Abstract

We study the properties of broadband optical fields produced by two classes of axicons: reflective axicons creating fields with a frequency-independent cone angle, and diffractive axicons that generate fields with frequency-independent transverse scale. We also consider two different types of illumination: spectrally completely coherent pulses and spectrally incoherent (stationary) light assuming that the spectra are the same in both situations. In the former case we evaluate the spatiotemporal shape of the output field, and in the latter case its spatiotemporal coherence properties. Physical reasons for the substantially different fields produced by the two types of axicons are identified. Our results are useful for optical applications in which joint spatial and temporal field localization is desired.

© 2014 Optical Society of America

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [Crossref]
  2. Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon – the most important optical element,” Opt. Photon. News 14, 34–39 (2005).
    [Crossref]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  4. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  5. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009).
    [Crossref]
  6. M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [Crossref]
  7. 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]
  8. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [Crossref]
  9. 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).
    [Crossref] [PubMed]
  10. J. Turunen, “Space–time coherence of polychromatic propagation-invariant fields,” Opt. Express 16, 20283–20294 (2008).
    [Crossref] [PubMed]
  11. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
    [Crossref]
  12. A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
    [Crossref]
  13. K. Reivelt and P. Saari, “Linear-optical generation of localized waves,” in Localized Waves, H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recamieds, eds. (Wiley, 2008).
    [Crossref]
  14. P. Saari, “X-type waves in ultrafast optics,” in Non-Diffracting Waves, H. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2014).
  15. C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. A 19, 2218–2222 (2002).
    [Crossref]
  16. I. Bialynicki-Birula and Z. Bialynicki-Birula, “Exponential beams of electromagnetic radiation,” J. Phys. B 39, S545 (2006).
    [Crossref]
  17. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [Crossref]
  18. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Append. III.
    [Crossref]
  19. J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977).
    [Crossref]
  20. K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
    [Crossref]
  21. K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
    [Crossref]
  22. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
    [Crossref] [PubMed]
  23. M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett. 37, 1238–1240 (2012).
    [Crossref] [PubMed]
  24. H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).
    [Crossref]
  25. H. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-Diffracting Waves (Wiley-VCH, 2014).
  26. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
    [Crossref]
  27. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [Crossref]

2012 (1)

2011 (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

2009 (3)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009).
[Crossref]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
[Crossref]

2006 (1)

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Exponential beams of electromagnetic radiation,” J. Phys. B 39, S545 (2006).
[Crossref]

2005 (2)

2002 (2)

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[Crossref]

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. A 19, 2218–2222 (2002).
[Crossref]

2000 (1)

A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
[Crossref]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

1996 (1)

1989 (1)

1987 (2)

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, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1986 (1)

M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[Crossref]

1982 (1)

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

1977 (1)

1954 (1)

Besieris, I. M.

A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
[Crossref]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Exponential beams of electromagnetic radiation,” J. Phys. B 39, S545 (2006).
[Crossref]

Bialynicki-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Exponential beams of electromagnetic radiation,” J. Phys. B 39, S545 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Append. III.
[Crossref]

Bowlan, P.

Brenner, K. H.

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

Brown, C. T. A.

Burvall, A.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon – the most important optical element,” Opt. Photon. News 14, 34–39 (2005).
[Crossref]

Cuadrado, J. M.

M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[Crossref]

Dholakia, K.

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, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977).
[Crossref]

Fischer, P.

Friberg, A. T.

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009).
[Crossref]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon – the most important optical element,” Opt. Photon. News 14, 34–39 (2005).
[Crossref]

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[Crossref]

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]

Gomez-Reino, C.

M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[Crossref]

Jaroszewicz, Z.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon – the most important optical element,” Opt. Photon. News 14, 34–39 (2005).
[Crossref]

Lõhmus, M.

López-Mariscal, C.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Morris, J. E.

Perez, M. V.

M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[Crossref]

Piksarv, P.

Porras, M. A.

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[Crossref]

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

K. Reivelt and P. Saari, “Linear-optical generation of localized waves,” in Localized Waves, H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recamieds, eds. (Wiley, 2008).
[Crossref]

Saari, P.

M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett. 37, 1238–1240 (2012).
[Crossref] [PubMed]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
[Crossref] [PubMed]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

K. Reivelt and P. Saari, “Linear-optical generation of localized waves,” in Localized Waves, H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recamieds, eds. (Wiley, 2008).
[Crossref]

P. Saari, “X-type waves in ultrafast optics,” in Non-Diffracting Waves, H. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2014).

Saastamoinen, K.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
[Crossref]

Shaarawi, A. M.

A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. A 19, 2218–2222 (2002).
[Crossref]

Sibbett, W.

Trebino, R.

Turunen, J.

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009).
[Crossref]

J. Turunen, “Space–time coherence of polychromatic propagation-invariant fields,” Opt. Express 16, 20283–20294 (2008).
[Crossref] [PubMed]

K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
[Crossref]

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]

Vahimaa, P.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
[Crossref]

Valtna-Lukner, H.

Vasara, A.

Wódkiewicz, K.

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Append. III.
[Crossref]

Wright, E. M.

J. Mod. Opt. (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

J. Opt. Soc. A (1)

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. A 19, 2218–2222 (2002).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
[Crossref]

J. Phys. B (1)

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Exponential beams of electromagnetic radiation,” J. Phys. B 39, S545 (2006).
[Crossref]

Opt. Acta (1)

M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[Crossref]

Opt. Commun. (2)

K. H. Brenner and K. Wódkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[Crossref]

K. Saastamoinen, J. Turunen, and P. Vahimaa, “Time-dependent physical spectra of Gaussian Schell-model pulses,” Opt. Commun. 271, 309–315 (2007).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Photon. News (1)

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon – the most important optical element,” Opt. Photon. News 14, 34–39 (2005).
[Crossref]

Phys. Rev. A (1)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

Phys. Rev. E (1)

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[Crossref]

Phys. Rev. Lett. (2)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Prog. Opt. (1)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009).
[Crossref]

Other (5)

H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).
[Crossref]

H. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-Diffracting Waves (Wiley-VCH, 2014).

K. Reivelt and P. Saari, “Linear-optical generation of localized waves,” in Localized Waves, H. Hernández-Figueroa, M. Zamboni-Rached, and E. Recamieds, eds. (Wiley, 2008).
[Crossref]

P. Saari, “X-type waves in ultrafast optics,” in Non-Diffracting Waves, H. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2014).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Append. III.
[Crossref]

Supplementary Material (2)

» Media 1: AVI (11469 KB)     
» Media 2: AVI (11431 KB)     

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Figures (8)

Fig. 1
Fig. 1 (a) Normalized spectral and (b) temporal intensity profiles of the incident field at distances ρ′ = 0 (blue), ρ ′ = w / 2 (black), ρ′ = w (magenta), and ρ ′ = 2 w (red).

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Fig. 2
Fig. 2 Operation of a reflective axicon A, which converts a plane wave into a conical wave with cone angle θ. The geometrical propagation-invariant range L is defined by the crossing point of the marginal rays with the optical axis. A beam splitter B may be inserted in the system to separate the axicon field and the incident field spatially.

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Fig. 3
Fig. 3 Amplitude of the field I ( ρ ; t ) after reflective axicon at distances (a) z = L/2, and (b) z = L. The parameters here are n = 1, w = 15 μm, λ̄ = 550 nm, and θ = 5°. The video animation Media 1 illustrates the temporal evolution of the pulse shape on propagation. The bar on the left shows the change of the axial peak intensity in relation to its maximum value.

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Fig. 4
Fig. 4 Intensity profiles of the field after the diffractive axicon on the optical axis (ρ = 0) at distances z = L/2 (black), z = L (blue), and off-axis at point ρ/d = 2.5 (red) at the distance z = L (these curves are scaled to the same maximum value to aid comparisons). Here n = 1, w = 15 μm, λ̄ = 550 nm, d = 1.5 μm, and θ(ω̄) = 21.5°.

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Fig. 5
Fig. 5 Amplitude of the field I ( ρ ; t ) after diffractive axicon at distances (a) z = L/2, and (b) z = L. The parameters are same as in Fig. 4. The change in the intensity of the pulse (bar on the left) as well as the evolution of its shape on propagation are shown in Media 2.

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Fig. 6
Fig. 6 (a) Geometrical explanation of the tailing effect in pulses generated by diffractive axicons. (b) The time-dependent physical spectrum of the field after diffractive axicon. We assume that the measurement instrument is a tunable Fabry–Perot filter with bandwidth Bf = 0.5 · 1015s−1.

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Fig. 7
Fig. 7 Temporal coherence profiles for the incident field and the fields after the axicons at (a) z = L/2 and at (b) z = L. Here the black curves indicate the incident field, blue curves for diffractive axicons, and the red curves for reflective axicons.

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Fig. 8
Fig. 8 Axial power spectra (in arbitrary units) of the incident field and the fields after the axicons at distances (a) z = L/2, and (b) z = L. Here the black curves stand for the incident field, the blue curves for diffractive axicons, and the red curves for reflective axicons. The different power spectra are scaled for ease of comparison.

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Equations (48)

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

U ( ρ ′ ; ω ) = ( 2 π w 2 ) 1 / 2 ( ω ω ¯ ) 1 / 2 S ( ω ) exp ( − ω ω ¯ ρ ′ 2 w 2 ) .
S ( ω ) = S 0 Γ ( 2 n ) ω ¯ ( 2 n ω ω ¯ ) 2 n exp ( − 2 n ω ω ¯ ) ,
S ( ρ ′ ; ω ) = 2 ( 2 n ) 2 n S 0 π w 2 Γ ( 2 n ) ω ¯ ( ω ω ¯ ) 2 n + 1 exp [ − 2 ω ω ¯ ( n + ρ ′ 2 w 2 ) ] .
ω ¯ ρ = ω ¯ n + 1 / 2 n + ρ ′ 2 / w 2 .
U ( ρ ′ ; t ) = ∫ 0 ∞ U ( ρ ′ ; ω ) exp ( − i ω t )   d ω .
U ( ρ ′ ; t ) = U 0 ( n + ρ ′ 2 / w 2 + i ω ¯ t ) n + 3 / 2
U 0 = ( 2 S 0 ω ¯ π w 2 Γ ( 2 n ) ) 1 / 2 ( 2 n ) n Γ ( n + 3 / 2 ) .
I ( ρ ′ ; t ) = I 0 [ ( n + ρ ′ 2 / w 2 ) 2 + ( ω ¯ t ) 2 ] n + 3 / 2 ,
ω ¯ T = 2 1 / ( n + 3 / 2 ) − 1 ( n + ρ ′ 2 / w 2 ) .
Γ ( ρ ′ ; τ ) = ∫ 0 ∞ S ( ρ ′ ; ω ) exp ( − i ω τ )   d ω ,
γ ( ρ ′ ; τ ) = ( n + ρ ′ 2 / w 2 n + ρ ′ 2 / w 2 + i ω ¯ τ / 2 ) 2 ( n + 1 ) .
ω ¯ Θ i = 2 2 1 / ( n + 1 ) − 1 ( n + ρ ′ 2 w 2 ) .
sin θ ( ω ) = 2 π c d ω
t ( ρ ′ ; ω ) = η ( ω ) exp [ − i ( ω / c ) sin θ ( ω ) ρ ′ ] ,
U ( ρ , z ; ω ) = ω i c z exp [ i ω c ( z + ρ 2 2 z ) ] × ∫ 0 ∞ ρ ′ t ( ρ ′ ; ω ) U ( ρ ′ ; ω ) J 0 ( ω c z ρ ρ ′ ) exp ( i ω 2 c z ρ ′ 2 )   d ρ ′ ,
U ( ρ , z ; ω ) = − i η ( ω ) π z z R ω c sin θ ( ω ) S ( ω ) exp [ − ω ω ¯ z 2 w 2 sin 2 θ ( ω ) ] × J 0 [ ω c ρ sin θ ( ω ) ] exp { i ω [ z v ( ω ) + ρ 2 2 c z ] } .
S ( ρ , z ; ω ) = − η ( ω ) π z z R ( ω c ) 2 sin 2 θ S ( ω ) exp ( − 2 ω ω ¯ z 2 w 2 sin 2 θ ) J 0 2 ( ω c ρ sin θ ) .
U ( ρ , z ; ω ) = − 2 π i d η ( ω ) π z z R S ( ω ) exp ( − 2 π 2 c z 2 d 2 z R ω ) × J 0 ( 2 π ρ d ) exp [ i ω c ( z + ρ 2 2 z ) ] exp ( − i 2 π 2 c z d 2 ω ) ,
S ( ρ , z ; ω ) = − 4 π 2 d 2 η ( ω ) π z z R S ( ω ) exp ( − 4 π 2 c z 2 d 2 z R ω ) J 0 2 ( 2 π ρ d ) .
U ( ρ , z ; t r ) = − i ( 2 n ) n [ S 0 π Γ ( 2 n ) z z R ω ¯ c ] 1 / 2 sin θ ∫ 0 ∞ x n + 1 J 0 ( ω ¯ c ρ sin θ x ) exp ( − a x ) d x .
a = n + ( z L ) 2 + i ω ¯ ( t r − ρ 2 2 c z ) ,
I ( 0 , z ; t r ) = I ( 0 , z ; 0 ) ( n + z 2 / L 2 ) 2 ( n + 2 ) [ ( n + z 2 / L 2 ) 2 + ( ω ¯ t r ) 2 ] n + 2 ,
I ( 0 , z ; 0 ) = ( 2 n ) 2 n Γ 2 ( n + 2 ) sin 2 θ ( n + z 2 / L 2 ) 2 ( n + 2 ) S 0 π Γ ( 2 n ) ω ¯ c z z R .
ω ¯ T r = 2 1 / ( n + 2 ) − 1 [ n + ( z / L ) 2 ] .
U ( ρ , z ; t r ) = − 4 π i d ( 2 n ) n [ S 0 z π ω ¯ Γ ( 2 n ) z R ] 1 / 2 J 0 ( 2 π ρ d ) × [ b ( z ) a n ( ρ , z ; t r ) ] ( n + 1 ) / 2 K n + 1 [ 2 a n ( ρ , z ; t r ) b ( z ) ]
a n ( ρ , z ; t r ) = n + i [ ω ¯ t r − π w / d ( ρ / d ) 2 z / L ] ,
b ( z ) = ( z L ) 2 + i π z L w d .
t = z c 1 − ( 2 π c / ω d ) 2 .
γ ( 0 , z ; τ ) = [ n + ( z / L ) 2 n + ( z / L ) 2 + i ω ¯ τ / 2 ] 2 n + 3 .
ω ¯ Θ r = 2 2 1 / ( 2 n + 3 ) − 1 [ n + ( z / L ) 2 ] .
γ ( ρ , z ; τ ) = ( 2 n 2 n + i ω ¯ τ ) n + 1 / 2 K 2 n + 1 ( 2 a n b ) K 2 n + 1 ( 2 2 n b ) ,
b ( z ) = 8 π 2 c 2 z 2 d 2 w 2 ω ¯ 2 .
S ( 0 , z ; ω ) = − η ( ω ) π z z R ( ω c ) 2 sin 2 θ S ( ω ) exp [ − 2 ω ω ¯ z 2 w 2 sin 2 θ ] ,
S ( 0 , z ; ω ) = − 4 π 2 d 2 η ( ω ) π z z R S ( ω ) exp ( − 4 π 2 c z 2 d 2 z R ω ) .
I ( ω ) = ∫ 0 ∞ f ( ρ ′ ) exp [ i ω g ( ρ ′ ) ]   d ρ ′ .
I ( ω ) = f ( ρ c ) exp [ i ω g ( ρ c ) ] ω g ″ ( ρ c ) ,
U ( ρ , z ; ω ) = C 0 F ( ω ) ∫ 0 ∞ ρ ′ J 0 ( ω c z ρ ρ ′ ) exp ( − ω ρ ′ 2 ω ¯ w 2 ) exp { i ω [ ρ ′ 2 2 c z − sin θ ( ω ) ρ ′ c ] } d ρ ′ ,
F ( ω ) = ω i c z exp [ i ω c ( z + ρ 2 2 z ) ] η ( ω ) ( 2 n ) n ( ω ω ¯ ) n + 1 / 2 exp ( − n ω ω ¯ ) .
f ( ρ ′ ) = ρ ′ J 0 ( ω c z ρ ρ ′ ) exp ( − ω ρ ′ 2 ω ¯ w 2 )
g ( ρ ′ ) = ρ ′ 2 2 c z − sin θ ( ω ) ρ ′ c .
g ′ ( ρ ′ ) = ρ ′ c z − sin θ ( ω ) c ,
g ″ ( ρ ′ ) = 1 c z .
G ( t , ω f , B f ) = ∫ ∫ − ∞ ∞ H * ( t − t 1 , ω f , B f ) H ( t − t 2 , ω f , B f ) Γ ( t 1 , t 2 ) d t 1 d t 2 ,
W ( ω 1 , ω 2 ) = ∫ ∫ − ∞ ∞ Γ ( t 1 , t 2 ) exp [ − i ( ω 1 t 1 − ω 2 t 2 ) ] d t 1 d t 2
H ˜ ( ω , ω f , B f ) = ∫ − ∞ ∞ H ( t , ω f , B f ) exp ( i ω t ) d t .
G ( ω , ω f , B f ) = 1 ( 2 π ) 2 ∫ ∫ − ∞ ∞ H ˜ * ( ω 1 , ω f , B f ) H ˜ ( ω 2 , ω f , B f ) W ( ω 1 , ω 2 ) exp [ i ( ω 1 − ω 2 ) t ] d ω 1 d ω 2 ,
H ˜ ( ω , ω f , B f ) = B f B f − i ( ω − ω f ) ,
G ( t , ω f , B f ) = B f 2 ( 2 π ) 2 ∫ ∫ − ∞ ∞ exp [ i ( ω 1 − ω f ) t ] B f + i ( ω 1 − ω f ) exp [ − i ( ω 2 − ω f ) t ] B f − i ( ω 2 − ω f ) W ( ω 1 , ω 2 ) d ω 1 d ω 2 .

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