Abstract

An analytical expression was derived for light amplification by stimulated emission in arbitrary photonic crystals, which showed an enhancement due to small group velocity. This enhancement was evaluated quantitatively for a two-dimensional crystal with a finite thickness, and an extremely large enhancement due to group-velocity anomaly peculiar to two- and three-dimensional crystals was found even for quite a thin crystal.

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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. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  6. S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
    [CrossRef]
  7. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
    [CrossRef] [PubMed]
  8. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
    [CrossRef] [PubMed]
  9. K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
    [CrossRef]
  10. K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
    [CrossRef]
  11. K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 84, 1210-1214 (1998).
    [CrossRef]
  12. T. Ueta, K. Ohtaka, N. Kawai, and K. Sakoda, "Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss," J. Appl. Phys. 84, 6299-6304 (1998).
    [CrossRef]
  13. D. L. Mills and S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B 36, 947-952 (1987).
    [CrossRef]
  14. S. John and N. Akozbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).
    [CrossRef] [PubMed]
  15. S. John and T. Quang, "Spontaneous emission near the edge of a photonic band gap," Phys. Rev. A 50, 1764-1769 (1994).
    [CrossRef] [PubMed]
  16. S. John and T. Quang, "Localization of superradiance near a photonic band gap," Phys. Rev. Lett. 74, 3419-3422 (1995).
    [CrossRef] [PubMed]
  17. 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]
  18. K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
    [CrossRef]
  19. M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
    [CrossRef]
  20. 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]
  21. K. Sakoda, "Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices," Phys. Rev. B 52, 7982-7986 (1995).
    [CrossRef]
  22. K. Sakoda, "Group-theoretical classification of eigenmodes in three-dimensional photonic lattices," Phys. Rev. B 55, 15345-15348 (1997).
    [CrossRef]
  23. K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. III. Group-theoretical treatment," J. Phys. Soc. Jpn. 65, 2670-2684 (1996).
    [CrossRef]
  24. 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]
  25. 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]
  26. 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.
  27. H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers: Part A, Fundamental Principles (Academic, New York, 1978), Sec. 2.10.
  28. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  29. K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
    [CrossRef]
  30. K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
    [CrossRef]
  31. 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]

Other (31)

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]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
[CrossRef] [PubMed]

K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
[CrossRef]

K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
[CrossRef]

K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 84, 1210-1214 (1998).
[CrossRef]

T. Ueta, K. Ohtaka, N. Kawai, and K. Sakoda, "Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss," J. Appl. Phys. 84, 6299-6304 (1998).
[CrossRef]

D. L. Mills and S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B 36, 947-952 (1987).
[CrossRef]

S. John and N. Akozbek, "Nonlinear optical solitary waves in a photonic band gap," Phys. Rev. Lett. 71, 1168-1171 (1993).
[CrossRef] [PubMed]

S. John and T. Quang, "Spontaneous emission near the edge of a photonic band gap," Phys. Rev. A 50, 1764-1769 (1994).
[CrossRef] [PubMed]

S. John and T. Quang, "Localization of superradiance near a photonic band gap," Phys. Rev. Lett. 74, 3419-3422 (1995).
[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]

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]

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.

H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers: Part A, Fundamental Principles (Academic, New York, 1978), Sec. 2.10.

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

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]

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]

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Figures (6)

Figure 1.
Figure 1.

The dispersion relation for E polarization in a two-dimensional photonic crystal composed of a regular square array of circular air-rods 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 abscissa represents the wave vector in the two-dimensional Brillouin zone where Γ, X, and M stand for (0, 0), (π/a, 0), and (π/a, π/a). The radius of the air-rods was assumed to be 0.28 times the lattice constant. Symmetric and antisymmetric modes are denoted by S and A, respectively. Note that the third lowest S mode and the lowest A mode in the Γ-X direction has a small group velocity over their whole frequency ranges.

Figure 2.
Figure 2.

The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for E polarization in the Γ-X direction. The same parameters as Fig. 1 were assumed for numerical calculation. The number of the lattice layers was assumed to be 16 for the calculation of the transmission spectrum. The dashed line in the left-hand side represents an A (antisymmetric) mode.

Figure 3.
Figure 3.

The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1-0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The negative imaginary part of the dielectric constant stands for the inverted population of the impurity atoms. The incident light was assumed to be propagated in the Γ-X direction. Note that the sum can be greater than unity because of the stimulated emission.

Figure 4.
Figure 4.

The dispersion relation for H polarization in the two-dimensional photonic crystal composed of a regular square array of circular air-rods formed in a dielectric material. The same parameters as Fig. 1 were assumed for numerical calculation. Symmetric and antisymmetric modes are denoted by S and A as before. Note that the third lowest S mode and the lowest A mode in the Γ-X direction show the group-velocity anomaly as E polarization.

Figure 5.
Figure 5.

The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for H polarization in the Γ-X direction. The same parameters as Fig. 2 were assumed for numerical calculation. The dashed line in the left-hand side represents an A mode.

Figure 6.
Figure 6.

The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1- 0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The incident light was assumed to be propagated in the Γ-X direction.

Equations (7)

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P ex r t α ( ω ) n ( r ) E k μ ( r ) e iωt .
E ex r t = 4 π P ex r t ( r ) + 4 π V k ' μ ' ω k μ E k μ ( r )
× V d r t dt E k μ * ( r ) · P ex ( r , t ) sin ω k μ ( t t ) ,
V d r ( r ) E k μ * ( r ) · E k μ ( r ) = V δ kk δ μμ .
E ex r t α ( ω ) π ω k μ ilF ( k μ ) 2 v g ( k μ ) E k μ ( r ) e iωt ,
F ( k μ ) = 1 V 0 V 0 d r n ( r ) E k μ ( r ) 2 ,
E r t exp { α ( ω k μ ) π ω k μ ilF ( k μ ) 2 v g ( k μ ) } E k μ ( r ) exp ( i ω k μ t ) .

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