Abstract

We present the design of an all-optical router based on the properties of both propagation and interaction of Gaussian beams in lenslike planar guides. Variational results of single co- and counterpropagation are derived and used to design three integrated optical devices, that is, a header extraction device, an optical bistable device and a data routing device, which perform an ultrafast, phase-insensitive and fiber compatible routing operation in the optical domain.

©2005 Optical Society of America

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References

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  1. G. Stix, “The triumph of light,” Scientific American, January (1998).
  2. B. Olsson, L. Rau, and J. Blumenthal, “WDM to OTDM multiplexing using an ultrafast all-optical wavelength converter,” IEEE Photon. Technol. Lett. 13, 1905 (2001).
    [Crossref]
  3. J. Blumenthal, B. Olsson, G. Rossi, T. E. Dimmick, L. Rau, M. Masanovic, O. Lavrova, R. Doshi, O. Jerphagnon, J. E. Bowers, V. Kaman, L. A. Coldren, and J. Barton, “All-optical label swapping networks and technologies,” J. Lightwave Technol. 18, 2058 (2000).
    [Crossref]
  4. I. Glesk, K.I. Kang, and P. R. Prucnal, “Ultrafast photonic packet switching with optical control,” Opt. Express 1, 126 (1997).
    [Crossref] [PubMed]
  5. K. H. Park and T. Mizumoto, “All-optical address extraction for optical routing,” Opt. Eng.,  38, 1848 (1999).
    [Crossref]
  6. V. W. S. Chan, K. L. Hall, E. Modiano, and K. A. Rauschenbach, “Architectures and technologies for high-speed optical data networks,” J. Lightwave Technol. 16, 2146 (1998).
    [Crossref]
  7. H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
    [Crossref]
  8. D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270 (1985).
    [Crossref] [PubMed]
  9. A. E. Kaplan, “Optical bistability that is due to mutual self-action of counterpropagating beams of light,” Opt. Lett. 6, 360 (1981).
    [Crossref] [PubMed]
  10. E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
    [Crossref]
  11. F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
    [Crossref]
  12. J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry-Perot interferometer,” Phys. Rev. A 17, 335 (1978).
    [Crossref]
  13. R. A. Sammut, C. Pask, and Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B 10, 485 (1993).
    [Crossref]
  14. R. G. Hunsperger, Integrated optics: Theory and technology (Springer-Verlag, Berlin, 1991).
  15. E. F. Mateo and J. Liñares, “All-optical integrated logic gates based on intensity-dependent transverse modal coupling,” Opt. Quantum Electron. 35, 1221 (2003).
    [Crossref]
  16. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys.Rev. A 27, 3135 (1983).
    [Crossref]
  17. J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
    [Crossref]
  18. J. Liñares and M. C. Nistal, “Single local mode propagation though ion-exchanged waveguide elements with quasi-abrupt transitions,” Jpn. J. Appl. Phys. 35, L1596 (1996).
    [Crossref]
  19. M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
    [Crossref]
  20. R. J. Gehr, G. L. Fisher, and R. W. Boyd, “Nonlinear-optical response of porous-glass-based composite materials,” J. Opt. Soc. Am. B 14, 2310 (1997).
    [Crossref]
  21. E. F. Mateo and J. Liñares, “Third order nonlinear integrated device based on an effective graded-index waveguide for all-optical multistability,” Fiber and Int. Optics. To be published (2005).
    [Crossref]
  22. J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
    [Crossref]

2003 (1)

E. F. Mateo and J. Liñares, “All-optical integrated logic gates based on intensity-dependent transverse modal coupling,” Opt. Quantum Electron. 35, 1221 (2003).
[Crossref]

2002 (1)

E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
[Crossref]

2001 (2)

B. Olsson, L. Rau, and J. Blumenthal, “WDM to OTDM multiplexing using an ultrafast all-optical wavelength converter,” IEEE Photon. Technol. Lett. 13, 1905 (2001).
[Crossref]

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

2000 (3)

J. Blumenthal, B. Olsson, G. Rossi, T. E. Dimmick, L. Rau, M. Masanovic, O. Lavrova, R. Doshi, O. Jerphagnon, J. E. Bowers, V. Kaman, L. A. Coldren, and J. Barton, “All-optical label swapping networks and technologies,” J. Lightwave Technol. 18, 2058 (2000).
[Crossref]

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
[Crossref]

J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
[Crossref]

1999 (1)

K. H. Park and T. Mizumoto, “All-optical address extraction for optical routing,” Opt. Eng.,  38, 1848 (1999).
[Crossref]

1998 (1)

1997 (2)

1996 (1)

J. Liñares and M. C. Nistal, “Single local mode propagation though ion-exchanged waveguide elements with quasi-abrupt transitions,” Jpn. J. Appl. Phys. 35, L1596 (1996).
[Crossref]

1993 (1)

1992 (1)

J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
[Crossref]

1991 (1)

M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
[Crossref]

1985 (1)

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270 (1985).
[Crossref] [PubMed]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys.Rev. A 27, 3135 (1983).
[Crossref]

1981 (1)

1978 (1)

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry-Perot interferometer,” Phys. Rev. A 17, 335 (1978).
[Crossref]

Alvarellos, J. E.

J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
[Crossref]

Anderson, D.

M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
[Crossref]

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270 (1985).
[Crossref] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys.Rev. A 27, 3135 (1983).
[Crossref]

Barton, J.

Bertolotti, M.

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
[Crossref]

Blumenthal, J.

Bowers, J. E.

Boyd, R. W.

Chan, V. W. S.

Coldren, L. A.

Desaix, M.

M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
[Crossref]

Dimmick, T. E.

Doshi, R.

Felber, F. S.

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry-Perot interferometer,” Phys. Rev. A 17, 335 (1978).
[Crossref]

Fisher, G. L.

Fong, K. H.

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

Garzia, F.

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
[Crossref]

Gehr, R. J.

Glesk, I.

Hall, K. L.

Hunsperger, R. G.

R. G. Hunsperger, Integrated optics: Theory and technology (Springer-Verlag, Berlin, 1991).

Jerphagnon, O.

Kaman, V.

Kang, K.I.

Kaplan, A. E.

Lavrova, O.

Lee, H. J.

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

Li, Q. Y.

Liñares, J.

E. F. Mateo and J. Liñares, “All-optical integrated logic gates based on intensity-dependent transverse modal coupling,” Opt. Quantum Electron. 35, 1221 (2003).
[Crossref]

E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
[Crossref]

J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
[Crossref]

J. Liñares and M. C. Nistal, “Single local mode propagation though ion-exchanged waveguide elements with quasi-abrupt transitions,” Jpn. J. Appl. Phys. 35, L1596 (1996).
[Crossref]

J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
[Crossref]

E. F. Mateo and J. Liñares, “Third order nonlinear integrated device based on an effective graded-index waveguide for all-optical multistability,” Fiber and Int. Optics. To be published (2005).
[Crossref]

Lisak, M.

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270 (1985).
[Crossref] [PubMed]

Lisak, M. J.

M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
[Crossref]

Marburger, J. H.

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry-Perot interferometer,” Phys. Rev. A 17, 335 (1978).
[Crossref]

Masanovic, M.

Mateo, E. F.

E. F. Mateo and J. Liñares, “All-optical integrated logic gates based on intensity-dependent transverse modal coupling,” Opt. Quantum Electron. 35, 1221 (2003).
[Crossref]

E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
[Crossref]

E. F. Mateo and J. Liñares, “Third order nonlinear integrated device based on an effective graded-index waveguide for all-optical multistability,” Fiber and Int. Optics. To be published (2005).
[Crossref]

Mizumoto, T.

K. H. Park and T. Mizumoto, “All-optical address extraction for optical routing,” Opt. Eng.,  38, 1848 (1999).
[Crossref]

Modiano, E.

Montero, C.

E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
[Crossref]

J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
[Crossref]

Nistal, M. C.

J. Liñares and M. C. Nistal, “Single local mode propagation though ion-exchanged waveguide elements with quasi-abrupt transitions,” Jpn. J. Appl. Phys. 35, L1596 (1996).
[Crossref]

Olsson, B.

Park, K. H.

K. H. Park and T. Mizumoto, “All-optical address extraction for optical routing,” Opt. Eng.,  38, 1848 (1999).
[Crossref]

Pask, C.

Prucnal, P. R.

Rau, L.

Rauschenbach, K. A.

Righini, G. C.

J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
[Crossref]

Rossi, G.

Sammut, R. A.

Sibilia, C.

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
[Crossref]

Sotelo, D.

J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
[Crossref]

Stix, G.

G. Stix, “The triumph of light,” Scientific American, January (1998).

Tsui, V. K.

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

Yoo, J. B.

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

App. Opt. (1)

J. Liñares, G. C. Righini, and J. E. Alvarellos, “Modal coupling analysis for integrated optical components in glass and lithium niobate,” App. Opt. 31, 5292 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (2)

B. Olsson, L. Rau, and J. Blumenthal, “WDM to OTDM multiplexing using an ultrafast all-optical wavelength converter,” IEEE Photon. Technol. Lett. 13, 1905 (2001).
[Crossref]

H. J. Lee, J. B. Yoo, V. K. Tsui, and K. H. Fong, “A simple all-optical label detection and swapping technique incorporating a fiber Bragg grating filter,” IEEE Photon. Technol. Lett. 13, 635 (2001).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. A: Pure Appl. Opt. (1)

E. F. Mateo, J. Liñares, and C. Montero, “Intrinsic bistability achieved by transverse modal coupling in a nonlinear integrated device,” J. Opt. A: Pure Appl. Opt. 4, 562 (2002).
[Crossref]

J. Opt. Soc. Am. B (2)

Jpn. J. Appl. Phys. (1)

J. Liñares and M. C. Nistal, “Single local mode propagation though ion-exchanged waveguide elements with quasi-abrupt transitions,” Jpn. J. Appl. Phys. 35, L1596 (1996).
[Crossref]

Opt. Commun. (1)

J. Liñares, C. Montero, and D. Sotelo, “Theory and design of an integrated optical sensor based on planar waveguiding lenses,” Opt. Commun. 180, 29 (2000).
[Crossref]

Opt. Eng. (1)

K. H. Park and T. Mizumoto, “All-optical address extraction for optical routing,” Opt. Eng.,  38, 1848 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

E. F. Mateo and J. Liñares, “All-optical integrated logic gates based on intensity-dependent transverse modal coupling,” Opt. Quantum Electron. 35, 1221 (2003).
[Crossref]

F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical serial switcher,” Opt. Quantum Electron. 32, 781 (2000).
[Crossref]

Opt. Soc. Am. B (1)

M. Desaix, D. Anderson, and M. J. Lisak, “Variational approach to collapse of optical pulses,” Opt. Soc. Am. B 8, 2082 (1991).
[Crossref]

Phys. Rev. A (2)

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry-Perot interferometer,” Phys. Rev. A 17, 335 (1978).
[Crossref]

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270 (1985).
[Crossref] [PubMed]

Phys.Rev. A (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys.Rev. A 27, 3135 (1983).
[Crossref]

Other (3)

R. G. Hunsperger, Integrated optics: Theory and technology (Springer-Verlag, Berlin, 1991).

G. Stix, “The triumph of light,” Scientific American, January (1998).

E. F. Mateo and J. Liñares, “Third order nonlinear integrated device based on an effective graded-index waveguide for all-optical multistability,” Fiber and Int. Optics. To be published (2005).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Sketch of the intensity-dependent routing operation where HE-D represents the Header Extraction Device, OB-D is the Optical Bistable Device and DR-D is the Data Routing Device. In figure (A) is represented the operation under the presence of a 1-valued header bit whereas in figure (B) is represented the operation for a 0-valued header bit.

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Fig. 2.
Fig. 2. Transverse view of the waveguide.

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Fig. 3.
Fig. 3. Top view of the header extraction device where the variational evolution of the header (solid) and data (dashed) Gaussian beam widths are shown.

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Fig. 4.
Fig. 4. Variational results for the data (A) and header (B) beam propagation.

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Fig. 5.
Fig. 5. Sketch of the bistable device showing the propagation behaviour of the counterpropagating beams in cases of low (A) and high (B) transmission output.

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Fig. 6.
Fig. 6. Optical bistability plot showing the bi-valued points for routing operation at pin =0.55, where A is the output power value for a header bit “0”, and B is for the header value “1”.

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Fig. 7.
Fig. 7. Integrated device performing the data routing under the absence (A) and presence (B) of the pump wave. The variational evolution of the pump (dashed) and data (solid) are shown for each case.

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Fig. 8.
Fig. 8. Propagation plots (top view) for the data routing operation. (A) Data wave propagation under the absence of pump (header bit “0”); (B) Data and pump wave propagation under the presence of pump (header bit “1”).

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Fig. 9.
Fig. 9. Scheme of the cascaded configuration for multi-bit header routing where a [0,1] header routes the data packet onto its correspondent output channel.

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Fig. 10.
Fig. 10. Sketch of a multilens with two EFT modules.

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Tables (1)

Tables Icon

Table 1. Transverse coupling efficiencies of the data and pump beams onto the output optical fibers.

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Equations (48)

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∇ 2 ℰ → ( r → , t ) = μ 0 ∂ 2 ∂ t 2 [ ε 0 n l 2 ( x , y ) + 3 ε 0 χ Re ( 3 ) ( x , y ) ℰ 2 ] ℰ → .
ℰ ( x , y , z , t ) = 1 2 φ ( y ) [ ψ 1 ( x , z ) exp ( i β 1 z ) + ψ 2 ( x , z ) exp ( i β 2 z ) ] exp ( − i ω 0 t ) + c c .
2 i β α ∂ ψ α ∂ z + ∂ 2 ψ α ∂ x 2 + k 2 [ ∫ Δ n ( x , y ) φ 2 d y ] ψ α +
3 4 k 2 [ ∫ χ Re ( 3 ) ( x , y ) φ 4 d y ] ( ∣ ψ α ∣ 2 + J ∣ ψ 3 − α ∣ 2 ) ψ α = 0 ,
2 i β α ∂ ψ α ∂ z + ∂ 2 ψ α ∂ x 2 − G 2 x 2 ψ α + k 2 n ˜ k 0 ( 1 + Q 2 x 2 ) ( ∣ ψ α ∣ 2 + J ∣ ψ 3 − α ∣ 2 ) ψ α = 0 ,
ψ α ( x , z ) = E 0 α exp [ − ( x − x α ) 2 a 2 w α 2 ] exp [ i β α ρ α ( x − x α ) 2 + i V α ( x − x α ) ] .
d 2 w α d τ 2 − 1 w α 3 + g w α + 1 2 p α 1 w α 2 ( 1 − q w α 2 + Q 2 x α 2 ) + 2 J p 3 − α 1 w α 2 ×
( 1 + w 3 − α 2 w α 2 ) − 3 ⁄ 2 ( K 0 α − q K 1 α + Q 2 K 2 α ) exp [ − 2 ( x α − x 3 − α ) 2 a 2 w α 2 + a 2 w 3 − α 2 ] = 0 ,
d 2 x α d τ 2 + g x α − 2 p α q x α 1 w α + J 2 p 3 − α ( D 0 α − q D 1 α ) exp [ − 2 ( x α − x 3 − α ) 2 a 2 w α 2 + a 2 w 3 − α 2 ] = 0 ,
V α = β α d x α d τ ,
K 0 α = 1 − 4 ( x α − x 3 − α ) 2 a 2 w α 2 + a 2 w 3 − α 2 , K 1 α = 2 w 3 − α 2 − 3 w α 2 w 3 − α 2 w α 2 + w 3 − α 2 ,
K 2 α = − 4 x 3 − α ( w α 2 x 3 − α + w 3 − α 2 x α ) w α 2 + w 3 − α 2 + 4 ( w α 2 x 3 − α + w 3 − α 2 x α ) 2 ( x α − x 3 − α ) 2 a 2 ( w α 2 + w 3 − α 2 ) 3
+ 5 ( w α 2 x 3 − α + w 3 − α 2 x α ) + w α 2 w 3 − α 2 ( x α − x 3 − α ) 2 ( w α 2 + w 3 − α 2 ) 2 .
D 0 α = ( x α − x 3 − α ) ( w α 2 + w 3 − α 2 ) 3 ⁄ 2 , D 1 α = 8 w 3 − α 2 ( w α 2 x 3 − α + w 3 − α 2 x α ) ( w α 2 + w 3 − α 2 ) 5 ⁄ 2
− 4 ( x α − x 3 − α ) ( w α 2 + w 3 − α 2 ) 5 ⁄ 2 − 4 ( w α 2 x 3 − α + w 3 − α 2 x α ) 2 ( x α − x 3 − α ) a 2 ( w α 2 + w 3 − α 2 ) 7 ⁄ 2 .
d 2 w h d τ 2 − 1 w h 3 + g w h + 1 2 p h 1 w h 2 ( 1 − q w h 2 + Q 2 x h 2 ) = 0 ,
d 2 x h d τ 2 + g x h − 2 p q x h 1 w h = 0 .
d 2 w d d τ 2 − 1 w d 3 + g w d = 0 ,
d 2 x d d τ 2 + g x d = 0 .
p st = 2 ( 1 − g ) 1 − q = 16 − 4 G 2 α 4 8 − Q 2 α 2 .
Λ l = 2 π g 1 ⁄ 2 = 4 π G a 2 .
d 2 w + 1 d τ 2 − 1 w + 1 3 + 1 2 p + 1 1 w + 1 2 + 2 2 p − 1 1 w + 1 2 ( 1 + w − 1 2 w + 1 2 ) − 3 ⁄ 2 = 0 ,
d 2 w − 1 d τ 2 − 1 w − 1 3 + 1 2 p − 1 1 w − 1 2 + 2 2 p + 1 1 w − 1 2 ( 1 + w + 1 2 w − 1 2 ) − 3 ⁄ 2 = 0 .
η 3 ( p + 1 , p − 1 ) = 2 w + 1 ( z 2 , p + 1 , p − 1 ) 1 + w + 1 2 ( z 2 , p + 1 , p − 1 ) ,
p − 1 = η 3 ( p + 1 , p − 1 ) R p + 1 .
η 0 ( p − 1 , p + 1 ) = 2 w − 1 ( z 1 , p − 1 , p + 1 ) 1 + w − 1 2 ( z 1 , p − 1 , p + 1 ) ,
p out = p − 1 η 0 ( p + 1 , p − 1 ) .
d 2 w d 0 d τ 2 − 1 w d 0 3 + g w d 0 = 0 , d 2 x d 0 d τ 2 + g x d 0 = 0 , V d 0 = β 0 d x d 0 d τ .
d 2 w p 1 d τ 2 − 1 w p 1 3 + g w p 1 + 1 2 p p 1 1 w p 1 2 ( 1 − q w p 1 2 + Q 2 x p 1 2 ) ,
d 2 x p 1 d τ 2 + g x p 1 − 2 p p 1 q x p 1 1 w p 1 = 0 , V p 1 = β 0 d x p 1 d τ .
d 2 w d 1 d τ 2 − 1 w d 1 3 + g w d 1 +
+ 2 p p 1 1 w d 1 2 ( 1 + w p 1 2 w d 1 2 ) − 3 ⁄ 2 ( K 0 + q K 1 − Q K 2 ) exp [ − 2 ( x d 1 − x p 1 ) 2 a 2 w d 1 2 + a 2 w p 1 2 ] = 0 ,
d 2 x d 1 d τ 2 + g x d 1 + 1 2 p p 1 ( D 0 − q D 1 ) exp [ − 2 ( x d 1 − x p 1 ) 2 a 2 w d 1 2 + a 2 w p 1 2 ] = 0 ,
V d 1 = β 0 d x d 1 d τ ,
a w d 1 ( z 3 ) = w g , x d 1 ( z 3 ) = − δ .
a w d 1 ( z 2 ) = 2 B k 1 w g , V d 1 ( z 2 ) = k δ B .
a w d 1 ( z 2 ) V d 1 ( z 2 ) = − 2 δ w g = − 10 , B / k = 1 2 a w d 1 ( z 2 ) w g .
a w d 0 ( z 3 ) = 2 B k 1 a w d 0 ( z 2 ) = w d 1 ( z 2 ) w g w d 0 ( z 2 ) , x d 0 ( z 3 ) = 0 , V d 0 ( z 3 ) = 0 ,
a w p 1 ( z 3 ) = 2 B k 1 a w p 1 ( z 2 ) = w d 1 ( z 2 ) w g w p 1 ( z 2 ) , x p 1 ( z 3 ) = B k x p 1 ( z 2 ) = 1 2 a w d 1 ( z 2 ) w g V p 1 ( z 2 ) ,
V p 1 ( z 3 ) = − k B x p 1 ( z 2 ) = 2 a w d 1 ( z 2 ) w g x p 1 ( z 2 ) .
ψ σ ( x ) = 2 1 ⁄ 4 ( π a 2 w σ 2 ) 1 ⁄ 4 exp [ − ( x − x σ ) 2 a 2 w σ 2 ] exp [ i V σ ( x − x σ ) ] ,
φ g ( x ) = 2 1 ⁄ 4 ( π w g 2 ) 1 ⁄ 4 exp [ − ( x − Δ ) 2 w g 2 ] .
η σ = ∣ ∫ − ∞ ∞ ψ σ ( x ) φ g ( x ) d x ∣ 2 = 2 a w σ w g a 2 w σ 2 + w g 2 exp [ − 1 2 a 2 w σ 2 w g 2 V σ 2 + 4 ( x σ − Δ ) 2 a 2 w σ 2 + w g 2 ] ,
N o t j = N i l j , m j cos − 1 ( 1 − t j N i f j ) = π 2 ,
f j = N o N i − N o 1 t j [ t j 2 + ( A j ⁄ 2 ) 2 ] .
a w j = 2 B j k 1 a w j − 1 , x j = B j k V j − 1 and V j = − k B j x j − 1 ,
B j = [ 2 t j f j N i − t j 2 N o 2 ] .
a w 2 = B 2 B 1 a w 0 , x 2 = − B 2 B 1 x 0 and V 2 = − B 1 B 2 V 0 .

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