Abstract

Polychromatic stationary propagation-invariant fields with complete spatial coherence in the space-frequency domain are considered. In general, the field is shown to be spatially partially coherent in the space-time domain, apart from transversely achromatic fields with complete transverse coherence. Particular attention is paid to fields that possess the same cone angle at each frequency; these are stationary counterparts of pulsed conical fields known as X waves. It is shown that, for such fields, the radius of the space-time-domain transverse coherence area depends critically on the bandwidth of the power spectrum and can be comparable to the central-lobe radius of the monochromatic field component at the peak frequency of the spectrum.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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  32. A. T. Friberg, J. Fagerholm, and M. M. Salomaa, "Space-frequency analysis of nondiffracting pulses," Opt. Commun. 136, 207-212 (1997).
    [CrossRef]
  33. E. Recami, "On localized ‘X-shaped’ superluminal solutions to Maxwell’s equations," Physica A 252, 586-601 (1999).
    [CrossRef]
  34. C. H. Gan, G. Gbur, and T. D. Visser, "Surface plasmons modulate the spatial coherence of light in Young’s interference experiment," Phys. Rev. Lett. 98, 043908 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2007 (2)

2006 (3)

2005 (3)

2004 (1)

2003 (2)

2002 (2)

W. Hu and H. Guo, "Ultrashort pulsed Bessel beams and spatially induuced group-velovcity dispersion," J. Opt. Soc. Am. A 19, 49-53 (2002).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

2001 (1)

1999 (1)

E. Recami, "On localized ‘X-shaped’ superluminal solutions to Maxwell’s equations," Physica A 252, 586-601 (1999).
[CrossRef]

1997 (3)

A. T. Friberg, J. Fagerholm, and M. M. Salomaa, "Space-frequency analysis of nondiffracting pulses," Opt. Commun. 136, 207-212 (1997).
[CrossRef]

A. M. Shaarawi, "Comparison of two localized wave fields generated from dynamic apertures," J. Opt. Soc. Am. A 14, 1804-1816 (1997).
[CrossRef]

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

1996 (1)

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

1995 (1)

Z. Bouchal and M. Olivík, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1666 (1995).
[CrossRef]

1993 (1)

J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

1992 (1)

J. Lu and J. F. Greenleaf, "Nondiffractiong X waves — Exact solutions to free-space wave equation and their finite-aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 90, 19-31 (1992).
[CrossRef]

1991 (4)

P. L. Overfelt, "Bessel-Gauss pulses," Phys. Rev. A 44, 3941-3947 (1991).
[CrossRef] [PubMed]

A. T. Friberg, A. Vasara, and J. Turunen, "Partially coherent propagation-invariant fields: passage through paraxial optical systems," Phys. Rev. A 43, 7079-7082 (1991).
[CrossRef] [PubMed]

S. R. Mishra, "A vector wave analysis of a bessel beam," Opt. Commun. 85, 159-161 (1991).
[CrossRef]

J. Turunen, A. Vasara, and A. T. Friberg, "Propagation-invariance and self-imaging in variable-coherence optics," J. Opt. Soc. Am. A 8, 282-289 (1991).
[CrossRef]

1990 (1)

J. A. Campbell and S. Soloway, "Generation of a nondiffracting beam with frequency- independent beamwidth," J. Acoust. Soc. Am. 88, 2467-2477 (1990).
[CrossRef]

1989 (3)

1988 (1)

K. B. Wolf, "Diffraction-free beams remain diffraction free under all paraxial optical transformations," Phys. Rev. Lett. 60, 757-759 (1988).
[CrossRef] [PubMed]

1987 (3)

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr. and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1985 (1)

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

1983 (2)

J. N. Brittingham; "Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

E. Wolf, "Youngs interference fringes with narrow-band light," Opt. Lett. 8, 250-252 (1983).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

L. Mandel and E. Wolf, "Complete coherence in the space-frequency domain," Opt. Commun. 36, 247-249 (1981).
[CrossRef]

Besieris, I. M.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Bouchal, Z.

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, ans applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

Z. Bouchal and M. Olivík, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1666 (1995).
[CrossRef]

Brittingham, J. N.

J. N. Brittingham; "Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

Brown, C. T. A.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6657.
[CrossRef] [PubMed]

Campbell, J. A.

J. A. Campbell and S. Soloway, "Generation of a nondiffracting beam with frequency- independent beamwidth," J. Acoust. Soc. Am. 88, 2467-2477 (1990).
[CrossRef]

Dholakia, K.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6657.
[CrossRef] [PubMed]

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr. and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr. and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Esposito, E.

Fagerholm, J.

A. T. Friberg, J. Fagerholm, and M. M. Salomaa, "Space-frequency analysis of nondiffracting pulses," Opt. Commun. 136, 207-212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Fischer, P.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6657.
[CrossRef] [PubMed]

Friberg, A. T.

A. T. Friberg, H. Lajunen, and V. Torres-Company, "Spectral elementary-coherence-function representation for partially coherent light pulses," Opt. Express 15, 5160-5165 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5160.
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

A. T. Friberg, J. Fagerholm, and M. M. Salomaa, "Space-frequency analysis of nondiffracting pulses," Opt. Commun. 136, 207-212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

J. Turunen, A. Vasara, and A. T. Friberg, "Propagation-invariance and self-imaging in variable-coherence optics," J. Opt. Soc. Am. A 8, 282-289 (1991).
[CrossRef]

A. T. Friberg, A. Vasara, and J. Turunen, "Partially coherent propagation-invariant fields: passage through paraxial optical systems," Phys. Rev. A 43, 7079-7082 (1991).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams using computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
[CrossRef] [PubMed]

Gan, C. H.

C. H. Gan, G. Gbur, and T. D. Visser, "Surface plasmons modulate the spatial coherence of light in Young’s interference experiment," Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

Gbur, G.

C. H. Gan, G. Gbur, and T. D. Visser, "Surface plasmons modulate the spatial coherence of light in Young’s interference experiment," Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

Gibson, G. M.

Girkin, J. M.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Greenleaf, J. F.

J. Lu and J. F. Greenleaf, "Nondiffractiong X waves — Exact solutions to free-space wave equation and their finite-aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 90, 19-31 (1992).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guo, H.

Hu, W.

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Indebetouw, G.

Lajunen, H.

Leach, J.

Little, H.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

López-Mariscal, C.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6657.
[CrossRef] [PubMed]

Lu, J.

J. Lu and J. F. Greenleaf, "Nondiffractiong X waves — Exact solutions to free-space wave equation and their finite-aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 90, 19-31 (1992).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, "Complete coherence in the space-frequency domain," Opt. Commun. 36, 247-249 (1981).
[CrossRef]

McDonnell, G.

McGloin, D.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr. and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Mishra, S. R.

S. R. Mishra, "A vector wave analysis of a bessel beam," Opt. Commun. 85, 159-161 (1991).
[CrossRef]

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Morris, J. E.

Olivík, M.

Z. Bouchal and M. Olivík, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1666 (1995).
[CrossRef]

Overfelt, P. L.

P. L. Overfelt, "Bessel-Gauss pulses," Phys. Rev. A 44, 3941-3947 (1991).
[CrossRef] [PubMed]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Padgett, M. J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Porras, M. A.

Recami, E.

E. Recami, "On localized ‘X-shaped’ superluminal solutions to Maxwell’s equations," Physica A 252, 586-601 (1999).
[CrossRef]

Reivelt, K.

K. Reivelt and P. Saari, "Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves," Opt. Lett. 29, 1176-1178 (2004).
[CrossRef] [PubMed]

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Saari, P.

K. Reivelt and P. Saari, "Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves," Opt. Lett. 29, 1176-1178 (2004).
[CrossRef] [PubMed]

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Salomaa, M. M.

A. T. Friberg, J. Fagerholm, and M. M. Salomaa, "Space-frequency analysis of nondiffracting pulses," Opt. Commun. 136, 207-212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular spectrum representation of nondiffarcting X waves, Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Sezginer, A.

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

Shaarawi, A. M.

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[CrossRef]

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P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
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P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6657.
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Smith, R. L.

P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

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J. A. Campbell and S. Soloway, "Generation of a nondiffracting beam with frequency- independent beamwidth," J. Acoust. Soc. Am. 88, 2467-2477 (1990).
[CrossRef]

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Vasara, A.

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C. H. Gan, G. Gbur, and T. D. Visser, "Surface plasmons modulate the spatial coherence of light in Young’s interference experiment," Phys. Rev. Lett. 98, 043908 (2007).
[CrossRef] [PubMed]

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Wolf, K. B.

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[CrossRef] [PubMed]

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Wright, E. M.

Wyrowski, F.

Ziolkowski, R. W.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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J. A. Campbell and S. Soloway, "Generation of a nondiffracting beam with frequency- independent beamwidth," J. Acoust. Soc. Am. 88, 2467-2477 (1990).
[CrossRef]

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J. N. Brittingham; "Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
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[CrossRef]

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P. Fischer, H. Little, R. L. Smith, C. López-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Wavelength dependent propagation and reconstruction of white light Bessel beams," J. Opt. A 8, 477-482 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

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A. T. Friberg, A. Vasara, and J. Turunen, "Partially coherent propagation-invariant fields: passage through paraxial optical systems," Phys. Rev. A 43, 7079-7082 (1991).
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Figures (7)

Fig. 1.
Fig. 1.

Spectral weight given by Eq. (11) for n=1 (solid line), n=10 (dashed line), and n=100 (dotted line), plotted for λ̄=0.55 µm.

Fig. 2.
Fig. 2.

Fig. 2. Square root of the radial intensity distribution for conical waves given by Eq. (28) if n=1 (solid line), n=10 (dashed line), and n=100 (dotted line).

Fig. 3.
Fig. 3.

The absolute value of the complex degree of transverse spatial coherence given by Eq. (29) for n = 1 (solid line), n = 10 (dashed line), and n = 100 (dotted line).

Fig. 4.
Fig. 4.

Longitudinal and temporal coherence given by Eq. (30) for n=10. Absolute value of the complex degree of coherence (a) as a function of longitudinal/temporal delay gz;τ) at radial positions α(ω̄)ρ=0 (solid line), α(ω̄)ρ=2.405 (dashed line), and α(ω̄)ρ=3.83 (dotted line) and (b) as a function of radial position for gz;τ)=2.5 (solid line), gz;τ)=5 (dashed line), and gz;τ)=10 (dotted line).

Fig. 5.
Fig. 5.

Intensity distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 as seen by a logarithmic intensity detector with an ideal, flat spectral response.

Fig. 6.
Fig. 6.

Intensity distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 as perceived by the human eye.

Fig. 7.
Fig. 7.

Spectral distributions of polychromatic fields with (a) n=1, (b) n=10, (c) n=100 with colors as perceived by the human eye in normal circumstances.

Equations (37)

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W ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; ω ) = U * ( x 1 , y 1 , z 1 ; ω ) U ( x 2 , y 2 , z 2 ; ω ) .
U ( x , y , z ; ω ) = s ( ω ) V ( x , y ; ω ) exp [ i β ( ω ) z ]
V ( x , y ; ω ) = 0 2 π A ( ϕ ) exp [ i α ( ω ) ( x cos ϕ + y sin ϕ ) ] d ϕ
α 2 ( ω ) + β 2 ( ω ) = ω 2 c 2 ,
Γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = 0 W ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; ω ) exp ( i ω τ ) d ω .
γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = Γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) Γ ( x 1 , y 1 , z 1 , x 1 , y 1 , z 1 ; 0 ) Γ ( x 2 , y 2 , z 2 , x 2 , y 2 , z 2 ; 0 )
Γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = 0 s ( ω ) V * ( x 1 , y 1 ; ω ) V ( x 2 , y 2 ; ω ) exp { i [ β ( ω ) Δ z ω τ ] } d ω ,
I ( x , y , z ) = 0 S ( x , y ; ω ) d ω ,
S ( x , y ; ω ) = s ( ω ) V ( x , y ; ω ) 2
s ( ω ) = 1 Γ ( 2 n ) ω ¯ ( 2 n ω ω ¯ ) 2 n exp ( 2 n ω ω ¯ )
s ( λ ) = 1 2 n Γ ( 2 n 1 ) λ ( 2 n λ ¯ λ ) 2 n exp ( 2 n λ ¯ λ )
β ( ω ) = ( ω c ) 2 α 2 .
sin θ ( ω ) = c α ω .
v g = c 1 ( c α ω ¯ ) 2 = c cos θ ( ω ¯ )
Γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = V * ( x 1 , y 1 ) V ( x 2 , y 2 )
× 0 s ( ω ) exp { i [ Δ z ( ω c ) 2 α 2 ω τ ] } d ω .
γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = γ ( x 1 , y 1 , z , x 2 , y 2 , z ; 0 ) γ ( x , y , z 1 , x , y , z 2 ; τ )
γ ( x 1 , y 1 , z , x 2 , y 2 , z ; 0 ) = V * ( x 1 , y 1 ) V ( x 2 , y 2 ) V ( x 1 , y 1 ) V ( x 2 , y 2 )
γ ( x , y , z 1 , x , y , z 2 ; τ ) = 0 s ( ω ) exp { i [ Δ z ( ω c ) 2 α 2 ω τ ] } d ω
α ( ω ) = ω c sin θ ,
β ( ω ) = ω c cos θ .
v g = c cos θ
Γ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ; τ ) = 0 2 π d ϕ A * ( ϕ ) A ( ϕ )
× s ˜ [ τ cos θ Δ z c + sin θ ( x 1 cos ϕ x 2 cos ϕ + y 1 sin ϕ y 2 sin ϕ ) c ] ,
s ˜ ( u ) = 0 s ( ω ) exp ( i ω u ) d ω
Γ ( 0 , 0 , z 1 , 0 , 0 , z 2 ; τ ) = s ˜ ( τ cos θ Δ z c ) .
V ( x , y ; ω ) V ( ρ ; ω ) = J 0 [ α ( ω ¯ ) ρ ω ω ¯ ] ,
S ( ρ ; ω ) = 1 Γ ( 2 n ) ω ¯ ( 2 n ω ω ¯ ) 2 n exp ( 2 n ω ω ¯ ) J 0 2 ( α ( ω ¯ ) ρ ω ω ¯ ) .
Γ ( ρ 1 , z 1 , ρ 2 , z 2 ; τ ) = 1 Γ ( 2 n ) 0 ( 2 n Ω ) 2 n exp ( 2 n Ω )
× J 0 [ α ( ω ¯ ) ρ 1 Ω ] J 0 [ α ( ω ¯ ) ρ 0 Ω ] exp { i [ β ( ω ¯ ) Δ z ω ¯ τ ] Ω } d Ω .
I ( ρ ) = 3 F 2 [ 1 2 , n + 1 2 , n + 1 ; 1 , 1 ; α 2 ( ω ¯ ) ρ 2 n 2 ] ,
γ ( ρ , z , 0 , z ; 0 ) = I ( ρ ) 1 2 2 F 1 [ n + 1 2 , n + 1 ; 1 ; α 2 ( ω ¯ ) ρ 2 4 n 2 ] .
γ ( ρ , z 1 , ρ , z 2 ; τ ) = I ( ρ ) 1 [ 1 i g ( Δ z ; τ ) ] ( 2 n + 1 )
× 3 F 2 [ 1 2 , n + 1 2 , n + 1 ; 1 , 1 ; 4 α 2 ( ω ¯ ) ρ 2 [ 2 n i g ( Δ z ; τ ) ] 2 ] ,
γ ( 0 , z 1 , 0 , z 2 ; τ ) = [ 1 i g ( Δ z ; τ ) 2 n ] ( 2 n + 1 ) ,
Γ ( ρ 1 , z 1 , ρ 2 , z 2 ; τ ) = J 0 [ α ( ω ¯ ) ρ 1 ] J 0 [ α ( ω ¯ ) ρ 2 ] exp { i [ β ( ω ¯ ) Δ z ω ¯ τ ] } .
γ ( ρ 1 , z , 0 , z ; 0 ) = J 0 [ α ( ω ¯ ) ρ ] J 0 [ α ( ω ¯ ) ρ ] = arg { J 0 [ α ( ω ¯ ) ρ ] }

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