Abstract

We present the design, theory and experimental implementation of a low modal volume microlaser based on a line-defect 2D-photonic crystal waveguide. The lateral confinement of low-group velocity modes is controlled by the post-processing of 1 to 3µm wide PMMA strips on top of two dimensional photonic crystal waveguides. Modal volume around 1.3 (λ/n)3 can be achieved using this scheme. We use this concept to fabricate microlaser devices from an InP-based heterostructure including InAs0.65P0.35 quantum wells emitting around 1550nm and bonded onto a fused silica wafer. We observe stable, room-temperature laser operation with an effective lasing threshold around 0.5mW.

© 2008 Optical Society of America

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  1. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
    [CrossRef]
  2. C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
    [CrossRef]
  3. A. Sugitatsu and S. Noda, "Room temperature operation of a two dimensional photonic crystal slab defect-waveguide-laser with optical pump," Electron. Lett. 39, 123-125 (2003).
    [CrossRef]
  4. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
    [CrossRef] [PubMed]
  5. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
    [CrossRef]
  6. F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, "Confinement of band edge modes in Photonic Crystals," Opt. Express 15, 10890-10902 (2007).
    [CrossRef] [PubMed]
  7. F. Bordas, C. Seassal, E. Dupuy, P. Regreny, M. Gendry, M. J. Steel, and A. Rahmani, "Room-Temperature InAs/InP Quantum-Dot Photonic Crystal Microlasers Using Cavity-Confined Slow Light," CLEO/QELS May 7-11 2007, Baltimore.
  8. M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
    [CrossRef]
  9. S. Tomljenovic-Hanic, C. M. de Sterke, M. J. Steel, B. J. Eggleton, Y. Tanaka, and S. Noda, "High-Q cavities in multilayer photonic crystal slabs," Opt. Express 15, 17248-17253 (2007).
    [CrossRef] [PubMed]
  10. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  11. B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
    [CrossRef]
  12. M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
    [CrossRef]
  13. http://alioth.debian.org/projects/tessa/.
  14. C. Seassal, C. Monat, J. Mouette, E . Touraille, B. Ben Bakir, H. Hattori, J. L. Leclercq, X. Letartre, P. Rojo-Romeo, and P. Viktorovitch, "InP bonded membrane photonics components and circuits: Toward 2.5 dimensional micro-nano-photonics," IEEE J. Sel. Top. in Quantum Electron. 11, 395-407 (2005).
    [CrossRef]
  15. X. Letartre, C. Monat, C. Seassal, and P. Viktorovitch, "An analytical modeling and an experimental investigation of 2D Photonic Crystal Micro-lasers : defect state (micro-cavity) versus band edge state (distributed feed-back) structures," J. Opt. Soc. Am. B 22, 2581-2595 (2005).
    [CrossRef]

2007 (2)

2006 (3)

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

2005 (1)

2003 (2)

A. Sugitatsu and S. Noda, "Room temperature operation of a two dimensional photonic crystal slab defect-waveguide-laser with optical pump," Electron. Lett. 39, 123-125 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

2002 (3)

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

2001 (1)

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Allard, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Asano, T.

B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Bagheri, M.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Bordas, F.

Charbonneau-Lefort, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

Dapkus, P. D.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

de Sterke, C. M.

Eggleton, B. J.

Gendry, M.

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Hwang, E. H.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Imada, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

Istrate, E.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kuang, W.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Letartre, X.

X. Letartre, C. Monat, C. Seassal, and P. Viktorovitch, "An analytical modeling and an experimental investigation of 2D Photonic Crystal Micro-lasers : defect state (micro-cavity) versus band edge state (distributed feed-back) structures," J. Opt. Soc. Am. B 22, 2581-2595 (2005).
[CrossRef]

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Mochizuki, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

Mock, A.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Monat, C.

X. Letartre, C. Monat, C. Seassal, and P. Viktorovitch, "An analytical modeling and an experimental investigation of 2D Photonic Crystal Micro-lasers : defect state (micro-cavity) versus band edge state (distributed feed-back) structures," J. Opt. Soc. Am. B 22, 2581-2595 (2005).
[CrossRef]

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Noda, S.

S. Tomljenovic-Hanic, C. M. de Sterke, M. J. Steel, B. J. Eggleton, Y. Tanaka, and S. Noda, "High-Q cavities in multilayer photonic crystal slabs," Opt. Express 15, 17248-17253 (2007).
[CrossRef] [PubMed]

B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
[CrossRef]

A. Sugitatsu and S. Noda, "Room temperature operation of a two dimensional photonic crystal slab defect-waveguide-laser with optical pump," Electron. Lett. 39, 123-125 (2003).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

Notomi, M.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

O'Brien, J. D.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Poon, J.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Rahmani, A.

Regreny, P.

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Rojo-Romeo, P.

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Sargent, E. H.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Seassal, C.

Shih, M. H.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Shinya, A.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Song, B. S.

B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Steel, M. J.

Sugitatsu, A.

A. Sugitatsu and S. Noda, "Room temperature operation of a two dimensional photonic crystal slab defect-waveguide-laser with optical pump," Electron. Lett. 39, 123-125 (2003).
[CrossRef]

Tanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Tanaka, Y.

Tomljenovic-Hanic, S.

Viktorovitch, P.

X. Letartre, C. Monat, C. Seassal, and P. Viktorovitch, "An analytical modeling and an experimental investigation of 2D Photonic Crystal Micro-lasers : defect state (micro-cavity) versus band edge state (distributed feed-back) structures," J. Opt. Soc. Am. B 22, 2581-2595 (2005).
[CrossRef]

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

Watanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Appl. Phys. Lett (1)

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett 88, 041112 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

C. Monat, C. Seassal, X. Letartre, P. Regreny, M. Gendry, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d??Yerville, D. Cassagne, et J. P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett. 815102-5104 (2002).
[CrossRef]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O'Brien, and P. D. Dapkus, "High-quality-factor photonic crystal heterostructure laser," Appl. Phys. Lett. 89, 101104 (2006).
[CrossRef]

Electron. Lett. (1)

A. Sugitatsu and S. Noda, "Room temperature operation of a two dimensional photonic crystal slab defect-waveguide-laser with optical pump," Electron. Lett. 39, 123-125 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

New J. Phys. (1)

B. S. Song, T. Asano, and S. Noda, "Physical origin of the small modal volume of ultra-high-Q photonic double-heterostructure nanocavities," New J. Phys. 8, 209 (2006).
[CrossRef]

Opt. Express (3)

Phys. Rev. B (2)

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306, 2002.
[CrossRef]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Other (3)

http://alioth.debian.org/projects/tessa/.

C. Seassal, C. Monat, J. Mouette, E . Touraille, B. Ben Bakir, H. Hattori, J. L. Leclercq, X. Letartre, P. Rojo-Romeo, and P. Viktorovitch, "InP bonded membrane photonics components and circuits: Toward 2.5 dimensional micro-nano-photonics," IEEE J. Sel. Top. in Quantum Electron. 11, 395-407 (2005).
[CrossRef]

F. Bordas, C. Seassal, E. Dupuy, P. Regreny, M. Gendry, M. J. Steel, and A. Rahmani, "Room-Temperature InAs/InP Quantum-Dot Photonic Crystal Microlasers Using Cavity-Confined Slow Light," CLEO/QELS May 7-11 2007, Baltimore.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of the experimental procedure necessary to obtain the W1+PMMA structure: (a) A 110 nm thick resist film is spin-coated on the sample surface; (b) the PC W1 structure is realized on the resist by Electron Beam Lithography (EBL) and then c) is transferred on the InP layer by means of Reactive Ion Etching (RIE); finally d) a PMMA strip is patterned above the PC with EBL.

Fig. 2.
Fig. 2.

(a) Calculated band diagram for the 2D photonic crystal slab with a triangular lattice and containing a line defect formed by a missing row of air holes. The dark grey area correspods to the light cone, while the green zones correspond to the conduction and valence bands. The blue curve is the odd guided mode, while the red one corresponds to the even mode. (b) The PMMA strip over the PC waveguide locally lowers the energy of the waveguided mode creating a band gap: photons with this lower energy are trapped inside the region under the PMMA strip.

Fig. 3.
Fig. 3.

Calculation of envelopes in the (x,z) plane at the middle of the waveguide for three different configurations (see text) extending from -20a to +20a.

Fig. 4.
Fig. 4.

Simulated Hz field spectrum (a) and (b) map of the eg1 mode at 1607 nm for the W1+PMMA configuration, the dashed rectangles correspond to the location.of the PMMA strip.

Fig. 5.
Fig. 5.

SEM image of the W1 waveguide (a) and of the W1+PMMA cavity (b). The lattice period and hole radius are 478 nm and 154 nm respectively.

Fig. 6.
Fig. 6.

(a) Spectrum of the spontaneous emission (brown line) and of the resonant peaks of four different structures (small changes in the holes radius). (b) laser intensity plotted against the incident pump power (7 ns pulse, 6% duty cycle) for one of the structures (indicated with the black arrow). The inset shows the peaks shape for the W1 (blue) and W1+PMMA (red) structures below lasing threshold.

Fig. 7.
Fig. 7.

Power in versus power out curve showing the power at the laser wavelength versus the incident pump power. The width of the PMMA strip is around 1 µm, and the pump pulse is around 7 ns long (6% duty cycle).

Equations (5)

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

1 2 m * d 2 f n k 0 ( x ) d x 2 = [ ω n 2 ( k 0 ) ω λ 2 [ 1 + Δ ( x ) ] ] f n k 0 ( x )
f n k 0 = { cos K x e γ ( L 2 x ) cos K L 2 } if { x < L 2 x > L 2 }
K = 2 m * ( ω λ 2 ( 1 + Δ ( x ) ) ω n 2 ( k 0 ) )
γ = 2 m * ( ω n 2 ( k 0 ) ω λ 2 )
1 m * = 2 ω n 2 k x 2 k 0 = 2 ω 0 α

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