Abstract

Backward Rayleigh scattering in optical fibers due to the fluctuations that are “frozen-in” to the fiber during the manufacturing process may limit the performance of optical sensors and bidirectional coherent optical communication systems. In this manuscript we describe a comprehensive model for studying intensity noise induced by spontaneous Rayleigh backscattering in optical systems that are based on self-homodyne detection. Our model includes amplitude and frequency noise of the laser source, random distribution of the scatterers along the fiber, and phase noise induced in fibers due to thermal and mechanical fluctuations. The model shows that at frequencies above about 10 kHz the noise spectrum is determined by the laser white frequency noise. The laser flicker frequency noise becomes the dominant effect at lower frequencies. The noise amplitude depends on the laser polarization. A very good agreement between theory and experiment is obtained for fibers with a length between 500 m to 100 km and for a laser with a linewidth below 5 kHz.

© 2015 Optical Society of America

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References

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    [Crossref]
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2015 (1)

2013 (2)

A. E. Alekseev and V. T. Potapov, “Noise power spectral density of a fibre scattered-light interferometer with a semiconductor laser source,” Quantum Electron. 43(10), 968–973 (2013).
[Crossref]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

2012 (1)

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012).
[Crossref]

2011 (2)

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

O. Llopis, P. H. Merrer, H. Brahimi, K. Saleh, and P. Lacroix, “Phase noise measurement of a narrow linewidth CW laser using delay line approaches,” Opt. Lett. 36(14), 2713–2715 (2011).
[Crossref] [PubMed]

2010 (4)

G. D. Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49(25), 4801–4807 (2010).
[Crossref] [PubMed]

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

L. Z. Duan, “Intrinsic thermal noise of optical fibres due to mechanical dissipation,” Electron. Lett. 46(22), 1515–1516 (2010).
[Crossref]

S. A. Diddams, “The evolving optical frequency comb [Invited],” J. Opt. Soc. Am. B 27(11), B51–B62 (2010).
[Crossref]

2008 (3)

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
[Crossref] [PubMed]

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

2005 (1)

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

2004 (1)

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

2003 (1)

T. Watanabe, K. Saito, and A. J. Ikushima, “Fictive temperature dependence of density fluctuation in SiO2 glass,” J. Appl. Phys. 94(8), 4824–4827 (2003).
[Crossref]

2001 (1)

1998 (2)

F. Corsi, A. Galtarossa, and L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightw. Technol. 16(10), 1832–1843 (1998).
[Crossref]

M. Froggatt and J. Moore, “High-spatial-resolution distributed strain measurement in optical fiber with Rayleigh scatter,” Appl. Opt. 37(10), 5162–5164 (1998).
[Crossref]

1997 (1)

K. Saito and A. J. Ikushima, “Reduction of light-scattering loss in silica glass by the structural relaxation of frozen-in density fluctuations,” Appl. Phys. Lett. 70(26), 3504–3506 (1997).
[Crossref]

1996 (1)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

1994 (1)

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurement based on transmission spectra through a polarizer,” J. Lightw. Technol. 12(6), 917–929 (1994).
[Crossref]

1993 (1)

M. O. Van Deventer, “Polarization properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 11(12), 1895–1899 (1993).
[Crossref]

1992 (1)

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28(1), 53–54 (1992).
[Crossref]

1991 (2)

R. K. Staubli and P. Gysel, “Crosstalk penalties due to coherent Rayleigh noise in bidirectional optical communication systems,” J. Lightwave Technol. 9(3), 375–380 (1991).
[Crossref]

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991).
[Crossref]

1990 (1)

P. Gysel and R. K. Staubli, “Statistical properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 8(4), 561–567 (1990).
[Crossref]

1989 (1)

K. Kikuchi, “Effect of l/f-Type FM Noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25(4), 684–688 (1989).
[Crossref]

1987 (1)

P. Healey, “Statistics of Rayleigh backscatter from a single-mode fiber,” IEEE Trans. Commun. Technol. 35(2), 210–214 (1987).
[Crossref]

1983 (2)

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers - part I: theoretical analysis,” IEEE J. Quantum Electron. 19(1), 34–46 (1983).
[Crossref]

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “Local-oscillator excess-noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8(8), 419–421 (1983).
[Crossref] [PubMed]

1973 (2)

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

1967 (1)

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

1962 (1)

I. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inform. Theory 8(3), 194–195 (1962).
[Crossref]

1941 (1)

Abbas, G. L.

Aksenov, S. B.

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

Alekseev, A. E.

A. E. Alekseev and V. T. Potapov, “Noise power spectral density of a fibre scattered-light interferometer with a semiconductor laser source,” Quantum Electron. 43(10), 968–973 (2013).
[Crossref]

Babin, S. A.

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

Bao, X.

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Barros, D. J. F.

Bartolo, R. E.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).

Brahimi, H.

Cahill, J.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

Cahill, J. P.

J. P. Cahill, O. Okusaga, W. Zhou, C. R. Menyuk, and G. M. Carter, “Superlinear growth of Rayleigh scattering-induced intensity noise in single-mode fibers,” Opt. Express 23(5), 6400–6407 (2015).
[Crossref] [PubMed]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

Carter, G. M.

Champagnon, B.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Chan, V. W. S.

Chen, L.

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Cho, K. Y.

U. H. Hong, K. Y. Cho, Y. Takushima, and Y. C. Chung, “Effects of Rayleigh backscattering in long-reach RSOA-based WDM PON,” in Optical Fiber Communication Conference, 2010 OSA NFOEC (Optical Society of America, 2010), paper OThG1.
[Crossref]

Chung, Y. C.

U. H. Hong, K. Y. Cho, Y. Takushima, and Y. C. Chung, “Effects of Rayleigh backscattering in long-reach RSOA-based WDM PON,” in Optical Fiber Communication Conference, 2010 OSA NFOEC (Optical Society of America, 2010), paper OThG1.
[Crossref]

Corsi, F.

F. Corsi, A. Galtarossa, and L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightw. Technol. 16(10), 1832–1843 (1998).
[Crossref]

Dandridge, A.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012).
[Crossref]

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

David, L.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

de Riedmatten, H.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Diddams, S. A.

Docherty, A.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

Domenico, G. D.

Dong, Y.

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Duan, L. Z.

L. Z. Duan, “Intrinsic thermal noise of optical fibres due to mechanical dissipation,” Electron. Lett. 46(22), 1515–1516 (2010).
[Crossref]

Faivre, A.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Favin, D. L.

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurement based on transmission spectra through a polarizer,” J. Lightw. Technol. 12(6), 917–929 (1994).
[Crossref]

Flammer, I.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

Froggatt, M.

Froggatt, M. E.

M. E. Froggatt and D. K. Gifford, “Rayleigh backscattering signatures of optical fibers - their properties and applications,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW1K.6.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

Galtarossa, A.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26(13), 962–964 (2001).
[Crossref]

F. Corsi, A. Galtarossa, and L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightw. Technol. 16(10), 1832–1843 (1998).
[Crossref]

Gifford, D. K.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

M. E. Froggatt and D. K. Gifford, “Rayleigh backscattering signatures of optical fibers - their properties and applications,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW1K.6.

Gisin, N.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Gray, M. A.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

Gysel, P.

R. K. Staubli and P. Gysel, “Crosstalk penalties due to coherent Rayleigh noise in bidirectional optical communication systems,” J. Lightwave Technol. 9(3), 375–380 (1991).
[Crossref]

P. Gysel and R. K. Staubli, “Statistical properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 8(4), 561–567 (1990).
[Crossref]

Hazemann, J. L.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

Hazemann, J.-L.

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Healey, P.

P. Healey, “Statistics of Rayleigh backscatter from a single-mode fiber,” IEEE Trans. Commun. Technol. 35(2), 210–214 (1987).
[Crossref]

Herman, R. M.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

Hong, U. H.

U. H. Hong, K. Y. Cho, Y. Takushima, and Y. C. Chung, “Effects of Rayleigh backscattering in long-reach RSOA-based WDM PON,” in Optical Fiber Communication Conference, 2010 OSA NFOEC (Optical Society of America, 2010), paper OThG1.
[Crossref]

Ikushima, A. J.

T. Watanabe, K. Saito, and A. J. Ikushima, “Fictive temperature dependence of density fluctuation in SiO2 glass,” J. Appl. Phys. 94(8), 4824–4827 (2003).
[Crossref]

K. Saito and A. J. Ikushima, “Reduction of light-scattering loss in silica glass by the structural relaxation of frozen-in density fluctuations,” Appl. Phys. Lett. 70(26), 3504–3506 (1997).
[Crossref]

Ip, E.

Jones, R. C.

Kahn, J. M.

Kikuchi, K.

K. Kikuchi, “Effect of l/f-Type FM Noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25(4), 684–688 (1989).
[Crossref]

Kirkendall, C. K.

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

Kreger, S. T.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

Laberge, N. L.

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

Lacroix, P.

Lau, A. P. T.

Le Parc, R.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Levelut, C.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Liang, H.

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Llopis, O.

Macedo, P. B.

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

Martinez, V.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

Menyuk, C. R.

J. P. Cahill, O. Okusaga, W. Zhou, C. R. Menyuk, and G. M. Carter, “Superlinear growth of Rayleigh scattering-induced intensity noise in single-mode fibers,” Opt. Express 23(5), 6400–6407 (2015).
[Crossref] [PubMed]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

Mercer, L. B.

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991).
[Crossref]

Merrer, P. H.

Minár, J.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Mohr, R.

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

Montrose, C. J.

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

Moore, J.

Okusaga, O.

J. P. Cahill, O. Okusaga, W. Zhou, C. R. Menyuk, and G. M. Carter, “Superlinear growth of Rayleigh scattering-induced intensity noise in single-mode fibers,” Opt. Express 23(5), 6400–6407 (2015).
[Crossref] [PubMed]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

Palmieri, L.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26(13), 962–964 (2001).
[Crossref]

F. Corsi, A. Galtarossa, and L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightw. Technol. 16(10), 1832–1843 (1998).
[Crossref]

Podivilov, E. V.

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

Poole, C. D.

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurement based on transmission spectra through a polarizer,” J. Lightw. Technol. 12(6), 917–929 (1994).
[Crossref]

Potapov, V. T.

A. E. Alekseev and V. T. Potapov, “Noise power spectral density of a fibre scattered-light interferometer with a semiconductor laser source,” Quantum Electron. 43(10), 968–973 (2013).
[Crossref]

Reed, I.

I. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inform. Theory 8(3), 194–195 (1962).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Saito, K.

T. Watanabe, K. Saito, and A. J. Ikushima, “Fictive temperature dependence of density fluctuation in SiO2 glass,” J. Appl. Phys. 94(8), 4824–4827 (2003).
[Crossref]

K. Saito and A. J. Ikushima, “Reduction of light-scattering loss in silica glass by the structural relaxation of frozen-in density fluctuations,” Appl. Phys. Lett. 70(26), 3504–3506 (1997).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Saleh, K.

Schiano, M.

Schilt, S.

Schroeder, J.

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

Simon, C.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Simon, J. P.

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

Simon, J.-P.

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Soller, B. J.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

Staubli, R. K.

R. K. Staubli and P. Gysel, “Crosstalk penalties due to coherent Rayleigh noise in bidirectional optical communication systems,” J. Lightwave Technol. 9(3), 375–380 (1991).
[Crossref]

P. Gysel and R. K. Staubli, “Statistical properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 8(4), 561–567 (1990).
[Crossref]

Takushima, Y.

U. H. Hong, K. Y. Cho, Y. Takushima, and Y. C. Chung, “Effects of Rayleigh backscattering in long-reach RSOA-based WDM PON,” in Optical Fiber Communication Conference, 2010 OSA NFOEC (Optical Society of America, 2010), paper OThG1.
[Crossref]

Tambosso, T.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Thomann, P.

Tosoni, O.

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

Tveten, A. B.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012).
[Crossref]

Van Deventer, M. O.

M. O. Van Deventer, “Polarization properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 11(12), 1895–1899 (1993).
[Crossref]

Vasilescu, V. V.

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Wanser, K. H.

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28(1), 53–54 (1992).
[Crossref]

Watanabe, T.

T. Watanabe, K. Saito, and A. J. Ikushima, “Fictive temperature dependence of density fluctuation in SiO2 glass,” J. Appl. Phys. 94(8), 4824–4827 (2003).
[Crossref]

Wolfe, M. S.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

Yamamoto, Y.

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers - part I: theoretical analysis,” IEEE J. Quantum Electron. 19(1), 34–46 (1983).
[Crossref]

Yee, T. K.

Zbinden, H.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Zhou, W.

J. P. Cahill, O. Okusaga, W. Zhou, C. R. Menyuk, and G. M. Carter, “Superlinear growth of Rayleigh scattering-induced intensity noise in single-mode fibers,” Opt. Express 23(5), 6400–6407 (2015).
[Crossref] [PubMed]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

Zhu, T.

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. Saito and A. J. Ikushima, “Reduction of light-scattering loss in silica glass by the structural relaxation of frozen-in density fluctuations,” Appl. Phys. Lett. 70(26), 3504–3506 (1997).
[Crossref]

Electron. Lett. (2)

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28(1), 53–54 (1992).
[Crossref]

L. Z. Duan, “Intrinsic thermal noise of optical fibres due to mechanical dissipation,” Electron. Lett. 46(22), 1515–1516 (2010).
[Crossref]

IEEE J. Quantum Electron. (3)

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012).
[Crossref]

K. Kikuchi, “Effect of l/f-Type FM Noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25(4), 684–688 (1989).
[Crossref]

Y. Yamamoto, “AM and FM quantum noise in semiconductor lasers - part I: theoretical analysis,” IEEE J. Quantum Electron. 19(1), 34–46 (1983).
[Crossref]

IEEE Photonics J. (1)

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photonics J. 5(2), 5500514 (2013).
[Crossref]

IEEE Trans. Commun. Technol. (1)

P. Healey, “Statistics of Rayleigh backscatter from a single-mode fiber,” IEEE Trans. Commun. Technol. 35(2), 210–214 (1987).
[Crossref]

IRE Trans. Inform. Theory (1)

I. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inform. Theory 8(3), 194–195 (1962).
[Crossref]

J. American Ceramic Society (1)

J. Schroeder, R. Mohr, P. B. Macedo, and C. J. Montrose, “Rayleigh and Brillouin scattering in K2O – SiO2 glasses,” J. American Ceramic Society 56(10), 510–514 (1973).
[Crossref]

J. Appl. Phys. (2)

T. Watanabe, K. Saito, and A. J. Ikushima, “Fictive temperature dependence of density fluctuation in SiO2 glass,” J. Appl. Phys. 94(8), 4824–4827 (2003).
[Crossref]

R. Le Parc, B. Champagnon, C. Levelut, V. Martinez, L. David, A. Faivre, I. Flammer, J. L. Hazemann, and J. P. Simon, “Density and concentration fluctuations in SiO2 – GeO2 optical fiber glass investigated by small angle x-ray scattering,” J. Appl. Phys. 103(9), 094917 (2008).
[Crossref]

J. Lightw. Technol. (2)

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurement based on transmission spectra through a polarizer,” J. Lightw. Technol. 12(6), 917–929 (1994).
[Crossref]

F. Corsi, A. Galtarossa, and L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightw. Technol. 16(10), 1832–1843 (1998).
[Crossref]

J. Lightwave Technol. (5)

M. O. Van Deventer, “Polarization properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 11(12), 1895–1899 (1993).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

R. K. Staubli and P. Gysel, “Crosstalk penalties due to coherent Rayleigh noise in bidirectional optical communication systems,” J. Lightwave Technol. 9(3), 375–380 (1991).
[Crossref]

P. Gysel and R. K. Staubli, “Statistical properties of Rayleigh backscattering in single-mode fibers,” J. Lightwave Technol. 8(4), 561–567 (1990).
[Crossref]

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991).
[Crossref]

J. of the American Ceramic Society (1)

N. L. Laberge, V. V. Vasilescu, C. J. Montrose, and P. B. Macedo, “Equilibrium compressibilities and density fluctuations in K2O – SiO2 glasses,” J. of the American Ceramic Society 56(10), 506–509 (1973).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Phys. D: Appl. Phys. (1)

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Phys. 37(18), R197–R216 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Phys. Rev. B (1)

C. Levelut, A. Faivre, R. Le Parc, B. Champagnon, J.-L. Hazemann, and J.-P. Simon, “In situ measurements of density fluctuations and compressibility in silica glasses as a function of temperature and thermal history,” Phys. Rev. B 72(22), 22421 (2005).
[Crossref]

Phys. Rev. Lett. (1)

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

Proc. SPIE (1)

T. Zhu, X. Bao, L. Chen, H. Liang, and Y. Dong, “Characteristics of stimulated Rayleigh scattering in optical fibers,” Proc. SPIE 7753, 77532R (2011).
[Crossref]

Quantum Electron. (2)

O. Tosoni, S. B. Aksenov, E. V. Podivilov, and S. A. Babin, “Model of a fibreoptic phase-sensitive reflectometer and its comparison with the experiment,” Quantum Electron. 40(10), 887–892 (2010).
[Crossref]

A. E. Alekseev and V. T. Potapov, “Noise power spectral density of a fibre scattered-light interferometer with a semiconductor laser source,” Quantum Electron. 43(10), 968–973 (2013).
[Crossref]

Other (10)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).

M. E. Froggatt and D. K. Gifford, “Rayleigh backscattering signatures of optical fibers - their properties and applications,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW1K.6.

S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single- and multi-mode fiber using swept-wavelength interferometry,” in Optical Fiber Sensors Conference, OSA Technical Digest (CD) (Optical Society of America, 2006), paper ThE42.
[Crossref]

U. H. Hong, K. Y. Cho, Y. Takushima, and Y. C. Chung, “Effects of Rayleigh backscattering in long-reach RSOA-based WDM PON,” in Optical Fiber Communication Conference, 2010 OSA NFOEC (Optical Society of America, 2010), paper OThG1.
[Crossref]

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Optical scattering induced noise in RF-photonic systems,” in Proceedings of IEEE Conference on Frequency Control (IEEE, 2011) pp. 1–6.

O. Okusaga, W. Zhou, J. Cahill, A. Docherty, and C. R. Menyuk, “Fiber-induced degradation in RF-over-fiber links,” Proceedings of IEEE Conference on Frequency Control (IEEE, 2012) pp. 1–5.

Teraxion white paper, “Narrow-linewidth semiconductor lasers: one technique does not fit all,” (Teraxion, 2011), http://www.teraxion.com .

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Corning product information, “Corning single-mode optical fiber,” (Corning, 2002), http://www.corning.com .

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Figures (4)

Fig. 1.
Fig. 1. Schematic description of a system for self-homodyne detection of Rayleigh backscattering induced noise. A continuous wave laser is split by a coupler with power coupling fractions of r and q. One of the two waves (q) passes through a variable optical attenuator that controls the power that is injected into a long optical fiber with a length L. The fiber is terminated by an optical isolator to minimize reflections from the end. The backscattered wave from the optical fiber passes through an optical circulator, interferes with a part of the laser wave (r), and is detected by a pair of balanced photo-detectors. The output signal is measured with an RF spectrum analyzer.
Fig. 2.
Fig. 2. Power spectral density of the detected backward Rayleigh wave that is calculated by numerically solving Eq. (27) (black curve) for a 2-km fiber in a self-homodyne detection system. The polarization factor TP(z) was approximated by its spatially averaged value 2/3. The figure also shows the spectrum in case that the laser flicker noise is set to zero (red curve). The coherence length of the laser, as defined by a laser white frequency noise source (76 km) is longer than the fiber length (2 km). The laser linewidth for a 1 ms observation time is approximately 3 kHz. The fiber parameters correspond to Corning SMF-28 fiber.
Fig. 3.
Fig. 3. Comparison between theory (dashed green curves) and experiment (solid blue curves) for the noise power spectral density for various fiber lengths: (a) 2 km, (b) 6 km, (c) 10 km, (d) 20 km, (e) 58 km and (f) 100 km. The input optical power was 0 dBm. The parameters of the fiber and coupling fractions that were used in the numerical calculations are the same as given in section 2.1. A balanced photo-detector was used. The coherence length of the laser that was used in the calculations is 76 km, assuming white frequency laser noise and a total linewidth of 3 kHz for a 1-ms observation time. Good agreement between theory and experiments is obtained for fiber lengths greater than 6 km and all the fiber lengths at frequencies above 200 Hz.
Fig. 4.
Fig. 4. Power spectral density of the detected noise vs. fiber length (L). We compare theoretical results (blue curves), experimental results (red dots), and empirical fits (green curves). In (a), we allow the frequency to vary, so that f = c/(2nL). At L = 5 km, the frequency is 21 kHz, at L = 20 km, the frequency is 5 kHz. The power spectral density is approximately proportional to L3. In (b), we set f = 1 kHz. The power spectral density is approximately proportional to L2.4.

Equations (44)

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E s ( t ) = p s E s ( t ) ,
E s ( t ) = E 0 [ 1 + m ( t ) ] exp [ j ψ ( t ) + j ω 0 t ] ,
p s ( z ) = P ( z ) T 1 p s ,
E F ( z , t ) = P ( z ) T 1 p s E s ( t n z c ) exp [ j 0 z d z ω 0 c Δ n e ( z , t n ( z z ) c z α 2 ) ] ,
E R z ( t ) = T 2 P T ( z ) E F ( z , t n z c ) κ ( z ) exp [ j z 0 d z ω 0 c Δ n e ( z , t n z c z α 2 ) ] ,
E R ( t ) = 0 L dz T 2 P T ( z ) P ( z ) T 1 p s E s ( t 2 n z c ) κ ( z ) exp ( α z ) exp [ j ϕ ( z , t ) ] ,
ϕ ( z , t ) = 0 z d z ω 0 c Δ n e ( z , t n ( 2 z z ) c ) z 0 d z ω 0 c Δ n e ( z , t n z c ) .
E ( t ) = exp ( j γ 0 ) r T ref E s ( t ) + q l E R ( t ) ,
V RF ( t ) = 1 2 A eff R η ε 0 n c | E ( t ) | 2 b | E ( t ) | 2 ,
r ( τ ) = V RF ( t ) V RF * ( t τ ) = b 2 | E ( t ) | 2 | E ( t τ ) | 2 .
S ϕ ( L , ω ) = 4 L [ α 1 F ( ω ) + α 2 / | ω | ] .
κ ( z 1 ) κ * ( z 2 ) σ κ 2 δ ( z 1 z 2 ) ,
τ ( τ ) / b 2 = | r E s ( t ) | 2 | r E s ( t τ ) | 2 + | l q E R ( t ) | 2 | l q E R ( t τ ) | 2 + | r E s ( t ) | 2 | l q E R ( t τ ) | 2 + | r E s ( t τ ) | 2 | l q E R ( t ) | 2 + 2 | r | 2 | q | 2 l 2 Re E R ( t ) p ref E s ( t ) E s * ( t τ ) p ref E R ( t τ ) .
| r E s ( t ) | 2 | r E s ( t τ ) | 2 2 | E 0 | 4 | r | 4 [ r m ( τ ) + O ( r m 2 ) ] ,
| r E s ( t ) | 2 | l q E R ( t τ ) | 2 + | r E s ( t τ ) | 2 | l q E R ( t ) | 2 2 | r | 2 | q | 2 l 2 | E 0 | 4 σ κ 2 0 L dz [ r m ( 2 n z / c + τ ) + r m ( 2 n z / c τ ) + O ( r m 2 ) ] .
E R ( t ) p ref E s ( t ) E s * ( t τ ) p ref E R ( t τ ) = l 2 | q | 2 | r | 2 σ κ 2 0 L dz exp ( 2 α z ) R fiber ( z , τ ) T P ( z ) × E s ( t ) E s * ( t τ ) E s * ( t 2 n z c ) E s ( t 2 n z c τ ) ,
R fiber ( z , τ ) = exp [ j ϕ ( z , t τ ) j ϕ ( z , t ) ]
T P ( z ) = | p ref T 2 P T ( z ) P ( z ) T 1 p s | 2
T P ( z ) = 1 2 [ 1 + s ref s ( z ) ] ,
E s ( t ) E s * ( t τ ) E s * ( t 2 n z c ) E s ( t 2 n z c τ ) = | E 0 | 4 [ 1 + m ( t ) ] [ 1 + m * ( t τ ) ] [ 1 + m * ( t 2 n z c ) ] [ 1 + m ( t 2 n z c τ ) ] × R S Δ ψ ( ω ) ( z , τ ) ,
R S Δ ψ ( ω ) ( z , τ ) = exp { j [ ψ ( t ) ψ ( t τ ) ψ ( t 2 n z c ) + ψ ( t 2 n z c τ ) ] } ,
S Δ ψ ( ω ) = d τ Δ ψ ( t ) Δ ψ ( t + τ ) exp ( j ω τ ) ,
R S Δ ψ ( ω ) ( z , τ ) = exp [ 4 π d ω sin 2 ( ω τ 2 ) sin 2 ( ω τ z 2 ) S Δ ψ ( ω ) ω 2 ] ,
τ z = 2 n z c
E s ( t ) E s * ( t τ ) E s * ( t 2 n z c ) E s ( t 2 n z c τ ) | E 0 | 4 [ 1 + 2 r m ( τ ) + 2 r m ( τ z ) + O ( r m 2 ) ] R S Δ ψ ( ω ) ( z , τ ) .
r ( τ ) = 2 | r | 4 P 0 2 R 2 η 2 r m ( τ ) + 2 σ κ 2 | r | 2 | P 0 R in R 2 η 2 × 0 L dz exp ( 2 α z ) T P ( z ) R fiber ( z , τ ) R S Δ ψ ( ω ) ( z , τ ) [ 1 + 2 r m ( τ ) + 2 r m ( 2 n z / c ) ] + 2 σ κ 2 | r | 2 P 0 P in R 2 η 2 0 L dz exp ( 2 α z ) [ r m ( 2 n z / c + τ ) + r m ( 2 n z / c τ ) ] + O ( σ κ 4 ) ,
r BPD ( τ ) / b 2 2 | r | 2 | q | 2 l 2 Re E R ( t ) p ref E s ( t ) E s * ( t τ ) p ref E R ( t τ ) ,
r BPD ( τ ) 2 σ κ 2 | r | 2 | P 0 P in R 2 η 2 × 0 L dz exp ( 2 α z ) T P ( z ) R fiber ( z , τ ) R S Δ ψ ( ω ) ( z , τ ) [ 1 + 2 r m ( τ ) + 2 r m ( 2 n z / c ) ] .
S Δ ψ ( ω ) = S 0 + k | ω | .
R S Δ ψ ( ω ) ( z , τ ) = R S 0 ( z , τ ) R k / | ω | ( z , τ ) ,
R S 0 ( z , τ ) = exp [ 4 π d ω sin 2 ( ω τ 2 ) sin 2 ( ω τ z 2 ) S 0 ω 2 ] , R k / | ω | ( z , τ ) = exp [ 4 π d ω sin 2 ( ω τ 2 ) sin 2 ( ω τ z 2 ) k | ω | 3 ] .
R S 0 ( z , τ ) = { exp ( S 0 | τ | ) | τ | τ z exp ( S 0 τ z ) | τ | > τ z ,
R k / | ω | ( z , τ ) = { exp [ 1 2 π τ 2 τ z 2 k v ( τ z , | τ | ) ] | τ | < τ z exp [ 1 2 π τ 2 τ z 2 k v ( | τ | , τ z ) ] | τ | > τ z exp [ 1 2 π k τ 2 log 16 ] | τ | = τ z 1 τ = 0 ,
v ( χ , χ 0 ) = 4 tanh 1 ( χ 0 / χ ) χ χ 0 + 2 tanh 1 ( 1 2 χ 0 2 / χ 2 ) χ 2 + log ( 1 χ 0 2 / χ 2 ) χ 0 2 , χ 0 χ .
S Δ ψ ( ω ) = S 0 + k | ω | + S HF ( ω ) ,
S HF ( ω ) = a | ω | 3 , | ω | ω 1 .
R S Δ ψ ( ω ) ( z , τ ) = R S 0 ( z , τ ) R k / | ω | ( z , τ ) R HF ( z , τ ) ,
T P 0 = 1 2 [ 1 + s ref s ( z ) z ] ,
s ( z ) z = 1 d z z + d d z s ( z )
S ( ω ) = 2 d τ r ( τ ) exp ( j ω τ ) ,
S ( ω ) / u = T P 0 π c δ ( ω ) [ 1 exp ( 2 L n S 0 / c ) ] n S 0 + T P 0 S 0 n ω ( S 0 2 + ω 2 ) 2 × [ 2 ω ( L n ( S 0 2 + ω 2 ) c S 0 ) + exp ( 2 L n S 0 c ) c ( S 0 2 ω 2 ) sin ( 2 L n ω c ) ] + T P 0 2 S 0 2 n ( S 0 2 + ω 2 ) 2 × exp ( 2 L n S 0 c ) c cos ( 2 L n ω c ) ,
S ( ω L ) / u T P 0 2 L 3 n 2 S 0 c 2 π 2
S ( ω 0 ) / S ( ω L ) 2 π 2 / 3 .
S ( ω ) / u T P 0 2 S 0 L S 0 2 + ω 2 .

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