Abstract

We address the following problem: Can two wave fields with different coherence properties produce the same optical intensity everywhere in the space? Limiting ourselves to paraxial propagation, we prove that in the one-dimensional case the answer is negative. On the other hand, in the two-dimensional case we show through examples that the answer is affirmative. Some consequences are discussed.

© 1993 Optical Society of America

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  1. E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef] [PubMed]
  2. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  3. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [CrossRef]
  4. B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
    [CrossRef]
  5. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  6. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  7. F. Gori, G. Guattari, “A new type of optical field,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  8. A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
    [CrossRef]
  9. A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
    [CrossRef]
  10. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  11. Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
    [CrossRef]
  12. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  13. Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [CrossRef]
  14. 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).
    [CrossRef]
  15. J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.
  16. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980), Chap. 8.
  17. M. Bertero, in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75.
    [CrossRef]
  18. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16.
  19. See Ref. 18, Chap. 17.
  20. See Ref. 15, Chap. 3.
  21. J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  22. H. A. Ferwerda, “The phase reconstruction problems for wave amplitudes and coherent functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, Germany, 1978).
    [CrossRef]
  23. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  24. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  25. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  26. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef] [PubMed]
  27. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 952.

1988

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
[CrossRef] [PubMed]

1987

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

1986

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983

1982

1980

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

1979

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1978

1976

Bertero, M.

M. Bertero, in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980), Chap. 8.

Collett, E.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef] [PubMed]

Dacic, Z.

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Durnin, J.

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

Eberly, J. H.

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

Farina, J. D.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problems for wave amplitudes and coherent functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, Germany, 1978).
[CrossRef]

Friberg, A. T.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gamliel, A.

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

Goodman, J. W

J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

Gori, F.

F. Gori, G. Guattari, “A new type of optical field,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 952.

Guattari, G.

F. Gori, G. Guattari, “A new type of optical field,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

He, Q.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

Ichikawa, K.

Lohmann, A. W.

Mandel, L.

Miceli, J.

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

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Narducci, L. M.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 952.

Saleh, B. E. A.

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16.

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Takeda, M.

Teague, M. R.

Turunen, J.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

Wolf, E.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical field,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Eng.

A. T. Friberg, J. Turunen, “Algebraic and graphical propagation methods for Gaussian Schell-model beams,” Opt. Eng. 25, 857–864 (1986).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

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

Other

H. A. Ferwerda, “The phase reconstruction problems for wave amplitudes and coherent functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Heidelberg, Germany, 1978).
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 952.

J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980), Chap. 8.

M. Bertero, in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75.
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16.

See Ref. 18, Chap. 17.

See Ref. 15, Chap. 3.

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Figures (2)

Fig. 1
Fig. 1

Linear relationship between t and p for any fixed value of z.

Fig. 2
Fig. 2

Accessible region in the (p, t) plane where the optical intensity is known for a finite interval of z values (z1zz2).

Equations (51)

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W ( r 1 , r 2 ; ν ) = V ( r 1 ; ν ) V * ( r 2 ; ν ) ,
I ( r ; ν ) = | V ( r ; ν ) | 2 ,
μ ( r 1 , r 2 ; ν ) = W ( r 1 , r 2 ; ν ) / [ I ( r 1 ; ν ) I ( r 2 ; ν ) ] 1 / 2 ,
0 | μ ( r 1 , r 2 ; ν ) | 1
U ( r , z ) = i λ z [ exp ( ikz ) ] U ( ρ , 0 ) exp [ i π λ z ( r ρ ) 2 ] d ρ ,
W ( r 1 , r 2 , z ) = 1 ( λ z ) 2 W 0 ( ρ 1 , ρ 2 ) × exp { i π λ z [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 ] } d ρ 1 d ρ 2 .
I ( x , z ) = 1 λ | z | W 0 ( ξ 1 , ξ 2 ) × exp { 2 π i λ z [ ξ 1 2 ξ 2 2 2 x ( ξ 1 ξ 2 ) ] } d ξ 1 d ξ 2 .
σ = ( ξ 1 + ξ 2 ) / 2 , τ = ( ξ 1 + ξ 2 ) / λ z .
I ( x , z ) = L 0 ( σ , λ z τ / 2 ) exp [ 2 π i ( σ x ) τ ] d σ d τ ,
L 0 ( s , t ) = W 0 ( s t , s + t ) .
I ( p , z ) = I ( x , z ) exp ( 2 π ipx ) d x .
I ( p , z ) = L 0 ( σ , λ z p / 2 ) exp ( 2 π i σ p ) d σ .
L 0 ( p , t ) = L 0 ( s , t ) exp ( 2 π ips ) d s .
I ( p , z ) = L 0 ( p , λ z p / 2 ) .
L 0 ( p , 0 ) = W 0 ( s , s ) exp ( 2 π ips ) d s .
L 0 ( p , 0 ) = I ( p , 0 ) ,
V n ( + ) ( r , θ , 0 ) = F n ( r , 0 ) exp ( i n θ ) ,
V n ( + ) ( r , θ , z ) = F n ( r , z ) exp ( i n θ ) ,
F n ( r , z ) = k | z | ( i ) n + 1 { exp [ i k ( z + r 2 2 z ) ] } 0 F n ( ρ , 0 ) × [ exp ( i k 2 z ρ 2 ) ] J n ( k ρ r z ) ρ d ρ .
V n ( ) ( r , θ , 0 ) = F n ( r , 0 ) exp ( i n θ ) .
V n ( ) ( r , θ , z ) = F n ( r , z ) exp ( i n θ ) .
V n ( r , θ , z ) = a n ( + ) V n ( + ) ( r , θ , z ) + a n ( ) V n ( ) ( r , θ , z ) ,
W n ( r 1 , θ 1 , r 2 , θ 2 ; z ) = F n ( r 1 , z ) F n * ( r 2 , z ) × { ( P n + M n ) cos [ n ( θ 1 θ 2 ) ] + i ( P n M n ) sin [ n ( θ 1 θ 2 ) ] } ,
P n = | a n ( + ) | 2 , M n = | a n ( ) | 2 .
T n = P n + M n , n = ( P n M n ) / ( P n + M n ) ,
W n ( r 1 , θ 1 , r 2 , θ 2 ; z ) = T n F n ( r 1 , z ) F n * ( r 2 , z ) { exp [ i δ n ( θ 1 θ 2 ) ] } × { cos 2 [ n ( θ 1 θ 2 ) ] + n 2 sin 2 [ n ( θ 1 θ 2 ) ] } 1 / 2 ,
δ ( θ 1 θ 2 ) = tan 1 { n tan [ n ( θ 1 θ 2 ) ] } .
I n ( r , z ) = T n | F n ( r , z ) | 2 .
μ n ( r 1 , θ 1 , r 2 , θ 2 ; z ) = ( exp { i [ Φ n ( r 1 , z ) Φ n ( r 2 , z ) + δ n ( θ 1 θ 2 ) ] } ) × { cos 2 [ n ( θ 1 θ 2 ) ] + n 2 sin 2 [ n ( θ 1 θ 2 ) ] } 1 / 2 ,
| μ n ( r 1 , θ 1 , r 2 , θ 2 ; z ) | = | cos [ n ( θ 1 θ 2 ) ] | .
V I ( ± ) ( r , θ , 0 ) = A r exp ( r 2 / w 0 2 ) exp ( ± i θ ) ,
V 1 ( + ) ( r , θ , 0 ) + V 1 ( ) ( r , θ , 0 ) = 2 A r exp ( r 2 / w 0 2 ) cos θ ,
V 1 ( + ) ( r , θ , 0 ) V 1 ( ) ( r , θ , 0 ) = 2 i A r exp ( r 2 / w 0 ) sin θ
I 1 ( r , z ) = T 1 A 2 r 2 exp ( 2 r 2 / w 0 2 ) ,
V ( r , θ , z ) = n V n ( r , θ , z ) .
a n ( + ) a m ( + ) * = δ n m P n , a n ( ) a m ( ) * = δ n m M n , a n ( + ) a m ( ) * = 0 ( n , m = ± 1 , ± 2 , ) .
W ( r 1 , θ 1 ; r 2 , θ 2 ; z ) = n W n ( r 1 , θ 1 ; r 2 , θ 2 ; z ) ,
I ( r , θ ) = n I n ( r , θ ) ,
V ( + ) ( x , y , 0 ) = A x cos ( β x ) + i A y cos ( β y ) ,
V ( ) ( x , y , 0 ) = A x cos ( β x ) i A y cos ( β y ) ,
V ( ) ( x , y , 0 ) = V ( + ) * ( x , y , 0 ) .
V ( ± ) ( x , y , z ) = { exp [ i z ( k 2 β 2 ) 1 / 2 ] } V ( ± ) ( x , y , 0 ) ,
W ( x 1 , y 1 ; x 2 , y 2 ) = ( P + M ) [ A x 2 cos ( β x 1 ) cos ( β x 2 ) + A y 2 cos ( β y 1 ) cos ( β y 2 ) ] i ( P M ) A x A y [ cos ( β x 1 ) cos ( β y 2 ) cos ( β y 1 ) cos ( β x 2 ) ] ,
P = | a ( + ) | 2 , M = | a ( ) | 2 .
I ( x , y ) = ( P + M ) [ A x 2 cos 2 ( β x ) + A y 2 cos 2 ( β y ) ] .
μ ( x 1 , y 1 ; x 2 , y 2 ) = cos ( β x 1 ) cos ( β x 2 ) + cos ( β y 1 ) cos ( β y 2 ) { [ cos 2 ( β x 1 ) + cos 2 ( β y 1 ) ] [ cos 2 ( β x 2 ) + cos 2 ( β y 2 ) ] } 1 / 2 .
y 2 = x 2 + ( 2 n + 1 ) ( π / β ) ,
V ( r , θ , z ) = i λ z [ exp ( ikz ) ] 0 2 π d ϕ 0 V ( ρ , θ , 0 ) × exp ( { i k 2 z [ r 2 + ρ 2 2 r ρ cos ( ϕ θ ) ] } ) ρ d ρ ,
V n ( + ) ( r , θ , z ) = i λ z exp { [ i k ( z + r 2 2 z ) ] } × 0 F n ( ρ , 0 ) [ exp ( i k 2 z ρ 2 ) ] ρ d ρ × 0 2 π exp [ i n ϕ i k r ρ z cos ( ϕ θ ) ] d ϕ .
V n ( + ) ( r , θ , z ) = 1 λ z [ exp ( i n θ ) ] ( i ) n + 1 { exp [ i k ( z + r 2 2 z ) ] } × 0 F n ( ρ , 0 ) [ exp ( i k 2 z ρ 2 ) ] ρ d ρ × 0 2 π exp [ i n τ i k r ρ z sin ( τ ) ] d τ .
V n ( + ) ( r , θ , z ) = [ exp ( i n θ ) ] ( i ) n + 1 2 π λ z { exp [ i k ( z + r 2 2 z ) ] } × 0 F n ( ρ , 0 ) [ exp ( i k 2 z ρ 2 ) ] J n ( k r ρ z ) ρ d ρ .

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