Abstract

Equations for the propagation of phase and irradiance are derived, and a Green’s function solution for the phase in terms of irradiance and perimeter phase values is given. A measurement scheme is discussed, and the results of a numerical simulation are given. Both circular and slit pupils are considered. An appendix discusses the local validity of the parabolic-wave equation based on the factorized Helmholtz equation approach to the Rayleigh–Sommerfeld and Fresnel diffraction theories. Expressions for the diffracted-wave field in the near-field region are given.

© 1983 Optical Society of America

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  1. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. R. H. T. Bates and W. R. Fright, "Reconstructing images from their Fourier intensities," in Signal and Image Reconstruction from Incomplete Data. Theory & Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.
  3. H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978); Inverse Scattering Problems in Optics (Springer-Verlag, New York, 1980).
    [CrossRef]
  4. M. R. Teague, "Irradiance moments: their propagation and use for unique retrieval of phase," J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  5. B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math. Phys. 16, 1719–1725 (1975).
    [CrossRef]
  6. A. M. Huiser and H. A. Ferweda, "The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images," Opt. Acta 23, 445–456 (1976).
    [CrossRef]
  7. J. G. Walker, "The phase retrieval problem: a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  8. J. M. Wood, M. A. Fiddy, and R. E. Burge, "Phase retrieval using two intensity measurements in the complex plane," Opt. Lett. 6, 514–516 (1981).
    [CrossRef] [PubMed]
  9. P. Kiedron, "Phase retrieval methods using differential filters," Proc. Soc. Photo-Opt. Instrum. Eng. 413 (to be published).
  10. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1918). The parabolic equation is treated in problem 9.9.
  11. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chaps. I, II, IX.
  12. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), formula (2.553), no. 3.
  13. J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  15. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  16. P. A. M. Dirac, "The quantum theory of the electron," Proc. R. Soc. London Ser. A 117, 610–624 (1928).
    [CrossRef]
  17. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley-Interscience, New York, 1959), Sec. 6-1.
  18. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), p. 5.
  19. M. D. Feit and J. A. Fleck, "Light propagation in graded index optical fibers," Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  20. M. J. Bastiaans, "Transport equations for the Wigner distribution functions," Opt. Acta 26, 1265–1272 (1979).
    [CrossRef]
  21. A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
    [CrossRef]
  22. F. D. Tappert, "The parabolic approximation method," in Wave Propagation and Underwater Acoustics, J. Ehlers et al., eds. (Springer-Verlag, Berlin, 1977), p. 272.
  23. L. Fishman and J. J. McCoy, "Direct and inverse wave propagation theories and the factorized Helmholtz equation. Path integral representations," Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).
  24. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  25. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec. 45.
  26. E. Lalor, "Conditions for the validity of the angular spectrum of plane waves," J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [CrossRef]
  27. W. H. Southwell, "Validity of the Fresnel approximation in the near field," J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]

1983

A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
[CrossRef]

1982

1981

1979

M. J. Bastiaans, "Transport equations for the Wigner distribution functions," Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

1978

1976

A. M. Huiser and H. A. Ferweda, "The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images," Opt. Acta 23, 445–456 (1976).
[CrossRef]

1975

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

1968

1928

P. A. M. Dirac, "The quantum theory of the electron," Proc. R. Soc. London Ser. A 117, 610–624 (1928).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, "Transport equations for the Wigner distribution functions," Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates and W. R. Fright, "Reconstructing images from their Fourier intensities," in Signal and Image Reconstruction from Incomplete Data. Theory & Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

Bjorken, J. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), p. 5.

Bogoliubov, N. N.

N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley-Interscience, New York, 1959), Sec. 6-1.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Burge, R. E.

Dirac, P. A. M.

P. A. M. Dirac, "The quantum theory of the electron," Proc. R. Soc. London Ser. A 117, 610–624 (1928).
[CrossRef]

Drell, S. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), p. 5.

Feit, M. D.

Ferweda, H. A.

A. M. Huiser and H. A. Ferweda, "The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images," Opt. Acta 23, 445–456 (1976).
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

Fishman, L.

L. Fishman and J. J. McCoy, "Direct and inverse wave propagation theories and the factorized Helmholtz equation. Path integral representations," Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

Fleck, J. A.

Fright, W. R.

R. H. T. Bates and W. R. Fright, "Reconstructing images from their Fourier intensities," in Signal and Image Reconstruction from Incomplete Data. Theory & Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

Goodman, J. W.

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

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), formula (2.553), no. 3.

Hardy, J. W.

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Hoenders, B. J.

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

Huiser, A. M.

A. M. Huiser and H. A. Ferweda, "The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images," Opt. Acta 23, 445–456 (1976).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chaps. I, II, IX.

Kiedron, P.

P. Kiedron, "Phase retrieval methods using differential filters," Proc. Soc. Photo-Opt. Instrum. Eng. 413 (to be published).

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978).
[CrossRef]

Lalor, E.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec. 45.

McCoy, J. J.

L. Fishman and J. J. McCoy, "Direct and inverse wave propagation theories and the factorized Helmholtz equation. Path integral representations," Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

Ojeda-Castaneda, J.

A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1918). The parabolic equation is treated in problem 9.9.

Ryzhik, I. M.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), formula (2.553), no. 3.

Shirkov, D. V.

N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley-Interscience, New York, 1959), Sec. 6-1.

Southwell, W. H.

Streibl, N.

A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
[CrossRef]

Tappert, F. D.

F. D. Tappert, "The parabolic approximation method," in Wave Propagation and Underwater Acoustics, J. Ehlers et al., eds. (Springer-Verlag, Berlin, 1977), p. 272.

Teague, M. R.

Walker, J. G.

J. G. Walker, "The phase retrieval problem: a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Wood, J. M.

Appl. Opt.

J. Math. Phys.

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

A. M. Huiser and H. A. Ferweda, "The problem of phase retrieval in light and electron microscopy of strong objects II. On the uniqueness and stability of object reconstruction procedures using two defocused images," Opt. Acta 23, 445–456 (1976).
[CrossRef]

J. G. Walker, "The phase retrieval problem: a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).
[CrossRef]

M. J. Bastiaans, "Transport equations for the Wigner distribution functions," Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, and N. Streibl, "Symmetries in coherent and partially coherent fields," Opt. Acta 30, 399–402 (1983).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London Ser.

P. A. M. Dirac, "The quantum theory of the electron," Proc. R. Soc. London Ser. A 117, 610–624 (1928).
[CrossRef]

Other

N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley-Interscience, New York, 1959), Sec. 6-1.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), p. 5.

F. D. Tappert, "The parabolic approximation method," in Wave Propagation and Underwater Acoustics, J. Ehlers et al., eds. (Springer-Verlag, Berlin, 1977), p. 272.

L. Fishman and J. J. McCoy, "Direct and inverse wave propagation theories and the factorized Helmholtz equation. Path integral representations," Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec. 45.

R. H. T. Bates and W. R. Fright, "Reconstructing images from their Fourier intensities," in Signal and Image Reconstruction from Incomplete Data. Theory & Experiment, Vol. I of Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, Greenwich, Conn., 1983), Chap. 5.

H. P. Baltes, ed., Inverse Source Problems in Optics (Springer-Verlag, New York, 1978); Inverse Scattering Problems in Optics (Springer-Verlag, New York, 1980).
[CrossRef]

P. Kiedron, "Phase retrieval methods using differential filters," Proc. Soc. Photo-Opt. Instrum. Eng. 413 (to be published).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1918). The parabolic equation is treated in problem 9.9.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chaps. I, II, IX.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), formula (2.553), no. 3.

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651–697 (1978).
[CrossRef]

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

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Beam compressor. Output plane is located at z2 = z1 + (1 + f/F)f. The output-plane wave amplitude is uz2(r) = (−1/m)u0(−r/m), where m = f/F.

Fig. 2
Fig. 2

Phase error for one wave of coma. Both here and in Fig. 3 a plot of δϕ(x, 0) = ϕactual(x, 0) − ϕcalculated(x, 0) is given. The solid curve has simulated noise added to the calculation. The dotted curve is the noise-free calculation and is nonzero because the number of points used in the numerical integrations was finite. The pupil radius is 1.0.

Fig. 3
Fig. 3

Phase error for one wave of spherical aberration.

Equations (74)

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( i z + 2 2 k + k ) u z ( r ) = 0 ,
I z ( r ) u z ( r ) 2
u z ( r ) = [ I z ( r ) ] 1 / 2 exp [ i ϕ z ( r ) ] .
2 π λ z I = - · I ϕ ,
4 π λ I 2 z ϕ = 1 2 I 2 I - 1 4 ( I ) 2 - I 2 ( ϕ ) 2 + k I 2 .
ψ = I ϕ ,
2 ψ = - 2 π λ z I ,
ψ z ( r ) = - 2 π λ R d r G ( r , r ) z I z ( r ) - P d s [ G ( r , r ) ψ z ( r ) n - ψ z ( r ) n G ( r , r ) ] ,
G ( r , r ) = 0 ,
2 G ( r , r ) = δ ( r - r )
G ( r , r ) = 1 4 π ln ( r 2 + r 2 - 2 r · r r 2 r 2 a 2 + a 2 - 2 r · r ) .
G ( r , r ) n = 1 2 π a ( a 2 - r 2 a 2 + r 2 - 2 r · r ) ,
ψ = I 0 ( ϕ + constant ) .
ϕ 0 ( r ) = R d r G ( r , r ) [ - 2 π λ I 0 I 0 ( r ) z ] + P d s ϕ 0 ( r ) G n ( r , r ) .
P d s n G ( r , r ) = 1
I 0 ( r ) z I δ z ( r ) - I 0 δ z ,
ϕ 0 ( x ) = - a a d x G ( x , x ) [ - 2 π λ I 0 z I 0 ( x ) ] + ( x + a 2 a ) ϕ 0 ( a ) - ( x - a 2 a ) ϕ 0 ( - a ) ,
G ( x , x ) = 1 2 x - x - 1 2 | x x a - a |
d 2 d x 2 G ( x , x ) = δ ( x - x ) ,
G ( x , a ) = G ( x , - a ) = 0.
z 2 = z 1 + ( 1 + m ) f z 1 + f ,
u z 2 ( r ) = - 1 m u 0 ( - r / m ) .
I z 2 ( r ) = 1 m 2 I 0 ( r / m )
ϕ z 2 ( r ) = ϕ 0 ( r / m ) + π ,
1 I z 2 I z 2 ( r ) z = 1 m 2 1 I 0 I 0 ( r / m ) z .
I z 2 + δ z ( r ) = I z 2 [ 1 + b 1 δ z + b 2 ( δ z ) 2 ] + Δ I z 2 ,
b 1 = - 2 ( λ 4 π ) 2 ϕ z 2
b 2 = 4 ( λ 4 π ) 2 [ ( ϕ x x ) 2 + ϕ x ϕ x x x + ϕ x x ϕ y y + ϕ y ϕ y y y + ( ϕ y y ) 2 ] z = z 2 .
( Δ I z 2 ) rms = I z 2 ( h c / λ A d Δ t I z 2 ) 1 / 2 = I z 2 / N z 2 ,
I z 2 + δ z - I z 2 δ z I z 2 = b 1 + b 2 δ z + Δ I z 2 δ z I z 2 .
δ z b 1 / b 2 ,
1 b 1 N z 2 δ z .
Δ ϕ ( x , 0 ) = ϕ ( x , 0 ) actual - ϕ ( x , 0 ) calculated .
[ 2 z 2 + 2 + ( 2 π λ ) 2 ] Ψ z ( r ) = 0
L + L - Ψ z ( r ) = 0 ,
L ± = z i 2 π λ [ 1 + ( λ 2 π ) 2 ] 1 / 2 .
L + u z ( r ) = 0
L - v z ( r ) = 0.
u z ( r ) = exp [ i k z ( 1 + 2 k 2 ) 1 / 2 ] u 0 ( r ) ,
U z ( ρ ) = exp [ i k z ( 1 - λ 2 ρ 2 ) 1 / 2 ] U 0 ( ρ ) ,
U z ( ρ ) = d r exp ( - i 2 π ρ · r ) u z ( r ) FT [ u z ( r ) ] ρ .
FT - 1 { exp [ i k z ( 1 - λ 2 ρ 2 ) 1 / 2 ] } = - 1 2 π z e i k R R
R ( z 2 + r 2 ) 1 / 2 .
u z ( r ) = - 1 2 π z [ u 0 ( r ) * * e i k R R ] ,
f ( r ) * * g ( r ) d r f ( r ) g ( r - r ) .
u z , F ( r ) = exp ( i k z ) exp [ ( i λ z 2 ) 4 π ] u 0 ( r )
U z , F ( ρ ) = exp ( i k z ) exp ( - i π λ z ρ 2 ) U 0 ( ρ ) ,
FT - 1 [ exp ( - i π λ z ρ 2 ) ] = e x p ( i π r 2 / λ z ) i λ z ,
u z , F ( r ) = exp ( i k z ) [ u 0 ( r ) * * exp ( i π r 2 / λ z ) i λ z ] ,
U z ( 0 ) = exp ( i k z ) - z exp [ i k ( z 2 + a 2 ) 1 / 2 ] ( z 2 + a 2 ) 1 / 2
U z , F ( 0 ) = exp ( i k z ) - exp [ i k ( z + 1 2 a 2 z ) ] .
( i z + 2 2 k + k ) u z , F ( r ) = 0
u z ( r ) = d ρ exp ( i 2 π ρ · r ) exp [ i k z ( 1 - λ 2 ρ 2 ) 1 / 2 ] U 0 ( ρ )
( i z + 2 k + k ) u z ( r ) = d ρ exp ( i 2 π ρ · r ) exp [ i k z ( 1 - λ 2 ρ 2 ) 1 / 2 ] × U 0 ( ρ ) [ - k ( 1 - λ 2 ρ 2 ) 1 / 2 - π λ ρ 2 + k ] .
J 1 ( 2 π a ρ ) ( 2 π a ρ ) ,
U 0 ( 1 3 λ ) ( λ a ) 3 / 2 U 0 ( 0 ) ,
( i z + 2 2 k + k - 1 8 4 k 3 ) u z ( r ) 0.
η 42 1 4 k 2 4 u z ( r ) 2 u z ( r )
η 42 0 [ λ 2 a 2 × ( number of waves of aberration ) 2 ] ,
η 42 0 [ 1 ( f # ) 2 ] ,
η 42 0 [ r 2 / λ 2 ( f # ) 4 ] .
exp ( i λ z 4 π 2 ) u 0 ( r ) = ? N = 0 1 N ! ( i λ z 2 4 π ) N u 0 ( r ) .
exp ( i λ z 4 π 2 ) u 0 ( r ) d ρ exp ( i 2 π ρ · r ) × exp ( - i π λ z ρ 2 ) U 0 ( ρ )
= ? N = 0 1 N ! d ρ exp ( i 2 π ρ · r ) × ( - i π λ z ρ 2 ) N U 0 ( ρ ) .
u 0 ( r ) = p 0 ( r ) f 0 ( r ) ,
p 0 ( r ) = [ 1 , r a 0 , otherwise
f 0 ( r ) = exp [ i k w ( r ) ] exp ( - r 2 ) ,
exp ( i λ z 2 4 π ) u 0 ( r ) = d ρ exp ( i 2 π ρ · r ) exp ( - i π λ z ρ 2 ) × d ρ P 0 ( ρ ) F 0 ( ρ - ρ ) = d ρ P 0 ( ρ ) d ρ exp ( i 2 π ρ · r ) × exp ( - i π λ z ρ 2 ) F 0 ( ρ - ρ ) = d ρ P 0 ( ρ ) exp ( i 2 π ρ · r ) × exp ( - i π λ z ρ 2 ) × { d ρ exp [ i 2 π ρ · ( r - λ z ρ ) ] × exp ( - i π λ z ρ 2 ) F 0 ( ρ ) } ,
exp ( i λ z 4 π 2 ) f 0 ( r - λ z ρ ) = exp ( i λ z 4 π 2 ) M = 0 N = 0 × f 0 ( m , n ) ( r ) ( - λ z ξ ) m ( - λ z η ) n m ! n ! ,
u z ( r ) = exp ( i k z ) m - 0 n = 0 1 m ! n ! [ exp ( i λ z 4 π 2 ) f 0 m , n ( r ) ] × { ( - λ z 2 π i x ) m ( - λ z 2 π i y ) n d ρ P 0 ( ρ ) × exp ( i 2 π ρ · r ) exp ( - i π λ z ρ 2 ) } .
u z ( r ) p z , F ( r ) [ exp ( i λ z 4 π 2 ) f 0 ( r ) ] ,
[ 1 - λ 2 ( ρ + ρ ) 2 ] 1 / 2 ( 1 - λ 2 ρ 2 ) 1 / 2 - λ 2 ρ · ρ + ( 1 - λ 2 ρ 2 ) 1 / 2 - 1
u z ( r ) m - 0 n = 0 1 m ! n ! { exp [ i k z ( 1 + 2 / k 2 ) 1 / 2 ] f 0 ( m , n ) ( r ) } × { ( - λ z 2 π i x ) m ( - λ z 2 π i y ) n d ρ P 0 ( ρ ) exp ( i 2 π ρ · r ) × exp [ i k z ( 1 - λ 2 ρ 2 ) 1 / 2 ] } exp ( i k z ) .
u z ( r ) p z ( r ) [ exp ( i λ z 4 π 2 ) f 0 ( r ) ] ,