Abstract

Using full three-dimensional analysis we show that coupled periodic optical waveguides can exhibit a giant slow light resonance associated with a degenerate photonic band edge. We consider the silicon-on-insulator material system for implementation in silicon photonics at optical telecommunications wavelengths. The coupling of the resonance mode with the input light can be controlled continuously by varying the input power ratio and the phase difference between the two input arms. Near unity transmission efficiency through the degenerate band edge structure can be achieved, enabling exploitation of the advantages of the giant slow wave resonance.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]

2012

N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A85(3), 033804 (2012).
[CrossRef]

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

N. Gutman, W. H. Dupree, Y. Sun, A. A. Sukhorukov, and C. M. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express20(4), 3519–3528 (2012).
[CrossRef] [PubMed]

N. Gutman, A. A. Sukhorukov, F. Eilenberger, and C. M. de Sterke, “Bistability suppression and low threshold switching using frozen light at a degenerate band edge waveguide,” Opt. Express20(24), 27363–27368 (2012).
[CrossRef] [PubMed]

2011

N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36(16), 3257–3259 (2011).
[CrossRef] [PubMed]

S. Kim, B.-H. Ahn, J.-Y. Kim, K.-Y. Jeong, K. S. Kim, and Y.-H. Lee, “Nanobeam photonic bandedge lasers,” Opt. Express19(24), 24055–24060 (2011).
[CrossRef] [PubMed]

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011).
[CrossRef]

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev.5(2), 201–213 (2011).
[CrossRef]

2010

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

2009

2008

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical signal processing using tunable delay elements based on slow light,” IEEE J. Sel. Top. Quantum Electron.14(3), 691–705 (2008).
[CrossRef]

A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B25(12), C65–C74 (2008).
[CrossRef]

2007

2006

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

2005

M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express13(18), 7145–7159 (2005).
[CrossRef] [PubMed]

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(3), 036619 (2005).
[CrossRef] [PubMed]

2001

1996

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

1994

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994).
[CrossRef] [PubMed]

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(4), 1896–1899 (1994).
[CrossRef]

Aamer, M.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Ahn, B.-H.

Bendickson, J. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Bloemer, M. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

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(4), 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994).
[CrossRef] [PubMed]

Botten, L.

N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A85(3), 033804 (2012).
[CrossRef]

Botten, L. C.

Bowden, C. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

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(4), 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994).
[CrossRef] [PubMed]

Brimont, A.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Cao, H.

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011).
[CrossRef]

Chigrin, D. N.

de Sterke, C. M.

Dowling, J. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

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(4), 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994).
[CrossRef] [PubMed]

Dündar, M. A.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

Dupree, W. H.

Eilenberger, F.

Fazal, I.

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical signal processing using tunable delay elements based on slow light,” IEEE J. Sel. Top. Quantum Electron.14(3), 691–705 (2008).
[CrossRef]

Fedeli, J.-M.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev.5(2), 201–213 (2011).
[CrossRef]

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A76(5), 053839 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(3), 036619 (2005).
[CrossRef] [PubMed]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Gardes, F. Y.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Gutierrez, A. M.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Gutman, N.

Ha, S.

Handmer, C. J.

He, S.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

Jeong, K.-Y.

Joannopoulos, J. D.

Johnson, S. G.

Karouta, F.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

Kim, J.-Y.

Kim, K. S.

Kim, S.

Kivshar, Y. S.

Lavrinenko, A. V.

Leavitt, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Ledbetter, H. S.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Lee, Y.-H.

Martí, J.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Noh, H.

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011).
[CrossRef]

Nötzel, R.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

Pelinovsky, D. E.

Povinelli, M. L.

Powell, D. A.

Reano, R. M.

Reed, G. T.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

Rooks, M. J.

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011).
[CrossRef]

Sanchis, P.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Scalora, M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

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(4), 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994).
[CrossRef] [PubMed]

Shadrivov, I. V.

Solomon, G. S.

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011).
[CrossRef]

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

Steel, M. J.

Sukhorukov, A.

N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A85(3), 033804 (2012).
[CrossRef]

Sukhorukov, A. A.

Sun, P.

Sun, Y.

Thomson, D. J.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012).
[CrossRef]

Tocci, M. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996).
[CrossRef] [PubMed]

van der Heijden, R. W.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev.5(2), 201–213 (2011).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Coupled periodic silicon optical waveguides designed to exhibit a DBE at telecommunications wavelengths; (a) Top view; (b) Cross-section.

Fig. 2
Fig. 2

(a)-(f) Dispersion diagrams. (a) Single mode waveguide; (b) Two coupled single mode waveguides; (c) Single periodic dielectric waveguide; (d) Two coupled periodic dielectric waveguides; (e) Two coupled periodic waveguides with a 0.5a lateral shift; (f) Lateral shift of 0.37a to realize a DBE; (g) Frequency difference versus wavenumber difference on loglog scale for RBE; (h) Frequency difference versus wavenumber difference on loglog scale for DBE (red) and the lower spectral branch (blue).

Fig. 3
Fig. 3

Outgoing power from all ports for excitation at port 1 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4. The band diagram in the inset indicates the three lowest order modes. The second mode (indicated in red) exhibits a DBE. The yellow region indicates the frequency range of the plotted s-parameters.

Fig. 4
Fig. 4

Outgoing power from all ports for excitation at port 3 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4.

Fig. 5
Fig. 5

Outgoing power from all ports at the fundamental resonance (λ = 1508.3 nm), for excitation at both ports 1 and 3, parameterized by |α|2 and ϕ (N = 65). (a)-(c) Reflected power from ports 1 and 3, respectively. (b)-(d) Transmitted power from ports 2 and 4, respectively.

Fig. 6
Fig. 6

Outgoing power versus wavelength for excitation at both ports 1 and 3 (N = 65). Red lines correspond to the excitation condition for maximum transmission from ports 2 and 4 (condition 1). Blue lines correspond to the excitation condition for maximum reflection from ports 1 and 3 (condition 2). (a) Sum of the reflected power from ports 1 and 3. (b) Sum of the transmitted powers from ports 2 and 4.

Fig. 7
Fig. 7

(a) Quality factor scaling as a function of number of periods for 50 nm gap DBE. Fitted dashed line corresponds to RBE-like behavior. Fitted solid line corresponds to DBE-like behavior. (b) Quality factor scaling as a function of the number of periods for two DBE structures with 50 and 150 nm gaps. True DBE-like behavior is only observed for the 50 nm gap structure in the plotted region.

Fig. 8
Fig. 8

(a) The envelopes of the fields inside the waveguide at the resonance frequency for N = 65. The Ex component (black line) is decomposed into two evanescent Bloch modes (yellow and blue lines) and two propagating Bloch modes (green and red lines); (b)-(c) Resonance frequency versus the number of periods fitted for DBE and RBE, respectively.

Equations (8)

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

( ω ω D )= ( k k D ) 4 / ξ 4
[ E 1 E 2 E 3 E 4 ]=[ S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ][ α 0 β 0 ]
β 1 α 2 e jϕ , 0α1, 0ϕ2π
k j = k D + ξ 4 ( ω ω D ) 4 = k D +Δk e i π 2 j ,j=0,1,2,3 ,
k 0,2 = k D ±Δk and k 1,3 = k D ±iΔk.
2πm=2φ+2kL,
2πm=2φ+2 [ ξ 4 ( ω ω D ) ] 1/4 aN.
ω= ω D ( πmφ ) 4 / ξ 4 (aN) 4 .

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