Wigner matrix formalism for phase-modulated signals

H. Coïc; C. Rouyer; N. Bonod

doi:10.1364/JOSAA.408363

Wigner matrix formalism for phase-modulated signals

H. Coïc, C. Rouyer, and N. Bonod

Journal of the Optical Society of America A
Vol. 38,
Issue 1,
pp. 124-139
(2021)
•https://doi.org/10.1364/JOSAA.408363

Get Citation
Copy Citation Text
H. Coïc, C. Rouyer, and N. Bonod, "Wigner matrix formalism for phase-modulated signals," J. Opt. Soc. Am. A 38, 124-139 (2021)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (7324 KB)
Set citation alerts
Save article

Accessible

Open Access

Abstract

Laser beams can carry multi-scale properties in space and time that ultimately impact their quality. The study of their evolution along complex optical sequences is of crucial interest, especially in high-intensity laser chains. For such analysis, results obtained with standard numerical methods strongly depend on the sampling. In this paper, we develop an analytic model for a sinusoidal phase modulation inside a sequence of first-order optics elements based on the Wigner matrix formalism. A Bessel decomposition of the Wigner function gives pseudo-Wigner functions that obey the general ABCD matrix law transformation without approximations and sampling considerations. Applied to a Gaussian beam, explicit expressions are obtained for the projections of the Wigner function in the sub-spaces and give a powerful tool for analyzing the laser beam properties. The formalism is established in the spatial and temporal domains and can be used to evaluate the impact of the phase noise on the beam properties and is not limited to small modulation depths. For the sake of illustration, the model is applied to the Talbot effect with the analysis of the propagation in the spatial and phase-space domains. A comparison with full numerical calculations evidences the high accuracy of the analytic model that retrieves all the features of the diffracted beam.

Full Article | PDF Article

More Like This

Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium

Harold T. Yura, Lars Thrane, and Peter E. Andersen
J. Opt. Soc. Am. A 17(12) 2464-2474 (2000)

David Mendlovic, Zeev Zalevsky, and Haldun M. Ozaktas
J. Opt. Soc. Am. A 17(12) 2339-2354 (2000)

Linear canonical transform as a tool to analyze coherence properties of electromagnetic beams propagating in a quadratic phase system

Arvind Yelashetty, Nitika Gupta, Devnath Dhirhe, and Unnikrishnan Gopinathan
J. Opt. Soc. Am. A 37(8) 1350-1360 (2020)

References

View by:
Article Order
|
Year
|
Author
|
Publication

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]
M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]
M. Nicolaizeau and J. L. Miquel, “LMJ status: Fifth bundle commissioning and PW class laser coupling,” Proc. SPIE 10898, 1089802 (2019).
[Crossref]
M. L. Spaeth, K. R. Manes, D. H. Kalantar, P. E. Miller, J. E. Heebner, E. S. Bliss, D. R. Spec, T. G. Parham, P. K. Whitman, P. J. Wegner, P. A. Baisden, J. A. Menapace, M. W. Bowers, S. J. Cohen, T. I. Suratwala, J. M. Di Nicola, M. A. Newton, J. J. Adams, J. B. Trenholme, R. G. Finucane, R. E. Bonanno, D. C. Rardin, P. A. Arnold, S. N. Dixit, G. V. Erbert, A. C. Erlandson, J. E. Fair, E. Feigenbaum, W. H. Gourdin, R. A. Hawley, J. Honig, R. K. House, K. S. Jancaitis, K. N. LaFortune, D. W. Larson, B. J. Le Galloudec, J. D. Lindl, B. J. MacGowan, C. D. Marshall, K. P. McCandless, R. W. McCracken, R. C. Montesanti, E. I. Moses, M. C. Nostrand, J. A. Pryatel, V. S. Roberts, S. B. Rodriguez, A. W. Rowe, R. A. Sacks, J. T. Salmon, M. J. Shaw, S. Sommer, C. J. Stolz, G. L. Tietbohl, C. C. Widmayer, and R. Zacharias, “Description of the NIF laser,” Fusion Sci. Technol. 69, 25–145 (2016).
[Crossref]
C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmari, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, “Petawatt and exawatt class lasers worldwide,” High Power Laser Sci. Eng. 7, 1–54 (2019).
[Crossref]
F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]
T. Alieva and J. Bastiaans, “Properties of linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[Crossref]
P. Vahimaa and J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
[Crossref]
J. R. Murray, J. Ray Smith, R. B. Ehrlich, D. T. Kyrazis, C. E. Thompson, T. L. Weiland, and R. B. Wilcox, “Experimental observation and suppression of transverse stimulated Brillouin scattering in large optical components,” J. Opt. Soc. Am. B 6, 2402–2411 (1989).
[Crossref]
J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
[Crossref]
J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]
R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]
D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
[Crossref]
J. Paye and A. Migus, “Space-time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
[Crossref]
M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[Crossref]
H. Harold Szu and J. A. Blodgett, “Wigner distribution and ambiguity function,” AIP Conf. Proc. 65, 355–381 (1980).
[Crossref]
R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]
R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]
G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]
M. J. Bastiaans, “Second order-moments of the Wigner distribution function in first order optical systems,” Optik 88, 163–168 (1991).
A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. B 18, 1897–1904 (2001).
[Crossref]
L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]
M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18, 22503–22514 (2010).
[Crossref]
M. J. Baastians, “ABCD law for partially coherent Gaussian light propagating through first order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[Crossref]
J. Bromage, C. Dorrer, and R. K. Jungquist, “Temporal contrast degradation at the focus of ultrafast pulses from high-frequency spectral phase modulation,” J. Opt. Soc. Am. B 29, 1125–1135 (2012).
[Crossref]
C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]
V. Arrizon and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
[Crossref]
M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics, Fundamentals and Applications (McGraw-Hill, 2010).
M. Testorf and J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. B 13, 119–125 (1996).
[Crossref]
S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]
O. Morice, “Miró: complete modelling and software for pulse amplification and propagation in high power laser systems,” Opt. Eng. 42, 1530–1541 (2003).
[Crossref]

2019 (2)

M. Nicolaizeau and J. L. Miquel, “LMJ status: Fifth bundle commissioning and PW class laser coupling,” Proc. SPIE 10898, 1089802 (2019).
[Crossref]

C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. Hillier, N. W. Hopps, Y. Kato, E. A. Khazanov, R. Kodama, G. Korn, R. Li, Y. Li, J. Limpert, J. Ma, C. H. Nam, D. Neely, D. Papadopoulos, R. R. Penman, L. Qian, J. J. Rocca, A. A. Shaykin, C. W. Siders, C. Spindloe, S. Szatmari, R. M. G. M. Trines, J. Zhu, P. Zhu, and J. D. Zuegel, “Petawatt and exawatt class lasers worldwide,” High Power Laser Sci. Eng. 7, 1–54 (2019).
[Crossref]

2018 (1)

C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]

2016 (1)

M. L. Spaeth, K. R. Manes, D. H. Kalantar, P. E. Miller, J. E. Heebner, E. S. Bliss, D. R. Spec, T. G. Parham, P. K. Whitman, P. J. Wegner, P. A. Baisden, J. A. Menapace, M. W. Bowers, S. J. Cohen, T. I. Suratwala, J. M. Di Nicola, M. A. Newton, J. J. Adams, J. B. Trenholme, R. G. Finucane, R. E. Bonanno, D. C. Rardin, P. A. Arnold, S. N. Dixit, G. V. Erbert, A. C. Erlandson, J. E. Fair, E. Feigenbaum, W. H. Gourdin, R. A. Hawley, J. Honig, R. K. House, K. S. Jancaitis, K. N. LaFortune, D. W. Larson, B. J. Le Galloudec, J. D. Lindl, B. J. MacGowan, C. D. Marshall, K. P. McCandless, R. W. McCracken, R. C. Montesanti, E. I. Moses, M. C. Nostrand, J. A. Pryatel, V. S. Roberts, S. B. Rodriguez, A. W. Rowe, R. A. Sacks, J. T. Salmon, M. J. Shaw, S. Sommer, C. J. Stolz, G. L. Tietbohl, C. C. Widmayer, and R. Zacharias, “Description of the NIF laser,” Fusion Sci. Technol. 69, 25–145 (2016).
[Crossref]

2015 (1)

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

2012 (1)

J. Bromage, C. Dorrer, and R. K. Jungquist, “Temporal contrast degradation at the focus of ultrafast pulses from high-frequency spectral phase modulation,” J. Opt. Soc. Am. B 29, 1125–1135 (2012).
[Crossref]

2011 (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[Crossref]

2010 (1)

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18, 22503–22514 (2010).
[Crossref]

2009 (1)

F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]

2007 (1)

T. Alieva and J. Bastiaans, “Properties of linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[Crossref]

2003 (2)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

O. Morice, “Miró: complete modelling and software for pulse amplification and propagation in high power laser systems,” Opt. Eng. 42, 1530–1541 (2003).
[Crossref]

2001 (1)

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. B 18, 1897–1904 (2001).
[Crossref]

1997 (3)

P. Vahimaa and J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
[Crossref]

J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
[Crossref]

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]

1996 (1)

M. Testorf and J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. B 13, 119–125 (1996).
[Crossref]

1995 (2)

D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
[Crossref]

J. Paye and A. Migus, “Space-time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
[Crossref]

1992 (2)

M. J. Baastians, “ABCD law for partially coherent Gaussian light propagating through first order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[Crossref]

V. Arrizon and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
[Crossref]

1991 (1)

M. J. Bastiaans, “Second order-moments of the Wigner distribution function in first order optical systems,” Optik 88, 163–168 (1991).

1989 (1)

J. R. Murray, J. Ray Smith, R. B. Ehrlich, D. T. Kyrazis, C. E. Thompson, T. L. Weiland, and R. B. Wilcox, “Experimental observation and suppression of transverse stimulated Brillouin scattering in large optical components,” J. Opt. Soc. Am. B 6, 2402–2411 (1989).
[Crossref]

1988 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

1987 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]

1984 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

1980 (1)

H. Harold Szu and J. A. Blodgett, “Wigner distribution and ambiguity function,” AIP Conf. Proc. 65, 355–381 (1980).
[Crossref]

1979 (1)

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Adams, J. J.

Alieva, T.

T. Alieva and J. Bastiaans, “Properties of linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[Crossref]

Alonso, M. A.

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[Crossref]

Arnold, P. A.

Arrizon, V.

V. Arrizon and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
[Crossref]

Audo, F.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Baastians, M. J.

M. J. Baastians, “ABCD law for partially coherent Gaussian light propagating through first order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[Crossref]

Baisden, P. A.

Bastiaans, J.

T. Alieva and J. Bastiaans, “Properties of linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “Second order-moments of the Wigner distribution function in first order optical systems,” Optik 88, 163–168 (1991).

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

Bliss, E. S.

Blodgett, J. A.

H. Harold Szu and J. A. Blodgett, “Wigner distribution and ambiguity function,” AIP Conf. Proc. 65, 355–381 (1980).
[Crossref]

Bonanno, R. E.

Bouillet, S.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Bowers, M. W.

Bromage, J.

Broscolo, S.

C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]

Butcher, T.

Cai, Y.

Chanteloup, J.-C.

Chaussard, F.

C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

Chowdhury, E. A.

Cohen, S. J.

Danson, C. N.

Daurios, J.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

Di Nicola, J. M.

Dixit, S. N.

Dorrer, C.

Dragoman, D.

D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
[Crossref]

Ehrlich, R. B.

Erbert, G. V.

Erlandson, A. C.

Eupherte, L.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Fair, J. E.

Feigenbaum, E.

Feuer, T.

F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]

Finot, C.

C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]

Finucane, R. G.

Frei, F.

F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]

Fréville, S.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Galler, A.

F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]

Galvanauskas, A.

Garnier, J.

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]

Gizzi, L. A.

Gouedard, C.

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]

Gourdin, W. H.

Haefner, C.

Harold Szu, H.

H. Harold Szu and J. A. Blodgett, “Wigner distribution and ambiguity function,” AIP Conf. Proc. 65, 355–381 (1980).
[Crossref]

Hawley, R. A.

Heebner, J. E.

Hein, J.

Hennelly, B.

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics, Fundamentals and Applications (McGraw-Hill, 2010).

Hillier, D.

Honig, J.

Hopps, N. W.

House, R. K.

Jancaitis, K. S.

Jungquist, R. K.

Kalantar, D. H.

Kato, Y.

Khazanov, E. A.

Kodama, R.

Korn, G.

Korotkova, O.

Kyrazis, D. T.

LaFortune, K. N.

Larson, D. W.

Le Galloudec, B. J.

Li, R.

Li, Y.

Limpert, J.

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

Lindl, J. D.

Ma, J.

MacGowan, B. J.

Manes, K. R.

Marshall, C. D.

McCandless, K. P.

McCracken, R. W.

Menapace, J. A.

Migus, A.

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]

J. Paye and A. Migus, “Space-time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
[Crossref]

Miller, P. E.

Miquel, J. L.

M. Nicolaizeau and J. L. Miquel, “LMJ status: Fifth bundle commissioning and PW class laser coupling,” Proc. SPIE 10898, 1089802 (2019).
[Crossref]

Montesanti, R. C.

Morice, O.

O. Morice, “Miró: complete modelling and software for pulse amplification and propagation in high power laser systems,” Opt. Eng. 42, 1530–1541 (2003).
[Crossref]

Moses, E. I.

Mukunda, N.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Murray, J. R.

Nam, C. H.

Neely, D.

Newton, M. A.

Nicolaizeau, M.

M. Nicolaizeau and J. L. Miquel, “LMJ status: Fifth bundle commissioning and PW class laser coupling,” Proc. SPIE 10898, 1089802 (2019).
[Crossref]

Nostrand, M. C.

Ojeda-Castañeda, J.

M. Testorf and J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. B 13, 119–125 (1996).
[Crossref]

V. Arrizon and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
[Crossref]

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics, Fundamentals and Applications (McGraw-Hill, 2010).

Papadopoulos, D.

Parham, T. G.

Paye, J.

J. Paye and A. Migus, “Space-time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
[Crossref]

Penman, R. R.

Pryatel, J. A.

Qian, L.

Rardin, D. C.

Ray Smith, J.

Roberts, V. S.

Rocca, J. J.

Rodriguez, S. B.

Rothenberg, J. E.

J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
[Crossref]

Rouyer, C.

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Rowe, A. W.

Sacks, R. A.

Salmon, J. T.

Shaw, M. J.

Shaykin, A. A.

Siders, C. W.

Simon, R.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Sommer, S.

Spaeth, M. L.

Spec, D. R.

Spindloe, C.

Stolz, C. J.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Suratwala, T. I.

Szatmari, S.

Testorf, M.

M. Testorf and J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. B 13, 119–125 (1996).
[Crossref]

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics, Fundamentals and Applications (McGraw-Hill, 2010).

Thompson, C. E.

Tietbohl, G. L.

Tovar, A. A.

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. B 18, 1897–1904 (2001).
[Crossref]

Trenholme, J. B.

Trines, R. M. G. M.

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

Wang, Z.

Wegner, P. J.

Weiland, T. L.

Whitman, P. K.

Widmayer, C. C.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Wilcox, R. B.

Yao, M.

Zacharias, R.

Zhu, J.

Zhu, P.

Zhu, S. Y.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

Zuegel, J. D.

Adv. Opt. Photonics (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[Crossref]

AIP Conf. Proc. (1)

H. Harold Szu and J. A. Blodgett, “Wigner distribution and ambiguity function,” AIP Conf. Proc. 65, 355–381 (1980).
[Crossref]

Appl. Opt. (1)

D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
[Crossref]

Eur. J. Phys. (1)

C. Finot, F. Chaussard, and S. Broscolo, “Impact of a temporal sinusoidal phase modulation on the optical spectrum,” Eur. J. Phys. 39, 055303 (2018).
[Crossref]

Fusion Sci. Technol. (1)

High Power Laser Sci. Eng. (1)

J. Chem. Phys. (1)

F. Frei, A. Galler, and T. Feuer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[Crossref]

T. Alieva and J. Bastiaans, “Properties of linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).
[Crossref]

P. Vahimaa and J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
[Crossref]

V. Arrizon and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
[Crossref]

J. Opt. Soc. Am. B (6)

M. Testorf and J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. B 13, 119–125 (1996).
[Crossref]

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. B 18, 1897–1904 (2001).
[Crossref]

J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
[Crossref]

J. Paye and A. Migus, “Space-time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
[Crossref]

Opt. Eng. (1)

O. Morice, “Miró: complete modelling and software for pulse amplification and propagation in high power laser systems,” Opt. Eng. 42, 1530–1541 (2003).
[Crossref]

Opt. Express (1)

Opt. Quantum Electron. (1)

M. J. Baastians, “ABCD law for partially coherent Gaussian light propagating through first order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[Crossref]

Optik (1)

M. J. Bastiaans, “Second order-moments of the Wigner distribution function in first order optical systems,” Optik 88, 163–168 (1991).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Phys. Rev. A (3)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian-Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Phys. Rev. E (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulses beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[Crossref]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[Crossref]

Proc. SPIE (2)

M. Nicolaizeau and J. L. Miquel, “LMJ status: Fifth bundle commissioning and PW class laser coupling,” Proc. SPIE 10898, 1089802 (2019).
[Crossref]

S. Bouillet, F. Audo, S. Fréville, L. Eupherte, C. Rouyer, and J. Daurios, “Optical diffraction interpretation: an alternative to interferometers,” Proc. SPIE 9575, 9575A (2015).
[Crossref]

Other (1)

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics, Fundamentals and Applications (McGraw-Hill, 2010).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1. Phase-space representation of the Talbot effect with five contributions for Gaussian beams for

${u} = {50}$

and

${m} = {0.5}$

. On the left of each figure is represented the spatial spectrum (integration over

${X}$

), and on the bottom, is represented the spatial profile (integration over

${Kx}$

). The case

${Z} = {0}$

represents the beam spatially phase modulated with no propagation (no spatial fluence modulations). The case

${Z} = {0.1Zt}$

represents a propagation over a fraction of the Talbot distance. The case

${Z} = {0.25Zt}$

presents a maximal spatial modulation, when the shift of the periodic is half-period. When the shift is

${Z} = {0.5Zt}$

, modulations are minimized but not canceled because the finite size of the beam.

Download Full Size | PPT Slide | PDF

Fig. 2. Evolution of the normalized fluence with

${Z}$

for a Gaussian beam: (a) Bessel decomposition with five contributions and (b) numerical calculus (beam propagation method) for

${u} = {50}$

and

${m} = {0.5}$

. Normalized fluence in linear scale.

Download Full Size | PPT Slide | PDF

Fig. 3. Wigner representation of the Wigner function

${\rm W}({x},{{k}_x})$

in the phase-space for 10 different spatial frequencies

${{K}_d}$

randomly given with the modulation depth

${ m}({{K}_d})$

following a Gaussian distribution given by the GSM model (log scale).

Download Full Size | PPT Slide | PDF

Fig. 4. (a) Spatial envelope of the beam obtained by the integration of the Wigner functions over the spatial frequencies (full blue line: GSM model; dashed red line: Bessel decomposition). (b) Spatial spectra obtained by the integration of the Wigner functions over the spatial coordinate (full blue line: GSM model; dashed red line: Bessel decomposition).

Download Full Size | PPT Slide | PDF

Tables (2)

Table 1. Wigner Decomposition for Five Contributions

View Table | View all tables in this article

Table 2. Random Choice Following the Gaussian Statistics of the GSM Beam for the Corresponding Values of $m (u)$

View Table | View all tables in this article

Equations (101)

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

\begin{aligned} W_{0} (x, t, k_{x}, ω) & = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (x + \frac{1}{2} x_{1}, t + \frac{1}{2} t_{1}) \\ \times E^{*} (x - \frac{1}{2} x_{1}, t - \frac{1}{2} t_{1}) \\ \times \exp [i x_{1} k_{x} + i t_{1} ω] d x_{1} d t_{1} . \end{aligned}

\begin{aligned} W^{S e q} (x, t, k_{x}, ω) & = W_{0} (X (x, t, k_{x}, ω), T (x, t, k_{x}, ω), \\ K_{x} (x, t, k_{x}, ω), Ω (x, t, k_{x}, ω)), \end{aligned}

E_{0} (x, t) = \sqrt{\frac{2 E_{t}}{Δ x Δ t π}} \exp [- \frac{x^{2}}{Δ x^{2}} - \frac{t^{2}}{Δ t^{2}}] .

\begin{aligned} W_{0} (x, t, k_{x}, ω) \\ = \frac{E_{t}}{π^{2}} \exp [- \frac{2}{Δ x^{2}} x^{2} - \frac{2}{Δ t^{2}} t^{2} - \frac{Δ x^{2}}{2} k_{x}^{2} - \frac{Δ t^{2}}{2} ω^{2}] . \end{aligned}

\begin{aligned} E (x, t) & = E_{0} (x, t) \exp [i m \sin (K_{d} x)] \\ = E_{0} (x, t) \sum_{n = - \infty}^{n = + \infty} J_{n} (m) \exp [in K_{d} x], \end{aligned}

G_{n} (x, t) = E_{0} (x, t) \exp [in K_{d} x],

E (x, t) = \sum_{n = - \infty}^{n = + \infty} J_{n} (m) G_{n} (x, t) .

\begin{aligned} W_{n, l} (x, t, k_{x}, ω) & = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} G_{n} (x + \frac{1}{2} x_{1}, t + \frac{1}{2} t_{1}) \\ \times G_{l}^{*} (x - \frac{1}{2} x_{1}, t - \frac{1}{2} t_{1}) \\ \times \exp [i x_{1} k_{x} + i t_{1} ω] d x_{1} d t_{1} . \end{aligned}

W_{n, l} (x, t, k_{x}, ω) = W_{l, n}^{*} (x, t, k_{x}, ω)

\begin{aligned} W (x, t, k_{x}, ω) = \sum_{n = - \infty}^{n = + \infty} \sum_{l = - \infty}^{l = + \infty} J_{n} (m) J_{l} (m) W_{n, l} (x, t, k_{x}, ω) . \end{aligned}

\begin{aligned} W (x, t, k_{x}, ω) = \sum_{n = - \infty}^{n = + \infty} \sum_{l = - \infty}^{l = + \infty} J_{n} (m) J_{l} (m) Re (W_{n, l} (x, t, k_{x}, ω)) . \end{aligned}

\begin{aligned} W_{n, l} (x, t, k_{x}, ω) & = \frac{E_{t}}{π^{2}} \exp [- \frac{2}{Δ x^{2}} x^{2} - \frac{2}{Δ t^{2}} t^{2} \\ - \frac{Δ x^{2}}{2} {(k_{x} + (\frac{n + l}{2}) K_{d})}^{2} - \frac{Δ t^{2}}{2} ω^{2}] \\ \times \exp [i (n - l) K_{d} x] . \end{aligned}

\begin{aligned} W_{n, l} (x, t, k_{x}, ω) & = W_{0} (x, t, k_{x} + (\frac{n + l}{2}) K_{d}, ω) \\ \times e x p [i (n - l) K_{d} x], \end{aligned}

\begin{aligned} W_{n, l} (x, t, k_{x}, ω) & = W_{0} (x, t, k_{x}, ω) e x p [- \frac{Δ x^{2}}{2} (\frac{n + l}{2}) \\ \times K_{d} (2 k_{x} + (\frac{n + l}{2}) K_{d})] \\ \times e x p [i (n - l) K_{d} x] . \end{aligned}

r = (\begin{matrix} x \\ t \\ k_{x} \\ ω \end{matrix}) .

W_{0} (r) = \frac{E_{t}}{π^{2} \sqrt{det (Q^{- 1})}} \exp (- r^{T} Q r),

E_{t} = \int W_{0} (r) d r,

W_{n, l} (r) = \frac{E_{t}}{π^{2}} \exp (- {(r + S^{n, l})}^{T} Q (r + S^{n, l})) \exp (i {(D^{n, l})}^{T} r),

| \begin{matrix} S^{n, l} = \frac{(n + l)}{2} K_{d} I_{3} \\ D^{n, l} = (n - l) K_{d} I_{1} \end{matrix},

I_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}), I_{1} = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) .

\begin{aligned} W_{n, l}^{M_{1}} (r) & = W_{n, l} (M_{1} r) \\ = \frac{E_{t}}{π^{2}} \exp (- {(M_{1} r + S^{n, l})}^{T} Q (M_{1} r + S^{n, l})) \\ \times \exp (i {(D^{n, l})}^{T} M_{1} r) . \end{aligned}

r \to M_{1} r .

W_{0}^{M_{0}} (r) = W_{0} (M_{0} r) = \frac{E_{t}}{π^{2}} \exp (- {(M_{0} r)}^{T} Q (M_{0} r)) .

\begin{aligned} W_{n, l}^{M_{0} + φ (x)} (r) & = \frac{E_{t}}{π^{2}} \exp (- {(M_{0} (r + S^{n, l}))}^{T} Q (M_{0} (r + S^{n, l}))) \\ \times \exp (i {(D^{n, l})}^{T} r) . \end{aligned}

\begin{aligned} W_{n, l}^{S e q} (r) & = \frac{E_{t}}{π^{2}} \exp (- {(M_{0} (M_{1} r + S^{n, l}))}^{T} \\ \times Q (M_{0} (M_{1} r + S^{n, l}))) \exp (i {(D^{n, l})}^{T} M_{1} r), \end{aligned}

\begin{aligned} W_{n, l}^{S e q} (r) & = \frac{E_{t}}{π^{2}} \exp (- {(M_{0} M_{1} r + M_{0} S^{n, l})}^{T} \\ \times Q (M_{0} M_{1} r + M_{0} S^{n, l})) \\ \times \exp (i {(D^{n, l})}^{T} M_{1} r) . \end{aligned}

W_{n, l}^{S e q} (r) = W_{0}^{S e q} (r) \exp (A_{n, l} (r) + i Φ_{n, l} (r)),

\begin{aligned} | \begin{array}{l} W_{0}^{S e q} (r) = W_{0} (M_{0} M_{1} r) = \frac{E_{t}}{π^{2}} \\ \times \exp (- {(M_{0} M_{1} r)}^{T} Q (M_{0} M_{1} r)) \\ A_{n, l} (r) = - (\frac{n + l}{2}) K_{d} [{(M_{0} I_{3})}^{T} Q (M_{0} M_{1} r) \\ + {(M_{0} M_{1} r)}^{T} Q (M_{0} I_{3})] - {(\frac{n + l}{2})}^{2} K_{d}^{2} (M_{0} I_{3})^{T} Q (M_{0} I_{3}) \\ Φ_{n, l} (r) = (D^{n, l})^{T} M_{1} r = (n - l) K_{d} (I_{1})^{T} M_{1} r \end{array} . \end{aligned}

\begin{aligned} W^{S e q} (r) & = W_{0}^{S e q} (r) \sum_{n = - \infty}^{n = + \infty} \sum_{l = - \infty}^{l = + \infty} J_{n} (m) J_{l} (m) \\ \times \exp (A_{n, l} (r) + i Φ_{n, l} (r)) . \end{aligned}

\begin{aligned} Γ_{GSM} (x_{1}, t_{1}, x_{2}, t_{2}) = \frac{E_{t}}{2 π Δ x Δ t} \\ \times \exp [- \frac{x_{1}^{2} + x_{2}^{2}}{Δ x^{2}} - \frac{{(x_{1} - x_{2})}^{2}}{2 δ_{x}^{2}} - \frac{t_{1}^{2} + t_{2}^{2}}{Δ t^{2}} - \frac{{(t_{1} - t_{2})}^{2}}{2 δ_{t}^{2}}], \end{aligned}

\begin{aligned} W_{GSM} (x, t, k_{x}, ω) = \\ \frac{1}{4 π^{2}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \begin{array}{l} Γ_{GSM} (x + \frac{1}{2} x_{1}, t + \frac{1}{2} t_{1}, x - \frac{1}{2} x_{1}, t - \frac{1}{2} t_{1}) \\ \times \exp [i x_{1} k_{x} + i t_{1} ω] d x_{1} d t_{1} . \end{array} \end{aligned}

W_{GSM} (r) = E_{t} \frac{p q}{π^{2}} \exp (- r^{T} Q_{G S M} r),

| \begin{array}{l} Q_{G S M} = (\begin{matrix} \frac{2}{Δ x^{2}} & 0 & 0 & 0 \\ 0 & \frac{2}{Δ t^{2}} & 0 & 0 \\ 0 & 0 & p^{2} \frac{Δ x^{2}}{2} & 0 \\ 0 & 0 & 0 & q^{2} \frac{Δ t^{2}}{2} \end{matrix}) \\ p = \frac{1}{\sqrt{1 + \frac{Δ x^{2}}{δ_{x}^{2}}}} \\ q = \frac{1}{\sqrt{1 + \frac{Δ t^{2}}{δ_{t}^{2}}}} \end{array} .

| \begin{array}{l} p \approx \frac{δ_{x}}{Δ x} \\ q \approx \frac{δ_{t}}{Δ t} \end{array},

W_{GSM} (r) = E_{t} \frac{p q}{π^{2}} \exp (- {(M r)}^{T} Q_{G S M} (M r)) .

\begin{aligned} W_{GSM} (r) & = E_{t} \frac{p q}{π^{2}} \exp (- r^{T} Q_{G S M} r) \\ = E_{t} \frac{p q}{π^{2}} \exp (- {(M_{G S M} r)}^{T} Q_{0} (M_{G S M} r)), \end{aligned}

Q_{0} = (\begin{matrix} 2 / Δ x^{2} & 0 & 0 & 0 \\ 0 & 2 / Δ t^{2} & 0 & 0 \\ 0 & 0 & Δ x^{2} / 2 & 0 \\ 0 & 0 & 0 & Δ t^{2} / 2 \end{matrix}) .

M_{G S M} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & q \end{matrix}) .

W (r) = (1 - P_{m}^{2}) W_{0} (r) + P_{m}^{2} W_{GSM} (r),

\begin{aligned} W (r) = \frac{E_{t}}{π^{2}} {\begin{cases} (1 - P_{m}^{2}) \exp (- {(M r)}^{T} Q_{0} (M r)) \\ + P_{m}^{2} p q \exp (- {(M r)}^{T} Q_{G S M} (M r)) \end{cases}} . \end{aligned}

\begin{aligned} W (r) = \frac{E_{t}}{π^{2}} \\ \times {\begin{cases} (1 - P_{m}^{2}) \exp (- {(M r)}^{T} Q_{0} (M r)) \\ + P_{m}^{2} p q \exp (- {(M_{0} M_{G S M} M_{1} r)}^{T} Q_{0} (M_{0} M_{G S M} M_{1} r)) \end{cases}} . \end{aligned}

Q_{0} = (\begin{matrix} 2 / Δ x^{2} & 0 & 0 & 0 \\ 0 & 2 / Δ t^{2} & 0 & 0 \\ 0 & 0 & Δ x^{2} / 2 & 0 \\ 0 & 0 & 0 & Δ t^{2} / 2 \end{matrix}) .

| \begin{array}{l} I_{n, l} (x, t) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{n, l} (x, t, k_{x}, ω) d k_{x} d ω \\ F_{n, l} (x) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{n, l} (x, t, k_{x}, ω) d k_{x} d ω d t = \int_{- \infty}^{\infty} I_{n, l} (x, t) d t \\ P_{n, l} (t) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{n, l} (x, t, k_{x}, ω) d k_{x} d ω d x = \int_{- \infty}^{\infty} I_{n, l} (x, t) d x \end{array} .

\begin{aligned} W_{0}^{S e q} (x, t, k_{x}, ω) \\ = \exp [\begin{array}{l} C_{xx}^{W_{0}} x^{2} + C_{tt}^{W_{0}} t^{2} + C_{k_{x} k_{x}}^{W_{0}} {k_{x}}^{2} + C_{ω ω}^{W_{0}} ω^{2} \\ + C_{x k_{x}}^{W_{0}} x k_{x} + C_{xt}^{W_{0}} x t + C_{x ω}^{W_{0}} x ω \\ + C_{k_{x} t}^{W_{0}} k_{x} t + C_{k_{x} ω}^{W_{0}} k_{x} ω + C_{t ω}^{W_{0}} t ω \end{array}], \end{aligned}

\begin{aligned} W_{n, l}^{S e q} (x, t, k_{x}, ω) = W_{0}^{S e q} (x, t, k_{x}, ω) \\ \times \exp [\begin{array}{l} (n + l) K_{d} (A_{x}^{L} x + A_{t}^{L} t + A_{k_{x}}^{L} k_{x} + A_{ω}^{L} ω) \\ + i (n - l) K_{d} (Φ_{x}^{L} x + Φ_{t}^{L} t + Φ_{k_{x}}^{L} k_{x} + Φ_{ω}^{L} ω) \\ + (n + l)^{2} K_{d}^{2} A_{0} \end{array}], \end{aligned}

I_{0}^{S e q} (x, t) = \exp [C_{xx}^{I_{0}} x^{2} + C_{tt}^{I_{0}} t^{2} + C_{xt}^{I_{0}} x t],

\begin{aligned} I_{n, l}^{S e q} (x, t) & = I_{0}^{S e q} (x, t) \\ \times \exp [\begin{array}{l} (n + l) K_{d} [{A_{I}}_{x}^{L} x + {A_{I}}_{t}^{L} t] \\ + i (n - l) K_{d} [{Φ_{I}}_{x}^{L} x + {Φ_{I}}_{t}^{L} t] \\ + {K_{d}}^{2} [{(n + l)}^{2} A {_{I}^{S}}_{0} + {(n - l)}^{2} A {_{I}^{D}}_{0}] \\ + i (n^{2} - l^{2}) {K_{d}}^{2} Φ_{0}^{I} \end{array}] . \end{aligned}

| \begin{array}{l} F_{n, l}^{S e q} (x) = \exp [\begin{array}{l} C^{F_{0}} x^{2} + K_{d} ((n + l) A_{F}^{L} + i (n - l) Φ_{F}^{L}) x \\ + K_{d}^{2} ((n + l)^{2} A_{F 0}^{S} + (n - l)^{2} A_{F 0}^{D} + i (n^{2} - l^{2}) Φ_{0}^{F}) \end{array}] \\ P_{n, l}^{S e q} (t) = \exp [\begin{array}{l} C^{P_{0}} t^{2} + K_{d} ((n + l) A_{P}^{L} + i (n - l) Φ_{P}^{L}) t \\ + K_{d}^{2} ((n + l)^{2} A_{P 0}^{S} + (n - l)^{2} A_{P 0}^{D} + i (n^{2} - l^{2}) Φ_{0}^{P}) \end{array}] \end{array} .

K (x) = \frac{d ϕ (x)}{d x} = m K_{d} \cos (K_{d} x),

| \begin{array}{l} M_{0} = (\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}) \\ M_{1} = M = (\begin{array}{cccc} 1 & 0 & - \frac{z}{k_{0}} & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}) \end{array} .

| \begin{array}{l} C_{xx}^{W_{0}} = - \frac{2}{Δ x^{2}} \\ C_{tt}^{W_{0}} = - \frac{2}{Δ t^{2}} \\ C_{ω ω}^{W_{0}} = - \frac{Δ t^{2}}{2} \end{array} .

C_{k_{x} k_{x}}^{W_{0}} = - \frac{Δ x^{2}}{2} - \frac{2}{Δ x^{2}} \times \frac{z^{2}}{k_{0}^{2}} .

| \begin{array}{l} C_{x k_{x}}^{W_{0}} = \frac{2}{Δ x^{2}} \times \frac{2 z}{k_{0}} \\ A_{k_{x}}^{L} = - \frac{Δ x^{2}}{2} \\ Φ_{x}^{L} = 1 \\ Φ_{k_{x}}^{L} = - \frac{z}{k_{0}} \end{array},

A_{0} = - \frac{Δ x^{2}}{8} .

\begin{aligned} W_{n, l}^{S e q} (X, K_{x}) = \exp [\begin{array}{l} - 2 (X - Z K_{x})^{2} - \frac{1}{2} {(K_{x} + (\frac{n + l}{2}) u)}^{2} \\ + i (n - l) u (X - Z K_{x}) \end{array}] . \end{aligned}

\begin{aligned} W_{n, l}^{S e q} (K_{x}) = \exp [- \frac{1}{2} {(K_{x} + (\frac{n + l}{2}) u)}^{2} - \frac{1}{8} {(n - l)}^{2} u^{2}] . \end{aligned}

\begin{aligned} F_{n, l}^{S e q} (X) & = \exp [- \frac{2 [{(X + (\frac{n + l}{2}) u Z)}^{2} + \frac{1}{4} {(n - l)}^{2} u^{2} Z^{2}]}{1 + 4 Z^{2}}] \\ \times \exp [i (n - l) u \frac{(X + (\frac{n + l}{2}) u Z)}{1 + 4 Z^{2}}] . \end{aligned}

| \begin{array}{l} C^{F_{0}} \approx - \frac{2}{Δ x^{2}} \times \frac{1}{1 + 4 Z^{2}} \\ A_{F}^{L} \approx - \frac{2 Z}{1 + 4 Z^{2}} \\ Φ_{F}^{L} \approx \frac{1}{1 + 4 Z^{2}} \\ A {_{F}^{S}}_{0} = A {_{F}^{D}}_{0} = - \frac{Δ x^{2}}{2} \times \frac{Z^{2}}{1 + 4 Z^{2}} \\ Φ_{0}^{F} \approx \frac{Δ x^{2}}{2} \times \frac{Z}{1 + 4 Z^{2}} \end{array}

\begin{aligned} F (X) & = \exp [- 2 X^{2}] \sum_{n = - \infty}^{n = + \infty} \sum_{l = - \infty}^{l = + \infty} J_{n} (m) J_{l} (m) \\ \times \cos [(n - l) u (X + (\frac{n + l}{2}) u Z)] . \end{aligned}

\begin{aligned} F (X) & = \exp [- 2 X^{2}] \sum_{n = - \infty}^{n = + \infty} \sum_{l = - \infty}^{l = + \infty} J_{n} (m) J_{l} (m) \\ \times \cos [(n - l) (u X + 2 π \frac{2 Z}{Z_{t}} (\frac{n + l}{2}))] . \end{aligned}

| \begin{array}{l} J_{0} (m) \approx 1 - \frac{1}{4} m^{2} \\ J_{1} (m) = - J_{- 1} (m) \approx \frac{1}{2} m \\ J_{2} (m) = J_{- 2} (m) \approx \frac{1}{8} m^{2} \end{array} .

\sum_{n} \sum_{l} J_{n} J_{l}^{*} = 1.

\begin{aligned} W^{S e q} (X, K_{x}) & = W_{n + l = 0} (X, K_{x}) + W_{n + l = 1} (X, K_{x}) \\ + W_{n + l = - 1} (X, K_{x}) + W_{n + l = 2} (X, K_{x}) \\ + W_{n + l = - 2} (X, K_{x}) . \end{aligned}

X^{'} = X - Z K_{x},

| \begin{array}{l} W_{n + l = 0} (K_{x}) = \exp [- \frac{1}{2} K_{x}^{2}] (1 - \frac{m^{2}}{2} (1 + \exp [- \frac{u^{2}}{2}])) \\ W_{n + l = 1} (K_{x}) = m \exp [- \frac{u^{2}}{8}] \exp [- \frac{1}{2} {(K_{x} + \frac{u}{2})}^{2}] \\ W_{n + l = - 1} (K_{x}) = - m \exp [- \frac{u^{2}}{8}] \exp [- \frac{1}{2} {(K_{x} - \frac{u}{2})}^{2}] \\ W_{n + l = 2} (K_{x}) = \frac{m^{2}}{4} {(1 + \exp [- \frac{u^{2}}{2}]) \exp [- \frac{1}{2} {(K_{x} + u)}^{2}]} \\ W_{n + l = - 2} (K_{x}) = \frac{m^{2}}{4} {(1 + \exp [- \frac{u^{2}}{2}]) \exp [- \frac{1}{2} {(K_{x} - u)}^{2}]} \end{array} .

\begin{aligned} W^{S e q} (K_{x}) \approx \\ {\begin{cases} \exp [- \frac{1}{2} K_{x}^{2}] (1 - \frac{m^{2}}{2}) \\ + \frac{m^{2}}{4} {(\exp [- \frac{1}{2} {(K_{x} - u)}^{2}] + \exp [- \frac{1}{2} {(K_{x} + u)}^{2}])} \end{cases}} . \end{aligned}

| \begin{array}{l} F_{n + l = 0} (X) = \exp [- \frac{2 X^{2}}{1 + 4 Z^{2}}] \\ \times {1 - \frac{m^{2}}{2} (1 + \exp [- \frac{2 u^{2} Z^{2}}{1 + 4 Z^{2}}] \cos (\frac{2 X u}{1 + 4 Z^{2}}))} \\ F_{n + l = 1} (X) = m \exp [- \frac{2 {(X + \frac{u}{2} Z)}^{2} + \frac{u^{2}}{2} Z^{2}}{1 + 4 Z^{2}}] \cos (\frac{u (X + \frac{u}{2} Z)}{1 + 4 Z^{2}}) \\ F_{n + l = - 1} (X) = - m \exp [- \frac{2 {(X - \frac{u}{2} Z)}^{2} + \frac{u^{2}}{2} Z^{2}}{1 + 4 Z^{2}}] \cos (\frac{u (X - \frac{u}{2} Z)}{1 + 4 Z^{2}}) \\ F_{n + l = 2} (X) = \frac{m^{2}}{4} \exp [- \frac{2 {(X + u Z)}^{2}}{1 + 4 Z^{2}}] \\ \times {1 + \exp [- \frac{2 u^{2} Z^{2}}{1 + 4 Z^{2}}] \cos [\frac{2 u (X + u Z)}{1 + 4 Z^{2}}]} \\ F_{n + l = - 2} (X) = \frac{m^{2}}{4} \exp [- \frac{2 {(X - u Z)}^{2}}{1 + 4 Z^{2}}] \\ \times {1 + \exp [- \frac{2 u^{2} Z^{2}}{1 + 4 Z^{2}}] \cos [\frac{2 u (X - u Z)}{1 + 4 Z^{2}}]} \end{array},

\begin{aligned} F^{S e q} (X) & = F_{n + l = 0} (X) + F_{n + l = 1} (X) + F_{n + l = - 1} (X) \\ + F_{n + l = 2} (X) + F_{n + l = - 2} (X) . \end{aligned}

\begin{aligned} F (X) & = 1 - 2 m \sin (u X) \sin (\frac{u^{2}}{2} Z) \\ = 1 - 2 m \sin (u X) \sin (2 π \frac{Z}{Z_{t}}) . \end{aligned}

W (x, k_{x}) = (1 - P_{m}^{2}) W_{0} (x, k_{x}) + P_{m}^{2} W_{GSM} (x, k_{x}),

\begin{aligned} W_{GSM} (X, K_{x}) = \frac{E_{t} p}{π} \exp [- 2 {(X - Z K_{x})}^{2} - \frac{1}{2} p^{2} K_{x}^{2}] . \end{aligned}

\begin{aligned} W_{GSM} (K_{x}) & = \int_{- \infty}^{+ \infty} W_{GSM} (X, K_{x}) d X \\ = \frac{E_{t} p Δ x}{\sqrt{2 π}} \exp [- \frac{1}{2} p^{2} K_{x}^{2}] . \end{aligned}

\begin{aligned} W_{GSM} (X) & = \int_{- \infty}^{+ \infty} W_{GSM} (X, K_{x}) d K_{x} \\ = \sqrt{\frac{2}{π}} \frac{E_{t}}{\sqrt{1 + \frac{4 Z^{2}}{p^{2}}}} \exp [- \frac{2 X^{2}}{1 + \frac{4 Z^{2}}{p^{2}}}] . \end{aligned}

\begin{aligned} W_{GSM} (X) = \int_{_\infty}^{+ \infty} W_{GSM} (X, K_{x}) d K_{x} \approx \sqrt{\frac{2}{π}} E_{t} \exp [- 2 X^{2}] . \end{aligned}

\begin{aligned} W_{n o i s e} (K_{x}, u, m) = W (K_{x}, u, m) - (1 - \frac{m^{2}}{2}) W_{0} (K_{x}) . \end{aligned}

P_{m}^{2} = \frac{m^{2}}{2} .

\begin{aligned} W_{n o i s e} (X, K_{x}) & = \sum_{i = 1}^{N} W (X, K_{x}, u [i], m (u [i])) \\ - W_{0} (X, K_{x}) \sum_{i = 1}^{N} (1 - \frac{m {(u [i])}^{2}}{2}), \end{aligned}

P_{m}^{2} \approx \sum_{i = 1}^{N} \frac{m {(u [i])}^{2}}{2} .

m (u [i]) = P_{0} \exp [- \frac{1}{2} p^{2} u {[i]}^{2}],

| \begin{array}{l} M_{0} = (\begin{matrix} p_{11} & p_{12} & p_{13} & p_{14} \\ p_{21} & p_{22} & p_{23} & p_{24} \\ p_{31} & p_{32} & p_{33} & p_{34} \\ p_{41} & p_{42} & p_{43} & p_{44} \end{matrix}) \\ M_{1} = (\begin{matrix} q_{11} & q_{12} & q_{13} & q_{14} \\ q_{21} & q_{22} & q_{23} & q_{24} \\ q_{31} & q_{32} & q_{33} & q_{34} \\ q_{41} & q_{42} & q_{43} & q_{44} \end{matrix}) \end{array},

M = M_{0} M_{1} = (\begin{matrix} s_{11} & s_{12} & s_{13} & s_{14} \\ s_{21} & s_{22} & s_{23} & s_{24} \\ s_{31} & s_{32} & s_{33} & s_{34} \\ s_{41} & s_{42} & s_{43} & s_{44} \end{matrix}),

\begin{aligned} W_{0}^{S e q} (x, t, k_{x}, ω) \\ = \exp [\begin{array}{l} C_{xx}^{W_{0}} x^{2} + C_{tt}^{W_{0}} t^{2} + C_{k_{x} k_{x}}^{W_{0}} {k_{x}}^{2} + C_{ω ω}^{W_{0}} ω^{2} \\ + C_{x k_{x}}^{W_{0}} x k_{x} + C_{xt}^{W_{0}} x t + C_{x ω}^{W_{0}} x ω \\ + C_{k_{x} t}^{W_{0}} k_{x} t + C_{k_{x} ω}^{W_{0}} k_{x} ω + C_{t ω}^{W_{0}} t ω \end{array}] . \end{aligned}

| \begin{array}{l} C_{xx}^{W_{0}} = - (\frac{2}{Δ x^{2}} s_{11}^{2} + \frac{2}{Δ t^{2}} s_{21}^{2} + \frac{Δ x^{2}}{2} s_{31}^{2} + \frac{Δ t^{2}}{2} s_{41}^{2}) \\ C_{tt}^{W_{0}} = - (\frac{2}{Δ x^{2}} s_{12}^{2} + \frac{2}{Δ t^{2}} s_{22}^{2} + \frac{Δ x^{2}}{2} s_{32}^{2} + \frac{Δ t^{2}}{2} s_{42}^{2}) \\ C_{k_{x} k_{x}}^{W_{0}} = - (\frac{2}{Δ x^{2}} s_{13}^{2} + \frac{2}{Δ t^{2}} s_{23}^{2} + \frac{Δ x^{2}}{2} s_{33}^{2} + \frac{Δ t^{2}}{2} s_{43}^{2}) \\ C_{ω ω}^{W_{0}} = - (\frac{2}{Δ x^{2}} s_{14}^{2} + \frac{2}{Δ t^{2}} s_{24}^{2} + \frac{Δ x^{2}}{2} s_{34}^{2} + \frac{Δ t^{2}}{2} s_{44}^{2}) \end{array},

| \begin{array}{l} C_{xt}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{12} s_{11} + \frac{2}{Δ t^{2}} s_{22} s_{21} + \frac{Δ x^{2}}{2} s_{32} s_{31} + \frac{Δ t^{2}}{2} s_{42} s_{41}) \\ C_{x k_{x}}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{13} s_{11} + \frac{2}{Δ t^{2}} s_{23} s_{21} + \frac{Δ x^{2}}{2} s_{33} s_{31} + \frac{Δ t^{2}}{2} s_{43} s_{41}) \\ C_{x ω}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{14} s_{11} + \frac{2}{Δ t^{2}} s_{24} s_{21} + \frac{Δ x^{2}}{2} s_{34} s_{31} + \frac{Δ t^{2}}{2} s_{44} s_{41}) \\ C_{k_{x} t}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{13} s_{12} + \frac{2}{Δ t^{2}} s_{23} s_{22} + \frac{Δ x^{2}}{2} s_{33} s_{32} + \frac{Δ t^{2}}{2} s_{43} s_{42}) \\ C_{t ω}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{14} s_{12} + \frac{2}{Δ t^{2}} s_{24} s_{22} + \frac{Δ x^{2}}{2} s_{34} s_{32} + \frac{Δ t^{2}}{2} s_{44} s_{42}) \\ C_{k_{x} ω}^{W_{0}} = - 2 (\frac{2}{Δ x^{2}} s_{14} s_{13} + \frac{2}{Δ t^{2}} s_{24} s_{23} + \frac{Δ x^{2}}{2} s_{34} s_{33} + \frac{Δ t^{2}}{2} s_{44} s_{43}) \end{array} .

\begin{aligned} W_{n, l}^{S e q} (x, t, k_{x}, ω) = W_{0}^{S e q} (x, t, k_{x}, ω) \\ \times \exp [\begin{array}{l} (n + l) K_{d} (A_{x}^{L} x + A_{t}^{L} t + A_{k_{x}}^{L} k_{x} + A_{ω}^{L} ω) \\ + i (n - l) K_{d} (Φ_{x}^{L} x + Φ_{t}^{L} t + Φ_{k_{x}}^{L} k_{x} + Φ_{ω}^{L} ω) \\ + (n + l)^{2} K_{d}^{2} A_{0} \end{array}] . \end{aligned}

| \begin{array}{l} A_{x}^{L} = - (\frac{2}{Δ x^{2}} p_{13} s_{11} + \frac{2}{Δ t^{2}} p_{23} s_{21} + \frac{Δ x^{2}}{2} p_{33} s_{31} + \frac{Δ t^{2}}{2} p_{43} s_{41}) \\ A_{t}^{L} = - (\frac{2}{Δ x^{2}} p_{13} s_{12} + \frac{2}{Δ t^{2}} p_{23} s_{22} + \frac{Δ x^{2}}{2} p_{33} s_{32} + \frac{Δ t^{2}}{2} p_{43} s_{42}) \\ A_{k_{x}}^{L} = - (\frac{2}{Δ x^{2}} p_{13} s_{13} + \frac{2}{Δ t^{2}} p_{23} s_{23} + \frac{Δ x^{2}}{2} p_{33} s_{33} + \frac{Δ t^{2}}{2} p_{43} s_{43}) \\ A_{ω}^{L} = - (\frac{2}{Δ x^{2}} p_{13} s_{14} + \frac{2}{Δ t^{2}} p_{23} s_{24} + \frac{Δ x^{2}}{2} p_{33} s_{34} + \frac{Δ t^{2}}{2} p_{43} s_{44}) \end{array} .

| \begin{array}{l} Φ_{x}^{L} = q_{11} \\ Φ_{t}^{L} = q_{12} \\ Φ_{k_{x}}^{L} = q_{13} \\ Φ_{ω}^{L} = q_{14} \end{array} .

A_{0} = - \frac{1}{4} (\frac{2}{Δ x^{2}} p_{13}^{2} + \frac{2}{Δ t^{2}} p_{23}^{2} + \frac{Δ x^{2}}{2} p_{33}^{2} + \frac{Δ t^{2}}{2} p_{43}^{2}) .

I_{0}^{S e q} (x, t) = \exp [C_{xx}^{I_{0}} x^{2} + C_{tt}^{I_{0}} t^{2} + C_{xt}^{I_{0}} x t],

| \begin{array}{l} C_{xx}^{I_{0}}, C_{tt}^{I_{0}} < 0 \\ 4 C_{xx}^{I_{0}} C_{tt}^{I_{0}} - {(C_{xt}^{I_{0}})}^{2} > 0 \end{array} .

| \begin{array}{l} C_{xx}^{I_{0}} = \frac{1}{g^{2}} [\begin{array}{l} 4 C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} C_{xx}^{W_{0}} + C_{x ω}^{W_{0}} C_{x k_{x}}^{W_{0}} C_{k_{x} ω}^{W_{0}} \\ - C_{xx}^{W_{0}} {(C_{k_{x} ω}^{W_{0}})}^{2} - C_{ω ω}^{W_{0}} {(C_{x k_{x}}^{W_{0}})}^{2} - C_{k_{x} k_{x}}^{W_{0}} {(C_{x ω}^{W_{0}})}^{2} \end{array}] \\ C_{tt}^{I_{0}} = \frac{1}{g^{2}} [\begin{array}{l} 4 C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} C_{tt}^{W_{0}} + C_{t ω}^{W_{0}} C_{k_{x} t}^{W_{0}} C_{k_{x} ω}^{W_{0}} \\ - C_{tt}^{W_{0}} {(C_{k_{x} ω}^{W_{0}})}^{2} - C_{ω ω}^{W_{0}} {(C_{k_{x} t}^{W_{0}})}^{2} - C_{k_{x} k_{x}}^{W_{0}} {(C_{t ω}^{W_{0}})}^{2} \end{array}] \\ C_{xt}^{I_{0}} = \frac{1}{g^{2}} [\begin{array}{l} - 4 C_{xt}^{W_{0}} C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} - C_{t ω}^{W_{0}} C_{x k_{x}}^{W_{0}} C_{k_{x} ω}^{W_{0}} + C_{xt}^{W_{0}} {(C_{k_{x} ω}^{W_{0}})}^{2} \\ + 2 C_{x k_{x}}^{W_{0}} C_{k_{x} t}^{W_{0}} C_{ω ω}^{W_{0}} + 2 C_{x ω}^{W_{0}} C_{k_{x} k_{x}}^{W_{0}} C_{t ω}^{W_{0}} - C_{x ω}^{W_{0}} C_{k_{x} t}^{W_{0}} C_{k_{x} ω}^{W_{0}} \end{array}] \end{array},

g = \sqrt{4 C_{ω ω}^{W_{0}} C_{k_{x} k_{x}}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2}} .

4 C_{ω ω}^{W_{0}} C_{k_{x} k_{x}}^{W_{0}} > {(C_{k_{x} ω}^{W_{0}})}^{2} .

\begin{aligned} I_{n, l}^{S e q} (x, t) = I_{0}^{S e q} (x, t) \\ \times \exp [\begin{array}{l} (n + l) K_{d} [{A_{I}}_{x}^{L} x + {A_{I}}_{t}^{L} t] \\ + i (n - l) K_{d} [{Φ_{I}}_{x}^{L} x + {Φ_{I}}_{t}^{L} t] \\ + {K_{d}}^{2} [{(n + l)}^{2} A {_{I}^{S}}_{0} + {(n - l)}^{2} A {_{I}^{D}}_{0}] \\ + i (n^{2} - l^{2}) {K_{d}}^{2} Φ_{0}^{I} \end{array}] . \end{aligned}

| \begin{array}{l} A_{Ix}^{L} = \frac{1}{g^{2}} [\begin{array}{l} C_{k_{x} ω}^{W_{0}} A_{k_{x}}^{L} C_{x ω}^{W_{0}} + C_{k_{x} ω}^{W_{0}} A_{ω}^{L} C_{x k_{x}}^{W_{0}} - 2 C_{k_{x} k_{x}}^{W_{0}} A_{ω}^{L} C_{x ω}^{W_{0}} \\ + 4 C_{k_{x} k_{x}}^{W_{0}} A_{x}^{L} C_{ω ω}^{W_{0}} - 2 C_{ω ω}^{W_{0}} A_{k_{x}}^{L} C_{x k_{x}}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2} A_{x}^{L} \end{array}] \\ Φ_{Ix}^{L} = \frac{1}{g^{2}} [\begin{array}{l} Φ_{k_{x}}^{L} C_{k_{x} ω}^{W_{0}} C_{x ω}^{W_{0}} + Φ_{ω}^{L} C_{k_{x} ω}^{W_{0}} C_{x k_{x}}^{W_{0}} - 2 Φ_{k_{x}}^{L} C_{ω ω}^{W_{0}} C_{x k_{x}}^{W_{0}} \\ + 4 Φ_{x}^{L} C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} - 2 Φ_{ω}^{L} C_{k_{x} k_{x}}^{W_{0}} C_{x ω}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2} Φ_{x}^{L} \end{array}] \end{array},

| \begin{array}{l} A_{It}^{L} = \frac{1}{g^{2}} [\begin{array}{l} C_{k_{x} ω}^{W_{0}} A_{k_{x}}^{L} C_{t ω}^{W_{0}} + C_{k_{x} t}^{W_{0}} A_{ω}^{L} C_{k_{x} ω}^{W_{0}} - 2 C_{k_{x} k_{x}}^{W_{0}} A_{ω}^{L} C_{t ω}^{W_{0}} \\ + 4 C_{k_{x} k_{x}}^{W_{0}} A_{t}^{L} C_{ω ω}^{W_{0}} - 2 C_{k_{x} t}^{W_{0}} A_{k_{x}}^{L} C_{ω ω}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2} A_{t}^{L} \end{array}] \\ Φ_{It}^{L} = \frac{1}{g^{2}} [\begin{array}{l} Φ_{k_{x}}^{L} C_{k_{x} ω}^{W_{0}} C_{t ω}^{W_{0}} + Φ_{ω}^{L} C_{k_{x} t}^{W_{0}} C_{k_{x} ω}^{W_{0}} - 2 Φ_{k_{x}}^{L} C_{k_{x} t}^{W_{0}} C_{ω ω}^{W_{0}} \\ + 4 Φ_{t}^{L} C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} - 2 Φ_{ω}^{L} C_{k_{x} k_{x}}^{W_{0}} C_{t ω}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2} Φ_{t}^{L} \end{array}] \end{array},

| \begin{array}{l} A_{I 0}^{S} = \frac{1}{g^{2}} [\begin{array}{l} (A_{k_{x}}^{L} A_{ω}^{L} C_{k_{x} ω}^{W_{0}} - {(A_{ω}^{L})}^{2} C_{k_{x} k_{x}}^{W_{0}} - {(A_{k_{x}}^{L})}^{2} C_{ω ω}^{W_{0}}) \\ + 4 A_{0} (4 C_{k_{x} k_{x}}^{W_{0}} C_{ω ω}^{W_{0}} - {(C_{k_{x} ω}^{W_{0}})}^{2}) \end{array}] \\ A_{I 0}^{D} = \frac{1}{g^{2}} [{(Φ_{k_{x}}^{L})}^{2} C_{ω ω}^{W_{0}} - Φ_{k_{x}}^{L} Φ_{ω}^{L} C_{k_{x} ω}^{W_{0}} + {(Φ_{ω}^{L})}^{2} C_{k_{x} k_{x}}^{W_{0}}] \\ Φ_{0}^{I} = \frac{1}{g^{2}} [\begin{array}{l} Φ_{k_{x}}^{L} A_{ω}^{L} C_{k_{x} ω}^{W_{0}} + Φ_{ω}^{L} A_{k_{x}}^{L} C_{k_{x} ω}^{W_{0}} \\ - 2 Φ_{k_{x}}^{L} A_{k_{x}}^{L} C_{ω ω}^{W_{0}} - 2 Φ_{ω}^{L} A_{ω}^{L} C_{k_{x} k_{x}}^{W_{0}} \end{array}] \end{array} .

F_{n, l}^{S e q} (x) = \exp [\begin{array}{l} C^{F_{0}} x^{2} + K_{d} ((n + l) A_{F}^{L} \\ + i (n - l) Φ_{F}^{L}) x \\ + K_{d}^{2} ((n + l)^{2} A_{F 0}^{S} + (n - l)^{2} A_{F 0}^{D} \\ + i (n^{2} - l^{2}) Φ_{0}^{F}) \end{array}],

| \begin{array}{l} C^{F_{0}} = C_{xx}^{I_{0}} - \frac{{(C_{xt}^{I_{0}})}^{2}}{C_{tt}^{I_{0}}} \\ A_{F}^{L} = A_{Ix}^{L} - A_{It}^{L} \frac{C_{xt}^{I_{0}}}{C_{tt}^{I_{0}}} \\ Φ_{F}^{L} = Φ_{Ix}^{L} - Φ_{It}^{L} \frac{C_{xt}^{I_{0}}}{C_{tt}^{I_{0}}} \\ A_{F 0}^{S} = A_{I 0}^{S} - \frac{{(A_{It}^{L})}^{2}}{4 C_{tt}^{I_{0}}} \\ A_{F 0}^{D} = A_{I 0}^{D} + \frac{{(Φ_{It}^{L})}^{2}}{4 C_{tt}^{I_{0}}} \\ Φ_{0}^{F} = Φ_{0}^{I} - \frac{A_{It}^{L} Φ_{It}^{L}}{2 C_{tt}^{I_{0}}} \end{array} .

P_{n, l}^{S e q} (t) = \exp [\begin{array}{l} C^{P_{0}} t^{2} + K_{d} ((n + l) A_{P}^{L} \\ + i (n - l) Φ_{P}^{L}) t \\ + K_{d}^{2} ((n + l)^{2} A_{P 0}^{S} + (n - l)^{2} A_{P 0}^{D} \\ + i (n^{2} - l^{2}) Φ_{0}^{P}) \end{array}],

| \begin{array}{l} C^{P_{0}} = C_{tt}^{I_{0}} - \frac{{(C_{xt}^{I_{0}})}^{2}}{C_{xx}^{I_{0}}} \\ A_{P}^{L} = A_{It}^{L} - A_{Ix}^{L} \frac{C_{xt}^{I_{0}}}{C_{xx}^{I_{0}}} \\ Φ_{P}^{L} = Φ_{It}^{L} - Φ_{Ix}^{L} \frac{C_{xt}^{I_{0}}}{C_{xx}^{I_{0}}} \\ A_{P 0}^{S} = A_{I 0}^{S} - \frac{{(A_{Ix}^{L})}^{2}}{4 C_{xx}^{I_{0}}} \\ A_{P 0}^{D} = A_{I 0}^{D} + \frac{{(Φ_{Ix}^{L})}^{2}}{4 C_{xx}^{I_{0}}} \\ Φ_{0}^{P} = Φ_{0}^{I} - \frac{A_{Ix}^{L} Φ_{Ix}^{L}}{2 C_{xx}^{I_{0}}} \end{array} .

( $n, l$ )	$J_{n} J_{l}$	Resulting Wigner Function
(0, 0)	$1 - \frac{m^{2}}{2}$	$\begin{array}{l} W_{n + l = 0} (X, K_{x}) \\ = \exp [- 2 {(X - Z K_{x})}^{2}] \exp [- \frac{1}{2} K_{x}^{2}] \\ \times {1 - \frac{m^{2}}{2} (1 + \cos [2 u (X - Z K_{x})])} \end{array}$
(−1, 1)	$- \frac{m^{2}}{4}$
(1, −1)	$- \frac{m^{2}}{4}$
(0, 1)	$\frac{m}{2}$	$\begin{array}{l} W_{n + l = 1} (X, K_{x}) = m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(1, 0)	$\frac{m}{2}$
(−1, 0)	$- \frac{m}{2}$	$\begin{array}{l} W_{n + l = - 1} (X, K_{x}) = - m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(0, −1)	$- \frac{m}{2}$
(1, 1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + u)}^{2}] \\ \times {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, 2)	$\frac{m^{2}}{8}$
(2, 0)	$\frac{m^{2}}{8}$
(−1, −1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = - 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - u)}^{2}] \\ {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, −2)	$\frac{m^{2}}{8}$
(−2, 0)	$\frac{m^{2}}{8}$

$u [i] (m)^{- 1}$	3.1	21.6	32.9	33.5	34.0	43.5	46.0	53.48	57.7	59.6
$m (u [i])$ $(\times 100)$	9.93	7.46	5.08	4.95	4.48	3.05	2.65	1.67	1.24	1.08
$Φ [i]$ (rad)	5.69	2.22	3.76	0.455	2.06	0.27	1.61	4.79	1.24	5.26

( $n, l$ )	$J_{n} J_{l}$	Resulting Wigner Function
(0, 0)	$1 - \frac{m^{2}}{2}$	$\begin{array}{l} W_{n + l = 0} (X, K_{x}) \\ = \exp [- 2 {(X - Z K_{x})}^{2}] \exp [- \frac{1}{2} K_{x}^{2}] \\ \times {1 - \frac{m^{2}}{2} (1 + \cos [2 u (X - Z K_{x})])} \end{array}$
(−1, 1)	$- \frac{m^{2}}{4}$
(1, −1)	$- \frac{m^{2}}{4}$
(0, 1)	$\frac{m}{2}$	$\begin{array}{l} W_{n + l = 1} (X, K_{x}) = m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(1, 0)	$\frac{m}{2}$
(−1, 0)	$- \frac{m}{2}$	$\begin{array}{l} W_{n + l = - 1} (X, K_{x}) = - m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(0, −1)	$- \frac{m}{2}$
(1, 1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + u)}^{2}] \\ \times {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, 2)	$\frac{m^{2}}{8}$
(2, 0)	$\frac{m^{2}}{8}$
(−1, −1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = - 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - u)}^{2}] \\ {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, −2)	$\frac{m^{2}}{8}$
(−2, 0)	$\frac{m^{2}}{8}$

$u [i] (m)^{- 1}$	3.1	21.6	32.9	33.5	34.0	43.5	46.0	53.48	57.7	59.6
$m (u [i])$ $(\times 100)$	9.93	7.46	5.08	4.95	4.48	3.05	2.65	1.67	1.24	1.08
$Φ [i]$ (rad)	5.69	2.22	3.76	0.455	2.06	0.27	1.61	4.79	1.24	5.26