## Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects |

Optics Express, Vol. 20, Issue 14, pp. 15945-15961 (2012)

http://dx.doi.org/10.1364/OE.20.015945

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### Abstract

We develop a coupled-wave model that is capable of treating finite-size square-lattice photonic crystal surface emitting lasers with transverse-electric polarization. Various properties of interest including threshold gain, mode frequency, field intensity envelope within the device, far-field pattern, as well as polarization and divergence angle of the output beam for the band-edge modes are calculated. Theoretical predictions of the lowest threshold mode and the output beam profile are in good agreement with our experimental findings. In particular, we show that, contrary to the infinite periodic case, the finite length of the device significantly affects surface emission and mode selection properties of the laser device.

© 2012 OSA

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(160.5298) Materials : Photonic crystals

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 6, 2012

Revised Manuscript: May 22, 2012

Manuscript Accepted: June 10, 2012

Published: June 28, 2012

**Citation**

Yong Liang, Chao Peng, Kyosuke Sakai, Seita Iwahashi, and Susumu 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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15945

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### References

- 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]
- 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,” Science293, 1123–1125 (2001). [CrossRef] [PubMed]
- M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002). [CrossRef]
- I. 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]
- D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express12, 1562–1568 (2004). [CrossRef] [PubMed]
- E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature441, 946 (2006). [CrossRef] [PubMed]
- 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]
- G. Xu, Y. Chassagneux, R. Colombelli, G. Beaudoin, and I. Sagnes, “Polarized single-lobed surface emission in mid-infrared, photonic-crystal, quantum-cascade lasers,” Opt. Lett.35, 859 (2010). [CrossRef] [PubMed]
- L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, and J. Faist, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express16, 5206–5217 (2008). [CrossRef] [PubMed]
- 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,” Nature457, 174–178 (2009). [CrossRef] [PubMed]
- L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photon. Rev.5, 647–658 (2011).
- H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008). [CrossRef]
- Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010). [CrossRef]
- M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B44, 8565–8571 (1991). [CrossRef]
- S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002). [CrossRef]
- 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. B68, 045209 (2003). [CrossRef]
- M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005). [CrossRef] [PubMed]
- H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972). [CrossRef]
- M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662 (1992). [CrossRef]
- 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]
- K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010). [CrossRef]
- 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. B84, 195119 (2011). [CrossRef]
- 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 Commun.23, 1335–1340 (2005). [CrossRef]
- W. Kunishi, D. Ohnishi, E. Miyai, K. Sakai, and S. Noda, “High-power single-lobed surface-emitting photonic-crystal laser,” Conference on Lasers and Electro-Optics (CLEO), CMKK1, Long Beach, May, 2006.
- W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron.13, 134–141 (1977). [CrossRef]
- 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. Express19, 24672–24686 (2011). [CrossRef] [PubMed]
- M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlGl-1N/InxG1-xN laser diodes,” J. Appl. Phys.84, 1196–1203 (1998). [CrossRef]
- D. H. Sheen, K. Tuncay, C. E. Baag, and P. J. Ortoleva, “Parallel implemantation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation,” Comput. Geosci.32, 1182–1191 (2006). [CrossRef]
- E. Miyai and S. Noda, “Phase-shift effect on a two-dimensional surface-emitting photonic-crystal laser,” Appl. Phys. Lett.86, 111113 (2005). [CrossRef]
- A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).
- Fundamentally, radiation fields emitted from the center of the laser cavity have similar properties to those emitted from an infinite periodic structure described in Ref. [22]. Unlike CC air holes, Fourier coefficients (ξm,n) of the dielectric function ε(r) for ET air holes are complex numbers. Therefore, the radiation field intensity is proportional to |ξ−1,0Rx + ξ1,0Sx|2 [see Eq. (A4) in Appendix A and Eq. (B11) in Appendix B]. These complex Fourier coefficient terms multiplied to basic waves may change the phase difference of the waves diffracted vertically, resulting in a suppression of the destructive interference.
- H. A. Haus, “Gain saturation in distributed feedback lasers,” Appl. Opt.14, 2650–2652 (1975). [CrossRef] [PubMed]
- S. H. Macomber, “Nonlinear analysis of surface-emitting distributed feedback lasers,” IEEE J. Quantum Electron.26, 2065–2074 (1990). [CrossRef]

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