Abstract

The application of grating-coupled waveguides to spread spectrum fiber-optic communications systems is described. It is first shown that such a device can act as its own matched filter for impulsive excitation, which allows its use as an encoder and decoder of ultrashort pulses. This is true even in the case of strong coupling, although considerable pulse distortion occurs. A typical performance that might be expected from existing technology is then outlined, and a number of numerical examples are presented to illustrate typical signal processing operations.

© 1991 Optical Society of America

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References

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  1. R. C. Dixon, Spread Spectrum Systems (Wiley, New York, 1984).
  2. H. Taub, D. L. Schilling, Principles of Communications Systems (McGraw-Hill, New York, 1986).
  3. P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
    [CrossRef]
  4. P. R. Prucnal, D. J. Blumenthal, A. Perrier, “Self-routing optical switch with optical processing,” in Technical Digest, Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper ThB4-1.
  5. A. S. Holmes, R. R. A. Syms, “A self-aligning guided wave system for delay line signal processing,” presented at the Seventh International Conference on Fibre Optics and Optoelectronics, London, 25–27 April 1989.
  6. E. Marom, O. G. Ramer, “Encoding-decoding optical fibre network,” Electron. Lett. 14, 48–49 (1978).
    [CrossRef]
  7. E. Marom, “Optical delay line matched filters,” IEEE Trans. Circuits Syst. CAS-25, 360–364 (1978).
    [CrossRef]
  8. D. P. Morgan, Surface Wave Devices for Signal Processing (Elsevier, Amsterdam, 1985).
  9. R. C. Alferness, L. L. Buhl, “Tunable electro-optic TE–TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
    [CrossRef]
  10. R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE–TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
    [CrossRef]
  11. J. S. Wilkinson, M. G. F. Wilson, “The experimental evaluation of a directional coupler wavelength filter having a periodic electrode,” in Proceedings, IEEE International Workshop on Integrated Optical and Related Technologies for Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 107–110.
  12. D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 139–145 (1986).
  13. W.-P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” IEEE J. Lightwave Technol. LT-7, 920–924 (1989).
    [CrossRef]
  14. A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
    [CrossRef]
  15. R. R. A. Syms, “Path integral formulation of nonuniform optical coupled wave problems,” Appl. Opt. 26, 4220–4230 (1987).
    [CrossRef] [PubMed]
  16. H. Kogelnik, R. V. Schmidt, “switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
    [CrossRef]
  17. A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  18. R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
    [CrossRef]

1989 (1)

W.-P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” IEEE J. Lightwave Technol. LT-7, 920–924 (1989).
[CrossRef]

1987 (1)

1986 (3)

D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 139–145 (1986).

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

1984 (1)

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

1982 (1)

R. C. Alferness, L. L. Buhl, “Tunable electro-optic TE–TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

1981 (1)

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE–TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

1978 (2)

E. Marom, O. G. Ramer, “Encoding-decoding optical fibre network,” Electron. Lett. 14, 48–49 (1978).
[CrossRef]

E. Marom, “Optical delay line matched filters,” IEEE Trans. Circuits Syst. CAS-25, 360–364 (1978).
[CrossRef]

1976 (1)

H. Kogelnik, R. V. Schmidt, “switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

1964 (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Alferness, R. C.

R. C. Alferness, L. L. Buhl, “Tunable electro-optic TE–TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE–TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

Blumenthal, D. J.

P. R. Prucnal, D. J. Blumenthal, A. Perrier, “Self-routing optical switch with optical processing,” in Technical Digest, Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper ThB4-1.

Booth, R. C.

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

Buhl, L. L.

R. C. Alferness, L. L. Buhl, “Tunable electro-optic TE–TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE–TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

Daymond-John, D. E.

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

Dixon, R. C.

R. C. Dixon, Spread Spectrum Systems (Wiley, New York, 1984).

Fan, T. R.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

Hardy, A.

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

Haus, H. A.

W.-P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” IEEE J. Lightwave Technol. LT-7, 920–924 (1989).
[CrossRef]

Holmes, A. S.

A. S. Holmes, R. R. A. Syms, “A self-aligning guided wave system for delay line signal processing,” presented at the Seventh International Conference on Fibre Optics and Optoelectronics, London, 25–27 April 1989.

Huang, W.-P.

W.-P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” IEEE J. Lightwave Technol. LT-7, 920–924 (1989).
[CrossRef]

Kogelnik, H.

H. Kogelnik, R. V. Schmidt, “switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

Marom, E.

E. Marom, O. G. Ramer, “Encoding-decoding optical fibre network,” Electron. Lett. 14, 48–49 (1978).
[CrossRef]

E. Marom, “Optical delay line matched filters,” IEEE Trans. Circuits Syst. CAS-25, 360–364 (1978).
[CrossRef]

Morgan, D. P.

D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 139–145 (1986).

D. P. Morgan, Surface Wave Devices for Signal Processing (Elsevier, Amsterdam, 1985).

Perrier, A.

P. R. Prucnal, D. J. Blumenthal, A. Perrier, “Self-routing optical switch with optical processing,” in Technical Digest, Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper ThB4-1.

Prucnal, P. R.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

P. R. Prucnal, D. J. Blumenthal, A. Perrier, “Self-routing optical switch with optical processing,” in Technical Digest, Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper ThB4-1.

Ramer, O. G.

E. Marom, O. G. Ramer, “Encoding-decoding optical fibre network,” Electron. Lett. 14, 48–49 (1978).
[CrossRef]

Santoro, M. A.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

Schilling, D. L.

H. Taub, D. L. Schilling, Principles of Communications Systems (McGraw-Hill, New York, 1986).

Schmidt, R. V.

H. Kogelnik, R. V. Schmidt, “switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

Sturges, P. E.

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, “Path integral formulation of nonuniform optical coupled wave problems,” Appl. Opt. 26, 4220–4230 (1987).
[CrossRef] [PubMed]

A. S. Holmes, R. R. A. Syms, “A self-aligning guided wave system for delay line signal processing,” presented at the Seventh International Conference on Fibre Optics and Optoelectronics, London, 25–27 April 1989.

Taub, H.

H. Taub, D. L. Schilling, Principles of Communications Systems (McGraw-Hill, New York, 1986).

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Wilkinson, J. S.

J. S. Wilkinson, M. G. F. Wilson, “The experimental evaluation of a directional coupler wavelength filter having a periodic electrode,” in Proceedings, IEEE International Workshop on Integrated Optical and Related Technologies for Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 107–110.

Wilson, M. G. F.

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

J. S. Wilkinson, M. G. F. Wilson, “The experimental evaluation of a directional coupler wavelength filter having a periodic electrode,” in Proceedings, IEEE International Workshop on Integrated Optical and Related Technologies for Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 107–110.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. C. Alferness, L. L. Buhl, “Tunable electro-optic TE–TM converter/wavelength filter,” Appl. Phys. Lett. 40, 861–862 (1982).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Polarization independent optical filter using interwaveguide TE–TM conversion,” Appl. Phys. Lett. 39, 131–134 (1981).
[CrossRef]

Electron. Lett. (2)

E. Marom, O. G. Ramer, “Encoding-decoding optical fibre network,” Electron. Lett. 14, 48–49 (1978).
[CrossRef]

R. C. Booth, D. E. Daymond-John, P. E. Sturges, M. G. F. Wilson, “Temperature tuning of LiNbO3 electro-optic waveguide TE/TM mode converters,” Electron. Lett. 20, 1045–1046 (1984).
[CrossRef]

IEEE J. Lightwave Technol. (3)

W.-P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” IEEE J. Lightwave Technol. LT-7, 920–924 (1989).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” IEEE J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Kogelnik, R. V. Schmidt, “switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

E. Marom, “Optical delay line matched filters,” IEEE Trans. Circuits Syst. CAS-25, 360–364 (1978).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. P. Morgan, “Spatially-variant coupling design for co-directional mode-converting bandpass filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 139–145 (1986).

Other (6)

D. P. Morgan, Surface Wave Devices for Signal Processing (Elsevier, Amsterdam, 1985).

J. S. Wilkinson, M. G. F. Wilson, “The experimental evaluation of a directional coupler wavelength filter having a periodic electrode,” in Proceedings, IEEE International Workshop on Integrated Optical and Related Technologies for Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 107–110.

P. R. Prucnal, D. J. Blumenthal, A. Perrier, “Self-routing optical switch with optical processing,” in Technical Digest, Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper ThB4-1.

A. S. Holmes, R. R. A. Syms, “A self-aligning guided wave system for delay line signal processing,” presented at the Seventh International Conference on Fibre Optics and Optoelectronics, London, 25–27 April 1989.

R. C. Dixon, Spread Spectrum Systems (Wiley, New York, 1984).

H. Taub, D. L. Schilling, Principles of Communications Systems (McGraw-Hill, New York, 1986).

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Figures (12)

Fig. 1
Fig. 1

All-optical encoding and decoding scheme for spread-spectrum communications.

Fig. 2
Fig. 2

Code generation using cascaded Mach–Zehnder interferometers.

Fig. 3
Fig. 3

(a) TE–TM mode converter filter, (b) an interwaveguide mode converter filter, and (c) a phase-matching diagram for grating-assisted coupling between codirectional modes.

Fig. 4
Fig. 4

Geometry assumed for theoretical analysis: (a) the forward-traveling case and (b) the backward-traveling case.

Fig. 5
Fig. 5

Electrically programmable filter based on independently controlled, cascaded sections: (a) the forward-traveling case and (b) the backward-traveling case.

Fig. 6
Fig. 6

Impulse response of a weakly coupled, single-section filter. The start of the trace is at tmin = n1dT/c = 36.667 ps; the trace ends at tmax = n2dT/c = 38.1 ps.

Fig. 7
Fig. 7

Impulse response of a weakly coupled, seven-section filter designed as a generator of the code 〈1000100010001〉. The start of the trace is at tmin = 476.66 ps; the trace ends at tmax = 495.3 ps.

Fig. 8
Fig. 8

Autocorrelation of the single pulse shown in Fig. 6. The start of the trace is at tmin = 2n1dT/c = 73.334 ps; the pulse ends at tmax = 2n2dT/c= 76.2 ps.

Fig. 9
Fig. 9

Autocorrelation of the pulse code shown in Fig. 7. The start of the trace is at tmin = 953.32 ps; the trace ends at tmax = 990.6 ps.

Fig. 10
Fig. 10

Autocorrelation of the pulse code corresponding to the sequence 〈 10001000 − 1000 − 1〉. The start of the trace is at tmin = 953.32 ps; the trace ends at tmax = 990.6 ps.

Fig. 11
Fig. 11

Impulse response of a strongly coupled, single-section device. The start of the trace is at tmin = 36.667 ps; the trace ends at tmax = 38.1 ps.

Fig. 12
Fig. 12

Impulse response of a strongly coupled, five-section filter, designed as a generator of the code 〈1000100010001〉. The start of the trace is at tmin = n1dT/c = 440 ps; the trace ends at tmax = n2dT/c = 457.2 ps.

Equations (41)

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n ( x , y , z ) = n T ( x , y ) + Δ n ( x , y , z ) cos [ K z + ϕ ( z ) ] .
2 E i ( x , y , z ) + n i 2 ( x , y ) k 0 2 E i ( x , y , z ) = 0 ,
E i ( x , y , z ) + E T i ( x , y , ) exp ( j β i z ) ,
x y 2 E T i ( x , y ) + [ n i 2 ( x , y ) k 0 2 β i 2 ] E T i ( x , y ) = 0 ,
E T i , E T i = all space E T i E T i * dxdy = 1 .
K = β 1 β 2 + Δ β ,
2 E ( x , y , z ) + { n T 2 ( x , y ) + 2 n T ( x , y ) Δ n ( x , y , z ) × cos [ K z + ϕ ( z ) ] } k 0 2 E ( x , y , z ) = 0 .
E ( x , y , z ) = A 1 ( z ) E T 1 ( x , y ) exp ( j β 1 z ) + A 2 ( z ) E T 2 ( x , y ) exp ( j β 2 z ) ,
d A 1 / d z + j κ ( z ) exp { j [ ϕ ( z ) + Δ β z ] } A 2 = 0 , d A 2 / d z + j κ ( z ) exp { + j [ ϕ ( z ) + Δ β z ] } A 1 = 0 ,
κ ( z ) = ( k 0 2 / 2 β 0 ) n T Δ n E 2 , E 1 * .
d A 1 / d z + j c ( z ) A 2 = 0 , d A 2 / d z + j c * ( z ) A 1 = 0 ,
E ( x , y , z ) = A 1 ( z ) E T 1 ( x , y ) exp ( + j β 1 z ) + A 2 ( z ) E T 2 ( x , y ) × exp ( + j β 2 z ) .
d A 1 / d z j c * ( z ) A 2 = 0 , d A 2 / d z j c ( z ) A 1 = 0 .
A 2 ( d ) A 1 ( 0 ) { j 0 d c * ( z 1 ) d z 1 + j 0 d 0 z 1 0 z 2 c * ( z 1 ) c ( z 2 ) c * ( z 3 ) d z 3 d z 2 d z 1 j 0 d 0 z 1 0 z 2 0 z 3 0 z 4 c * ( z 1 ) c ( z 2 ) c * ( z 3 ) c ( z 4 ) × c * ( z 5 ) d z 5 d z 4 d z 3 d z 2 d z 1 + } .
A 2 ( d ) A 1 ( 0 ) [ j 0 d c * ( z 1 ) d z 1 ] .
A 2 ( 0 ) = A 1 ( d ) { j 0 d c ( z 1 ) d z 1 + j 0 d z 1 d z 2 d c ( z 1 ) c * ( z 2 ) c ( z 3 ) d z 3 d z 2 d z 1 j 0 d z 1 d z 2 d z 3 d z 4 d c ( z 1 ) c * ( z 2 ) c ( z 3 ) × c * ( z 4 ) c ( z 5 ) d z 5 d z 4 d z 3 d z 2 d z 1 + } .
0 d 0 z 1 0 z 2 c * ( z 1 ) c ( z 2 ) c * ( z 3 ) d z 3 d z 2 d z 1 = 0 d z 1 d z 2 d c * ( z 1 ) c ( z 2 ) c * ( z 3 ) d z 3 d z 2 d z 1
A 2 ( 0 ) / A 1 ( d ) = [ A 2 ( d ) / A 1 ( 0 ) ] * .
a 1 ( z ) = A 1 ( z ) exp ( j β 1 z ) , a 2 ( z ) = A 2 ( z ) exp ( j β 2 z ) , a 1 ( z ) = A 1 ( z ) exp ( + j β 1 z ) , a 2 ( z ) = A 2 ( z ) exp ( + j β 2 z ) ,
E ( x , y , z ) = a 1 ( z ) E T 1 ( x , y ) + a 2 ( z ) E T 2 ( x , y ) , E ( x , y , z ) = a 1 ( z ) E T 1 ( x , y ) + a 2 ( z ) E T 2 ( x , y ) .
a 2 ( 0 ) / a 1 ( d ) = exp { j ( β 1 + β 2 ) d } [ a 2 ( d ) / a 1 ( 0 ) ] * .
A 1 = B 1 exp { j ( ϕ + Δ β z ) / 2 } ; A 2 = B 2 exp { + j ( ϕ + β z ) / 2 }
d B 1 / d z j ( Δ β / 2 ) B 1 + j κ B 2 = 0 , d B 2 / d z + j ( Δ β / 2 ) B 2 + j κ B 1 = 0 .
[ B 1 ( d ) B 2 ( d ) ] = [ Q 11 Q 12 Q 12 * Q 11 * ] [ B 1 ( 0 ) B 2 ( 0 ) ] ,
Q 11 = cos { d [ κ 2 + ( Δ β / 2 ) 2 ] 1 / 2 } + j ( Δ β / 2 ) sin { d [ κ 2 + ( Δ β / 2 ) 2 ] 1 / 2 } / [ κ 2 + ( Δ β / 2 ) 2 ] 1 / 2 , Q 12 = j κ sin { d [ κ 2 + ( Δ β / 2 ) 2 ] 1 / 2 } / [ κ 2 + ( Δ β / 2 ) 2 ] 1 / 2 .
[ a 1 ( d ) a 2 ( d ) ] = [ exp { j ( β 1 + Δ β / 2 ) d } Q 11 exp { j [ ( β 1 + Δ β / 2 ) d + ϕ ] } Q 12 exp { j [ ( β 2 Δ β / 2 ) d ϕ ] } Q 12 * exp { j [ ( β 2 Δ β / 2 ) d ] Q 11 * ] [ a 1 ( 0 ) a 2 ( 0 ) ] ,
M = M T .
M = M N M N 1 M 2 M 1 ,
M = M 1 M 2 M N 1 M N ,
a 2 ( d T ) / a 1 ( 0 ) = M 21 ,
a 2 ( 0 ) / a 1 ( d T ) = M 21 ,
a 2 ( 0 ) / a 1 ( d T ) = exp [ j ( β 1 + β 2 ) d T ] [ a 2 ( d T ) / a 1 ( 0 ) ] * .
a 2 ( d ) = j κ exp ( β 2 d ) 0 d exp ( + j Δ β z 1 ) d z 1 .
β 1 ( ω ) = β 10 + ( ω ω 0 ) β 1 / ω , β 2 ( ω ) = β 20 + ( ω ω 0 ) β 2 / ω .
a 2 ( ω , d ) = j κ d exp ( j β 20 d ) exp [ j α 1 ( ω ω 0 ) ] × sin [ α 2 ( ω ω 0 ) ] / [ α 2 ( ω ω 0 ) ] ,
α 1 = ( d / 2 ) [ β 2 / ω + β 1 / ω ] , α 2 = ( d / 2 ) [ β 2 / ω β 1 / ω ] .
a 2 ( t , d ) j κ d exp ( j β 20 d ) 2 α 2 for α 1 α 2 < t < α 1 + α 2 = 0 otherwise .
τ = d ( 1 / υ g 2 1 / υ g 1 ) .
ω ω 0 = 2 π c Δ λ / λ 0 2 .
τ = λ 0 2 / c Δ λ .
[ 0 τ ( κ d / τ ) 2 d t ] 2 = [ κ 2 c d / ( n 2 n 1 ) ] 2 ,

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