependent relative dielectric constants of HL and BG regions, respectively. Since the PWE method assumes that the structure has infinite length with uniformity along the

*x*direction, we use the effective relative dielectric constants for each region as

_{3}*c*is velocity of light. By the summation of all reciprocal vectors, we can obtainBy using Eqs. (2) and (3), Eq. (6) is modified asIn order to satisfy Eq. (7) with arbitrary

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter **44**(16), 8565–8571 (1991). [CrossRef] [PubMed]

*n*, of the EM waves propagating along the γ direction (

_{γ}**k'**) in the PC plane, is expressed aswhere

*n*and

_{α}*n*are refractive indices of the EM waves propagating along the orthogonal directions, α and β, respectively; φ is the angle between the α and γ directions (see Fig. 1). The refractive indices are expanded as: where,

_{β}*θ*is defined as the angle between the α direction and a primitive transformation vector of PC along the ΓX direction. The

*g,*is calculated as

## 3. Calculation model

1. 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**(5862), 445–447 (2008). [CrossRef]

_{0.04}Ga

_{0.96}N lower cladding layer, a GaN (0.1 μm) or SiO

_{2}(0.01 μm)/Air (0.09 μm) AROG layer, a GaN lower optical confinement layer (0.15 μm), a double QW active layer (In

_{0.26}Ga

_{0.74}N well (0.003 μm) and In

_{0.16}Ga

_{0.84}N barrier (0.015 μm)), a GaN upper optical confinement layer (0.06 μm), an Al

_{0.18}Ga

_{0.82}N electron blocking layer (0.02 μm), an Al

_{0.06}Ga

_{0.94}N upper cladding layer (0.6 μm), a GaN contact layer (0.05 μm), and a gold electrode. The air/SiO

_{2}region in the AROG layer forms a square-aligned circular hole so that it corresponds to HL in Fig. 1, while the other region corresponds to BG. The period,

*a,*and

*r/a*ratio of the PC is 207 nm and 0.2, respectively. The PC cavity is set to be infinity in plane.

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B **80**(11), 115320 (2009). [CrossRef]

^{12}cm

^{−2}, although the gain peak wavelength is shifted to 504 nm in this study. The propagation direction with larger gain (maximum material gain of approximate 3500 cm

^{−1}) is set to the α direction, while that with the smaller (approximate 680cm

^{−1}) is set to the β direction. The dielectric constant of GaN is the square of refractive indices in [30

30. T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. **82**(7), 3528–3535 (1997). [CrossRef]

31. M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer Al_{x}Ga_{1-x}N/In_{x}Ga_{1-x}N laser diodes,” J. Appl. Phys. **84**(3), 1196–1203 (1998). [CrossRef]

_{2}and gold are from [31

31. M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer Al_{x}Ga_{1-x}N/In_{x}Ga_{1-x}N laser diodes,” J. Appl. Phys. **84**(3), 1196–1203 (1998). [CrossRef]

32. L. Q. Zhang, D. S. Jiang, J. J. Zhu, D. G. Zhao, Z. S. Liu, S. M. Zhang, and H. Yang, “Confinement factor and absorption loss of AlInGaN based laser diodes emitting from ultraviolet to green,” J. Appl. Phys. **105**(2), 023104 (2009). [CrossRef]

## 4. Calculation results

### 4.1 Photonic band structure

_{2}). Although we calculate the PBS for isotropic gain and anisotropic gain with different

*θ*directions, no significant difference is observed in the axis range in Fig. 3. The left ordinate of Fig. 3 represents the normalized frequency (

*ω*), while the right one represents the corresponding wavelength. The photonic band gap is approximately 2 × 10

_{r}a/2πc^{−4}in

*ω*at the Γ

_{r}a/2πc_{2}point. The PBS shows four photonic modes, designated A, B, C, and D. Four similar modes can be seen in the conventional square-lattice PC [1

**319**(5862), 445–447 (2008). [CrossRef]

### 4.2 Near- field distributions

### 4.3 Gain dispersion characteristics

*ω*) of the frequency of a photonic mode provides the dispersion curve as a function of wavevectors, the imaginary part (

_{r}*ω*) gives the dispersion curve of the gain. Here,

_{i}*ω*is normalized as

_{i}*ω*Fig. 6 shows gain dispersion characteristics for isotropic gain and anisotropic gain with

_{i}a/2πc.*θ*= 0, 45, and 90 deg. Each graph has four lines, which are the A, B, C, and D photonic modes. The left axis is the normalized imaginary frequency (

*ω*) and the right axis is the corresponding gain. This gain is not the threshold gain, but the net modal gain. In anisotropic gain, at

_{i}a/2πc*θ*= 0 deg (Fig. 6(b)), the A and D modes possess the highest gain of

*ω*= 4.8 × 10

_{i}a/2πc^{−5}, or 71 cm

^{−1}. On the other hand, the value of

*ω*is the smallest for the B and C modes, giving a gain of 11 cm

_{i}a/2πc^{−1}. Thus the A and D modes, unlike the B and C modes, are able to oscillate. At

*θ*= 90 deg (Fig. 6(d)), the larger gain in the B and C modes lead to their lasing operation. At

*θ*= 45 deg (Fig. 6(c)), the gain is not high compared with other value of

*θ*and has the same value as in the case of isotropic gain (Fig. 6(a)). This is because photonic modes consist of plane waves whose directions are parallel or perpendicular only to the ΓX direction (see Sec. 5.1). Thus, when the direction of larger gain in the PC material is parallel or perpendicular to the ΓX direction, the directions of material gain and the plane waves coincide, so that the gains of corresponding photonic modes become large. When the direction of larger gain in the material is parallel to the ΓM direction, they do not coincide, so that the gains of photonic modes become small. This means that the lasing action is difficult, when

*θ*is 45 deg.

*θ*is summarized in Fig. 7 . For anisotropic gain, as

*θ*increases, the gain of A and D modes decreases, while the B and C modes obtain higher gain. This is because the resonant direction of A and D (B and C) modes become far from (close to) the direction of higher gain in the PC material. At

*θ*= 45 deg, the gain contribution to each mode is the same, so that the crossover between A-D and B-C modes occurs. Over the

*θ*= 45 deg, the gains of B and C modes are larger than those of A and D modes. Therefore, when

*θ*varies from 0 deg to 90 deg, the PC laser with anisotropic gain should function as follows: first, strong emission with the resonant direction parallel to the ΓX direction, then weak emission around

*θ*= 45 deg, and finally strong emission with the resonant direction perpendicular to the ΓX direction.

## 5. Discussions

### 5.1 PC symmetry change by anisotropic gain

*θ*= 45 deg in more detail. Here,

*Δθ*is defined as 45 minus

*θ*in units of degree.

_{2}point is shown in Fig. 8 as a function of

*Δθ*. The fields ofC and D modes are very sensitive to the

*Δθ*. At

*Δθ*= 10

^{−4}deg, the near-field distributions of the C and D modes are almost identical to those at

*θ*= 45 deg. At

*Δθ*= 10

^{−3}deg, the strong field portions of C and D modes affect the horizontal and vertical neighboring strong portions, respectively. At

*Δθ*= 10

^{−2}deg, the field distribution is similar to that at

*θ*= 0 or 90 deg. In contrast, such small

*Δθ*results in little variation for the A and B modes. At

*Δθ*= 0.8 deg, the near-field distributions of the A and B modes remain almost identical to those at

*θ*= 45 deg. At

*Δθ*= 1 deg, the strong field regions of A and B modes affect the vertical and horizontal neighboring strong regions, respectively. At

*Δθ*= 2 deg, the field distributions of A and B modes are similar to those at

*θ*= 0 or 90 deg. In addition, Fig. 8 clearly indicates that the A, B, C, and D photonic modes originate from the monopole, quadrupole, symmetric dipole, and asymmetric dipole modes, respectively.

_{2}point, i.e., Figure 9 shows the real and imaginary parts of the B-coefficients,

_{2}point, as a function of

*Δθ*. When

*Δθ*is less than 10

^{−4}deg, each mode consists of four reciprocal vectors,

**G**

_{1}to

**G**

_{4}. These four vectors cause 2D resonances, as shown in Fig. 8. When

*Δθ*increases from 10

^{−4}to 10

^{−2}deg, the amplitudes of the B-coefficients of

**G**

_{1}and

**G**

_{4}(

**G**

_{2}and

**G**

_{3}) reduce in C (D) modes monotonically, leading to 1D resonances. When

*Δθ*is around 0.8 deg, the real part of the B-coefficients of

**G**

_{2}and

**G**

_{3}(

**G**

_{1}and

**G**

_{4}) in A (B) modes decrease, while, in turn, the imaginary parts appear and fade away. This indicates that, when the resonances of A and B modes change from 2D to 1D, the phase of the B-coefficients of

**G**

_{2}and

**G**

_{3}(

**G**

_{1}and

**G**

_{4}) vary toward π/2.

*θ*≠ 45 deg have 2-fold rotational symmetry. Note that anisotropic gain with

*θ*= 45 deg produces 4-fold rotational symmetry despite there being geometrical 2-fold symmetry, because

*x*and

_{1}*x*components such as (1,1)2π/

_{2}*a*. In 2-fold rotational symmetry, the resonance occurs in 1D only. Therefore the resonance dimension changes around

*θ*= 45 deg, which causes an abrupt change in the resonance frequency and the near-field distribution. In addition, since the A and B modes inherently possess a symmetric field distribution with respect to the hole shape as shown in Fig. 5, these modes are relatively robust to disturbance by the anisotropic gain. On the other hand, the C and D modes have an inherently asymmetric distribution with respect to the hole shape, and thus they obtain an asymmetric nature easily caused by the anisotropic gain. Such difference explains the sensitivity of the field or B-coefficient variation to

*Δθ*variation, as shown in Figs. 8 and 9.

### 5.2 Radiation loss effects

^{−1}for C and D modes, while there is no loss for A and B modes. Taking these losses into consideration, the dependence of gain on

*θ*is calculated as shown in Fig. 10 . Negative gain indicates loss. Compared to Fig. 7, gains of C and D modes decrease drastically, since they suffer from the radiation loss. At

*θ*= 0 (90) deg, the gain of A and D (B and C) modes separates so that the PC laser will oscillate in a single photonic mode of A (B). In contrast, the no gain difference between A and B modes is observed for the isotropic gain, e.g., c-plane case, which leads to dual mode operation. Thus, Fig. 10 indicates that it is important to consider anisotropic gain when characterizing surface-emitting performance.

*H*fields of C and D give symmetric electric fields. Since the symmetric electric fields interfere constructively in the PC plane, the radiation becomes strong. On the other hand, symmetric

_{3}*H*fields of A and B forms an asymmetric electric field in the PC plane, leading to destructive interference and weak radiation. Sakai et al. studied a square-lattice PC laser and showed that each PC mode suffers from this radiation loss [16

_{3}16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **46**(5), 788–795 (2010). [CrossRef]

16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **46**(5), 788–795 (2010). [CrossRef]

## 6. Conclusions

## Appendix

34. 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**(2), 144–150 (1985). [CrossRef]

*H*in a square-lattice PC can be expressed as:

_{3}*x*is the axis normal to the PC plane;

_{3}*i*,

*j*= 1, 2. In Eq. (a.1),

*x*and is given by:

_{3}*x*, which is in-plane averaged for the PC layer. The inverse of the dielectric constant,

_{3}*g*and

_{1}*g*are defined as the

_{2}*x*range of the PC layer. From Eqs. (a.5) and (a.6), we can derive

_{3}34. 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**(2), 144–150 (1985). [CrossRef]

*x*>

_{3}*t*, negative sign for

*x*<

_{3}*t*); and

**G**

_{m}give

*d*=|

_{g}*g*-

_{2}*g*|.

_{1}**G**

_{1}and

**G**

_{2}reciprocal vectors, where

*x*only, so that we can use

_{1}*H*fields, we can set

_{3}*H*fields, respectively;

_{3}*a*and

_{m}*b*are

_{m}*x*components of

_{1}**G**

_{1}(

**G**

_{2}) component in Eq. (a.1) at the

*x*(−

_{1}*x*) direction. Note that

_{1}*H*fields,

_{3}*Q*and

_{s}*Q*, respectively, can be obtained as

_{a}*Q*and

_{A}*Q*, respectively, are as follows;

_{B}*Q*, and

_{C}*Q*, respectively, is equal to the radiation loss of the 1D asymmetric resonance mode,

_{D}*Q*. Note that the radiation loss, Eqs. (a.22) to (a.25), should be zero in the case of a negative value, because the radiation does not result in gain.

_{a}## Acknowledgments

## References and links

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5. | G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. |

6. | L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express |

7. | O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. |

8. | E. Matioli, B. Fleury, E. Rangel, T. Melo, E. Hu, J. Speck, and C. Weisbuch, “High extraction efficiency GaN-based photonic crystal light-emitting diodes: comparison of extraction lengths between surface and embedded photonic crystals,” Appl. Phys. Express |

9. | D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. |

10. | S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. |

11. | S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. |

12. | M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter |

13. | V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B |

14. | I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B |

15. | S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B |

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

17. | K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express |

18. | K. Okamoto, J. Kashiwagi, T. Tanaka, and M. Kubota, “Nonpolar m-plane InGaN multiple quantum well laser diodes with a lasing wavelength of 499.8nm,” Appl. Phys. Lett. |

19. | A. Tyagi, Y.-D. Lin, D. A. Cohen, M. Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Stimulated emission at blue-green (480nm) and green (514nm) wavelengths from nonpolar (m-plane) and semipolar (11-22) InGaN multiple quantum well laser diode structures,” Appl. Phys. Express |

20. | Y. Enya, Y. Yoshizumi, T. Kyono, K. Akita, M. Ueno, M. Adachi, T. Sumitomo, S. Tokuyama, T. Ikegami, K. Katayama, and T. Nakamura, “531 nm green lasing of InGaN based laser diodes on semipolar {20-21} free-standing GaN substrates,” Appl. Phys. Express |

21. | P. S. Hsu, K. M. Kelchner, A. Tyagi, R. M. Farrell, D. A. Haeger, K. Fujito, H. Ohta, S. P. DenBaars, J. S. Speck, and S. Nakamura, “InGaN/GaN blue laser diode grown on semipolar (30-31) free-standing GaN substrates,” Appl. Phys. Express |

22. | T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. |

23. | S.-H. Park, “Crystal orientation effects on many-body optical gain of wurtzite InGaN/GaN quantum well lasers,” Jpn. J. Appl. Phys. |

24. | W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B |

25. | K. Okamoto, H. Ohta, S. F. Chichibu, J. Ichihara, and H. Takasu, “Continuous-wave operation of |

26. | T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in |

27. | H. Kitagawa, T. Suto, M. Fujita, Y. Tanaka, T. Asano, and S. Noda, “Green photoluminescence from GaInN photonic crystals,” Appl. Phys. Express |

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

29. | A. Yariv, |

30. | T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. |

31. | M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer Al |

32. | L. Q. Zhang, D. S. Jiang, J. J. Zhu, D. G. Zhao, Z. S. Liu, S. M. Zhang, and H. Yang, “Confinement factor and absorption loss of AlInGaN based laser diodes emitting from ultraviolet to green,” J. Appl. Phys. |

33. | K. Inoue and K. Ohtaka, |

34. | 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. |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 4, 2011

Manuscript Accepted: April 20, 2011

Published: April 29, 2011

**Citation**

Shinichi Takigawa and Susumu Noda, "Analysis of two-dimensional photonic crystal with anisotropic gain," Opt. Express **19**, 9475-9491 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9475

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### References

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