Abstract

We use the periodic-signal ambiguity function for visualizing the intensity-spectrum evolution through propagation in a first-order dispersive medium. We show that the degree of temporal coherence of the optical source plays the role of a low-pass filter on the signal’s ambiguity function. Based on this, we present a condition on the temporal Lau effect for filtering harmonics at fractions of the Talbot length. This result allows one to increase the repetition rate of a pulse train obtained from a sinusoidally phase-modulated CW signal.

© 2007 Optical Society of America

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  1. E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848-3858 (1971).
    [CrossRef]
  2. J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262-2273 (1992).
    [CrossRef]
  3. C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
    [CrossRef]
  4. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, "Heterodyne measurement of Wigner distributions for classical optical fields," Opt. Lett. 24, 1370-1372 (1999).
    [CrossRef]
  5. T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
    [CrossRef]
  6. C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 26, 1481-1483 (2006).
  7. C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
    [CrossRef]
  8. T. Jannson and J. Jannson, "Temporal self-imaging effect in single-mode optical fiber," J. Opt. Soc. Am. 71, 1373-1376 (1981).
  9. J. Azaña and M. A. Muriel, "Temporal Talbot effect in fiber gratings and its applications," Appl. Opt. 38, 6700-6704 (1999).
    [CrossRef]
  10. J. Azaña and M. A. Muriel, "Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings," Opt. Lett. 24, 1672-1674 (1999).
    [CrossRef]
  11. N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
    [CrossRef]
  12. J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
    [CrossRef]
  13. 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]
  14. J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
    [CrossRef]
  15. J. Capmany, A. Martínez, B. Ortega, and D. Pastor, "Transfer function of analog fiber-optic systems driven by Fabry-Perot lasers," J. Opt. Soc. Am. B 22, 2099-2106 (2005).
    [CrossRef]
  16. D. Pastor, B. Ortega, J. Capmany, S. Sales, A. Martínez, and P. Muñoz, "Optical microwave filter based on spectral slicing by use of arrayed waveguide gratings," Opt. Lett. 28, 1802-1804 (2003).
    [CrossRef] [PubMed]
  17. A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779-788 (1974).
    [CrossRef]
  18. J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
    [CrossRef]
  19. K. H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
    [CrossRef]
  20. K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  21. E. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  22. A. Castro and J. Ojeda-Castaneda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
    [CrossRef] [PubMed]
  23. D. Marcuse, "Pulse distortion in single-mode fibers," Appl. Opt. 19, 1653-1660 (1980).
    [CrossRef] [PubMed]
  24. T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
    [CrossRef]
  25. V. Torres-Company, J. Lancis, and P. Andrés, "Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light," Opt. Express 14, 3171-3180 (2006).
    [CrossRef] [PubMed]

2006 (5)

C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 26, 1481-1483 (2006).

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (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]

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

V. Torres-Company, J. Lancis, and P. Andrés, "Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light," Opt. Express 14, 3171-3180 (2006).
[CrossRef] [PubMed]

2005 (3)

T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
[CrossRef]

J. Capmany, A. Martínez, B. Ortega, and D. Pastor, "Transfer function of analog fiber-optic systems driven by Fabry-Perot lasers," J. Opt. Soc. Am. B 22, 2099-2106 (2005).
[CrossRef]

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
[CrossRef]

2004 (1)

2003 (3)

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

D. Pastor, B. Ortega, J. Capmany, S. Sales, A. Martínez, and P. Muñoz, "Optical microwave filter based on spectral slicing by use of arrayed waveguide gratings," Opt. Lett. 28, 1802-1804 (2003).
[CrossRef] [PubMed]

1999 (3)

1998 (1)

C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
[CrossRef]

1995 (1)

1992 (1)

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262-2273 (1992).
[CrossRef]

1984 (1)

K. H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

1983 (1)

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

1981 (1)

1980 (1)

1978 (1)

J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
[CrossRef]

1974 (1)

1971 (1)

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848-3858 (1971).
[CrossRef]

Alieva, T.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

Andrés, P.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

V. Torres-Company, J. Lancis, and P. Andrés, "Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light," Opt. Express 14, 3171-3180 (2006).
[CrossRef] [PubMed]

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Atkins, S.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

Azaña, J.

Bali, S.

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

Berger, N. K.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

Brenner, K. H.

K. H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Capmany, J.

Caraquitena, J.

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Castro, A.

Cathey, W. T.

Chantada, L.

Constanzo-Caso, P.

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
[CrossRef]

Cuadrado-Laborde, C.

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
[CrossRef]

Dorrer, C.

Dowski, E.

Duchowicz, R.

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
[CrossRef]

Fernández-Pousa, C. R.

Fernández-Pouza, C.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

Fischer, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

Gómez-Reino, C.

Gómez-Sarabia, C. M.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

Guigay, J. P.

J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
[CrossRef]

Iaconis, C.

C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
[CrossRef]

Jannson, J.

Jannson, T.

Kang, I.

Kawanishi, S.

T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
[CrossRef]

Komukai, T.

T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
[CrossRef]

Lancis, J.

V. Torres-Company, J. Lancis, and P. Andrés, "Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light," Opt. Express 14, 3171-3180 (2006).
[CrossRef] [PubMed]

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
[CrossRef]

Lee, K. F.

Levit, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Marcuse, D.

Martínez, A.

Muñoz, P.

Muriel, M. A.

Ojeda-Castaneda, J.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

A. Castro and J. Ojeda-Castaneda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
[CrossRef] [PubMed]

K. H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Ortega, B.

Papoulis, A.

Pastor, D.

Paye, J.

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262-2273 (1992).
[CrossRef]

Reil, F.

Sales, S.

Sicre, E. E.

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
[CrossRef]

Stankovic, L.

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

Thomas, J. E.

Torres-Company, V.

Treacy, E. B.

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848-3858 (1971).
[CrossRef]

Walmsley, I. A.

C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
[CrossRef]

Wax, A.

Wong, V.

C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
[CrossRef]

Yamamoto, T.

T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262-2273 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Iaconis, V. Wong, and I. A. Walmsley, "Direct interferometric techniques for characterizing ultrashort optical pulses," IEEE J. Sel. Top. Quantum Electron. 4, 285-294 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005).
[CrossRef]

IEEE Trans. Signal Process. (1)

T. Alieva, M. J. Bastiaans, and L. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

J. Appl. Phys. (1)

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848-3858 (1971).
[CrossRef]

J. Eur. Opt. Soc.--Rapid Publications (1)

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castaneda, C. Fernández-Pouza, and P. Andrés, "Temporal Lau effect: noncoherent reconstruction of periodic pulse trains," J. Eur. Opt. Soc.--Rapid Publications 1, 06018 (2006).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

K. H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

Opt. Commun. (5)

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

J. P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136-138 (1978).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, "Repetition-rate multiplication of optical pulses using uniform fiber Bragg gratings," Opt. Commun. 221, 331-335 (2003).
[CrossRef]

J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetion rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005).
[CrossRef]

C. Cuadrado-Laborde, P. Constanzo-Caso, R. Duchowicz, and E. E. Sicre, "Pulse propagation analysis based on the temporal Radon-Wigner transform," Opt. Commun. 266, 32-38 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the proposed approach: (a) monochromatic case, (b) spectrally incoherent case.

Fig. 2
Fig. 2

Schematic diagram of the optical setup.

Fig. 3
Fig. 3

Modulus of the ambiguity function of a sinusoidal phase signal. The repetition rate is 20 GHz and the modulation index value is fixed to π 2 rad .

Fig. 4
Fig. 4

Plot of the complex degree of coherence of the multiwavelength source. For the plot we assume an ideal infinite number of spectral lines producing a comblike structure.

Fig. 5
Fig. 5

Output intensity at the Talbot distance of 1/4 obtained with sinusoidal phase modulation at 20 GHz repetition rate and modulation index of 2.1 rad with a (a) monochromatic source and (b) multiwavelength source satisfying Ω s = 2 3 Ω .

Equations (31)

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

g ( t ) = m = a m exp ( i m Ω t ) .
I ( t , z = 0 ) = m = { n = a m + n a n * } exp ( i m Ω t ) ,
I ̃ ( ω , 0 ) = m = { n = a m + n a n * } δ ( ω m Ω ) .
u ( τ , z ) = m = { a m exp [ i ( β 2 2 ) m 2 Ω 2 z ) ] } exp ( i m Ω τ ) .
I ( τ , z ) = m = { n = a m + n a n * exp [ ( i β 2 m n Ω 2 z ) ] } exp [ i ( β 2 2 ) m 2 Ω 2 z ) ] exp ( i m Ω τ ) .
I ̃ ( ω , z ) = m = { n = a m + n a n * exp [ i ( β 2 m n Ω 2 z ) ] } exp [ i ( β 2 2 ) m 2 Ω 2 z ) ] δ ( ω m Ω ) .
A ( ω , t ) = g ( t + t 2 ) g * ( t t 2 ) exp ( i ω t ) d t = ( 1 2 π ) G ( ω + ω 2 ) G * ( ω ω 2 ) exp ( i t ω ) d ω ,
A ( ω , t ) = m = { n = a m + n a n * exp ( i n Ω t ) } exp ( i m Ω t 2 ) δ ( ω m Ω ) .
I ( ω , z ) = m = A ( m Ω , β 2 z m Ω ) δ ( ω m Ω ) .
I ( τ , z ) = m = A ( m Ω , β 2 z m Ω ) exp ( i m Ω τ ) .
I ( t , z ) = ( 1 2 π ) S ( ω ) R ( t , z , ω ) 2 d ω ,
R ( t , z , ω ) = G ( ω ω ) exp [ i β ( ω ) z i ω t ] d ω .
R ( τ , z , ω ) 2 = m = { n = a m + n a n * exp [ i ( β 2 m n Ω 2 z ) ] } exp [ i ( β 2 2 ) m 2 Ω 2 z ) ] exp [ i m Ω τ + i β 2 m Ω z ( ω ω 0 ) ] ,
R ( τ , z , ω ) 2 = m = A ( m Ω , β 2 z m Ω ) exp [ i m Ω τ + i β 2 m Ω z ( ω ω 0 ) ] .
I ( τ , z ) = m = [ ( 1 2 π ) S ( ω ) exp [ i β 2 m Ω z ( ω ω 0 ) ] d ( ω ω 0 ) ] A ( m Ω , β 2 z m Ω ) exp ( i m Ω τ ) .
I ( τ , z ) = m = γ ( β 2 m Ω z ) A ( m Ω , β 2 z m Ω ) exp ( i m Ω τ ) .
g ( t ) = exp [ i Δ θ sin ( 2 π t T ) ] .
A ( ω , t ) = exp { i [ 2 Δ θ sin ( Ω t 2 ) ] cos ( Ω t ) } exp ( i ω t ) d t = n = ( i ) n J n [ 2 Δ θ sin ( Ω t 2 ) ] δ ( ω n Ω ) .
A ( m Ω , β 2 m Ω z ) = ( i ) m J m [ 2 Δ θ sin ( β 2 Ω 2 z m 2 ) ] .
I ( τ , z ) = m = γ ( m β 2 Ω z ) J m [ 2 Δ θ sin ( β 2 Ω 2 z m 2 ) ] exp [ i m ( Ω τ + π 2 ) ] .
I ( τ , z ) = 1 + 2 m = 1 γ ( m β 2 Ω z ) J m [ 2 Δ θ sin ( β 2 Ω 2 z m 2 ) ] cos [ m ( Ω τ + π 2 ) ] .
I ( τ , Z T M ) = 1 + 2 m = 1 γ ( 2 m T M ) J m [ 2 Δ θ sin ( 2 π m M ) ] cos [ m ( Ω τ + π 2 ) ] .
I ( τ , Z T 4 ) = 1 + 2 m = 1 γ ( m T 2 ) J m { 2 Δ θ sin [ ( π 2 ) m ] } cos [ m ( Ω τ + π 2 ) ] .
S ( ω ) = [ 1 ( 2 Q + 1 ) ] q = Q Q δ ( ω ω 0 q Ω s ) .
γ ( τ ) = [ 1 ( 2 Q + 1 ) ] [ 1 + 2 q = 1 Q cos ( q Ω s τ ) ] .
γ ( 2 m T M ) = [ 1 ( 2 Q + 1 ) ] [ 1 + 2 q = 1 Q cos ( 2 m q Ω s T M ) ] .
Ω s = Ω ( M 2 N ) ,
γ ( 2 m T M ) = [ 1 ( 2 Q + 1 ) ] [ 1 + 2 q = 1 Q cos ( 2 π m q N ) ] .
γ ( 2 m T M ) = δ m , n N .
I ( τ , Z T M ) = 1 + 2 m = 1 J m N [ 2 Δ θ sin ( 2 π m ( N M ) ) ] cos [ m N ( Ω τ + π 2 ) ] .
I ( τ , Z T 4 ) = 1 + 2 m = 1 J 3 m [ 2 Δ θ sin ( ( 3 π 2 ) m ) ] cos [ 3 m ( Ω τ + π 2 ) ] .

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