Abstract

An analytical expression of the lasing threshold for arbitrary photonic crystals was derived, which showed their reduction due to small group velocities of electromagnetic eigenmodes. The lasing threshold was also evaluated numerically for a two-dimensional photonic crystal by examining the divergence of its transmission and reflection coefficients numerically. A large reduction of lasing threshold caused by a group-velocity anomaly that is peculiar to two- and three-dimensional photonic crystals was found.

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
  2. C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).
  3. C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, 1996).
    [CrossRef]
  4. K. Sakoda, "Photonic crystals," in Optical Properties of Low-Dimensional Materials, Vol. 2, T. Ogawa and Y. Kanemitsu, ed. (World Scientific, Singapore, 1998).
    [CrossRef]
  5. K. Sakoda, "Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal," Opt. Express 4, 167-176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm
    [CrossRef] [PubMed]
  6. K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
    [CrossRef]
  7. K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
    [CrossRef]
  8. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: a new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
    [CrossRef]
  9. S. Nojima, "Enhancement of optical gain in two-dimensional photonic crystals with active lattice points," Jpn. J. Appl. Phys. 2, Lett. 37 L565-L567 (1998).
    [CrossRef]
  10. K. Sakoda, "Enhanced stimulated emission in a two-dimensional photonic crystal," Proc. 1998 Int. Conf. Appl. Phot. Tech., Applications of Photonic Technology 3, Vol. SPIE 3491, edited by G. A. Lampropoulos and R. A. Lessard (SPIE, Washington, D.C., 1998) 248-253.
  11. M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, "Laser oscillation from dye molecules in a 2D photonic crystals," Technical Digest of the Pacific Rim Conference on Lasers and Electro-Optics, 42-43 (1997).
  12. K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, "Laser action characteristic of a two-dimensional photonic lattice," 1998 OSA Technical Digest Series 7, 47-48 (1998).
  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. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Light-emitting devices with one- and two- dimensional air/semiconductor gratings embedded by wafer fusion technique," Conference Digest of IEEE International Semiconductor Laser Conference, 211-212 (1998).
  15. M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Surface-emitting laser with two-dimensional photonic band structure embedded by wafer fusion technique," 1999 OSA Technical Digest Series, in press.
  16. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  17. K. Sakoda, "Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice," J. Opt. Soc. Am. B 14, 1961-1966 (1997).
    [CrossRef]
  18. A. Yariv, Quantum Electronics (Wily, New York, 1967) Sec. 19.6.
  19. K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
    [CrossRef]
  20. K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
    [CrossRef]
  21. M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
    [CrossRef]
  22. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Measurement of photonic band structure in a two-dimensional periodic dielectric array," Phys. Rev. Lett. 68, 2023-2026 (1992).
    [CrossRef] [PubMed]
  23. K. Sakoda, "Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices," Phys. Rev. B 52, 7982-7986 (1995).
    [CrossRef]
  24. K. Sakoda, "Group-theoretical classification of eigenmodes in three-dimensional photonic lattices," Phys. Rev. B 55, 15345-15348 (1997).
    [CrossRef]
  25. K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. III. Group-theoretical treatment," J. Phys. Soc. Jpn. 65, 2670-2684 (1996).
    [CrossRef]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).

C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, 1996).
[CrossRef]

K. Sakoda, "Photonic crystals," in Optical Properties of Low-Dimensional Materials, Vol. 2, T. Ogawa and Y. Kanemitsu, ed. (World Scientific, Singapore, 1998).
[CrossRef]

K. Sakoda, "Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal," Opt. Express 4, 167-176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm
[CrossRef] [PubMed]

K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: a new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

S. Nojima, "Enhancement of optical gain in two-dimensional photonic crystals with active lattice points," Jpn. J. Appl. Phys. 2, Lett. 37 L565-L567 (1998).
[CrossRef]

K. Sakoda, "Enhanced stimulated emission in a two-dimensional photonic crystal," Proc. 1998 Int. Conf. Appl. Phot. Tech., Applications of Photonic Technology 3, Vol. SPIE 3491, edited by G. A. Lampropoulos and R. A. Lessard (SPIE, Washington, D.C., 1998) 248-253.

M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, "Laser oscillation from dye molecules in a 2D photonic crystals," Technical Digest of the Pacific Rim Conference on Lasers and Electro-Optics, 42-43 (1997).

K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, "Laser action characteristic of a two-dimensional photonic lattice," 1998 OSA Technical Digest Series 7, 47-48 (1998).

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]

M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Light-emitting devices with one- and two- dimensional air/semiconductor gratings embedded by wafer fusion technique," Conference Digest of IEEE International Semiconductor Laser Conference, 211-212 (1998).

M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Surface-emitting laser with two-dimensional photonic band structure embedded by wafer fusion technique," 1999 OSA Technical Digest Series, in press.

P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
[CrossRef]

K. Sakoda, "Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice," J. Opt. Soc. Am. B 14, 1961-1966 (1997).
[CrossRef]

A. Yariv, Quantum Electronics (Wily, New York, 1967) Sec. 19.6.

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
[CrossRef]

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Measurement of photonic band structure in a two-dimensional periodic dielectric array," Phys. Rev. Lett. 68, 2023-2026 (1992).
[CrossRef] [PubMed]

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

K. Sakoda, "Group-theoretical classification of eigenmodes in three-dimensional photonic lattices," Phys. Rev. B 55, 15345-15348 (1997).
[CrossRef]

K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. III. Group-theoretical treatment," J. Phys. Soc. Jpn. 65, 2670-2684 (1996).
[CrossRef]

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Figures (5)

Figure 1.
Figure 1.

The dispersion relation for E polarization (left-hand side) and the threshold of laser oscillation (right-hand side) of a two-dimensional photonic crystal composed of a regular square array of circular air cylinders formed in a dielectric material with a dielectric constant of 2.1. The ordinate is the normalized frequency where ω, a, and c denote the angular frequency of the radiation field, the lattice constant of the two-dimensional crystal, and the light velocity in vacuum, respectively. The radius of the air cylinders was assumed to be 0.28 times the lattice constant. The number of the lattice layers was assumed to be eight. The dispersion relation was presented from Γ to X points in the two-dimensional Brillouin zone. The solid lines represent the dispersion relation for symmetric modes that can be excited by an incident plane wave, whereas the dashed line represents that of an antisymmetric (uncoupled) mode that does not contribute to the light propagation. The threshold of laser oscillation is given by the imaginary part of the dielectric constant of the host material. Note that the third lowest symmetric mode has a small group velocity over its entire spectral range (group-velocity anomaly) and the lasing threshold for this mode is smaller than that for the lowest and the second lowest modes by about two orders of magnitude. Also note the decrease of the threshold at the upper band edges of the latter.

Figure 2.
Figure 2.

The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the third symmetric band of the two-dimensional photonic crystal as a function of the normalized frequency, ω a/2π c, and the imaginary part of the dielectric constant, ″. The same parameters as Fig. 1 were used for numerical calculation and the incident light was propagated in the Γ-X direction. We assumed that the front and the rear surfaces of the crystal were perpendicular to the propagation direction and that the distance between each surface and the center of the first air cylinder was half a lattice constant. Note that the sum is divergent for a certain combination of ωa/2πc and ″.

Figure 3.
Figure 3.

The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the second band of the two-dimensional photonic crystal. The same parameters as Fig. 2 were used for numerical calculation.

Figure 4.
Figure 4.

The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the first band of the two-dimensional photonic crystal. The same parameters as Fig. 2 were used for numerical calculation.

Figure 5.
Figure 5.

The dispersion relation for H polarization (left-hand side) and the threshold of laser oscillation (right-hand side) of the two-dimensional photonic crystal. The same parameters as Fig. 1 were used for numerical calculation.

Equations (7)

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β ( k μ ) α ( ω k μ ) π ω k μ iF ( k μ ) 2 υ g ( k μ ) ,
F ( k μ ) = 1 V 0 V 0 d r n ( r ) E k μ ( r ) 2 ,
V 0 d r ( r ) E k μ ( r ) 2 = V 0 ,
R 2 ( k μ ) exp [ 2 { β ( k μ ) + ik } L ] = 1 ,
R n eff 1 n eff + 1 .
( β + ik ) L log ( 1 + υ g c 1 υ g c ) + mπi ,
f th 4 π n ̅ α th 8 ̅ υ g ω L log ( 1 + υ g c 1 υ g c ) .

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