Analysis of two-dimensional photonic crystal with anisotropic gain

Shinichi Takigawa; Susumu Noda

doi:10.1364/OE.19.009475

Analysis of two-dimensional photonic crystal with anisotropic gain

Shinichi Takigawa and Susumu Noda

Optics Express
Vol. 19,
Issue 10,
pp. 9475-9491
(2011)
•https://doi.org/10.1364/OE.19.009475

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

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (1658 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Photonic Crystals and Devices
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Photonic crystals (050.5298)
- Laser theory (140.3430)

About this Article
History
- Original Manuscript: April 4, 2011
- Manuscript Accepted: April 20, 2011
- Published: April 29, 2011

Accessible

Open Access

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.

Full Article | PDF Article

OSA Recommended Articles

Mode analysis of two-dimensional photonic crystal terahertz lasers with gain/loss dispersion characteristics

Shinichi Takigawa and Susumu Noda
J. Opt. Soc. Am. B 27(12) 2556-2567 (2010)

Symmetry properties of two-dimensional anisotropic photonic crystals

G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen
J. Opt. Soc. Am. A 23(8) 2002-2013 (2006)

Engineering the bandgap of a two-dimensional anisotropic photonic crystal

G. Alagappan, X. W. Sun, P. Shum, and M. B. Yu
J. Opt. Soc. Am. B 23(7) 1478-1483 (2006)

References

View by:
Article Order
|
Year
|
Author
|
Publication

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]
M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]
R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref] [PubMed]
K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Comm. 23(7), 1335–1340 (2005).
[Crossref]
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. 101(8), 081726 (2007).
[Crossref]
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 16(8), 5206–5217 (2008).
[Crossref] [PubMed]
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]
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]
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]
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]
S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001).
[Crossref]
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]
V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60(8), 5359–5365 (1999).
[Crossref]
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]
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]
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]
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]
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]
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]
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]
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]
T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).
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]
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]
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]
T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in m-plane InxGa1-xN/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. 93(9), 091112 (2008).
[Crossref]
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]
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]
A. Yariv, Introduction to Optical Electronics (Holts, Rinehart and Winston. Inc, 1974).
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]
M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998).
[Crossref]
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]
K. Inoue and K. Ohtaka, Photonic Crystals (Springer-Verlag, 2004).
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]

2010 (3)

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]

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]

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]

2009 (4)

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]

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]

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]

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]

2008 (6)

T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in m-plane InxGa1-xN/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. 93(9), 091112 (2008).
[Crossref]

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]

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]

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 16(8), 5206–5217 (2008).
[Crossref] [PubMed]

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]

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]

2007 (5)

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]

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]

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]

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. 101(8), 081726 (2007).
[Crossref]

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]

2005 (1)

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Comm. 23(7), 1335–1340 (2005).
[Crossref]

2003 (2)

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref] [PubMed]

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]

2002 (1)

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

2001 (1)

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

2000 (1)

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]

1999 (1)

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

1998 (2)

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]

M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998).
[Crossref]

1997 (2)

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]

T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

1994 (1)

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]

1991 (1)

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]

1985 (1)

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]

Abram, R. A.

Adachi, M.

Adachi, S.

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]

Akita, K.

Amanti, M. I.

Apostolopoulos, V.

Asano, T.

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]

Beere, H.

Beere, H. E.

Bergmann, M. J.

M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998).
[Crossref]

Biswas, R.

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]

Brand, S.

Capasso, F.

Casey, H. C.

M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998).
[Crossref]

Chan, C. T.

Chen, X.

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]

Chichibu, S. F.

Cho, A. Y.

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

Cohen, D. A.

Colombelli, R.

DenBaars, S. P.

Dunbar, L. A.

El-Kady, I.

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]

Enya, Y.

Faist, J.

Farrell, R. M.

Fleury, B.

Freeman, J. R.

Fujita, M.

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]

Fujito, K.

Fuke, S.

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]

Giovannini, M.

Gmachl, C. F.

Haeger, D. A.

Hangleiter, A.

Henry, C. H.

Ho, K. H.

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]

Ho, K. M.

Houdre, R.

Houdré, R.

Hoyler, N.

Hsu, P. S.

Hu, E.

Ichihara, J.

Ikegami, T.

Imada, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

Jiang, D. S.

Jiang, X.

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]

Jianglin, Y.

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]

Kaliteevski, M. A.

Kashiwagi, J.

Katayama, K.

Kawashima, T.

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]

Kazarinov, R. F.

Kelchner, K. M.

Kitagawa, H.

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]

Kubota, M.

Kuroda, T.

T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

Kyono, T.

Lin, Y.-D.

Liu, Z. S.

Maradudin, A. A.

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]

Marshall, O. P.

Matioli, E.

Matsubara, H.

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]

Melo, T.

Miyai, E.

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]

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]

Mochizuki, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

Modinos, A.

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

Nakamura, S.

Nakamura, T.

Noda, S.

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]

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]

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]

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]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

Nojima, S.

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

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]

Ohnishi, D.

Ohta, H.

Ohtoshi, T.

T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

Ohtsuka, K.

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]

Okamoto, K.

Okano, T.

Onuma, T.

Painter, O.

Park, S.-H.

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]

Plihal, M.

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]

Rangel, E.

Ritchie, D.

Ritchie, D. A.

Rungsawang, R.

Sadowski, M. L.

Saito, H.

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]

Saito, M.

Sakaguchi, T.

Sakai, K.

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]

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]

Scalari, G.

Scheibenzuber, W.

Schwarz, U.

Sergent, A. M.

Sigalas, M. M.

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]

Sirigu, L.

Sivco, D. L.

Soukoulis, C. M.

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]

Speck, J.

Speck, J. S.

Srinivasan, K.

Stefanou, N.

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

Sumitomo, T.

Suto, T.

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]

Takasu, H.

Tanaka, T.

Tanaka, Y.

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]

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]

Tennant, D. M.

Terazzi, R.

Tokuyama, S.

Troccoli, M.

Tyagi, A.

Ueno, M.

Veprek, R.

Walther, C.

Weisbuch, C.

Witzigmann, B.

Yang, H.

Yannopapas, V.

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

Yao, P.

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]

Yoshikawa, H.

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]

Yoshimoto, S.

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]

Yoshizumi, Y.

Zhang, J.

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]

Zhang, L. Q.

Zhang, S. M.

Zhao, D.

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]

Zhao, D. G.

Zhu, J. J.

Appl. Phys. Express (5)

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]

Appl. Phys. Lett. (5)

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]

T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

IEEE J. Quantum Electron. (2)

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]

IEEE J. Sel. Areas Comm. (1)

J. Appl. Phys. (5)

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

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]

M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998).
[Crossref]

Jpn. J. Appl. Phys. (3)

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]

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]

Opt. Express (2)

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]

Phys. Rev. B (5)

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

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]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002).
[Crossref]

Phys. Rev. B Condens. Matter (2)

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]

Science (2)

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]

Other (2)

A. Yariv, Introduction to Optical Electronics (Holts, Rinehart and Winston. Inc, 1974).

K. Inoue and K. Ohtaka, Photonic Crystals (Springer-Verlag, 2004).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 Schematic diagram of a PC layer (period: a; radius:r). Here α and β are the　orthogonal gain directions of the PC material; k' is the wavevector for an expanded plane wave. The dashed line is parallel to a primitive transformation vector of PC. HL and BG represent the regions within or outside a circular cross-section.

Download Full Size | PPT Slide | PDF

Fig. 2 Cross-section of calculated model. AROG denotes “air holes retained over growth,” whose layers provide the PC configuration. HL and BG correspond the regions within or outside a circular cross-section in Fig. 1.

Download Full Size | PPT Slide | PDF

Fig. 3 Photonic band structure with four photonic modes, A, B, C, and D. Although calculation is undertaken for isotropic gain and anisotropic gain with θ = 0, 45, and 90 deg, no significant difference is observed in this axis range.

Download Full Size | PPT Slide | PDF

Fig. 4 Normalized frequencies of photonic modes at the Γ₂ point as a function of θ. Solid and dotted lines represent anisotropic and isotropic gains, respectively.

Download Full Size | PPT Slide | PDF

Fig. 5 Near-field distribution of photonic modes at Γ₂ point. Black circles represent the HL shape of the PC.

Download Full Size | PPT Slide | PDF

Fig. 6 Gain dispersion of photonic modes for (a) isotropic gain and anisotropic gain with (b) θ = 0, (c) 45, and (d) 90 deg.

Download Full Size | PPT Slide | PDF

Fig. 7 Gains of photonic modes at the Γ₂ point as a function of θ. Solid and dotted lines are anisotropic and isotropic gains, respectively.

Download Full Size | PPT Slide | PDF

Fig. 8 Near-field distribution of the anisotropic-gain photonic modes at Γ₂ point. Δθ is defined as Δθ = 45 deg - θ. Black circles represent the HL shape of the PC.

Download Full Size | PPT Slide | PDF

Fig. 9 Real and imaginary parts of B coefficients of expanded plane waves as a function of Δθ. (a), (b), (c), and (d) are A, B, C, and D modes, respectively. The parameters are reciprocal vectors, (G)₁ to (G)₄.

Download Full Size | PPT Slide | PDF

Fig. 10 Gains of photonic modes at the Γ₂ point as a function of θ with radiation loss. Solid and dotted lines are anisotropic and isotropic gains, respectively.

Download Full Size | PPT Slide | PDF

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

H_{3} (\bar{x}) = \sum_{G} h_{3} (k, G, \bar{x}) \cdot \exp (- i ω t),

h_{3} (k, G, \bar{x}) = B (k, G) \exp {i (k + G) \bar{x}},

1 / ε (k', \bar{x}) = \sum_{G'} \bar{κ} (k', G') \exp (i G' \cdot \bar{x}),

\bar{κ} (k', G') = {\begin{matrix} f / ε_{a} (k') + (1 - f) / ε_{b} (k') & (G' = 0), \\ 2 f [1 / ε_{a} (k') - 1 / ε_{b} (k')] J_{1} (| G' r |) / | G' r | & (G' \neq 0), \end{matrix}

\frac{\partial}{\partial x_{1}} {\frac{1}{ε (k + G, \bar{x})} \cdot \frac{\partial}{\partial x_{1}} h_{3} (k, G, \bar{x})} + \frac{\partial}{\partial x_{2}} {\frac{1}{ε (k + G, \bar{x})} \cdot \frac{\partial}{\partial x_{2}} h_{3} (k, G, \bar{x})} = - \frac{ω^{2}}{c^{2}} h_{3} (k, G, \bar{x}),

\frac{\partial}{\partial x_{1}} \sum_{G} {\frac{1}{ε (k + G, \bar{x})} \cdot \frac{\partial}{\partial x_{1}} h_{3} (k, G, \bar{x})} + \frac{\partial}{\partial x_{2}} \sum_{G} {\frac{1}{ε (k + G, \bar{x})} \cdot \frac{\partial}{\partial x_{2}} h_{3} (k, G, \bar{x})} = - \frac{ω^{2}}{c^{2}} \sum_{G} h_{3} (k, G, \bar{x}) .

\sum_{G} \sum_{G'} (k + G) \cdot (k + G') \bar{κ} (k + G', G - G') B (k, G') \exp {i (k + G) \bar{x}} = \frac{ω^{2}}{c^{2}} \sum_{G} h_{3} (k, G, \bar{x}) .

\sum_{G'} (k + G) \cdot (k + G') \bar{κ} (k + G', G - G') B (k, G') = (ω^{2} / c^{2}) B (k, G)

1 / n_{γ}^{2} = \cos^{2} φ / n_{α}^{2} + \sin^{2} φ / n_{β}^{2},

1 / n_{γ}^{2} = 1 / ε (k', \bar{x}) = \sum_{G'} \bar{κ} (k', G') \exp (i G' \cdot \bar{x}),

1 / n_{α}^{2} = 1 / ε_{α} (\bar{x}) = \sum_{G'} {\bar{κ}}_{α} (G') \exp (i G' \cdot \bar{x}),

1 / n_{β}^{2} = 1 / ε_{β} (\bar{x}) = \sum_{G'} {\bar{κ}}_{β} (G') \exp (i G' \cdot \bar{x}),

\bar{κ} (k', G') = {\bar{κ}}_{α} (G') \cos^{2} φ + {\bar{κ}}_{β} (G') \sin^{2} φ .

ω = ω_{r} + i \cdot ω_{i} .

H_{3} (\bar{x}) \approx \sum_{n = 1}^{4} B (k, G_{n}, ω) \cdot \exp i [(k + G_{n}) \bar{x} - ω t] .

H_{3} (\hat{x}) = \sum_{n = 1}^{4} B_{\bar{x}} (G_{n}) \exp [i (k + G_{n}) \bar{x}] φ (x_{3}) + Δ H (\hat{x}),

\partial^{2} Δ H (\hat{x}) / \partial x_{i} \partial x_{j} = \partial^{2} B_{\bar{x}} (G, ω) / \partial x_{i} \partial x_{j} = 0,

\partial^{2} φ (x_{3}) / \partial x_{3}^{2} = {(ω / c)}^{2} Δ ε (x_{3}) φ (x_{3}),

1 / ε (\hat{x}) = \sum_{G} \hat{κ} (G, x_{3}) \exp (i G \cdot \bar{x}),

\hat{κ} (G, x_{3}) = {\begin{matrix} \begin{matrix} {\bar{κ}}_{p} (G) & (g_{1} \leq x_{3} \leq g_{2}), \end{matrix} \\ \begin{matrix} 1 / ε (x_{3}) & (x_{3} < g_{1}, g_{2} \leq x_{3}, G = 0), \end{matrix} \\ \begin{matrix} 0 & (x_{3} < g_{1}, g_{2} \leq x_{3}, G \neq 0), \end{matrix} \end{matrix}

{\bar{κ}}_{p} (G) = {\begin{matrix} f / ε_{p a} + (1 - f) / ε_{p b} & (G = 0), \\ (1 / ε_{p a} - 1 / ε_{p b}) 2 f J_{1} (| G r |) / | G r | & (G \neq 0) . \end{matrix}

\frac{\partial}{\partial x_{1}} [\frac{1}{ε (\hat{x})} \frac{\partial H_{3} (\hat{x})}{\partial x_{1}}] + \frac{\partial}{\partial x_{2}} [\frac{1}{ε (\hat{x})} \frac{\partial H_{3} (\hat{x})}{\partial x_{2}}] + \frac{1}{ε (\hat{x})} \frac{\partial^{2} H_{3} (\hat{x})}{\partial x_{3}^{2}} = - \frac{ω^{2}}{c^{2}} H_{3} (\hat{x}) .

\begin{array}{l} \sum_{G} [\sum_{n = 1}^{4} \hat{κ} (G - G_{n}, x_{3}) {i (G + G_{n}) \cdot \bar{\nabla} - G \cdot G_{n} + {(ω / c)}^{2} Δ ε (x_{3})} B_{\bar{x}} (G_{n}) φ (x_{3}) \\ + \hat{κ} (G, x_{3}) (i G \cdot \bar{\nabla} + \partial^{2} / \partial x_{3}^{2}) Δ H (\hat{x})] \exp (i G \cdot \bar{x}) + {(ω / c)}^{2} [\sum_{n = 1}^{4} B_{\bar{x}} (G_{n}) φ (x_{3}) \exp (i G_{n} \cdot \bar{x}) + Δ H (\hat{x})] = 0, \end{array}

[\hat{κ} (0, x_{3}) \partial^{2} / \partial x_{3}^{2} + {(ω / c)}^{2}] Δ H (\hat{x}) = - \sum_{n = 1}^{4} \hat{κ} (G_{n}, x_{3}) [i G_{n} \cdot \bar{\nabla} + {(ω / c)}^{2} Δ ε (x_{3})] B_{\bar{x}} (G_{n}) φ (x_{3}) .

\begin{array}{l} Δ H (\hat{x}) = - \int_{- \infty}^{\infty} \sum_{n = 1}^{4} [\hat{κ} (G_{n}, t) / \hat{κ} (0, t)] [i G_{n} \cdot \bar{\nabla} + {(ω / c)}^{2} Δ ε (t)] B_{\bar{x}} (G_{n}) φ (t) \tilde{G} (x_{3}, t) d t \\ = - \sum_{n = 1}^{4} [{\bar{κ}}_{p} (G_{n}) / {\bar{κ}}_{p} (0)] [i G_{n} \cdot \bar{\nabla} + {(ω / c)}^{2} Δ ε_{p}] B_{\bar{x}} (G_{n}) \int_{g_{1}}^{g_{2}} φ (t) \tilde{G} (x_{3}, t) d t, \end{array}

\begin{array}{l} \sum_{n = 1}^{4} \hat{κ} (G_{m} - G_{n}, x_{3}) {i (G_{m} + G_{n}) \cdot \bar{\nabla} - G_{m} \cdot G_{n} + {(ω / c)}^{2} Δ ε (x_{3})} B_{\bar{x}} (G_{n}) φ (x_{3}) \\ + \hat{κ} (G_{m}, x_{3}) (i G_{m} \cdot \bar{\nabla} + \partial^{2} / \partial x_{3}^{2}) Δ H (\hat{x}) + {(ω / c)}^{2} B_{\bar{x}} (G_{m}) φ (x_{3}) = 0. \end{array}

\sum_{n = 1}^{4} [i (F (G_{n}, G_{m}) \cdot \bar{\nabla} + A (G_{n}, G_{m})] B_{\bar{x}} (G_{n}) + {(ω / c)}^{2} B_{\bar{x}} (G_{m}) = 0,

\begin{array}{l} F (G_{n}, G_{m}) = {\bar{κ}}_{p} (G_{m} - G_{n}) (G_{m} + G_{n}) \\ - [{\bar{κ}}_{p} (G_{m}) {\bar{κ}}_{p} (G_{n}) / {\bar{κ}}_{p} (0)] [ς {(ω / c)}^{2} Δ ε_{p} G_{m} + ξ G_{n}], \end{array}

\begin{array}{l} A (G_{n}, G_{m}) = {\bar{κ}}_{p} (G_{m} - G_{n}) [- G_{m} \cdot G_{n} + {(ω / c)}^{2} Δ ε_{p}] \\ - [{\bar{κ}}_{p} (G_{m}) {\bar{κ}}_{p} (G_{n}) / {\bar{κ}}_{p} (0)] ξ {(ω / c)}^{2} Δ ε_{p}, \end{array}

ς = (1 / Γ_{g}) \int_{g_{1}}^{g_{2}} \int_{g_{1}}^{g_{2}} φ (x_{3}) φ (t) \tilde{G} (x_{3}, t) d t d x_{3},

ξ = (1 / Γ_{g}) \int_{g_{1}}^{g_{2}} φ (x_{3}) [\frac{\partial^{2}}{\partial x_{3}^{2}} \int_{g_{1}}^{g_{2}} φ (t) \tilde{G} (x_{3}, t) d t] d x_{3} .

Γ_{g} = \int_{g_{1}}^{g_{2}} φ {(x_{3})}^{2} d x_{3} .

ς = [i K_{3} d_{g} + 1 - \exp (i K_{3} d_{g})] / i K_{3}^{3} d_{g},

ξ = i [1 - \exp (i K_{3} d_{g})] / K_{3} d_{g} .

i [\partial B_{x_{1}} (G_{1}) / \partial x_{1}] = - [(c_{m} \pm d_{m}) / (a_{m} \pm b_{m})] B_{x_{1}} (G_{1}),

i [\partial B_{x_{1}} (G_{2}) / \partial x_{1}] = [(c_{m} \pm d_{m}) / (a_{m} \pm b_{m})] B_{x_{1}} (G_{2}),

\begin{array}{l} Q_{s} = Im [Γ_{g} (c_{m} + d_{m}) / (a_{m} + b_{m})] \\ = Im {\frac{Γ_{g} | G_{m} |}{2} \cdot \frac{x^{2} [(1 + P_{2}) Δ ε + 1 / {\bar{κ}}_{p} (0) - 2 ξ P_{1}^{2} Δ ε] - (1 - P_{2})}{1 - ξ P_{1}^{2}}}, \end{array}

\begin{array}{l} Q_{a} = Im [Γ_{g} (c_{m} - d_{m}) / (a_{m} - b_{m})] \\ = Im {\frac{Γ_{g} | G_{m} |}{2} \cdot \frac{x^{2} [(1 - P_{2}) Δ ε + 1 / {\bar{κ}}_{p} (0)] - (1 + P_{2})}{1 - ς Δ ε {(x | G_{m} | P_{1})}^{2}}}, \end{array}

Q_{A} = Im {\frac{Γ_{g} | G_{m} |}{2} \cdot \frac{x^{2} [(1 + P_{2} + 2 P_{3}) Δ ε + 1 / {\bar{κ}}_{p} (0) - 4 ξ P_{1}^{2} Δ ε] - (1 - P_{2})}{1 + P_{3} - ξ P_{1}^{2}}},

Q_{B} = Im {\frac{Γ_{g} | G_{m} |}{2} \cdot \frac{x^{2} [(1 + P_{2} - 2 P_{3}) Δ ε - 1 / {\bar{κ}}_{p} (0)] - (1 - P_{2})}{1 + P_{3} - ξ P_{1}^{2}}},