Abstract

The spectrum of the intensity of dispersed waves obeying cyclostationary statistics is studied. The formalism is based on an exact formula by Marshall and Yariv [IEEE Photon. Technol. Lett. 12, 302 (2000)] relating the intensity spectrum after first-order dispersion to the Fourier transform of a certain restriction of the time-averaged fourth-order correlation of the optical wave e(t) before dispersion. The formalism permits a simple computation of the spectrum of composite models defined by the independent addition or multiplication of a stationary and a cyclostationary field. The computations are simplified by introducing the auxiliary field zτ(t)=e(t)*e(t+τ), whose power spectral density represents the basic building block for solving the spectrum of composite models. The results are illustrated by a number of examples, including the intensity spectrum after dispersion of analog-modulated, partially coherent carriers, or the complete spectrum of intensity fluctuations of multiwavelength dispersion-based microwave photonic filters.

© 2009 Optical Society of America

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    [Crossref]
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    [Crossref]
  4. F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11, 1937-1940 (1993).
    [Crossref]
  5. D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).
  6. D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 2041-217 (1986).
  7. C. Dorrer and D. N. Maywar, “RF spectrum analysis of optical signals using nonlinear optics,” J. Lightwave Technol. 22, 266-274 (2003).
    [Crossref]
  8. K. Peterman and E. Weidel, “Semiconductor laser noise in an interferometer system,” IEEE J. Quantum Electron. 17, 1251-1256 (1981).
    [Crossref]
  9. P. R. Morkel, R. I. Laming, and D. N. Payne, “Noise characteristics of high-power doped-fibre superluminiscent sources,” Electron. Lett. 26, 96-98 (1990).
    [Crossref]
  10. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394-396 (2004).
    [Crossref] [PubMed]
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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    [Crossref]
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  14. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).
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    [Crossref]
  17. M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
    [Crossref]
  18. W. K. Marshall and A. Yariv, “Spectrum of the intensity of modulated noisy light after propagation in dispersive fiber,” IEEE Photon. Technol. Lett. 12, 302-304 (2000).
    [Crossref]
  19. Formula in lacks the ensemble average because it was derived from a single realization, i.e., in the framework of the time series (see below). Our Eq. is the formula for the stochastic framework.
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    [Crossref]
  25. W. K. Marshall, B. Crosignani, and A. Yariv, “Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory,” Opt. Lett. 25, 165-167 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref]
  28. J. Azaña and M. A. Muriel, “Temporal self-imaging effect: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
    [Crossref]
  29. B. Picinbono and E. Boileau, “Higher-order coherence functions of optical fields and phase fluctuations,” J. Opt. Soc. Am. 58, 784-789 (1968).
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    [Crossref]
  33. H. Ogura, “Spectral representation of a periodic nonstationary random process,” IEEE Trans. Inf. Theory 17, 143-149 (1971).
    [Crossref]
  34. W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
    [Crossref]
  35. The general transformation rule of cyclic spectra after a linear, almost-periodically time-variant system can be consulted, for instance, in , Sect. 3.6.
  36. W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill, 1986).
  37. In the intensity is defined by low-pass filtering the square of the (real) electric field, Re[ẽ(t)]. Here we follow the standard notation in wave optics, p. 162. The definition of implies a difference of 12 in the intensity and a subsequent difference of 14 in the intensity spectrum.
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    [Crossref]
  39. L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  44. J. W. Goodman, Statistical Optics (Wiley, 1985).
  45. F. J. Fraile Peláez, “Analytical signal formalism in the description of optical pulse photodetection,” Microwave Opt. Technol. Lett. 37, 347-352 (2003).
    [Crossref]
  46. G. L. Pierobon and L. Tomba, “Moment characterization of phase noise in coherent optical systems,” J. Lightwave Technol. 9, 996-1005 (1991).
    [Crossref]
  47. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

2007 (2)

2006 (7)

2005 (1)

2004 (2)

2003 (2)

C. Dorrer and D. N. Maywar, “RF spectrum analysis of optical signals using nonlinear optics,” J. Lightwave Technol. 22, 266-274 (2003).
[Crossref]

F. J. Fraile Peláez, “Analytical signal formalism in the description of optical pulse photodetection,” Microwave Opt. Technol. Lett. 37, 347-352 (2003).
[Crossref]

2001 (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effect: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[Crossref]

2000 (2)

W. K. Marshall, B. Crosignani, and A. Yariv, “Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory,” Opt. Lett. 25, 165-167 (2000).
[Crossref]

W. K. Marshall and A. Yariv, “Spectrum of the intensity of modulated noisy light after propagation in dispersive fiber,” IEEE Photon. Technol. Lett. 12, 302-304 (2000).
[Crossref]

1999 (1)

E. Shafir and Y. Weissman, “Effects of chromatic dispersion on microwave and millimetre wave carrier transmission in optical fibres,” J. Mod. Opt. 46, 2143-2156 (1999).

1994 (1)

B. Picinbono, “On circularity,” IEEE Trans. Signal Process. 42, 3473-3482 (1994).
[Crossref]

1993 (1)

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11, 1937-1940 (1993).
[Crossref]

1991 (2)

G. L. Pierobon and L. Tomba, “Moment characterization of phase noise in coherent optical systems,” J. Lightwave Technol. 9, 996-1005 (1991).
[Crossref]

W. A. Gardner, “Exploitation of spectral redundancy in cyclostationary signals,” IEEE Signal Process. Mag. 8, 14-36 (1991).
[Crossref]

1990 (2)

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

P. R. Morkel, R. I. Laming, and D. N. Payne, “Noise characteristics of high-power doped-fibre superluminiscent sources,” Electron. Lett. 26, 96-98 (1990).
[Crossref]

1989 (2)

M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
[Crossref]

L. J. Wang, B. E. Magill, and L. Mandel, “Propagation of thermal light through a dispersive medium,” J. Opt. Soc. Am. B 6, 964-966 (1989).
[Crossref]

1986 (1)

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 2041-217 (1986).

1981 (1)

K. Peterman and E. Weidel, “Semiconductor laser noise in an interferometer system,” IEEE J. Quantum Electron. 17, 1251-1256 (1981).
[Crossref]

1977 (1)

1975 (1)

W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[Crossref]

1971 (1)

H. Ogura, “Spectral representation of a periodic nonstationary random process,” IEEE Trans. Inf. Theory 17, 143-149 (1971).
[Crossref]

1968 (2)

1965 (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[Crossref]

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Agrawal, G. P.

Andrés, P.

Azaña, J.

J. Azaña and M. A. Muriel, “Temporal self-imaging effect: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[Crossref]

Baney, C. M.

M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
[Crossref]

Boileau, E.

Capmany, J.

Chantada, L.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Spectral analysis of the temporal self-imaging phenomenon in fiber dispersive lines,” J. Lightwave Technol. 24, 2015-2025 (2006).
[Crossref]

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[Crossref]

Crosignani, B.

Davis, B. J.

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[Crossref]

Derickson, D.

D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).

Devaux, F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11, 1937-1940 (1993).
[Crossref]

Dorrer, C.

Eberly, J. H.

Edagawa, N.

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

Eyal, A.

Fernández-Pousa, C. R.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Spectral analysis of the temporal self-imaging phenomenon in fiber dispersive lines,” J. Lightwave Technol. 24, 2015-2025 (2006).
[Crossref]

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[Crossref]

Fortenberry, R. M.

R. M. Fortenberry and W. V. Sorin, “Apparatus for characterizing short optical pulses,” U.S. patent 5,684,568 (June 13, 1996).

Fraile Peláez, F. J.

F. J. Fraile Peláez, “Analytical signal formalism in the description of optical pulse photodetection,” Microwave Opt. Technol. Lett. 37, 347-352 (2003).
[Crossref]

Franks, L. E.

W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[Crossref]

Gardner, W. A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[Crossref]

W. A. Gardner, “Exploitation of spectral redundancy in cyclostationary signals,” IEEE Signal Process. Mag. 8, 14-36 (1991).
[Crossref]

W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random processes,” IEEE Trans. Inf. Theory 21, 4-14 (1975).
[Crossref]

Gómez-Reino, C.

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Theory of the partially coherent temporal Talbot effect,” Opt. Commun. 266, 393-398 (2006).
[Crossref]

L. Chantada, C. R. Fernández-Pousa, and C. Gómez-Reino, “Spectral analysis of the temporal self-imaging phenomenon in fiber dispersive lines,” J. Lightwave Technol. 24, 2015-2025 (2006).
[Crossref]

Gómez-Sarabia, C. M.

Goodman, J. W.

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

Gosnell, T. R.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[Crossref]

Ip, E.

Kerdiles, J. F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11, 1937-1940 (1993).
[Crossref]

Khan, J. M.

Laming, R. I.

P. R. Morkel, R. I. Laming, and D. N. Payne, “Noise characteristics of high-power doped-fibre superluminiscent sources,” Electron. Lett. 26, 96-98 (1990).
[Crossref]

Lancis, J.

Magill, B. E.

Mandel, L.

L. J. Wang, B. E. Magill, and L. Mandel, “Propagation of thermal light through a dispersive medium,” J. Opt. Soc. Am. B 6, 964-966 (1989).
[Crossref]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marshall, W. K.

W. K. Marshall and A. Yariv, “Spectrum of the intensity of modulated noisy light after propagation in dispersive fiber,” IEEE Photon. Technol. Lett. 12, 302-304 (2000).
[Crossref]

W. K. Marshall, B. Crosignani, and A. Yariv, “Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory,” Opt. Lett. 25, 165-167 (2000).
[Crossref]

Martínez, A.

Maywar, D. N.

Minasian, R. A.

Morkel, P. R.

P. R. Morkel, R. I. Laming, and D. N. Payne, “Noise characteristics of high-power doped-fibre superluminiscent sources,” Electron. Lett. 26, 96-98 (1990).
[Crossref]

Muriel, M. A.

J. Azaña and M. A. Muriel, “Temporal self-imaging effect: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[Crossref]

Napolitano, A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[Crossref]

Nazarathy, M.

M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
[Crossref]

Newton, S. A.

M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
[Crossref]

Nurmikko, A. V.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[Crossref]

Ogura, H.

H. Ogura, “Spectral representation of a periodic nonstationary random process,” IEEE Trans. Inf. Theory 17, 143-149 (1971).
[Crossref]

Ojeda-Castañeda, J.

Ortega, B.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

Pastor, D.

Paura, L.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[Crossref]

Payne, D. N.

P. R. Morkel, R. I. Laming, and D. N. Payne, “Noise characteristics of high-power doped-fibre superluminiscent sources,” Electron. Lett. 26, 96-98 (1990).
[Crossref]

Peterman, K.

K. Peterman and E. Weidel, “Semiconductor laser noise in an interferometer system,” IEEE J. Quantum Electron. 17, 1251-1256 (1981).
[Crossref]

Picinbono, B.

Pierobon, G. L.

G. L. Pierobon and L. Tomba, “Moment characterization of phase noise in coherent optical systems,” J. Lightwave Technol. 9, 996-1005 (1991).
[Crossref]

Ponomarenko, S. A.

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

Risk, W. P.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[Crossref]

Rudin, W.

W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill, 1986).

Shafir, E.

E. Shafir and Y. Weissman, “Effects of chromatic dispersion on microwave and millimetre wave carrier transmission in optical fibres,” J. Mod. Opt. 46, 2143-2156 (1999).

Sorel, Y.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11, 1937-1940 (1993).
[Crossref]

Sorin, W. V.

M. Nazarathy, W. V. Sorin, C. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083-1096 (1989).
[Crossref]

R. M. Fortenberry and W. V. Sorin, “Apparatus for characterizing short optical pulses,” U.S. patent 5,684,568 (June 13, 1996).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Taga, H.

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

Tomba, L.

G. L. Pierobon and L. Tomba, “Moment characterization of phase noise in coherent optical systems,” J. Lightwave Technol. 9, 996-1005 (1991).
[Crossref]

Torres-Company, V.

Toshida, Y.

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

von der Linde, D.

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 2041-217 (1986).

Wakabayashi, H.

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

Wang, L. J.

Wax, N.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944) and S. O. Rice, Bell Syst. Tech. J. 24, 46-156 (1945) [reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003)].

Weidel, E.

K. Peterman and E. Weidel, “Semiconductor laser noise in an interferometer system,” IEEE J. Quantum Electron. 17, 1251-1256 (1981).
[Crossref]

Weissman, Y.

E. Shafir and Y. Weissman, “Effects of chromatic dispersion on microwave and millimetre wave carrier transmission in optical fibres,” J. Mod. Opt. 46, 2143-2156 (1999).

Y. Weissman, Optical Network Theory (Artech House, 1992).

Wódkiewicz, K.

Wolf, E.

S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394-396 (2004).
[Crossref] [PubMed]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231-287 (1965).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Yamamoto, S.

S. Yamamoto, N. Edagawa, H. Taga, Y. Toshida, and H. Wakabayashi, “Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation and direct detection optical-fiber transmission,” J. Lightwave Technol. 8, 1716-1722 (1990).
[Crossref]

Yariv, A.

W. K. Marshall and A. Yariv, “Spectrum of the intensity of modulated noisy light after propagation in dispersive fiber,” IEEE Photon. Technol. Lett. 12, 302-304 (2000).
[Crossref]

W. K. Marshall, B. Crosignani, and A. Yariv, “Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory,” Opt. Lett. 25, 165-167 (2000).
[Crossref]

Yi, X.

Zadok, A.

Appl. Phys. B (1)

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 2041-217 (1986).

Bell Syst. Tech. J. (1)

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Other (12)

Formula in lacks the ensemble average because it was derived from a single realization, i.e., in the framework of the time series (see below). Our Eq. is the formula for the stochastic framework.

Y. Weissman, Optical Network Theory (Artech House, 1992).

The general transformation rule of cyclic spectra after a linear, almost-periodically time-variant system can be consulted, for instance, in , Sect. 3.6.

W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill, 1986).

In the intensity is defined by low-pass filtering the square of the (real) electric field, Re[ẽ(t)]. Here we follow the standard notation in wave optics, p. 162. The definition of implies a difference of 12 in the intensity and a subsequent difference of 14 in the intensity spectrum.

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Figures (6)

Fig. 1
Fig. 1

Scheme of the systems under consideration. Top, a general nonstationary wave is first-order dispersed and subsequently detected and analyzed in the RF domain. Bottom, in the simplest case, the nonstationary wave can be created by the combination of a stationary source (eventually partially coherent) and external modulation (eventually nondeterministic).

Fig. 2
Fig. 2

Schematic representation of the optical spectrum and the intensity spectrum after dispersion of an arrayed stationary source with two partially coherent, independent waves at ω 0 and ω 0 + Δ ω . (a) Optical spectrum, Eq. (39). (b) Two-sided intensity spectrum, obtained from Eq. (44) after the substitution τ = ϕ Ω . The numbers refer to the four terms in Eq. (44): 1, dc level; 2, excess intensity noise of the wave at ω 0 ; 3, excess intensity noise of the wave at ω 0 + Δ ω ; 4, beat between the two components of the array. In (a), the arrow indicates the beat tone between carriers.

Fig. 3
Fig. 3

Optical spectrum and intensity spectrum after dispersion for a SSB-modulated carrier with RF-phase noise. (a) Optical spectrum, Eq. (48), composed of two coherent waves at ω 0 and ω 0 + Ω 0 , and a noise skirt at ω 0 + Ω 0 (b) Intensity spectrum. The numbers refer to the three terms in Eq. (49).

Fig. 4
Fig. 4

Optical spectrum and intensity spectrum after dispersion for a DSB-modulated partially coherent carrier. (a) Optical spectrum, composed of a partially coherent carrier at ω 0 and its sidebands at ω 0 ± Ω 0 . (b) Intensity spectrum. The numbers refer to the four terms in Eq. (57).

Fig. 5
Fig. 5

Optical spectrum and intensity spectrum after dispersion of sum-frequency generated light. (a) Optical spectrum, whose linewidth is the convolution of the linewidths of the mixed waves. (b) Intensity spectrum. The numbers refer to the four terms in Eq. (60).

Fig. 6
Fig. 6

Optical spectrum and intensity noise spectrum of the two-tap photonic microwave filter. (a) Optical spectrum of the unmodulated stationary source. (b) Optical spectrum after DSB amplitude modulation with a tone at frequency Ω 0 . (c) Two-sided intensity spectrum, Eqs. (62, 63). In (b) and (c) the numbers refer to the five terms in Eq. (63). In (c) the letters refer to the two terms in Eq. (62). A, dc level; B, modulation sidebands.

Equations (89)

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G ( Ω ) = lim L 1 2 L E L ( Ω ) 2 .
G ( Ω ) = d τ Γ ¯ e * e ( τ ) e j Ω τ ,
Γ ¯ e * e ( τ ) Γ e * e ( t , t + τ ) ¯ = lim L 1 2 L L L d t Γ e * e ( t , t + τ ) .
S ( Ω ) = lim L 1 2 L I L ( Ω ) 2 = lim L 1 2 L 1 2 π E L ( Ω ) E L * ( Ω ) 2 ,
S ( Ω ) = d u Γ ¯ i i ( u ) e j Ω u = S i ( Ω ) + S Δ i ( Ω ) .
S ( Ω , ϕ ) = d u Γ e * e e * e ( t , t + ϕ Ω , t + ϕ Ω + u , t + u ) ¯ e j Ω u ,
Γ e * e e * e ( t 1 , t 2 , t 3 , t 4 ) = e * ( t 1 ) e ( t 2 ) e * ( t 3 ) e ( t 4 )
z τ ( t ) = e * ( t ) e ( t + τ )
Φ ( Ω , τ ) = d u z τ * ( t ) z τ ( t + u ) ¯ e j Ω u = d u e ( t ) e * ( t + τ ) e * ( t + u ) e ( t + u + τ ) ¯ e j Ω u .
Φ ( Ω , τ ) = d u e * ( t ) e ( t τ ) e * ( t τ + u ) e ( t + u ) ¯ e j Ω u .
S ( Ω , ϕ ) = Φ ( Ω , τ = ϕ Ω ) ,
Γ e * e ( t , t + τ ) = k Γ e * e ( k ) ( τ ) exp ( j Ω 0 k t ) ,
E * ( Ω 1 ) E ( Ω 2 ) = 2 π k G ( k ) ( Ω 2 ) δ ( Ω 2 Ω 1 Ω 0 k ) ,
E ( Ω ) = k C k ( Ω Ω 0 k ) ,
C n * ( Ω 1 ) C p ( Ω 2 ) = 2 π F n , p ( Ω 2 ) δ ( Ω 2 Ω 1 ) ,
G ( k ) ( Ω ) = n F n k , n ( Ω Ω 0 n ) ,
e ( t ) = k c k ( t ) exp ( j Ω 0 k t ) ,
Γ e * e ( t , t + τ ) = k exp ( j Ω 0 k t ) n c n k * ( t ) c n ( t + τ ) exp ( j Ω 0 n τ ) ,
Γ e * e ( k ) ( τ ) = n c n k * ( t ) c n ( t + τ ) exp ( j Ω 0 n τ ) .
z τ ( t ) = z τ ( t ) + Δ z τ ( t ) = Γ e * e ( t , t + τ ) + Δ z τ ( t ) ,
Φ ( Ω , τ ) = Φ z ( Ω , τ ) + Φ Δ z ( Ω , τ ) .
Φ z ( Ω , τ ) = d u z τ ( t ) * z τ ( t + u ) ¯ e j Ω u = k , p Γ e * e ( p ) * ( τ ) Γ e * e ( k ) ( τ ) exp [ j ( k p ) Ω 0 t ] ¯ d u e j ( Ω k Ω 0 ) u = 2 π k , p Γ e * e ( p ) * ( τ ) Γ e * e ( k ) ( τ ) δ k , p δ ( Ω k Ω 0 ) = 2 π k Γ e * e ( k ) ( τ ) 2 δ ( Ω k Ω 0 ) .
S i ( Ω , ϕ ) = Φ z ( Ω , ϕ Ω ) .
I ( Ω , ϕ ) = d t i ϕ ( t ) e j Ω t = A F e ( Ω , ϕ Ω ) ,
A F e ( Ω , u ) = d t e * ( t u 2 ) e ( t + u 2 ) exp ( j Ω t ) .
I ( Ω , ϕ ) = 2 π e j ϕ Ω 2 2 k Γ e * e ( k ) ( ϕ Ω ) δ ( Ω k Ω 0 ) ,
I ( Ω , ϕ ) = d Ω ̃ 2 π E ϕ * ( Ω ̃ Ω ) E ϕ ( Ω ̃ ) = 2 π k δ ( Ω Ω 0 k ) d Ω ̃ 2 π G ϕ ( k ) ( Ω ̃ ) ,
E ϕ * ( Ω ̃ Ω ) E ϕ ( Ω ̃ ) = E * ( Ω ̃ Ω ) E ( Ω ̃ ) e j ϕ Ω 2 2 e j ϕ Ω ̃ Ω ,
G ϕ ( k ) ( Ω ̃ ) = G ( k ) ( Ω ̃ ) exp ( j ϕ k 2 Ω 0 2 2 j ϕ k Ω 0 Ω ̃ ) .
S Δ i ( Ω , ϕ ) = Φ Δ z ( Ω , ϕ Ω ) .
lim u Δ z τ * ( t ) Δ z τ ( t + u ) = 0 .
lim u e * ( t ) e ( t τ ) e * ( t τ + u ) e ( t + u ) = Γ e * e * ( t τ , t ) Γ e * e ( t τ + u , t + u ) .
Φ ( Ω , τ ) = 2 π k Γ e * e ( k ) ( τ ) 2 δ ( Ω k Ω 0 ) + Φ Δ z ( Ω , τ ) ,
Γ e * e ( k ) ( τ ) = n c n k * c n exp ( j Ω 0 n τ ) ,
G ( Ω ) = G ( 0 ) ( Ω ) = 2 π n c n 2 δ ( Ω n Ω 0 ) .
Γ e * e ( k ) ( ϕ k Ω 0 ) = Γ e * e ( k ) ( ϕ k Ω 0 s k T ) = Γ e * e ( k ) ( ϕ k Ω 0 ) .
e ( t ) = a ( t ) + b ( t ) e j Δ ω t ,
Γ e * e ( k ) ( τ ) = Γ a * a ( k ) ( τ ) + δ k , 0 Γ b * b ( 0 ) ( τ ) e j Δ ω τ ,
G e ( Ω ) = G a ( Ω ) + G b ( Ω Δ ω ) ,
Γ e * e e * e ( t , t τ , t τ + u , t + u ) = Γ a * a a * a ( t , t τ , t τ + u , t + u ) + Γ b * b b * b ( τ , τ + u , u ) + Γ a * a ( t , t τ ) Γ b * b ( τ ) e j Δ ω τ + Γ a * a ( t τ + u , t + u ) Γ b * b ( τ ) e j Δ ω τ + Γ a * a ( t τ + u , t τ ) Γ b * b ( u ) e j Δ ω u + Γ a * a ( t , t + u ) Γ b * b ( u ) e j Δ ω u .
Γ ¯ e * e e * e ( τ , τ + u , u ) = Γ ¯ a * a a * a ( τ , τ + u , u ) + Γ b * b b * b ( τ , τ + u , u ) + Γ ¯ a * a ( τ ) Γ b * b ( τ ) e j Δ ω τ + Γ ¯ a * a ( τ ) Γ b * b ( τ ) e j Δ ω τ + Γ ¯ a * a ( u ) Γ b * b ( u ) e j Δ ω u + Γ ¯ a * a ( u ) Γ b * b ( u ) e j Δ ω u ,
Φ e ( Ω , τ ) = Φ a ( Ω , τ ) + Φ b ( Ω , τ ) + 2 π ( Γ ¯ a * a ( τ ) Γ b * b * ( τ ) e j Δ ω τ + c.c. ) δ ( Ω ) + 1 2 π ( G a ( Ω ) G b ( Ω ) Ω = Ω Δ ω + G a ( Ω ) G b ( Ω ) Ω = Ω + Δ ω ) ,
S e ( Ω , ϕ ) = S a ( Ω , ϕ ) + S b ( Ω , ϕ ) + 4 π Γ ¯ a * a ( 0 ) Γ b * b ( 0 ) δ ( Ω ) + 1 2 π G a ( Ω ) G b ( Ω ) Ω = Ω Δ ω + 1 2 π G a ( Ω ) G b ( Ω ) Ω = Ω + Δ ω .
Φ e ( Ω , τ ) = 2 π Γ a * a ( τ ) + Γ b * b ( τ ) e j Δ ω τ 2 δ ( Ω ) + Φ Δ z , a ( Ω , τ ) + Φ Δ z , b ( Ω , τ ) + 1 2 π G a ( ± Ω ) G b ( Ω ) Ω = Ω ± Δ ω ,
G a ( Ω ) G a ( Ω ) + G b ( Ω ) G b ( Ω ) + G a ( ± Ω ) G b ( Ω ) Ω = Ω ± Δ ω = G e ( Ω ) G e ( Ω ) ,
e ( t ) = 1 ( μ 2 8 ) + ( μ 2 ) exp ( j Ω 0 t + j φ ( t ) ) 1 ( μ 2 8 ) + ( μ 2 ) exp ( j Ω 0 t ) + j ( μ 2 ) φ ( t ) exp ( j Ω 0 t ) ,
Γ b * b ( u ) = b ( t ) * b ( t + u ) = ( μ 2 4 ) R φ ( u ) ,
G e ( Ω ) = 2 π ( 1 μ 2 4 ) δ ( Ω ) + 2 π ( μ 2 4 ) δ ( Ω Ω 0 ) + ( μ 2 4 ) S φ ( Ω Ω 0 ) ,
S e ( Ω , ϕ ) = 2 π 1 + ( μ 2 2 ) φ rms 2 δ ( Ω ) + 2 π ( μ 2 4 ) δ ( Ω ± Ω 0 ) + ( μ 2 4 ) S φ ( Ω ± Ω 0 ) .
e ( t ) = m ( t ) c ( t ) .
Γ e * e ( k ) ( τ ) = Γ m * m ( k ) ( τ ) Γ c * c ( 0 ) ( τ ) ,
G e ( Ω ) = 1 2 π G m ( Ω ) G c ( Ω ) .
Γ e * e e * e ( t , t τ , t τ + u , t + u ) = Γ m * m m * m ( t , t τ , t τ + u , t + u ) Γ c * c c * c ( τ , τ + u , u ) ,
Φ e ( Ω , τ ) = 1 2 π Φ m ( Ω , τ ) Ω Φ c ( Ω , τ ) ,
S e ( Ω , ϕ ) = Γ c * c ( ϕ Ω ) 2 S m ( Ω , ϕ ) + 1 2 π Φ m ( Ω , τ ) Ω Φ Δ z , c ( Ω , τ ) τ = ϕ Ω .
S e ( Ω , ϕ ) = 2 π k Γ c * c ( ϕ k Ω 0 ) 2 Γ m * m ( k ) ( ϕ k Ω 0 ) 2 δ ( Ω k Ω 0 ) + Γ c * c ( ϕ Ω ) 2 S Δ i , m ( Ω , ϕ ) + k Γ m * m ( k ) ( ϕ Ω ) 2 S Δ i , c ( Ω k Ω 0 , ϕ ) + 1 2 π Φ Δ z , m ( Ω , τ ) Ω Φ Δ z , c ( Ω , τ ) τ = ϕ Ω ,
S e ( Ω , ϕ ) = 2 π Γ c * c 2 ( 0 ) δ ( Ω ) + 2 π ( μ 2 4 ) cos 2 ( ϕ Ω 0 2 2 ) Γ c * c ( ϕ Ω 0 ) 2 δ ( Ω ± Ω 0 ) + [ 1 ( μ 2 2 ) sin 2 ( Ω 0 ϕ Ω 2 ) ] S Δ i , c ( Ω , ϕ ) + ( μ 2 4 ) cos 2 ( Ω 0 ϕ Ω 2 ) S Δ i , c ( Ω ± Ω 0 , ϕ ) .
[ 1 ( μ 2 2 ) sin 2 ( π ϕ Ω 0 f ) ] RIN ( f , ϕ ) d f + ( μ 2 4 ) cos 2 ( π ϕ Ω 0 f ) RIN ( f ± f 0 , ϕ ) d f ,
e ( t ) = a ( t ) b ( t ) = e j φ ̃ a ( t ) + j φ ̃ b ( t )
S e ( Ω , ϕ ) = 2 π ( Γ a * a ( 0 ) Γ b * b ( 0 ) ) 2 δ ( Ω ) + Γ a * a ( ϕ Ω ) 2 S Δ i , b ( Ω , ϕ ) + Γ b * b ( ϕ Ω ) 2 S Δ i , a ( Ω , ϕ ) + 1 2 π Φ Δ z , a ( Ω , τ ) Ω Φ Δ z , b ( Ω , τ ) τ = ϕ Ω .
Φ a ( Ω , τ ) = 2 π Γ a * a ( τ ) 2 1 + ξ e j Δ ω τ 2 δ ( Ω ) + ( 1 + ξ 2 ) Φ Δ z , a ( Ω , τ ) + 1 2 π ξ G a ( Ω ) G a ( Ω ) Ω = Ω ± Δ ω .
S i , e ( Ω , ϕ ) = 2 π ( 1 + ξ ) 2 Γ a * a 2 ( 0 ) δ ( Ω ) + 2 π ( μ 2 4 ) cos 2 ( ϕ Ω 0 2 2 ) Γ a * a ( ϕ Ω 0 ) 2 1 + ξ e j Δ ω ϕ Ω 0 2 δ ( Ω ± Ω 0 ) .
S Δ i , e ( Ω , ϕ ) = [ 1 ( μ 2 2 ) sin 2 ( Ω 0 ϕ Ω 2 ) ] ( 1 + ξ 2 ) S Δ i , a ( Ω , ϕ ) + ( μ 2 4 ) cos 2 ( Ω 0 ϕ Ω 2 ) ( 1 + ξ 2 ) S Δ i , a ( Ω ± Ω 0 , ϕ ) + 1 2 π [ 1 ( μ 2 2 ) sin 2 ( Ω 0 ϕ Ω 2 ) ] ξ G a ( Ω ) G a ( Ω ) Ω = Ω ± Δ ω + 1 2 π ( μ 2 4 ) cos 2 ( Ω 0 ϕ Ω 2 ) ξ G a ( Ω ) G a ( Ω ) Ω = Ω ± Δ ω ± Ω 0 + 1 2 π ( μ 2 4 ) cos 2 ( Ω 0 ϕ Ω 2 ) ξ G a ( Ω ) G a ( Ω ) Ω = Ω ± Δ ω Ω 0 .
e ( t ) = a ( t ) e j θ a + b ( t ) e j Δ ω 0 t + j θ b ,
e ( t ) = [ 1 ( μ 2 8 ) + ( μ 2 ) e j Ω 0 t + j φ ( t ) ] e j θ [ 1 ( μ 2 8 ) + ( μ 2 ) e j Ω 0 t ] e j θ + j ( μ 2 ) φ ( t ) e j Ω 0 t + j θ .
a * ( t ) b ( t τ ) a * ( t τ + u ) b ( t + u ) ( μ 2 ) 2 R φ ( u + τ ) e j Ω 0 ( 2 t + u τ ) ,
Φ ( Ω , τ ) = 2 π Γ e * e ( τ ) 2 δ ( Ω ) + Φ Δ z ( Ω , τ ) ,
Γ e * e 2 ( 0 ) RIN Δ z ( f , τ ) d f = Φ Δ z ( Ω = 2 π f , τ ) d Ω ,
e * ( t ) e ( t τ ) e * ( t τ + u ) e ( t + u ) = Γ e * e ( τ ) 2 + Γ e * e ( u ) 2 ,
Φ ( Ω , τ ) = 2 π Γ e * e ( τ ) 2 δ ( Ω ) + 1 2 π G ( Ω ) G ( Ω ) .
Φ ( Ω , τ ) = 2 π Γ e * e ( τ ) 2 δ ( Ω ) + 2 Δ ω Δ ω 2 + Ω 2 [ 1 exp ( Δ ω τ ) × ( cos ( Ω τ ) + Δ ω Ω sin ( Ω τ ) ) ] ,
lim u or τ e ( t ) e * ( t τ ) e * ( t τ + u ) e ( t + u ) = Γ e * e ( τ ) 2 + Γ e * e ( u ) 2 .
e ( t ) = m ( t ) = [ 1 + μ cos ( Ω 0 t ) ] 1 2 = 1 + ( μ 2 ) cos ( Ω 0 t ) ( μ 2 8 ) cos 2 ( Ω 0 t ) + O ( μ 3 ) ,
e ( t ) = m ( t ) = 1 ( μ 2 16 ) + ( μ 2 ) cos ( Ω 0 t ) ( μ 2 16 ) cos ( 2 Ω 0 t ) + O ( μ 3 ) .
Γ m * m ( 0 ) ( τ ) = 1 ( μ 2 8 ) + ( μ 2 8 ) cos ( Ω 0 τ ) ,
Γ m * m ( ± 1 ) ( τ ) = ( μ 2 ) e ± j Ω 0 τ 2 cos ( Ω 0 τ 2 ) .
Φ ( Ω , τ ) = 2 π 1 ( μ 2 2 ) sin 2 ( Ω 0 τ 2 ) δ ( Ω ) + 2 π ( μ 2 4 ) cos 2 ( Ω 0 τ 2 ) δ ( Ω ± Ω 0 ) .
G ( Ω ) = 2 π ( 1 μ 2 8 ) δ ( Ω ) + 2 π ( μ 2 16 ) δ ( Ω ± Ω 0 ) .
e ( t ) = m ( t ) = 1 ( μ 2 8 ) + ( μ 2 ) exp ( j Ω 0 t ) ,
Γ m * m ( 0 ) ( τ ) = 1 ( μ 2 4 ) + ( μ 2 4 ) e j Ω 0 τ ,
Γ m * m ( 1 ) ( τ ) = ( μ 2 ) e j Ω 0 τ , Γ m * m ( 1 ) ( τ ) = μ 2 .
Φ ( Ω , τ ) = 2 π 1 ( μ 2 2 ) sin 2 ( Ω 0 τ 2 ) δ ( Ω ) + 2 π ( μ 2 4 ) δ ( Ω ± Ω 0 ) ,
G ( Ω ) = 2 π ( 1 μ 2 4 ) δ ( Ω ) + 2 π ( μ 2 4 ) δ ( Ω Ω 0 ) .
e ( t ) = m ( t ) = exp [ j β cos ( Ω 0 t ) ] .
z τ ( t ) = m ( t ) * m ( t + τ ) = exp [ j β cos ( Ω 0 t ) + j β cos ( Ω 0 t + Ω 0 τ ) ] .
z τ ( t ) = exp [ j 2 β sin ( Ω 0 τ 2 ) sin ( Ω 0 t + Ω 0 τ 2 ) ] = k J k ( 2 β sin ( Ω 0 τ 2 ) ) exp ( j k Ω 0 τ 2 ) exp ( j k Ω 0 t ) ,
Γ m * m ( k ) ( τ ) = J k ( 2 β sin ( Ω 0 τ 2 ) ) e j k Ω 0 τ 2 ,
Φ ( Ω , τ ) = 2 π k J k 2 ( 2 β sin ( Ω 0 τ 2 ) ) δ ( Ω k Ω 0 ) .
G ( Ω ) = J 0 ( 2 β sin ( Ω 0 τ 2 ) ) e j Ω τ d τ = 2 π n J n 2 ( β ) δ ( Ω n Ω 0 ) ,

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