Abstract

Using the three-dimensional (3D) finite-difference time-domain (FDTD) method, we have investigated in detail the optical properties of a two-dimensional (2D) photonic crystal (PC) surface-emitting laser having a square-lattice structure. In this study we perform the 3D-FDTD calculation for the structure of an actual fabricated device. The device is based on band-edge resonance, and four band edges are present at the corresponding band edge point. For these band edges, we calculate the quality (Q) factor. The results show that the Q factor of a resonant mode labeled A1 is larger than that of other resonant modes; that is, lasing occurs easily in mode A1. The device can thus achieve single-mode lasing oscillation. To increase the Q factor, we also consider the optimization of device parameters. The results provide important guidelines for device fabrication.

© 2004 Optical Society of America

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References

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  1. E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, �??Two-Dimensional Photonic Band-Gap Defect Mode Laser,�?? Science 284, 1819-1821 (1999).
    [CrossRef] [PubMed]
  3. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, �??Full Three-Dimensional Photonic Crystals at Near-Infrared Wavelengths,�?? Science 289, 604-606 (2000).
    [CrossRef] [PubMed]
  4. S. Noda, M. Imada, and A. Chutinan, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608-610 (2000).
    [CrossRef] [PubMed]
  5. B. S. Song, S. Noda and T. Asano, �??Photonic devices based on in-plane hetero photonic crystals,�?? Science 300, 1537 (2003).
    [CrossRef] [PubMed]
  6. S. Noda and T. Baba, Eds., Roadmap on Photonic Crystals, (Kluwer Academic, New York, 2003).
  7. 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]
  8. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, �??Multidirectionally distributed feedback photonic crystal lasers,�?? Phys. Rev. B 65, 195306 (2002).
    [CrossRef]
  9. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, �??Polarization mode control of twodimensional photonic crystal laser by unit cell structure design,�?? Science 293, 1123-1125 (2001).
    [CrossRef] [PubMed]
  10. M. Yokoyama and S. Noda, �??Polarization mode control of two-dimensional photonic crystal laser having a square lattice structure,�?? IEEE J. Quantum Electron. 39, 1074-1080 (2003).
    [CrossRef]
  11. M. Yokoyama and S. Noda, �??Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,�?? IEICE Trans. Electron. E87-C, 386-392 (2004).
  12. 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. 74, 7-9 (1999).
    [CrossRef]
  13. K. Inoue, M. Sasada, J. Kawamata, K. Sakoda and J. W. Haus, �??A Two-Dimensional Photonic Crystal Laser,�?? Jpn. J. Appl. Phys. 38, L157-L159 (1999).
    [CrossRef]
  14. M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel and O. Nalamasu, �??Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,�?? J. Appl. Phys. 86, 3502-3507 (1999).
    [CrossRef]
  15. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. 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, 1374-1377 (2003).
    [CrossRef] [PubMed]
  16. K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D.M. Tennant, A.M. Sergent, D.L. Sivco, A.Y. Cho, M. Troccoli, and F. Capasso, �??Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,�?? App. Phys. Lett. 84, 4164-4166 (2004).
    [CrossRef]
  17. K. S. Yee, �??Numerical Solution of Initial Boundary Value Problem Involving Maxwell�??s Equations in Isotropic Media,�?? in Proceedings of IEEE Conference on Antennas and Propagat. AP-14 (Institute of Electrical and Electronics Engineers, New York, 1966), pp. 302-307.
  18. M. Okano and S. Noda, �??Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,�?? Phys. Rev. B 70, 125105 (2004).
    [CrossRef]
  19. M. Plihal, A. Shambrook, and A. A. Maradudin, �??Two-dimensional photonic band structures,�?? Opt. Comm. 80, 199-204 (1991).
    [CrossRef]
  20. K. Sakoda, �??Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,�?? Phys. Rev. B 52, 7982-7986 (1995).
    [CrossRef]
  21. K. Sakoda, Optical Properties of Photonic Crystals, (Springer Verlag, Berlin, 2001).
  22. G. Mur, �??Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,�?? in Proceedings of IEEE Conference on Electromagn. Compat. EMC-23 (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 377-382.
  23. O. J. Painter, J. Vuckovic, and A. Scherer, �??Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,�?? J. Opt. Soc. Am. B 16, 275-285 (1999).
    [CrossRef]
  24. 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. Express 12, 1562-1568 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562</a>.
    [CrossRef] [PubMed]
  25. T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107 (2001).
    [CrossRef]
  26. S. Fan and J. D. Joannopoulos, �??Analysis of guided resonances in photonic crystal slabs,�?? Phys. Rev. B 65, 235112 (2002).
    [CrossRef]
  27. M. Imada, S. Noda, H. Kobayashi, and G. Sasaki, �??Characterization of a Distributed Feedback Laser with Air/Semiconductor Gratings Embedded by the Wafer Fusion Technique,�?? IEEE J. Quantum Electron. 35, 1277-1283 (1999).
    [CrossRef]

App. Phys. Lett. (1)

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D.M. Tennant, A.M. Sergent, D.L. Sivco, A.Y. Cho, M. Troccoli, and F. Capasso, �??Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,�?? App. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Appl. Phys Lett. (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]

Appl. Phys. Lett. (1)

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. 74, 7-9 (1999).
[CrossRef]

IEEE Conf. on Antennas and Propagat. '96 (1)

K. S. Yee, �??Numerical Solution of Initial Boundary Value Problem Involving Maxwell�??s Equations in Isotropic Media,�?? in Proceedings of IEEE Conference on Antennas and Propagat. AP-14 (Institute of Electrical and Electronics Engineers, New York, 1966), pp. 302-307.

IEEE Conf. on Electromagn. Compat. 1981 (1)

G. Mur, �??Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,�?? in Proceedings of IEEE Conference on Electromagn. Compat. EMC-23 (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 377-382.

IEEE J. Quantum Electron. (2)

M. Imada, S. Noda, H. Kobayashi, and G. Sasaki, �??Characterization of a Distributed Feedback Laser with Air/Semiconductor Gratings Embedded by the Wafer Fusion Technique,�?? IEEE J. Quantum Electron. 35, 1277-1283 (1999).
[CrossRef]

M. Yokoyama and S. Noda, �??Polarization mode control of two-dimensional photonic crystal laser having a square lattice structure,�?? IEEE J. Quantum Electron. 39, 1074-1080 (2003).
[CrossRef]

IEICE Trans. Electron. (1)

M. Yokoyama and S. Noda, �??Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,�?? IEICE Trans. Electron. E87-C, 386-392 (2004).

J. Appl. Phys. (1)

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel and O. Nalamasu, �??Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,�?? J. Appl. Phys. 86, 3502-3507 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

K. Inoue, M. Sasada, J. Kawamata, K. Sakoda and J. W. Haus, �??A Two-Dimensional Photonic Crystal Laser,�?? Jpn. J. Appl. Phys. 38, L157-L159 (1999).
[CrossRef]

Nature (1)

S. Noda, M. Imada, and A. Chutinan, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Opt. Comm. (1)

M. Plihal, A. Shambrook, and A. A. Maradudin, �??Two-dimensional photonic band structures,�?? Opt. Comm. 80, 199-204 (1991).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (5)

T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107 (2001).
[CrossRef]

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

K. Sakoda, �??Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,�?? Phys. Rev. B 52, 7982-7986 (1995).
[CrossRef]

M. Okano and S. Noda, �??Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,�?? Phys. Rev. B 70, 125105 (2004).
[CrossRef]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, �??Multidirectionally distributed feedback photonic crystal lasers,�?? Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Science (5)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, �??Two-Dimensional Photonic Band-Gap Defect Mode Laser,�?? Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, �??Full Three-Dimensional Photonic Crystals at Near-Infrared Wavelengths,�?? Science 289, 604-606 (2000).
[CrossRef] [PubMed]

B. S. Song, S. Noda and T. Asano, �??Photonic devices based on in-plane hetero photonic crystals,�?? Science 300, 1537 (2003).
[CrossRef] [PubMed]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, �??Polarization mode control of twodimensional photonic crystal laser by unit cell structure design,�?? Science 293, 1123-1125 (2001).
[CrossRef] [PubMed]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. 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, 1374-1377 (2003).
[CrossRef] [PubMed]

Other (2)

K. Sakoda, Optical Properties of Photonic Crystals, (Springer Verlag, Berlin, 2001).

S. Noda and T. Baba, Eds., Roadmap on Photonic Crystals, (Kluwer Academic, New York, 2003).

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Figures (10)

Fig. 1.
Fig. 1.

(a), (b), (c), (d) Magnetic field distributions at band edges A1, B1, E, and E, respectively. The amplitudes of the magnetic fields in the direction perpendicular to the plane are indicated by white and black areas, which denote positive and negative regions, respectively. The thick black circles indicate the shapes and locations of lattice points.

Fig. 2.
Fig. 2.

Coordinates and symmetry of the square lattice structure having a circular unit cell structure

Fig. 3.
Fig. 3.

Schematic of 3D-FDTD calculations. The device is composed of an n-cladding layer, a p-cladding layer, and an active layer. The 2D PC is embedded at the cladding layer close to the active layer.

Fig. 4.
Fig. 4.

(a) Magnetic field distributions of A1 mode normal to the photonic crystal plane at the center of the active layer. The PC size is 100×100. (b) Wave number space (k-space) patterns of (a). The figure is obtained by Fourier transformation of the magnetic field of the photonic crystal area in Fig 4(a). The white dotted square represents the first Brillouin zone.

Fig. 5.
Fig. 5.

The resonant frequency of each resonant mode versus air filling factor. We can classify air filling factor into three regions depending on the by the relative height the resonant frequency. The horizontal line at a frequency of 0.3116 is the cladding light line, above which light leaks to the cladding layer.

Fig. 6.
Fig. 6.

(a) (b) (c) Typical band structures of regions A, B, and C around the Γ2 point, respectively. These band structures are calculated by the 2D PW expansion method. The dielectric constants inside the unit cell are 9.26, 9.86, and 10.01, respectively. The dielectric constants outside the unit cell are 10.60, 10.47, and 10.38 respectively. The air filling factors are 8.50%, 30.0%, and 50.0%, respectively.

Fig. 7.
Fig. 7.

(a)(b) The wave number space (k-space) patterns of the electric field (E x ) of the A1 mode, when the air filling factors are 10% and 60%, respectively. The circles indicated by a solid line, a broken line, and a dotted line are the light cones determined by the refractive index of the active layer, the cladding layer, and air, respectively.

Fig. 8.
Fig. 8.

Q factors as functions of air filling factor for A1, B1, and E modes. Size of the PC is 100×100. The Q factor of the A1 mode is larger than those of the other resonant modes. This result indicates that, among the resonant modes, the A1 mode allows relatively easy lasing.

Fig. 9.
Fig. 9.

(a)(b) Horizontal Q factor (Q) and vertical Q factor (Q ) as functions of the air filling factor for A1, B1, and E modes. Q and Q represent losses of guided mode and out-of-plane radiation, respectively, in the direction normal to the photonic crystal plane.

Fig. 10.
Fig. 10.

Frequency differences between maximum and minimum resonant frequencies for each of the A1, B1, and E resonant modes. The solid line is obtained by the 3D-FDTD calculation and the dotted line is obtained by the 2D PW expansion method using the effective refractive index. The characteristic trend is similar to that for Q.

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