Abstract

We recently reported experimental evidence for double enhancement of spontaneous emission due to increased photon density of states and small group velocity at photonic band-edge frequencies by observing angle-resolved emission and excitation spectra of photoluminescence [ K. Kuroda et al., Opt. Express 17, 13168 (2009) ]. The specimen we used was a one-dimensional photonic crystal composed of periodic multilayers of Ta2O5 and SiO2 with oxygen vacancies as light emitters. In the present study, we report on the lack of any excitation intensity dependence of the emission peak height and width, which excludes possibilities of nonlinear effects, the polarized emission spectra, and their comparison with theoretical calculations to further confirm our finding.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).
  3. K. Sakoda, “Optics of photonic crystals,” Opt. Rev. 6, 381-392 (1999).
    [CrossRef]
  4. M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
    [CrossRef]
  5. E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
    [CrossRef]
  6. K. Kuroda, T. Sawada, T. Kuroda, K. Watanabe, and K. Sakoda, “Doubly enhanced spontaneous emission due to increased photon density of states at photonic band edge frequencies,” Opt. Express 17, 13168-13177 (2009).
    [CrossRef] [PubMed]
  7. P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
    [CrossRef] [PubMed]
  8. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
    [CrossRef] [PubMed]
  9. 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]
  10. M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).
  11. Y. Matsuhisa, Y. Huang, Y. Zhou, S.-T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express 15, 616-622 (2007).
    [CrossRef] [PubMed]
  12. M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
    [CrossRef]

2009 (1)

2008 (1)

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

2007 (1)

2004 (2)

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

1999 (2)

K. Sakoda, “Optics of photonic crystals,” Opt. Rev. 6, 381-392 (1999).
[CrossRef]

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

1996 (1)

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

1994 (1)

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]

André, R.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Astic, M.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Belabas, N.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

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]

Bowden, C. M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

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]

Brick, P.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Delaye, Ph.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Deych, L. I.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Dowling, J. P.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

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]

Ell, C.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Erementchouk, M. V.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Frey, R.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Fujii, A.

Gibbs, H. M.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Haus, J. W.

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

Huang, Y.

Hübner, M.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Irman, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Ivchenko, E. L.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Khitrova, G.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Koch, S. W.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Kuroda, K.

Kuroda, T.

Lee, E. S.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Lisyansky, A. A.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Matsuhisa, Y.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Ozaki, M.

Prineas, J. P.

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Raj, R.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Roosen, G.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Sagnes, I.

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Sakoda, K.

Sawada, T.

Scalora, M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

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]

Takao, Y.

Tocci, M.

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

Tocci, M. D.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Voronov, M. M.

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Watanabe, K.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Wu, S.-T.

Zhou, Y.

Appl. Phys. B (1)

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B 60, S57-S61 (1995).

J. Appl. Phys. (1)

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]

J. Phys. D (1)

M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D 41, 224005 (2008).
[CrossRef]

Nature (1)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Rev. (1)

K. Sakoda, “Optics of photonic crystals,” Opt. Rev. 6, 381-392 (1999).
[CrossRef]

Phys. Rev. A (1)

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (1)

E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B 70, 195106 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. 83, 2841-2844 (1999).
[CrossRef]

Other (2)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

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

Schematic illustration of the experimental configuration. θ is the observation angle. ω and k denote the angular frequency and wave vector of emitted light.

Fig. 2
Fig. 2

Calculated emission spectra of s (thick curve) and p (thin curve) polarizations for various observation angles.

Fig. 3
Fig. 3

Emission spectra in the normal direction ( θ = 0 ) measured with various excitation powers from 0.45 mW to 36 mW . The scale of the vertical axis is proportional to the excitation power.

Fig. 4
Fig. 4

Curve fitting for the shorter wavelength peak with a Lorentzian function and (b) that for the longer wavelength peak. (c) Excitation intensity dependence of the emission intensity. Open circles, solid circles, and open squares denote the longer wavelength peak, the shorter wavelength peak, and the mean value between 680 and 700 nm . Solid lines are the best fit by the power law. Exponents of the power-law dependence obtained by the best fit are all close to unity ( 1.00 ± 0.08 ) . (d) Linewidths of the shorter (solid circles) and longer (open circles) wavelength peaks.

Fig. 5
Fig. 5

Polarized emission spectra for θ = 0 to 45°. Thick and thin curves show s- and p-polarized emissions, respectively.

Fig. 6
Fig. 6

Magnified view of the longer wavelength peaks of the s-(thick curve) and p-polarized (thin curve) emissions at θ = 20 ° , and (b) that of the shorter wavelength peaks. (c) Difference of peak wavelengths between the s- and p-polarized emissions as a function of detection angle θ. Solid circles, longer wavelength peak; Open squares, shorter wavelength peak; Solid line, theory.

Equations (20)

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

{ x 1 ϵ ( x ) x + y 1 ϵ ( x ) y } H z = ω 2 c 2 H z ,
1 ϵ ( x ) { 2 x 2 + 2 y 2 } E z = ω 2 c 2 E z ,
1 ϵ ( x ) = η 0 + η 1 e i G x + η 1 * e i G x ,
H z = m = A m e i ( k x + G m ) x + i k y y ,
E z = m = B m e i ( k x + G m ) x + i k y y ,
G m = 2 m π a
A m [ η 0 { ( k x + G m ) 2 + k y 2 } ω 2 c 2 ] + A m 1 η 1 { ( k x + G m 1 ) ( k x + G m ) + k y 2 } + A m + 1 η 1 * { ( k x + G m + 1 ) ( k x + G m ) + k y 2 } = 0 .
ω 2 c 2 η 0 ( k x 2 + k y 2 ) and k x π a
A 0 [ c 2 η 0 { ( k x + G 0 ) 2 + k y 2 } ω 2 ] + A 1 c 2 η 1 { ( k x + G 1 ) ( k x + G 0 ) + k y 2 } = 0 ,
A 1 [ c 2 η 0 { ( k x + G 1 ) 2 + k y 2 } ω 2 ] + A 0 c 2 η 1 * { ( k x + G 0 ) ( k x + G 1 ) + k y 2 } = 0 .
ω ± c a π 2 ( η 0 ± | η 1 | ) + a 2 k y 2 ( η 0 | η 1 | ) + a c h 2 2 η 0 | η 1 | ± 2 π 2 η 0 2 | η 1 | ( π 2 a 2 k y 2 ) π 2 ( η 0 ± | η 1 | ) + a 2 k y 2 ( η 0 | η 1 | ) ,
B m [ η 0 { ( k x + G m ) 2 + k y 2 } ω 2 c 2 ] + B m 1 η 1 { ( k x + G m 1 ) 2 + k y 2 } + B m + 1 η 1 * { ( k x + G m + 1 ) 2 + k y 2 } = 0 .
B 0 [ c 2 η 0 { ( k x + G 0 ) 2 + k y 2 } ω 2 ] + B 1 c 2 η 1 { ( k x + G 1 ) ( k x + G 0 ) + k y 2 } = 0 ,
B 1 [ c 2 η 0 { ( k x + G 1 ) 2 + k y 2 } ω 2 ] + B 0 c 2 η 1 * { ( k x + G 0 ) ( k x + G 1 ) + k y 2 } = 0 .
ω ± c a ( η 0 ± | η 1 | ) ( π 2 + a 2 k y 2 ) + a c h 2 η 0 ± | η 1 | 2 π 2 + a 2 k y 2 { 1 ± 2 π 2 ( η 0 | η 1 | ) | η 1 | ( π 2 + a 2 k y 2 ) } .
ω ± π c η 0 ± | η 1 | a { 1 ± a 2 h 2 ( 2 η 0 | η 1 | ) 2 π 2 | η 1 | } .
π c a η 0 | η 1 | < ω < π c a η 0 + | η 1 | .
ω c π c η 0 a and Δ ω g π c | η 1 | a η 0 .
c = d ω d k = ω k x d k x d k + ω k y sin θ .
d s d k = ( d k x d k ) 2 + ( d k y d k ) 2 = ( ω k x ) 2 ( c ω k y sin θ ) 2 + sin 2 θ .

Metrics