Abstract

We present a theoretical analysis of recently demonstrated ultrafast all-optical interferometric switching devices (based on Sagnac and Mach–Zehnder interferometers) that use a large optical nonlinearity in a resonant regime. These devices achieve ∼10-ps switching windows and do not require high-energy optical control pulses. We theoretically analyze and compare one Sagnac and two Mach–Zehnder switching configurations.

© 1996 Optical Society of America

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  1. A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
    [CrossRef]
  2. K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
    [CrossRef]
  3. H. Kawaguchi, “Proposal for a new all-optical waveguide functional device,” Opt. Lett. 10, 411–413 (1985).
    [CrossRef] [PubMed]
  4. I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
    [CrossRef]
  5. S. R. Friber, A. M. Weiner, Y. Silberberg, G. G. Sfez, P. S. Smith, “Femtosecond switching in a dual-core nonlinear coupler,” Opt. Lett. 13, 904–906 (1988).
    [CrossRef]
  6. K. Otsuka, “Nonlinear antiresonant ring interferometer,” Opt. Lett. 8, 471–473 (1983).
    [CrossRef] [PubMed]
  7. N. J. Doran, D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
    [CrossRef] [PubMed]
  8. N. J. Doran, D. S. Forrester, B. K. Nayar, “Experimental investigation of all-optical switching in fibre loop mirror device,” Electron. Lett. 25, 267–269 (1989).
    [CrossRef]
  9. K. J. Blow, N. J. Doran, B. K. Nayar, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756 (1989);M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
    [CrossRef] [PubMed]
  10. M. C. Farries, D. N. Payne, “Optical fiber switch employing a Sagnac interferometer,” Appl. Phys. Lett. 55, 25–26 (1989);K. J. Blow, N. J. Doran, B. K. Nayar, B. P. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
    [CrossRef] [PubMed]
  11. M. Eiselt, “Optical loop mirror with semiconductor laser amplifier,” Electron. Lett. 28, 1505–1507 (1992).
    [CrossRef]
  12. J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
    [CrossRef]
  13. I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
    [CrossRef]
  14. K. Tajima, “All-optical switch with switch-off time unrestricted by carrier lifetime,” Jpn. J. Appl. Phys. 32, L1746– L1749 (1993).
    [CrossRef]
  15. S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
    [CrossRef]
  16. K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
    [CrossRef]
  17. M. G. Kane, I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Asymmetric optical loop mirror analysis of an all-optical switch,” Appl. Opt. 33, 6833–6842 (1994).
    [CrossRef] [PubMed]
  18. This device has been named a Symmetric Mach–Zehnder (SMZ). However, the device can be operated in an asymmetric configuration as well.

1995 (1)

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

1994 (2)

M. G. Kane, I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Asymmetric optical loop mirror analysis of an all-optical switch,” Appl. Opt. 33, 6833–6842 (1994).
[CrossRef] [PubMed]

S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
[CrossRef]

1993 (3)

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
[CrossRef]

K. Tajima, “All-optical switch with switch-off time unrestricted by carrier lifetime,” Jpn. J. Appl. Phys. 32, L1746– L1749 (1993).
[CrossRef]

1992 (1)

M. Eiselt, “Optical loop mirror with semiconductor laser amplifier,” Electron. Lett. 28, 1505–1507 (1992).
[CrossRef]

1989 (3)

N. J. Doran, D. S. Forrester, B. K. Nayar, “Experimental investigation of all-optical switching in fibre loop mirror device,” Electron. Lett. 25, 267–269 (1989).
[CrossRef]

K. J. Blow, N. J. Doran, B. K. Nayar, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756 (1989);M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
[CrossRef] [PubMed]

M. C. Farries, D. N. Payne, “Optical fiber switch employing a Sagnac interferometer,” Appl. Phys. Lett. 55, 25–26 (1989);K. J. Blow, N. J. Doran, B. K. Nayar, B. P. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
[CrossRef] [PubMed]

1988 (3)

N. J. Doran, D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
[CrossRef] [PubMed]

I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
[CrossRef]

S. R. Friber, A. M. Weiner, Y. Silberberg, G. G. Sfez, P. S. Smith, “Femtosecond switching in a dual-core nonlinear coupler,” Opt. Lett. 13, 904–906 (1988).
[CrossRef]

1985 (2)

K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

H. Kawaguchi, “Proposal for a new all-optical waveguide functional device,” Opt. Lett. 10, 411–413 (1985).
[CrossRef] [PubMed]

1983 (2)

K. Otsuka, “Nonlinear antiresonant ring interferometer,” Opt. Lett. 8, 471–473 (1983).
[CrossRef] [PubMed]

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

Blow, K. J.

Boncek, R. K.

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

Chang, T. G.

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

Doran, N. J.

Eiselt, M.

M. Eiselt, “Optical loop mirror with semiconductor laser amplifier,” Electron. Lett. 28, 1505–1507 (1992).
[CrossRef]

Epworth, R. E.

I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
[CrossRef]

Farries, M. C.

M. C. Farries, D. N. Payne, “Optical fiber switch employing a Sagnac interferometer,” Appl. Phys. Lett. 55, 25–26 (1989);K. J. Blow, N. J. Doran, B. K. Nayar, B. P. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
[CrossRef] [PubMed]

Forrester, D. S.

N. J. Doran, D. S. Forrester, B. K. Nayar, “Experimental investigation of all-optical switching in fibre loop mirror device,” Electron. Lett. 25, 267–269 (1989).
[CrossRef]

Friber, S. R.

Glesk, I.

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

M. G. Kane, I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Asymmetric optical loop mirror analysis of an all-optical switch,” Appl. Opt. 33, 6833–6842 (1994).
[CrossRef] [PubMed]

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
[CrossRef]

Haus, H. A.

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

Ippen, E. P.

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

Kane, M.

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

Kane, M. G.

Kang, K. I.

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

Kawaguchi, H.

Kimura, Y.

K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Kitayama, K.

K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Lattes, A.

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

Leonberger, F. J.

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

Nakamura, S.

S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
[CrossRef]

Nayar, B. K.

Otsuka, K.

Payne, D. N.

M. C. Farries, D. N. Payne, “Optical fiber switch employing a Sagnac interferometer,” Appl. Phys. Lett. 55, 25–26 (1989);K. J. Blow, N. J. Doran, B. K. Nayar, B. P. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
[CrossRef] [PubMed]

Penty, R. V.

I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
[CrossRef]

Prucnal, I. P. R.

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

Prucnal, P. R.

M. G. Kane, I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Asymmetric optical loop mirror analysis of an all-optical switch,” Appl. Opt. 33, 6833–6842 (1994).
[CrossRef] [PubMed]

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
[CrossRef]

Seikai, S.

K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Sfez, G. G.

Silberberg, Y.

Smith, P. S.

Sokoloff, J. P.

M. G. Kane, I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Asymmetric optical loop mirror analysis of an all-optical switch,” Appl. Opt. 33, 6833–6842 (1994).
[CrossRef] [PubMed]

I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
[CrossRef]

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

Sugimoto, Y.

S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
[CrossRef]

Tajima, K.

S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
[CrossRef]

K. Tajima, “All-optical switch with switch-off time unrestricted by carrier lifetime,” Jpn. J. Appl. Phys. 32, L1746– L1749 (1993).
[CrossRef]

Weiner, A. M.

White, I. H.

I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
[CrossRef]

Wood, D.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

S. Nakamura, K. Tajima, Y. Sugimoto, “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach–Zehnder all-optical switch,” Appl. Phys. Lett. 65, 283–285 (1994).
[CrossRef]

M. C. Farries, D. N. Payne, “Optical fiber switch employing a Sagnac interferometer,” Appl. Phys. Lett. 55, 25–26 (1989);K. J. Blow, N. J. Doran, B. K. Nayar, B. P. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
[CrossRef] [PubMed]

K. Kitayama, Y. Kimura, S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Electron. Lett. (5)

N. J. Doran, D. S. Forrester, B. K. Nayar, “Experimental investigation of all-optical switching in fibre loop mirror device,” Electron. Lett. 25, 267–269 (1989).
[CrossRef]

I. H. White, R. V. Penty, R. E. Epworth, “Demonstration of the optical Kerr effect in an all-fibre Mach–Zehnder interferometer at laser diode powers,” Electron. Lett. 24, 340–341 (1988).
[CrossRef]

M. Eiselt, “Optical loop mirror with semiconductor laser amplifier,” Electron. Lett. 28, 1505–1507 (1992).
[CrossRef]

I. Glesk, J. P. Sokoloff, P. R. Prucnal, “Demonstration of all-optical demultiplexing of TDM data at 250 Gbit/s,” Electron. Lett. 30, 339–440 (1993).
[CrossRef]

K. I. Kang, I. Glesk, T. G. Chang, I. P. R. Prucnal, R. K. Boncek, “Demonstration of all optical Mach–Zehnder demultiplexer,” Electron. Lett. 31, 749–750 (1995);K. I. Kang, T. G. Chang, I. Glesk, P. R. Prucnal, R. K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integratable switch,” Appl. Phys. Lett. 67, 605–607 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Lattes, H. A. Haus, F. J. Leonberger, E. P. Ippen, “An ultrafast all-optical gate,” IEEE J. Quantum Electron. QE-19, 1718–1723(1983).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. P. Sokoloff, P. R. Prucnal, I. Glesk, M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett. 5, 787–790 (1993).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Tajima, “All-optical switch with switch-off time unrestricted by carrier lifetime,” Jpn. J. Appl. Phys. 32, L1746– L1749 (1993).
[CrossRef]

Opt. Lett. (5)

Other (1)

This device has been named a Symmetric Mach–Zehnder (SMZ). However, the device can be operated in an asymmetric configuration as well.

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Figures (8)

Fig. 1
Fig. 1

Switching device (a) is based on the Sagnac interferometer, and devices (b) and (c) are based on the Mach–Zehnder interferometer. The control and the data pulses travel in opposite directions in the CPMZ but travel in the same directions in the TPMZ. In the analysis presented in this paper, the nonlinear optical elements (NLE's) are the semiconductor optical amplifiers (SOA's).

Fig. 2
Fig. 2

Temporal responses for (a) the gain and (b) the phase for a nonlinear optical element. The assumed initial gain is 10, and the maximum gain compression is 5, with a recovery of 400 ps. The transition period is approximately 2 ps for copropagation (solid curves) and 11 ps for counterpropagation (dashed curves). A similar temporal dependence for the phase response has been assumed.

Fig. 3
Fig. 3

Switching windows for the TOAD with (b) output signals and (c) conjugate signal, for a given geometric configuration (a). The x axis in both graphs (b) and (c) is the time delay between the control and the data pulses, and the y axis is the magnitude of the transfer. The scales are shown at the top of the figure.

Fig. 4
Fig. 4

Switching windows for the CPMZ with (b) output signals and (c) a conjugate signal for a given geometric configuration (a). The scales are shown at the top of the figure.

Fig. 5
Fig. 5

Switching windows for the TPMZ with (b) output signals and (c) a conjugate signal for a given time delay of the control pulses (a). The scales are shown at the top of the figure.

Fig. 6
Fig. 6

Temporal response of the (a) gain and (b) phase at the time of zero switching-window offset for the TOAD. The dashed curves indicate counterpropagation of the control and data signals, and the solid curves indicate copropagation. The expected switching window is shown in (c), with the output signal represented by the solid curve and the conjugate signal by the dashed curve.

Fig. 7
Fig. 7

Temporal response of the (a) gain and (b) phase for a 4-ps switching-window offset for the CPMZ. The switching-window offset is determined by the two similar responses represented by the solid and the dashed curves. The expected switching window is shown in (c), with the output signal represented by the solid curve and the conjugate signal represented by the dashed curve.

Fig. 8
Fig. 8

Switching window of the output signal from the TOAD with a recovery time of 50 ps. Switching-window offsets are (a) 10 ps, (b) –10 ps, and (c) 0 ps.

Equations (9)

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E ( t ) = E 0 ( t ) 2 exp [ i β z i ω t + i ϕ ( t ) ] ,
E ( t ) = G ( t ) E 0 ( t ) 2 exp [ i β z i ω t + i ϕ ( t ) ] ,
I out ( t ) = I in ( t ) 4 { G 1 ( t ) + G 2 ( t ) ± 2 [ G 1 ( t ) G 2 ( t ) ] 1 / 2 × cos [ ϕ 1 ( t ) ϕ 2 ( t ) ] } ,
S ( t ) = 0.25 { G 1 ( t ) + G 2 ( t ) + 2 [ G 1 ( t ) G 2 ( t ) ] 1 / 2 × cos [ ϕ 1 ( t ) ϕ 2 ( t ) ] } .
G ( t ) = G 0 ( Δ G ) R ( t ) P ( t t ) d t ,
ϕ ( t ) = ( Δ ϕ ) R ( t ) P ( t t ) d t ,
P ( t t ) = 1 σ 2 π exp [ 1 2 ( t t σ ) 2 ] ,
R ( t ) = θ ( t ) exp ( t τ ) ,
R ( t ) = exp ( t / τ ) l l / 2 l / 2 θ ( x + υ t 2 ) exp [ 2 x / ( υ τ ) ] d x .

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