Abstract

After temporal amplitude modulation of a spectrally incoherent optical source the averaged intensity profile at the so-called temporal far-zone regime coalesces with a magnified replica of the spectral density function of the source. This has provided the basis for the generalization of the frequency-to-time mapping technique in the partially coherent case. Based on this fact, temporal intensity waveform generation is demonstrated by spectral filtering the incoherent source before the temporal modulation stage. We refer to this technique as full incoherent pulse shaping. Although only the average intensity of the output signal is properly shaped, intensity fluctuations between the different realizations of the output shaped waveform are shown to be small in the practical situation. Finally, we provide some computer simulations concerning arbitrary picosecond pulse generation from an amplified spontaneous emission source.

© 2007 Optical Society of America

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References

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  1. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
    [CrossRef]
  2. L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
    [CrossRef]
  3. J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
    [CrossRef]
  4. T. Jannson, "Real-time Fourier transformation in dispersive optical fibers," Opt. Lett. 8, 232-235 (1983).
    [CrossRef] [PubMed]
  5. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
    [CrossRef]
  6. Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
    [CrossRef]
  7. M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings," Opt. Lett. 24, 1-3 (1999).
    [CrossRef]
  8. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 1184-1193 (1986).
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    [CrossRef]
  14. 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]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. C. Dorrer and I. A. Wamsley, "Concepts for the temporal characterization of short optical pulses," EURASIP J. Appl. Sign. Process. 10, 1541-1553 (2005).
    [CrossRef]
  20. F. Gori, "Far-zone approximation for partially coherent sources," Opt. Lett. 30, 2840-2842 (2005).
    [CrossRef] [PubMed]
  21. A. C. Schell, "A technique for the determination of the radiation pattern of a partially coherent aperture," IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
    [CrossRef]
  22. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  23. V. Binjrajka, C. C. Chang, A. W. R. Emanuel, D. E. Leaird, and A. M. Weiner, "Pulse shaping of incoherent light by use of a liquid-crystal modulator array," Opt. Lett. 21, 1756-1758 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]

2006 (1)

2005 (4)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

C. Dorrer and I. A. Wamsley, "Concepts for the temporal characterization of short optical pulses," EURASIP J. Appl. Sign. Process. 10, 1541-1553 (2005).
[CrossRef]

F. Gori, "Far-zone approximation for partially coherent sources," Opt. Lett. 30, 2840-2842 (2005).
[CrossRef] [PubMed]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent plane-wave-type pulses," Opt. Lett. 30, 2973-2975 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (2)

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

2000 (3)

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

1999 (1)

1997 (1)

Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
[CrossRef]

1996 (1)

1993 (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

1986 (2)

R. M. Abdula and B. E. A. Saleh, "Dynamic spectra of pulsed laser-diodes and propagation in single-mode fibers," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 1184-1193 (1986).

1983 (1)

1981 (1)

1967 (1)

A. C. Schell, "A technique for the determination of the radiation pattern of a partially coherent aperture," IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

Abdula, R. M.

R. M. Abdula and B. E. A. Saleh, "Dynamic spectra of pulsed laser-diodes and propagation in single-mode fibers," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

Agrawal, G. P.

Andres, P.

Andrés, M. V.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Azaña, J.

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings," Opt. Lett. 24, 1-3 (1999).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

Barthélémy, A.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Bekker, A.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

Berger, N. K.

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

Binjrajka, V.

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

Capmany, J.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Carballar, A.

Chan, L. Y.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
[CrossRef]

Chang, C. C.

Cruz, J. L.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Dorrer, C.

C. Dorrer and I. A. Wamsley, "Concepts for the temporal characterization of short optical pulses," EURASIP J. Appl. Sign. Process. 10, 1541-1553 (2005).
[CrossRef]

C. Dorrer, "Temporal van Cittert-Zernike theorem and its application to the measurement of chromatic dispersion," J. Opt. Soc. Am. B 21, 1417-1423 (2004).
[CrossRef]

Emanuel, A. W. R.

Fischer, B.

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

Friberg, A. T.

Froehly, C.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

Gori, F.

Irshid, M. I.

B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 1184-1193 (1986).

Jannson, T.

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

Lajunen, H.

Lancis, J.

Leaird, D. E.

Levit, B.

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

Louradour, F.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Mandel, L.

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

Marcuse, D.

Messager, V.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Mora, J.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Mouradian, L. K.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Muriel, M. A.

Ortega, B.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Ostlund, P.

Pastor, D.

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Ponomarenko, S. A.

Saleh, B. E. A.

R. M. Abdula and B. E. A. Saleh, "Dynamic spectra of pulsed laser-diodes and propagation in single-mode fibers," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 1184-1193 (1986).

Schell, A. C.

A. C. Schell, "A technique for the determination of the radiation pattern of a partially coherent aperture," IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

Silvestre, E.

Tervo, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Tong, Y. C.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
[CrossRef]

Torres-Company, V.

Tsang, H. K.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
[CrossRef]

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Wamsley, I. A.

C. Dorrer and I. A. Wamsley, "Concepts for the temporal characterization of short optical pulses," EURASIP J. Appl. Sign. Process. 10, 1541-1553 (2005).
[CrossRef]

Weiner, A. M.

Wolf, E.

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270-272 (1993).
[CrossRef]

Electron. Lett. (3)

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Time-lens-based spectral analysis of optical pulses by electrooptic modulation," Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

Y. C. Tong, L. Y. Chan, and H. K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett. 33, 983-985 (1997).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Real-time optical spectrum analyser based on chirped fibre gratings," Electron. Lett. 36, 1189-1191 (2000).
[CrossRef]

EURASIP J. Appl. Sign. Process. (1)

C. Dorrer and I. A. Wamsley, "Concepts for the temporal characterization of short optical pulses," EURASIP J. Appl. Sign. Process. 10, 1541-1553 (2005).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. M. Abdula and B. E. A. Saleh, "Dynamic spectra of pulsed laser-diodes and propagation in single-mode fibers," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, "Spectro-temporal imaging of femtosecond events," IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, "Spectro-temporal imaging of optical pulses with a single time lens," IEEE Photon. Technol. Lett. 16, 882-884 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. C. Schell, "A technique for the determination of the radiation pattern of a partially coherent aperture," IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

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

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

Opt. Commun. (2)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

J. Mora, B. Ortega, J. L. Cruz, J. Capmany, D. Pastor, and M. V. Andrés, "White light sources filtered with fiber Bragg gratings for RF-photonics applications," Opt. Commun. 222, 221-225 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Other (2)

G. P. Agrawal, Fiber-Optic Communications Systems (Wiley, 1997).

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

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Figures (6)

Fig. 1
Fig. 1

Quasi-homogeneous light pulse: (a) modulus of the complex degree of coherence and (b) temporal averaged intensity.

Fig. 2
Fig. 2

Schematic of the mathematical 1D Fourier transformations associated with a quasi-homogeneous light pulse in the far zone.

Fig. 3
Fig. 3

Schematic of the principle of incoherent frequency-to-time mapping. The dispersive medium is represented by a LCFG working in reflection.

Fig. 4
Fig. 4

Block diagram of operation of the incoherent pulse shaping technique.

Fig. 5
Fig. 5

(a) Spectral density function of an ASE source (dashed curve) and the numerically calculated required filter (solid curve) and (b) averaged output intensity function consisting in a Gaussian pulse with two satellite peaks.

Fig. 6
Fig. 6

(a) Same as in Fig. 5a, but for a different spectral filter and (b) averaged output intensity function corresponding to a rectangular pulse.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

Γ out ( t 1 , t 2 ) = exp ( j t 1 2 t 2 2 2 Φ ̈ ) d t 1 d t 2 Γ in ( t 1 , t 2 ) exp ( j t 1 2 t 2 2 2 Φ ̈ ) exp ( j t 1 t 1 t 2 t 2 Φ ̈ ) .
Φ ̈ σ o min [ σ c , 2 σ o ] 4 π .
Γ out ( t 1 , t 2 ) = exp ( j t 1 2 t 2 2 2 Φ ̈ ) d t 1 d t 2 Γ in ( t 1 , t 2 ) exp ( j t 1 t 1 t 2 t 2 Φ ̈ ) .
Γ in ( t 1 , t 2 ) = I in ( t 1 + t 2 2 ) γ in ( t 2 t 1 ) .
Γ out ( t 1 , t 2 ) = exp ( j t 1 2 t 2 2 2 Φ ̈ ) F T [ I in ( t ) ] ω = ( t 2 t 1 ) ϕ ̈ F T [ γ in ( t ) ] ω = ( t 2 + t 1 ) 2 ϕ ̈ ,
γ out ( t 1 , t 2 ) = Γ out ( t 1 , t 2 ) Γ out ( t 1 , t 1 ) Γ out ( t 2 , t 2 ) = exp ( j t 1 2 t 2 2 2 Φ ̈ ) F T [ γ in ( t ) ] ω = ( t 2 + t 1 ) 2 ϕ ̈ F T [ γ in ( t ) ] ω = t 1 ϕ ̈ F T [ γ in ( t ) ] ω = t 2 ϕ ̈ F T [ I in ( t ) ] ω = ( t 2 t 1 ) ϕ ̈ ,
I out ( t ) = Γ out ( t , t ) = I o F T [ γ in ( t ) ] ω = t ϕ ̈ .
γ out ( t 1 , t 2 ) γ out ( t 2 t 1 ) = F T [ I in ( t ) ] ω = ( t 2 t 1 ) ϕ ̈ ,
Γ in ( t 1 , t 2 ) = M * ( t 1 ) M ( t 2 ) Γ s ( t 2 t 1 ) .
Γ s ( t 2 t 1 ) = d ω S ( ω ) exp [ j ω ( t 2 t 1 ) ] .
γ in ( t 2 t 1 ) = Γ s ( t 2 t 1 ) Γ s ( 0 ) .
Γ in ( t 1 , t 2 ) = M ( t 1 + t 2 2 ) 2 Γ s ( t 2 t 1 ) .
I in ( t ) = Γ s ( 0 ) M ( t ) 2 ,
I out ( t ) = S ( t Φ ̈ ) Γ s ( 0 ) .
Φ ̈ σ o ( 4 π σ ω ) .
Δ I ( t ) = J ( t ) J ( t ) = J ( t ) I ( t ) .
J ( t 1 ) J ( t 2 ) = J ( t 1 ) J ( t 2 ) + Γ ( t 1 , t 2 ) 2 .
Δ I ( t 1 ) Δ I ( t 2 ) J ( t 1 ) J ( t 2 ) = γ ( t 1 , t 2 ) 2 .
σ c , out σ c , in Φ ̈ σ ω σ o .

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