Abstract

We demonstrate both theoretically and numerically that slow light can enhance the stimulated emission of four-level atomic systems in photonic crystals. By applying the Bloch–Floquet formalism and the semiclassical physical model of harmonic oscillators coupled to electromagnetic fields, we develop a formalism that relates the group velocity of slow light to the conversion rate between the electric field and atomic potential energy. From our numerical study of the stimulated emission in fiber Bragg gratings (FBGs) and nongrating fiber, a ninefold enhancement of stimulated emission is observed in a FBG over the nongrating fiber when pumping at the band edge, while a 20-fold enhancement of is observed when the frequency of the stimulated emission approaches the band edge.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Win, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).
  4. H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
    [CrossRef] [PubMed]
  5. E. Hecht, Optics (Addison-Wesley, 1998).
  6. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, 1976).
  7. P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
    [CrossRef]
  8. R. K. Lee, O. Painter, B. Kitzke, A. Scherer, and A. Yariv, “Emission properties of a defect cavity in a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17, 629-633 (2000).
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science Books, 1986).
  10. K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
    [CrossRef]
  11. A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
    [CrossRef]
  12. C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991).
    [CrossRef]
  13. M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).
  14. S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light puse in Bragg-grating couplers,” J. Opt. Soc. Am. B 25, C15-C22 (2008).
    [CrossRef]
  15. V. Govindan and S. Blair, “Nonlinear pulse interaction in microresonator slow-light waveguides,” J. Opt. Soc. Am. B 25, C23-C30 (2008).
    [CrossRef]
  16. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167-176 (1999).
    [CrossRef] [PubMed]
  17. M. Soljačić, S. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052-2059 (2002).
    [CrossRef]
  18. J. McMillan, X. Yang, N. Panoiu, R. Osgood, and C. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett. 31, 1235-1237 (2006).
    [CrossRef] [PubMed]
  19. D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
    [CrossRef]
  20. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman-Hall, 1983).
  21. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334-340 (1998).
    [CrossRef]
  22. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70-73 (2000).
    [CrossRef] [PubMed]
  23. M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge Univ. Press,1999).
  24. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972-2974 (2006).
    [CrossRef] [PubMed]

2008 (2)

2006 (3)

2005 (1)

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

2002 (1)

2000 (2)

1999 (1)

1998 (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334-340 (1998).
[CrossRef]

1997 (1)

A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
[CrossRef]

1994 (1)

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

1991 (1)

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

1987 (2)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, 1976).

Bermel, P.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972-2974 (2006).
[CrossRef] [PubMed]

Bialio, D.

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

Biallo, D.

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

Blair, S.

Bogaerts, W.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge Univ. Press,1999).

Botey, M.

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

Burr, G.

De Sario, M.

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

Desurvire, E.

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

D'Orazio, A.

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

Engelen, R.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Fan, S.

Farjadpour, A.

Fink, Y.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Gersen, H.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Giles, C.

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

Govindan, V.

Ha, S.

Hattori, K.

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

Horiguchi, M.

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

Ibanescu, M.

Ippen, E.

Jiang, X.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70-73 (2000).
[CrossRef] [PubMed]

Joannopoulos, J.

Joannopoulos, J. D.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972-2974 (2006).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Win, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Johnson, S.

Johnson, S. G.

Karle, T.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kitagawa, T.

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

Kitzke, B.

Korterik, J.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Krauss, T.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kuipers, L.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Lee, R. K.

Lidorikis, E.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman-Hall, 1983).

Martorell, J.

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

Maymo, M.

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

McMillan, J.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Win, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, 1976).

Mori, A.

A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
[CrossRef]

Nagra, A. S.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334-340 (1998).
[CrossRef]

Oguma, M.

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

Ohishi, Y.

A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
[CrossRef]

Ohmori, Y.

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

Osgood, R.

Painter, O.

Panoiu, N.

Petruzzelli, L.

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

Prudenzano, F.

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

Rodriguez, A.

Roundy, D.

Sakoda, K.

Scherer, A.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman-Hall, 1983).

Soljacic, M.

Soukoulis, C. M.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70-73 (2000).
[CrossRef] [PubMed]

Sudo, S.

A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
[CrossRef]

Sukhorukov, A. A.

Van Hulst, N.

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Win, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Win, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Wolf, E.

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge Univ. Press,1999).

Wong, C.

Yablonovitch, E.

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

Yang, X.

Yariv, A.

York, R. A.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334-340 (1998).
[CrossRef]

Electron. Lett. (2)

K. Hattori, T. Kitagawa, M. Oguma, Y. Ohmori, and M. Horiguchi, “Erbium-doped silica-based waveguide amplifier integrated with a 980/1530 nm WDM coupler,” Electron. Lett. 30, 856-857 (1994).
[CrossRef]

A. Mori, Y. Ohishi, and S. Sudo, “Erbium-doped tellurite glass fibre laser and amplifier,” Electron. Lett. 33, 863-864 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334-340 (1998).
[CrossRef]

J. Lightwave Technol. (1)

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

J. Opt. Soc. Am. B (4)

Laser Phys. (1)

M. Botey, M. Maymo, D. Biallo, and J. Martorell, “Photon lifetime at the inner band edges of a 3-D photonic crystal,” Laser Phys. 14, 643-647 (2004).

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (1)

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70-73 (2000).
[CrossRef] [PubMed]

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

H. Gersen, T. Karle, R. Engelen, W. Bogaerts, J. Korterik, N. Van Hulst, T. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Other (7)

E. Hecht, Optics (Addison-Wesley, 1998).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, 1976).

J. D. Joannopoulos, R. D. Meade, and J. N. Win, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

D. Bialio, A. D'Orazio, M. De Sario, L. Petruzzelli, and F. Prudenzano, “Optical amplification in Er3+ doped SiO2-TiO2 photonic crystals,” in Transparent Optical Networks, 2005, Proceedings of 2005 7th International Conference (IEEE, 2005), Vol. 1, pp. 149-154.
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman-Hall, 1983).

A. E. Siegman, Lasers (University Science Books, 1986).

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge Univ. Press,1999).

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.


Figures (6)

Fig. 1
Fig. 1

Interactions between the energy levels for a four-level atomic system. Solid and dashed arrows correspond to the radioactive transition and nonradiative decay, respectively.

Fig. 2
Fig. 2

Schematic structure of the FBG studied in this paper. The total length is 305 a , where a is the lattice constant. The band diagram, the group velocity, and the transmittance of the FBG are also given. The bandgap is between 0.324 ( 2 π c a ) and 0.331 ( 2 π c a ) , and λ B here is estimated to be 0.328 ( 2 π c a ) .

Fig. 3
Fig. 3

Enhancement of stimulated emission of FBG over nongrating fiber at different pump frequencies. The solid red curve denotes the enhancement, while the dashed gray one denotes the transmittance of the FBG without active material in the simulation. The frequency of the stimulated emission is fixed at 0.20 ( 2 π c a ) .

Fig. 4
Fig. 4

Enhancement of stimulated emission of FBG over nongrating fiber at different stimulated emission frequencies. The frequency of the pump is fixed at 0.40 ( 2 π c a ) .

Fig. 5
Fig. 5

Relationship between the enhancement of the stimulated emission intensity and the group velocity of the pump. The blue curve denotes the frequencies at the lower edge of the bandgap, while the red one denotes the frequencies at the upper edge.

Fig. 6
Fig. 6

Relationship between the enhancement and group velocity of the stimulated emission. The blue line denotes the frequencies at the lower edge of the bandgap, and the red one denotes the frequencies at the upper edge.

Equations (13)

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

E ( r , t ) = E s ( r , ω s ) exp ( i ω s t ) + E p ( r , ω p ) exp ( i ω p t ) ,
v g k = P k Λ 1 2 ε 0 V 0 ε ( r ) | E k ( r , ω k ) | 2 d V ,
d 2 P k d t 2 + Γ k d P k d t + ω k 2 P k = σ k Δ N k E ,
d N 3 d t = 1 ω p E d P p d t N 3 τ 32 ,
d N 2 d t = 1 ω s E d P s d t + N 3 τ 32 N 2 τ 21 ,
d N 1 d t = 1 ω s E d P s d t + N 2 τ 21 N 1 τ 10 ,
d N 0 d t = 1 ω p E d P p d t + N 1 τ 10 .
P k ( ω ) = σ k Δ N k ω k 2 ω 2 i ω Γ k E ( ω ) = σ k Δ N k i ω k Γ k E ( ω k ) .
P k ( t ) = σ k Δ N k i ω k Γ k E k ( t ) ,
V 0 1 ω k E d P k d t d V = σ k Δ N k 2 ω k Γ k V 0 | E k ( r , ω k ) | 2 d V = σ k Δ N k Λ Ψ k 2 ω k Γ k v g k P k ,
Ψ k = V 0 | E k ( r , ω k ) | 2 d V 1 2 ε 0 V 0 ε ( r ) | E k ( r , ω k ) | 2 d V .
× E = B t ,
× H = J + ε E t + P t ,

Metrics