Abstract

The effects of twist phenomenon (beam rotation) of a partially coherent field are studied on the operation of two classes of uniform-intensity diffractive axicons. A general theory of axicon image formation is developed, discussed, and examined. We show that the intensity of the diffracted field is a multiple Bessel field, and only the energy of the zero-order Bessel field diffracts along the propagation axes. We also show that, at any twist strength in all correlation levels, the images can be evaluated by using the stationary-phase method. The three-dimensional stationary-phase formula of axicon images is derived. Such formula may be used in fast image evaluation, for designing diffractive axicons that perform a uniform axial intensity in a twisted partially coherent field.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. R. Staroński, J. Sochacki, Z. Jaroszewicz, and A. Kolodziejczyk, “ Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  2. I. Golub, B. Chebbi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830 (2010).
    [CrossRef]
  3. A. Thaning, A. T. Friberg, and Z. Jaroszewicz, “Synthesis of diffractive axicons for partially coherent light based on asymptotic wave theory,” Opt. Lett. 26, 1648–1650 (2001).
    [CrossRef]
  4. V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, and Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995).
    [CrossRef]
  5. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef]
  6. N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilenses: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
    [CrossRef]
  7. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef]
  8. S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  9. S. Yu. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
    [CrossRef]
  10. J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
    [CrossRef]
  11. A. Thaning, A.T. Friberg, S.Yu. Popov, and Z. Jaroszewicz, “Design of diffractive axicons producing uniform line images in Gaussian Schell-model illumination,” J. Opt. Soc. Am. A 19, 491–496 (2002).
    [CrossRef]
  12. A. T. Friberg and S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996).
    [CrossRef]
  13. A. T. Friberg and S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
    [CrossRef]
  14. A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
    [CrossRef]
  15. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  16. K. Saundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  17. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  18. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  20. G. Arfken, Mathematical Methods For Physicists, 6th ed.(Elsevier, 2005).
  21. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).
  22. M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
    [CrossRef]
  23. S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
    [CrossRef]
  24. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  25. W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
    [CrossRef]
  26. A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
    [CrossRef]
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

2011

A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
[CrossRef]

2010

2002

2001

1999

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

A. T. Friberg and S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

1998

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

1997

1996

1995

1994

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

1993

1992

1991

Alarify, Y. S.

A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
[CrossRef]

Alkelly, A. A.

A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods For Physicists, 6th ed.(Elsevier, 2005).

Bara, S.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chebbi, B.

Churin, E. G.

Davidson, N.

Friberg, A. T.

A. Thaning, A. T. Friberg, and Z. Jaroszewicz, “Synthesis of diffractive axicons for partially coherent light based on asymptotic wave theory,” Opt. Lett. 26, 1648–1650 (2001).
[CrossRef]

A. T. Friberg and S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
[CrossRef]

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

A. T. Friberg and S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[CrossRef]

Friberg, A.T.

Friesem, A. A.

Golub, I.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Hasman, E.

Honkanen, M.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Jaroszewicz, Z.

Kolodziejczyk, A.

Koronkevich, V. P.

Lautanen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Mandel, L.

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

Mikhaltsova, I. A.

Mukunda, N.

Nemoto, S.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Nowacki, D.

Perrone, M. R.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Popov, S. Yu.

A. T. Friberg and S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

A. T. Friberg and S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996).
[CrossRef]

Popov, S.Yu.

Pu, J.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

Saundar, K.

Scaglione, S.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Schnabel, B.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Shaw, D.

Shukri, M. A.

A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
[CrossRef]

Simon, R.

Sochacki, J.

Staronski, L. R.

Tervonen, E.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[CrossRef]

Thaning, A.

Turunen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[CrossRef]

Wang, W.

Wolf, E.

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yurlov, Yu. I.

Zhang, H.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Zhang, W.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

J. Opt. A

J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[CrossRef]

A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam,” Opt. Commun. 284, 4658–4662 (2011).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

S. Yu. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Other

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

G. Arfken, Mathematical Methods For Physicists, 6th ed.(Elsevier, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed. (Elsevier, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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.


Figures (6)

Fig. 1.
Fig. 1.

Geometry of the apodized annular-aperture diffractive axicons, and a notation of the corresponded on-axis profile of image intensity.

Fig. 2.
Fig. 2.

On-axis intensity distribution S(0,z) produced by annular-aperture logarithmic axicon [Eq. (17)] along the focal region d1zd2, in spatially uniform intensity with unit-amplitude twisted partially coherent illumination of different degrees of coherence: (a) σμ=, (b) σμ=4mm, (c) σμ=2mm, (d) σμ=1mm, (e) σμ=0.5mm, and (f) σμ=0.25mm for the same characteristic values of twist parameter η. All curves are calculated from Eq. (16) with super-Gaussian apodization. The system parameters are r1=2.5mm, r2=5mm, d1=100mm, d2=200mm, n=18, and wavelength λ=0.6328μm.

Fig. 3.
Fig. 3.

Same as in Fig. 2(d) but on making use of the phase function given by Eq. (18).

Fig. 4.
Fig. 4.

Lateral intensity distribution for the apodized annular-aperture logarithmic axicon along the focal region d1zd2 in spatially uniform intensity of different degrees of coherence and different strengths of twist phase: (I) σμ=4mm, (a) η=0, (b) η=0.3, (c) η=0.6; (II) σμ=2mm, (a) η=0, (b) η=0.3, (c) η=0.6; (III) σμ=1mm, (a) η=0, (b) η=0.2, (c) η=0.4; (IV) σμ=0.5mm, (a) η=0, (b) η=0.1, (c) η=0.2. All profiles are calculated with the help of Eq. (12). The system parameters are the same as in Fig. 2.

Fig. 5.
Fig. 5.

Lateral intensity distribution produced by the diffractive axicon given by Eq. (18) for σμ=1mm: (a) η=0, (b) η=0.2, and (c) η=0.4. The system parameters are the same as in Fig. 2.

Fig. 6.
Fig. 6.

3D intensity distribution computed from the asymptotic formula, Eq. (22), for σμ=0.5mm and η=0.1. The other parameters are as in Fig. 2.

Equations (31)

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

E(r,ν)=(k2iπ)AE0(r,ν)t(r)exp[ikφ(r)]×exp{iks}sd2r,
W(r1,r2;ν)=(k2π)2AW0(r1,r2;ν)t(r1)t(r2)×exp{ik[ϕ(r1)ϕ(r2)]}×exp[ik(s1s2)]s1s2d2r1d2r2,
sizi+ρi22zi+ρi22ziρiρizicos(ϕiϕi).
W0(r1,r2)=W0(ρ1,ρ2)=S0exp[(ρ1ρ2)2/2σμ2]×exp[ikuρ1.ϵρ21],
S(ρ,z)=S0(k2πz)2at(ρ1)t(ρ2)×CT(ρ1,ρ2;ρ,z;σμ,u)exp{ik[φ(ρ1)φ(ρ2)]}×exp[ik(ρ12ρ22)/2z]ρ1ρ2dρ1dρ2,
CT(ρ1,ρ2;ρ,z;σμ,u)=exp[(ρ12+ρ22)/2σμ2]×02πexp[ρ1ρ2cos(ϕ1ϕ2)/σμ2]×exp[ikuρ1ρ2sin(ϕ1ϕ2)]×exp{ikρ[ρ1cos(ϕϕ1)ρ2cos(ϕϕ2)]/z}dϕ1dϕ2.
eρ1ρ2σμ2cos(ϕ1ϕ2)=n=In(ρ1ρ2/σμ2)ein(ϕ1ϕ2),
eikuρ1ρ2sin(ϕ1ϕ2)==J(kuρ1ρ2)ei(ϕ1ϕ2),
02πexp[ixcos(ϕϕj)+iqϕj]dϕj=2πexp[iq(π2+ϕ)]Jq(x).
CT(ρ1,ρ2;ρ,z;σμ,u)=(2π)2exp[(ρ12+ρ22)/2σμ2]×nIn(ρ1ρ2/σμ2)J(kuρ1ρ2)×Jn+(kρρ1/z)Jn+(kρρ2/z).
(x+yx-y)m2Im(x2y2)=kJk(y)Imk(x),
CT(ρ1,ρ2;ρ,z;σμ,η)=(2π)2exp[(ρ12+ρ22)/2σμ2]×m=(1+η1-η)m2Im(ρ1ρ21η2/σμ2)×Jm(kρρ1/z)Jm(kρρ2/z),
S(ρ,z)=S0k2z2m=Tm(η)Gm(ρ,z;σμ,η),
Gm(ρ,z;σμ,η)=t(ρ1)t(ρ2)exp{ik[ψ(ρ1,z)ψ(ρ2,z)]}×exp[ρ12+ρ222σμ2]Im(ρ1ρ21η2σμ2)×Jm(kρρ1z)Jm(kρρ2z)ρ1ρ2dρ1dρ2,
Tm(η)=[(1+η)/(1η)]m2;
ψ(ρj,z)=φ(ρj)+ρj2/2z
S(0,z)=A0k2z2t(ρ1)t(ρ2)exp{ik[ψ(ρ1,z)ψ(ρ2,z)]}×exp[ρ12+ρ222σμ2]I0(ρ1ρ21η2σμ2)ρ1ρ2dρ1dρ2,
φ(ρ)=12aln[1+a(ρ2r12)/d1],
φ(1)(ρ)pc=2ρ/{C1ρ2exp(ρ2/σμ2)[I0(ρ2/σμ2)+I1(ρ2/σμ2)]+C2},
t(ρ)=exp[(ρr¯ω)n],
ρs=[r12+(zd1)/a]1/2,
ψ2(2)(ρs,z)=2aρs2/z2=ψ1(2)(ρs,z).
Ssp(ρ,z)=2πS0kρs2t2(ρs)exp[ρs2/σμ2]z2ψ(2)(ρs,z)×m=Tm(η)Im(ρs21η2/σμ2)Jm(kρszρ)2.
g1(y,t)=exp[y2(t1t)]=j=tjJj(y),
g2(x,t)=exp[x2(t+1t)]=k=tkIk(x),
g1(y,t)g2(x,t)=k=j=Jj(y)Ik(x)tj+k,
g1(y,t)g2(x,t)=exp[12((x+y)t+xyt)]=exp[x2y22(x+yxyt+xyx+y1t)],
g1(y,t)g2(x,t)=n=(x+yxy)ntnIn(x2y2).
n=(x+yxy)ntnIn(x2y2)=k=j=Jj(y)Ik(x)tj+k.
n=(x+yxy)ntnIn(x2y2)=nj=tnj=Jj(y)Inj(x),
(x+yxy)nIn(x2y2)=j=Jj(y)Inj(x).

Metrics