Abstract

The optical noise of low-coherence sources coupled to optical networks can make a significant contribution to the overall output noise in certain applications. The characteristics of this noise depend on both the statistical properties of the source and the architecture of the network. The case of constant-amplitude and random-phase sources has already been studied in considerable detail. The present study deals with the case of a source obeying Gaussian statistics coupled to a linear, single-mode optical network. Utilization of the well-known properties of Gaussian random processes allows us to derive a general expression for the spectral density of the output optical noise. For the special case of systems that have a periodic frequency response and are polarization degenerate, a factorization between the source and the system parameters is achieved, and an explicit expression for the system-dependent factor is derived. The intensity of the noise considered in this study increases with the source power, becoming comparable with the shot noise at source powers of a few microwatts. The general theory is applied to the special case of the fiber-optic Fabry–Perot resonator.

© 1990 Optical Society of America

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References

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  1. C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
    [CrossRef]
  2. K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
    [CrossRef]
  3. M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
    [CrossRef]
  4. M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
    [CrossRef]
  5. M. Tur, A. Arie, E. Shafir, “Recent studies of laser phase noise in optical systems with time delays,” in Fiber Optic and Laser Sensors IV, R. P. De Paula, ed., Proc. Soc. Photo-Opt. Instrum. Eng.718, 274–279 (1986).
    [CrossRef]
  6. E. Shafir, M. Tur, “Phase induced intensity noise in an incoherent Fabry–Perot interferometer and other recirculating devices,” J. Opt. Soc. Am. A 4, 77–81 (1987).
    [CrossRef]
  7. M. Tur, A. Arie, “Phase induced intensity noise in concatenated delay lines,” IEEE J. Lightwave Technol. LT-6, 120–130 (1988).
    [CrossRef]
  8. W. van Etten, “Coupling of LED light into a single-mode fiber,” J. Opt. Commun. 9, 100–101 (1988).
  9. K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
    [CrossRef]
  10. W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  12. M. Tur, E. Shafir, K. Blotekjaer, “Source induced noise in optical systems driven by low coherence sources,” submitted to IEEE J. Lightwave Technol.
  13. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [CrossRef]
  14. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am 53, 317–323 (1963).
    [CrossRef]
  15. W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).
  16. P. Urquhart, “Transversely coupled fiber Fabry–Perot resonator: theory,” Appl. Opt. 26, 456–462 (1987).
    [CrossRef] [PubMed]
  17. F. Zhang, J. W. Y. Lit, “Direct-coupling single-mode fiber ring resonator,” J. Opt. Soc. Am. A 5, 1347–1355 (1988).
    [CrossRef]
  18. L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” Ph.D. dissertation, submitted to Stanford University, Stanford, Calif. (1983).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  20. Y. Weissman, “Optical network calculus,” in 6th Meeting in Israel on Optical Engineering, R. Finkler, J. Shamir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1038, 350–362 (1989).
    [CrossRef]
  21. H. J. Carlin, “The scattering matrix in network theory,” IRE Trans. Circ. Theory CT-3, 88–97 (1956).
    [CrossRef]
  22. G. Schiffner, W. R. Leeb, H. Krammer, J. Wittman, “Reciprocity of birefringent single-mode fibers for optical gyros,” Appl. Opt. 18, 2096–2097 (1979).
    [CrossRef] [PubMed]

1988 (4)

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

M. Tur, A. Arie, “Phase induced intensity noise in concatenated delay lines,” IEEE J. Lightwave Technol. LT-6, 120–130 (1988).
[CrossRef]

W. van Etten, “Coupling of LED light into a single-mode fiber,” J. Opt. Commun. 9, 100–101 (1988).

F. Zhang, J. W. Y. Lit, “Direct-coupling single-mode fiber ring resonator,” J. Opt. Soc. Am. A 5, 1347–1355 (1988).
[CrossRef]

1987 (2)

1985 (1)

M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
[CrossRef]

1983 (1)

W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
[CrossRef]

1982 (1)

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

1981 (1)

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

1979 (1)

1963 (1)

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am 53, 317–323 (1963).
[CrossRef]

1956 (1)

H. J. Carlin, “The scattering matrix in network theory,” IRE Trans. Circ. Theory CT-3, 88–97 (1956).
[CrossRef]

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Arie, A.

M. Tur, A. Arie, “Phase induced intensity noise in concatenated delay lines,” IEEE J. Lightwave Technol. LT-6, 120–130 (1988).
[CrossRef]

M. Tur, A. Arie, E. Shafir, “Recent studies of laser phase noise in optical systems with time delays,” in Fiber Optic and Laser Sensors IV, R. P. De Paula, ed., Proc. Soc. Photo-Opt. Instrum. Eng.718, 274–279 (1986).
[CrossRef]

Barakat, R.

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am 53, 317–323 (1963).
[CrossRef]

Blotekjaer, K.

M. Tur, E. Shafir, K. Blotekjaer, “Source induced noise in optical systems driven by low coherence sources,” submitted to IEEE J. Lightwave Technol.

Bohm, K.

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Bowers, J. E.

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Burns, W. K.

W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
[CrossRef]

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Carlin, H. J.

H. J. Carlin, “The scattering matrix in network theory,” IRE Trans. Circ. Theory CT-3, 88–97 (1956).
[CrossRef]

Chen, C.

W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
[CrossRef]

Cheng, W. H.

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Cutler, C. C.

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Digonnet, M.

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

Fesler, K.

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

Goodman, J. W.

M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
[CrossRef]

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Hwang, C. J.

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Jackson, K. P.

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Kim, B. Y.

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

Krammer, H.

Leeb, W. R.

Lit, J. W. Y.

Liu, K.

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

Marten, P.

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

Moeller, R. P.

W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
[CrossRef]

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Moslehi, B.

M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
[CrossRef]

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Newton, S. A.

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Petermann, K.

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

Schiffner, G.

Shafir, E.

E. Shafir, M. Tur, “Phase induced intensity noise in an incoherent Fabry–Perot interferometer and other recirculating devices,” J. Opt. Soc. Am. A 4, 77–81 (1987).
[CrossRef]

M. Tur, A. Arie, E. Shafir, “Recent studies of laser phase noise in optical systems with time delays,” in Fiber Optic and Laser Sensors IV, R. P. De Paula, ed., Proc. Soc. Photo-Opt. Instrum. Eng.718, 274–279 (1986).
[CrossRef]

M. Tur, E. Shafir, K. Blotekjaer, “Source induced noise in optical systems driven by low coherence sources,” submitted to IEEE J. Lightwave Technol.

Shaw, H. J.

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

Stokes, L. F.

L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” Ph.D. dissertation, submitted to Stanford University, Stanford, Calif. (1983).

Tur, M.

M. Tur, A. Arie, “Phase induced intensity noise in concatenated delay lines,” IEEE J. Lightwave Technol. LT-6, 120–130 (1988).
[CrossRef]

E. Shafir, M. Tur, “Phase induced intensity noise in an incoherent Fabry–Perot interferometer and other recirculating devices,” J. Opt. Soc. Am. A 4, 77–81 (1987).
[CrossRef]

M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
[CrossRef]

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

M. Tur, A. Arie, E. Shafir, “Recent studies of laser phase noise in optical systems with time delays,” in Fiber Optic and Laser Sensors IV, R. P. De Paula, ed., Proc. Soc. Photo-Opt. Instrum. Eng.718, 274–279 (1986).
[CrossRef]

M. Tur, E. Shafir, K. Blotekjaer, “Source induced noise in optical systems driven by low coherence sources,” submitted to IEEE J. Lightwave Technol.

Ulrich, R.

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

Urquhart, P.

van Etten, W.

W. van Etten, “Coupling of LED light into a single-mode fiber,” J. Opt. Commun. 9, 100–101 (1988).

Wang, C. S.

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Weissman, Y.

Y. Weissman, “Optical network calculus,” in 6th Meeting in Israel on Optical Engineering, R. Finkler, J. Shamir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1038, 350–362 (1989).
[CrossRef]

Wittman, J.

Wolf, E.

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Zhang, F.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C. S. Wang, W. H. Cheng, C. J. Hwang, W. K. Burns, R. P. Moeller, “High-power low divergence superradiance diode,” Appl. Phys. Lett. 41, 587–589 (1982).
[CrossRef]

Electron. Lett. (2)

K. Liu, M. Digonnet, K. Fesler, B. Y. Kim, H. J. Shaw, “Broadband diode-pumped fibre-laser,” Electron. Lett. 24, 838–840 (1988).
[CrossRef]

K. Bohm, P. Marten, K. Petermann, R. Ulrich, “Low-drift fibre gyro using a superluminiscent diode,” Electron. Lett. 17, 352–353 (1981).
[CrossRef]

IEEE J. Lightwave Technol. (3)

W. K. Burns, C. Chen, R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” IEEE J. Lightwave Technol. LT-1, 98–105 (1983).
[CrossRef]

M. Tur, B. Moslehi, J. W. Goodman, “Theory of laser phase noise in recirculating fiber-optic delay lines,” IEEE J. Lightwave Technol. LT-3, 20–31 (1985).
[CrossRef]

M. Tur, A. Arie, “Phase induced intensity noise in concatenated delay lines,” IEEE J. Lightwave Technol. LT-6, 120–130 (1988).
[CrossRef]

IRE Trans. Circ. Theory (1)

H. J. Carlin, “The scattering matrix in network theory,” IRE Trans. Circ. Theory CT-3, 88–97 (1956).
[CrossRef]

J. Opt. Commun. (1)

W. van Etten, “Coupling of LED light into a single-mode fiber,” J. Opt. Commun. 9, 100–101 (1988).

J. Opt. Soc. Am (1)

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am 53, 317–323 (1963).
[CrossRef]

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

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Other (8)

L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” Ph.D. dissertation, submitted to Stanford University, Stanford, Calif. (1983).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Y. Weissman, “Optical network calculus,” in 6th Meeting in Israel on Optical Engineering, R. Finkler, J. Shamir, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1038, 350–362 (1989).
[CrossRef]

M. Tur, A. Arie, E. Shafir, “Recent studies of laser phase noise in optical systems with time delays,” in Fiber Optic and Laser Sensors IV, R. P. De Paula, ed., Proc. Soc. Photo-Opt. Instrum. Eng.718, 274–279 (1986).
[CrossRef]

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.412, 22–27 (1983).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. Tur, E. Shafir, K. Blotekjaer, “Source induced noise in optical systems driven by low coherence sources,” submitted to IEEE J. Lightwave Technol.

W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

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Figures (3)

Fig. 1
Fig. 1

Normalized structure function corresponding to the transmitted field in the fiber-optic Fabry–Perot resonator: a, |r|2 = 0.1; b, |r|2 = 0.3; c, |r|2 = 0.5; d, |r|2 = 0.7; e, |r|2 = 0.9.

Fig. 2
Fig. 2

Normalized structure function corresponding to the reflected field in the fiber-optic Fabry–Perot resonator: a, |r|2 = 0.9; b, |r|2 = 0.7; c, |r|2 = 0.5; d, |r|2 = 0.3; e, |r|2 = 0.1.

Fig. 3
Fig. 3

Schematic diagram of the fiber-optic Fabry–Perot resonator. The boxes denoted by M1 and M2 are fiber-optic partial reflectors.

Equations (78)

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I ( t ) = η j | E j ( t ) | 2 ,
P s ( f ) = exp ( 2 π if τ ) Cov ( τ ) d τ ,
Cov ( τ ) = I ( t + τ ) I ( t ) I 2 .
Cov ( τ ) = η 2 j , k | E j ( t + τ ) | 2 | E k ( t ) | 2 I 2 .
| u ( t 1 ) | 2 | u ( t 2 ) | 2 = | u ( t 1 ) | 2 | u ( t 2 ) | 2 + | u ( t 1 ) u * ( t 2 ) | 2 .
Cov ( τ ) = η 2 j , k | G jk ( τ ) | 2 ,
G jk ( τ ) = E j ( t + τ ) E k * ( t ) .
G jk * ( τ ) = G kj ( τ ) ,
Cov ( τ ) = η 2 Tr [ G ( τ ) G ( τ ) ] .
G ( τ ) = 0 exp ( 2 π i ν τ ) G ( ν ) d ν .
Cov ( τ ) = η 2 0 d ν 1 d ν 2 exp [ 2 π i τ ( ν 1 ν 2 ) Tr [ G ( ν 1 ) G ( ν 2 ) ] .
P s ( f ) = η 2 0 Tr [ G ( ν ) G ( ν + f ) ] d ν .
G ( ν ) = J ( ν ) G ( ν ) J ( ν ) .
P o ( f ) = η 2 0 Tr [ H ( ν , f ) G ( ν ) H ( ν , f ) G ( ν + f ) ] d ν ,
H ( ν , f ) = J ( ν + f ) J ( ν ) .
H ( ν , f ) = h ( ν , f ) I = j * ( ν + f ) j ( ν ) I ,
P o ( f ) = η 2 0 | h ( ν , f ) | 2 Tr [ G ( ν ) G ( ν + f ) ] d ν .
P o ( f ) = η 2 k = 0 k Δ f k Δ f + Δ f | h ( ν , f ) | 2 Tr [ G ( ν ) G ( ν + f ) ] d ν .
P o ( f ) = η 2 k = 0 Tr [ G ( k Δ f ) G ( k Δ f + f ) ] k Δ f k Δ f + Δ f | h ( ν , f ) | 2 d ν .
P o ( f ) = S ( f ) P s ( f ) ,
S ( f ) = 1 Δ f [ Δ f ] | h ( ν , f ) | 2 d ν .
N = 2 q η T W ,
P o ( f ) N = S ( f ) P s ( f ) 2 q η T W .
T ( ν ) = | j ( ν ) | 2 ,
T = 1 Δ f [ Δ f ] | j ( ν ) | 2 d ν .
P s η 2 G 2 Δ ν , W G Δ ν ,
P o N η 2 q S T W Δ ν .
W 3 μ W .
S ( f ) = 1 Δ f [ Δ f ] | j ( ν + f ) j ( ν ) | 2 d ν .
g ( z ) = | j ( ν ) | 2 ,
z = exp ( 2 π i ν / Δ f ) .
S ( f ) = 1 2 π i unity circle g ( z ) g [ exp ( 2 π if / Δ f ) z ] d z z .
S ( f ) = g 2 ( 0 ) + k = 1 M R k z k { g [ exp ( 2 π if / Δ f ) z k ] + g [ exp ( 2 π if / Δ f ) z k ] } .
1 2 π i unity circle ( g ) z d z z = R z 0 + g ( 0 ) .
1 2 π i unity circle g ( z ) d z z = 1 Δ f [ Δ f ] | j ( ν ) | 2 d ν = T ,
S ( f ) = g 2 ( 0 ) + [ T g ( 0 ) ] { g [ exp ( 2 π if / Δ f ) z 0 ] + g [ exp ( 2 π i f / Δ f ) z 0 ] } .
S ( f ) = T { g [ exp ( 2 π if / Δ f ) z 0 ] + g [ exp ( 2 π if / Δ f ) z 0 ] } .
W t = | j t | 2 W i , W r = | j r | 2 W i ,
| j t | 2 + | j r | 2 = 1.
g t ( z ) + g r ( z ) = 1 ,
S r ( f ) = g r 2 ( 0 ) + k = 1 M R k z k { g t [ exp ( 2 π if / Δ f ) z k ] + g t [ exp ( 2 π if / Δ f ) z k ] 2 } = S t ( f ) 2 k = 1 M R k z k + g r ( 0 ) g t ( 0 ) ,
T t = k = 1 M R k z k + g t ( 0 ) ,
T r = k = 1 M R k z k + g r ( 0 ) ,
S r ( f ) S t ( f ) = T r T t .
T r = 1 T t .
S r ( f ) = S t ( f ) + 1 2 T t .
j ( ν ) = A 1 exp ( 2 π i ν τ 1 ) + A 2 exp ( 2 π i ν τ 2 ) ,
g ( z ) = | A 1 | 2 + | A 2 | 2 + A 1 A 2 * z + A 1 * A 2 / z ,
S ( f ) = ( | A 1 | 2 + | A 2 | 2 ) 2 + 2 | A 1 A 2 | 2 cos [ 2 π f ( τ 1 τ 2 ) ] .
j t ( ν ) = exp ( π i ν τ ) t 2 1 r 2 exp ( 2 π i ν τ ) ,
g t ( z ) = | t | 4 z r 2 z 2 + ( 1 + | r | 4 ) z r * 2 .
T t = | t | 4 1 | r | 4 .
g t ( e i ϕ z 0 ) + g t ( e i ϕ z 0 ) = | t | 4 ( 1 + | r | 4 ) 1 + | r | 8 2 | r | 4 cos ( ϕ ) .
S t ( f ) = 1 + | r | 4 1 | r | 4 | t | 8 [ 1 + | r | 8 2 | r | 4 cos ( 2 π f ) ] 1 .
S ( f ) = poles p Res [ F ( p ) ] ,
F ( z ) = g ( z ) g [ exp ( i ϕ ) z ] / z ,
| z j | | z k |
z j 0
z j = exp ( i ϕ ) z j .
S ( f ) = Res [ F ( 0 ) ] + k = 1 M Res [ F ( z k ) ] + k = 1 M Res [ F ( z k ) ] .
Res [ F ( 0 ) ] = g 2 ( 0 ) ,
Res [ F ( z k ) ] = g [ exp ( i ϕ ) z k ] Res [ g ( z k ) ] z k ,
Res [ F ( z k ) ] = g [ exp ( i ϕ ) z k ] Res [ g ϕ ( z k ) ] z k .
g ( z ) = Res [ g ( z k ) ] z z k + f ( z ) ,
g ϕ ( z ) = Res [ g ( z k ) ] exp ( i ϕ ) z z k + f [ exp ( i ϕ ) z ] = exp ( i ϕ ) Res [ g ( z k ) ] z exp ( i ϕ ) z k + f [ exp ( i ϕ ) z ] .
Res [ g ϕ ( z k ) ] = exp ( i ϕ ) Res ( g ( z k ) ] ,
Res [ g ϕ ( z k ) ] z k = Res [ g ( z k ) ] z k .
S ( f ) = g 2 ( 0 ) + k = 1 M Res [ g ( z k ) ] z k { g [ exp ( i ϕ ) z k ] + g [ exp ( i ϕ ) z k ] } ,
S = [ r 1 t t r 2 ] ,
| r 1 | 2 + | t | 2 = | r 2 | 2 + | t | 2 = 1
r 1 t * + r 2 * t = 0 .
M 1 = [ u 1 t 1 t 1 u 2 ] , M 2 = [ υ 1 t 2 t 2 υ 2 ] .
E 2 = t 1 E 1 + T u 2 E 3 ,
E 3 = T υ 1 E 2 ,
E 4 = T t 2 E 2 ,
T = exp ( π i ν τ ) ,
j t ( ν ) = E 4 E 1 = T t 1 t 3 1 T 2 υ 1 u 2 .
| j t ( ν ) | 2 = | t | 4 T 2 r 2 T 4 + ( 1 + | r | 4 ) T 2 r * 2 .

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