Abstract

The Bessel-beam representation for partially coherent fields is introduced as an alternative to the familiar angular spectrum representation. This alternative representation is applied to far fields generated by partially coherent secondary sources, nondiffracting partially coherent fields, and statistically homogeneous fields. A scheme for generating nondiffracting partially coherent fields is also discussed.

© 1995 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  4. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  5. A. M. Belskii, “Self-reproducing beams and their relationship with nondiffracting beams,” Opt. Spectrosc. 73, 568–569 (1992).
  6. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [CrossRef]
  7. A. T. Friberg, A. Vasara, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
    [CrossRef] [PubMed]
  8. W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” submitted to Phys. Rev. E.
  9. T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Secs. 3.2 and 5.6.3.
  12. A collection of papers on the angular spectrum representation may be found in K. E. Oughstun, ed., Selected Papers on Scalar Wave Diffraction, Vol. MS51 of Milestone Series (SPIE—Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1992).
  13. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]
  14. S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 260–264.
  15. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 4.3.
  16. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Gaithersburg, Md., 1964), p. 360.
  17. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.
  18. L. Mandel, E. Wolf, “Complete coherence in the space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]
  19. Ref. 13, Eq. (32), with some obvious modifications.
  20. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948), p. 560; I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 979.
  21. The cross-spectral density is Hermitian if the coefficients Fq(u⊥) obey Eq. (3.14). However, this condition is not sufficient to ensure that the cross-spectral density be nonnegative definite.
  22. Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  23. See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1989), Chap. 20.
  24. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982); “New theory of partial coherence in the space–frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [CrossRef]
  25. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  26. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993); K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  27. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  28. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  29. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978), Eq. (5.3).
    [CrossRef]
  30. Brief discussions of the differential operator 1+∇⊥2/k2 may be found in M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979), Sec. 2, and in A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992), Appendix A.
    [CrossRef]

1994 (2)

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

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

1993 (3)

1992 (1)

A. M. Belskii, “Self-reproducing beams and their relationship with nondiffracting beams,” Opt. Spectrosc. 73, 568–569 (1992).

1991 (2)

J. Turunen, A. Vasara, 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, J. Turunen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

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

Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

1982 (1)

1981 (1)

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

1979 (1)

Brief discussions of the differential operator 1+∇⊥2/k2 may be found in M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979), Sec. 2, and in A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992), Appendix A.
[CrossRef]

1978 (1)

1972 (1)

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bastiaans, M. J.

Brief discussions of the differential operator 1+∇⊥2/k2 may be found in M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979), Sec. 2, and in A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992), Appendix A.
[CrossRef]

Belskii, A. M.

A. M. Belskii, “Self-reproducing beams and their relationship with nondiffracting beams,” Opt. Spectrosc. 73, 568–569 (1992).

Crosignani, B.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 260–264.

DiPorto, P.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 260–264.

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

Eberly, J. H.

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

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Guattari, G.

Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Herminghaus, S.

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Indebetouw, G.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.

Mandel, L.

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

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 4.3.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Secs. 3.2 and 5.6.3.

Marchand, E. W.

Miceli, J. J.

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

Mukunda, N.

Padovani, C.

Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Siegman, A. E.

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1989), Chap. 20.

Simon, R.

Solimeno, S.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 260–264.

Sundar, K.

Tervonen, E.

Turunen, J.

Vasara, A.

Wang, W.

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” submitted to Phys. Rev. E.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948), p. 560; I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 979.

Wolf, E.

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982); “New theory of partial coherence in the space–frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

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

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978), Eq. (5.3).
[CrossRef]

E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
[CrossRef]

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 4.3.

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” submitted to Phys. Rev. E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Secs. 3.2 and 5.6.3.

Wulle, T.

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Acta (1)

Brief discussions of the differential operator 1+∇⊥2/k2 may be found in M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979), Sec. 2, and in A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992), Appendix A.
[CrossRef]

Opt. Commun. (2)

Statistically homogeneous fields with J0correlation are discussed in Ref. 6 [see Eq. (20) in that reference]. The coherent mode representation of J0-correlated Schell-model sources is given in F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

Opt. Spectrosc. (1)

A. M. Belskii, “Self-reproducing beams and their relationship with nondiffracting beams,” Opt. Spectrosc. 73, 568–569 (1992).

Phys. Rev. A (1)

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

Phys. Rev. Lett. (2)

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

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Other (12)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Secs. 3.2 and 5.6.3.

A collection of papers on the angular spectrum representation may be found in K. E. Oughstun, ed., Selected Papers on Scalar Wave Diffraction, Vol. MS51 of Milestone Series (SPIE—Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1992).

Ref. 13, Eq. (32), with some obvious modifications.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1948), p. 560; I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 979.

The cross-spectral density is Hermitian if the coefficients Fq(u⊥) obey Eq. (3.14). However, this condition is not sufficient to ensure that the cross-spectral density be nonnegative definite.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 260–264.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 4.3.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Gaithersburg, Md., 1964), p. 360.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.

W. Wang, E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” submitted to Phys. Rev. E.

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1989), Chap. 20.

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Figures (1)

Fig. 1
Fig. 1

Illustration of the simple arrangement used to examine the generation of (paraxial) nondiffracting partially coherent fields. The source plane is at a distance d from a thin lens, which has focal length f.

Equations (62)

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Γ ( r 1 , r 2 ; τ ) V * ( r 1 , t ) V ( r 2 , t + τ ) .
W ( r 1 , r 2 ; ω ) 1 2 π - Γ ( r 1 , r 2 ; τ ) exp ( i ω τ ) d τ ,
W ( r 1 , r 2 ) = A ( u 1 , u 2 ) exp [ i k ( u 2 · ρ 2 - u 1 · ρ 1 ) ] × exp [ i k ( u 2 z * z 2 - u 1 z z 1 ) ] d 2 u 1 d 2 u 2 ,
u j z = 1 - u j 2             for u j 1 ,
u j z = i u j 2 - 1             for u j > 1 ,
A ( u 1 , u 2 ) = ( k 2 π ) 4 W ( 0 ) ( ρ 1 , ρ 2 ) × exp [ - i k ( u 2 · ρ 2 - u 1 · ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 .
A ( u 1 , u 2 ) = 1 ( 2 π ) 2 u 1 u 2 n , p = - C n p ( u 1 , u 2 ) i n - p × exp [ i ( p ψ 2 - n ψ 1 ) ] ,
C n p ( u 1 , u 2 ) = i p - n u 1 u 2 0 2 π 0 2 π A ( u 1 , u 2 ) × exp [ - i ( p ψ 2 - n ψ 1 ) ] d ψ 1 d ψ 2 .
W ( r 1 , r 2 ) = 1 ( 2 π ) 2 n , p = - 0 d u 1 0 d u 2 × C n p ( u 1 , u 2 ) i p - n exp [ i k ( u 2 z * z 2 - u 1 z z 1 ) ] × { 0 2 π exp ( - i n ψ 1 ) exp [ - i k u 1 ρ 1 cos ( ψ 1 - φ 1 ) ] d ψ 1 } × { 0 2 π exp ( i p ψ 2 ) exp [ i k u 2 ρ 2 cos ( ψ 2 - φ 2 ) ] d ψ 2 } .
J n ( v ) = i - n π 0 π cos ( n ψ ) exp ( i v cos ψ ) d ψ .
W ( r 1 , r 2 ) = n , p = - 0 0 C n p ( u 1 , u 2 ) × B u 1 ( n ) * ( r 1 ) B u 2 ( p ) ( r 2 ) d u 1 d u 2 ,
B u j ( n ) ( r ) J n ( k u j ρ ) exp ( i n φ ) exp ( i k u j z z )
j 2 B u j ( n ) ( r ) = - ( k u j ) 2 B u j ( n ) ( r ) .
B u j ( n ) ( ρ , z ) = B u j ( n ) ( ρ , 0 ) ,
B u j ( n ) ( ρ , z ) = B u j ( n ) ( ρ , 0 ) exp ( - k z u j 2 - 1 ) .
0 J n ( α ρ ) J n ( α ρ ) d ρ = δ ( α - α ) α .
C n p ( u 1 , u 2 ) = k 4 ( 2 π ) 2 u 1 u 2 0 0 ρ 1 ρ 2 d ρ 1 d ρ 2 × 0 2 π 0 2 π d φ 1 d φ 2 W ( 0 ) ( ρ 1 , φ 1 , ρ 2 , φ 2 ) × exp [ - i ( p φ 2 - n φ 1 ) ] J n ( k u 1 ρ 1 ) × J p ( k u 2 ρ 2 ) .
C p n ( u 2 , u 1 ) = C n p * ( u 1 , u 2 ) .
W ( z 1 , z 2 ) = 0 0 C 00 ( u 1 , u 2 ) × exp [ i k ( u 2 z * z 2 - u 1 z z 1 ) ] d u 1 d u 2 ,
C 00 ( u 1 , u 2 ) = k 4 ( 2 π ) 2 u 1 u 2 0 0 ρ 1 ρ 2 d ρ 1 d ρ 2 × 0 2 π 0 2 π d φ 1 d φ 2 W ( 0 ) ( ρ 1 , φ 1 , ρ 2 , φ 2 ) × J 0 ( k u 1 ρ 1 ) J 0 ( k u 2 ρ 2 ) .
C n p ( u 1 , u 2 ) = c n * ( u 1 ) c p ( u 2 ) .
U ( r ) = n = - 0 c n ( u ) B u ( n ) ( r ) d u .
W ( ) ( r 1 u 1 , r 2 u 2 ) = ( 2 π ) 2 u 1 z u 2 z exp [ i k ( r 2 - r 1 ) ] k 2 r 1 r 2 × A ( u 1 , u 2 ) .
W ( ) ( r 1 u 1 , r 2 u 2 ) = exp [ i k ( r 2 - r 1 ) ] k 2 r 1 r 2 u 1 z u 2 z u 1 u 2 × n , p = - C n p ( u 1 , u 2 ) i n - p × exp [ i ( p ψ 2 - n ψ 1 ) ] .
W ( ) ( r 1 u 1 , r 2 u 2 ) = exp [ i k ( r 2 - r 1 ) ] k 2 r 1 r 2 cot θ 1 cot θ 2 × n , p = - C n p ( sin θ 1 , sin θ 2 ) i n - p × exp [ i ( p ψ 2 - n ψ 1 ) ] .
S ( ) ( r u ) = ( cot θ k r ) 2 m , p = - C n p ( sin θ , cos θ ) i n - p × exp [ i ( p - n ) ψ ] .
W N D ( ρ 1 , z , ρ 2 , z ) = W N D ( ρ 1 , 0 , ρ 2 , 0 ) .
1 2 W N D ( ρ 1 , z , ρ 2 , z ) = 2 2 W N D ( ρ 1 , z , ρ 2 , z ) .
C n p ( u 1 , u 2 )             = D n p ( u 1 ) δ ( u 1 - u 2 ) for u 1 1 and u 2 1 = 0 otherwise .
W N D ( r 1 , r 2 ) = n , p = - 0 1 D n p ( u ) B u ( n ) * ( r 1 ) B u ( p ) ( r 2 ) d u = n , p = - 0 1 D n p ( u ) J n ( k u ρ 1 ) J p ( k u ρ 2 ) × exp [ i ( p φ 2 - n φ 1 ) ] × exp [ i k ( z 2 - z 1 ) 1 - u 2 ] d u .
D p n ( u ) = D n p * ( u ) .
A ( u 1 , u 2 )             = G ( u 1 ) δ ( 2 ) ( u 1 - u 2 ) for u 1 1 and u 2 1 = 0 otherwise ,
C n p ( u 1 , u 2 )             = F n - p ( u 1 ) δ ( u 1 - u 2 ) for u 1 1 and u 2 < 1 = 0 otherwise .
F - q ( u ) = F q * ( u ) .
W H ( r 1 - r 2 ) = p , q = - 0 1 F q ( u ) B u ( p + q ) * ( r 1 ) B u ( p ) ( r 2 ) d u = p , q = - 0 1 F q ( u ) J p + q ( k u ρ 1 ) J p ( k u ρ 2 ) × exp [ i p ( φ 2 - φ 1 ) ] exp ( - i q φ 1 ) × exp [ i k ( z 2 - z 1 ) 1 - u 2 ] d u .
J q ( s R ) [ ρ 1 - ρ 2 exp ( - i φ ) ρ 1 - ρ 2 exp ( i φ ) ] q / 2 = p = - J p + q ( s ρ 1 ) J p ( s ρ 2 ) × exp ( i p φ ) ,
R = ( ρ 1 2 + ρ 2 2 - 2 ρ 1 ρ 2 cos φ ) 1 / 2 .
W H ( r 1 - r 2 ) = q = - 0 1 F q ( u ) H u ( q ) ( r 1 - r 2 ) d u ,
H u ( q ) ( r 1 - r 2 ) J q ( k u ρ 1 - ρ 2 ) [ ( x 1 - x 2 ) - i ( y 1 - y 2 ) ] q ρ 1 - ρ 2 q × exp [ i k ( z 2 - z 1 ) 1 - u 2 ] .
H u ( q ) ( r 1 - r 2 ) = J q ( k u ρ 1 - ρ 2 ) exp ( - i q φ ) × exp [ i k ( z 2 - z 1 ) 1 - u 2 ] .
F q ( u ) = k 2 2 π u 0 0 2 π W H ( 0 ) ( ρ ) × exp ( i q φ ) J q ( k u ρ ) ρ d ρ d φ .
H u ( q ) ( r ) = B u ( q ) * ( r ) .
U ( r ) = n = - 0 1 c n ( u ) B u ( n ) ( r ) d u .
U ( r ) = K J m ( k α ρ ) exp ( i m φ ) exp ( i k z 1 - α 2 ) .
W H ( r 1 - r 2 ) = S J 0 ( k α ρ 1 - ρ 2 ) exp [ i k ( z 2 - z 1 ) 1 - α 2 ] .
U in ( ρ ) = n = - α n δ ( ρ - a ) exp ( i n φ ) ,
U out ( ρ ) = exp ( i k d ) 2 π i B U in ( ρ ) × exp [ i 2 B ( A ρ 2 - 2 ρ · ρ + D ρ 2 ) ] d 2 ρ ,
[ A B C D ] = [ 1 d k - k f 1 - d f ] .
U out ( ρ ) = k exp ( i k d ) 2 π i f U in ( ρ ) × exp ( i k 2 f ρ 2 - i k f ρ · ρ ) d 2 ρ .
U out ( ρ ) = k a exp ( i k d ) 2 π i f exp ( i k a 2 / 2 f ) × n = - α n 0 2 π exp ( i n φ ) × exp [ - i k a ρ cos ( φ - φ ) / f ] d φ ,
U out ( ρ ) = k a exp ( i k d ) i f exp ( i k a 2 / 2 f ) × n = - α n ( - i ) n J n ( k a ρ f ) exp ( i n φ ) .
W in ( ρ 1 , ρ 2 ) = U in * ( ρ 1 ) U in ( ρ 2 ) ω = n , p = - α n * α p ω δ ( ρ 1 - a ) δ ( ρ 2 - a ) × exp [ i ( p φ 2 - n φ 1 ) ] ,
W out ( ρ 1 , ρ 2 ) = U out * ( ρ 1 ) U out ( ρ 2 ) ω = k 2 a 2 f 2 n , p = - α n * α p ω i n - p J n ( k a ρ 1 f ) × J p ( k a ρ 2 f ) exp [ i ( p φ 2 - n φ 1 ) ] ,
W out ( ρ 1 , ρ 2 ) = n , p = - 0 β D n p ( a f ) J n ( k a ρ 1 f ) J n ( k a ρ 2 f ) × exp [ i ( p φ 2 - n φ 1 ) ] d ( a f ) ,
D n p ( a f ) = k 2 a 2 f 2 a n * α p i n - p
( 1 2 + 2 z 1 2 + k 2 ) W ( ρ 1 , z 1 , ρ 2 , z 2 ) = 0 ,
( 2 2 + 2 z 2 2 + k 2 ) W ( ρ 1 , z 1 , ρ 2 , z 2 ) = 0.
( z 1 + i k 1 + 1 2 / k 2 ) W ( ρ 1 , z 1 , ρ 2 , z 2 ) = 0 ,
( z 2 - i k 1 + 2 2 / k 2 ) W ( ρ 1 , z 1 , ρ 2 , z 2 ) = 0.
W ( ρ 1 , z 1 , ρ 2 , z 2 ) = exp ( i k z 2 1 + 2 2 / k 2 - i k z 1 1 + 1 2 / k 2 ) W ( 0 ) ( ρ 1 , ρ 2 ) .
exp ( i k z 1 + 1 2 / k 2 ) W N D ( ρ 1 , z , ρ 2 , z ) = exp ( i k z 1 + 2 2 / k 2 ) W N D ( ρ 1 , z , ρ , z )
1 2 W N D ( ρ 1 , z , ρ 2 , z ) = 2 2 W N D ( ρ 1 , z , ρ 2 , z ) .

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