Abstract

Multiple images with spots similar to those produced by a Gaussian Schell-model (GSM) source are realized in the observation plane of the Fresnel diffraction region at a distance 4NZT (N, a positive integer; ZT, Talbot distance) from the GSM source. In this formation the grating with apertures of a Gaussian amplitude transmittance is placed midway between the GSM source and the observation plane. We base our analysis on coherence theory, using the propagation of the ambiguity function from which the average intensity can be derived. It is found that the intensity distribution of the spots in the multiple images becomes wider with decreases in the source size, the spatial-coherence length of the source, and each aperture size of the grating used.

© 1994 Optical Society of America

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References

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  1. G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
    [CrossRef]
  2. D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” Proc. Inst. Electr. Eng. Part J 139, 14–23 (1992).
  3. P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
    [CrossRef]
  4. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  5. P. Szwaykowski, V. Arrizon, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  6. M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continuous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
    [CrossRef] [PubMed]
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
    [CrossRef]
  8. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  9. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  10. M. Bom, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.
  11. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  12. K. Dutta, J. W. Goodman, “Reconstructions of images of partially coherent objects from samples of mutual intensity,” J. Opt. Soc. Am. 67, 796–803 (1977).
    [CrossRef]
  13. J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  14. K.-H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [CrossRef]
  15. H. Yoshimura, N. Takai, T. Asakura, “Equiambiguity function ellipse of Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1136–1140 (1994).
    [CrossRef]
  16. H. Yoshimura, N. Takai, T. Asakura, “Effects of amplitude and phase fluctuations of the Ronchi grating on Talbot images under illumination of the partially coherent light,” J. Mod. Opt. 40, 825–839 (1993).
    [CrossRef]
  17. K. Patorski, “The self-imaging phenomenon and its application,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 1–108.
    [CrossRef]
  18. S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
    [CrossRef]

1994 (1)

1993 (3)

1992 (1)

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” Proc. Inst. Electr. Eng. Part J 139, 14–23 (1992).

1991 (1)

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1987 (1)

G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

1984 (1)

K.-H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

1983 (1)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

1982 (1)

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

1979 (1)

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

1978 (1)

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

1977 (2)

Arrizon, V.

Asakura, T.

H. Yoshimura, N. Takai, T. Asakura, “Equiambiguity function ellipse of Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1136–1140 (1994).
[CrossRef]

H. Yoshimura, N. Takai, T. Asakura, “Effects of amplitude and phase fluctuations of the Ronchi grating on Talbot images under illumination of the partially coherent light,” J. Mod. Opt. 40, 825–839 (1993).
[CrossRef]

Bom, M.

M. Bom, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.

Botez, D.

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” Proc. Inst. Electr. Eng. Part J 139, 14–23 (1992).

Brennan, T. M.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Carter, W. H.

Dutta, K.

Ehbets, P.

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

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

Gale, M. T.

Goodman, J. W.

K. Dutta, J. W. Goodman, “Reconstructions of images of partially coherent objects from samples of mutual intensity,” J. Opt. Soc. Am. 67, 796–803 (1977).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
[CrossRef]

Gourley, P. L.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Guigay, J.-P.

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Hadley, G. R.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Hammons, B. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Herzig, H. P.

Hohimer, J. P.

G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Ojeda-Castaneda, J.

K.-H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Owyoung, A.

G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

Patorski, K.

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its application,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 1–108.
[CrossRef]

Prongué, D.

Rossi, M.

Schütz, H.

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

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

Szapiel, S.

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Szwaykowski, P.

Takai, N.

H. Yoshimura, N. Takai, T. Asakura, “Equiambiguity function ellipse of Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1136–1140 (1994).
[CrossRef]

H. Yoshimura, N. Takai, T. Asakura, “Effects of amplitude and phase fluctuations of the Ronchi grating on Talbot images under illumination of the partially coherent light,” J. Mod. Opt. 40, 825–839 (1993).
[CrossRef]

Vawter, G. A.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Warren, M. E.

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

Wolf, E.

Yoshimura, H.

H. Yoshimura, N. Takai, T. Asakura, “Equiambiguity function ellipse of Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1136–1140 (1994).
[CrossRef]

H. Yoshimura, N. Takai, T. Asakura, “Effects of amplitude and phase fluctuations of the Ronchi grating on Talbot images under illumination of the partially coherent light,” J. Mod. Opt. 40, 825–839 (1993).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. L. Gourley, M. E. Warren, G. R. Hadley, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Coherent beams from high efficiency two-dimensional surface-emitting semiconductor laser arrays,” Appl. Phys. Lett. 58, 890–892 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. R. Hadley, J. P. Hohimer, A. Owyoung, “Free-running modes for gain-guided diode laser arrays,” IEEE J. Quantum Electron. QE-23, 765–774 (1987).
[CrossRef]

J. Mod. Opt. (2)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

H. Yoshimura, N. Takai, T. Asakura, “Effects of amplitude and phase fluctuations of the Ronchi grating on Talbot images under illumination of the partially coherent light,” J. Mod. Opt. 40, 825–839 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (3)

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

K.-H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

Opt. Commun. (2)

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

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” Proc. Inst. Electr. Eng. Part J 139, 14–23 (1992).

Other (3)

M. Bom, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its application,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 1–108.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Optical coordinates at the GSM source, the grating, and the observation planes.

Fig. 2
Fig. 2

Solid curves represent the intensity distributions 〈iz(x)〉 of the multiple images for the case of σ = 0.5, f = 0.3, and N = 1. (a), (b), and (c) correspond to the results for a/d = 0.1, a/d = 0.3, and a/d = 0.5, respectively. Dashed curves represent the normalized intensity of the GSM source with a period of 2d.

Fig. 3
Fig. 3

Dependence of Ns (circles) on a/d at an interval of 0.01 in 0 < a/d ≤ 0.5, for the case of σ=0.5, f = 0.3, and N = 1.

Fig. 4
Fig. 4

Solid curves represent the intensity distributions 〈iz(x)〉 of the multiple images for the case of a/d = 0.3, f = 0.3, and N = 1. (a), (b), (c), and (d) correspond to the results for σ = 0.2, σ = 0.5, σ=1.0, and σ = ∞, respectively. Dashed curves represent the normalized intensity of the GSM source with a period of 2d.

Fig. 5
Fig. 5

Dependence of Ns (circles) on σ in 0.01 ≤ σ ≤ 100, for the case of a/d = 0.3, f = 0.3, and N = 1.

Fig. 6
Fig. 6

Solid curves, the intensity distribution 〈iz(x)〉 of the multiple images for the case of a/d = 0.3, σ = 0.5, and N = 1. (a), (b), and (c), Results for f = 0.1, f = 0.3, and f = 0.5, respectively. Dashed curves, the normalized intensity of the GSM source with a period of 2d.

Fig. 7
Fig. 7

Dependence of Ns (circles) on f at an interval of 0.01 in 0 < f ≤0.5 for the case of a/d = 0.3, σ = 0.5, and N = 1.

Equations (67)

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A z = z 2 ( Δ ν , Δ x ) = A z = z 1 [ Δ ν , Δ x λ ( z 2 z 1 ) Δ ν ] ,
I z ( x ) = A z ( Δ ν , Δ x = 0 ) exp ( 2 π i Δ ν x ) d Δ ν .
W 0 ( x 1 , x 2 ) = [ I 0 ( x 1 ) I 0 ( x 2 ) ] 1 / 2 γ 0 ( x 2 x 1 ) ,
I 0 ( x ) = I s exp ( 2 x 2 / a 2 ) ,
γ 0 ( x ) = exp ( x 2 / 2 x s 2 ) .
σ = x s / a .
W 0 ( x , Δ x ) = I s exp ( 2 x 2 a 2 ) exp [ 1 2 a 2 ( 1 + 1 σ 2 ) Δ x 2 ] .
A ( Δ ν , Δ x ) = W ( x , Δ x ) exp ( 2 π i x Δ ν ) d x ,
A 0 ( Δ ν , Δ x ) = A 0 ( 0 , 0 ) exp [ ( π a Δ ν ) 2 2 ] × exp [ 1 2 a 2 ( 1 + 1 σ 2 ) Δ x 2 ] ,
A 0 ( 0 , 0 ) = π / 2 I s a .
A z 1 ( Δ ν , Δ x ) = A 0 ( Δ ν , Δ x λ z 1 Δ ν ) = A 0 ( 0 , 0 ) exp [ ( π a Δ ν ) 2 2 ] × exp [ 1 2 a 2 ( 1 + 1 σ 2 ) ( Δ x λ z 1 Δ ν ) 2 ] .
t g ( x ) = n a n exp [ i 2 π n ( x / d ) ] ,
W g ( x , Δ x ) = t g ( x + Δ x / 2 ) t g * ( x Δ x / 2 ) = n a n exp [ i 2 π n ( x + Δ x / 2 ) d ] × m a m * exp [ i 2 π m ( x Δ x / 2 ) d ] = n m a n a m * exp [ i π ( n + m ) Δ x / d ] × exp [ i 2 π ( n m ) x / d ] .
A g ( Δ ν , Δ x ) = n m a n a m * exp [ i π ( n + m ) Δ x d ] × δ ( Δ ν n m d ) .
A z 1 + ( Δ ν , Δ x ) = A z 1 ( Δ ν , Δ x ) A g ( Δ ν , Δ x ) A z 1 ( Δ ν , Δ x ) A g ( Δ ν Δ ν , Δ x ) d Δ ν = A 0 ( 0 , 0 ) n m a n a m * exp [ i π ( n + m ) Δ x / d ] × exp [ ( π a ) 2 2 ( Δ ν n m d ) 2 ] × exp { 1 2 a 2 ( 1 + 1 σ 2 ) × [ Δ x λ z 1 ( Δ ν n m d ) ] 2 } ,
A z ( Δ ν , Δ x ) = A z 1 + ( Δ ν , Δ x λ z 2 Δ ν ) = A 0 ( 0 , 0 ) n m a n a m * × exp [ i π ( n + m ) Δ x λ z 2 Δ ν d ] × exp [ ( π a ) 2 2 ( Δ ν n m d ) 2 ] × exp { 1 2 a 2 ( 1 + 1 σ 2 ) × ( Δ x λ z Δ ν + λ z 1 n m d ) 2 ] .
I z ( x ) = A z ( Δ ν , Δ x = 0 ) exp ( 2 π i Δ ν x ) d Δ ν = I 0 ( 0 ) n m a n a m * × exp { 2 w z 2 [ z 2 z T ( n m ) d ] 2 ( 1 + 1 σ 2 ) } × exp { 2 w z 2 [ x z 2 z T ( n + m ) d ] 2 } × exp [ i 2 π x n m d ( R z z 2 R z ) ] × exp [ i 2 π ( n 2 m 2 ) z 2 z T ( R z z 2 R z ) ] ,
w z = a [ 1 + ( λ z π a 2 ) 2 ( 1 + 1 σ 2 ) ] 1 / 2 ,
R z = z [ 1 + 1 ( λ z / π a 2 ) 2 ( 1 + 1 / σ 2 ) ] .
z T = 2 d 2 / λ .
I z ( 0 ) = I s a / w z .
z 2 z T ( R z z 2 R z ) = N ( N = 1 , 2 , 3 , ) ,
z 2 = M z T ( M = N + 1 , N + 2 , , 2 N ) ,
z 1 = 2 N z T [ 1 ( π 4 N ) 2 ( a d ) 4 σ 2 1 + σ 2 ] 1 / 2 ,
z 2 = 2 N z T .
0 < a / d < 4 / π .
I z ( x ) = I z ( 0 ) n m a n a m * × exp { 2 w z 2 [ 2 ( n m ) N d ] 2 ( 1 + 1 σ 2 ) } × exp { 2 w z 2 [ x 2 ( n + m ) N d ] 2 } × exp ( i 2 π x n m 2 d ) .
w z = 8 N d 2 π a ( 1 + 1 σ 2 ) 1 / 2 ,
t g ( x ) = exp ( x 2 a g 2 ) n δ ( x n d ) ,
f = 2 a g / d
a n = 1 d d / 2 d / 2 exp ( x 2 a g 2 ) exp ( i 2 π n x d ) d x 1 d exp ( x 2 a g 2 ) exp ( i 2 π n x d ) d x = π f 2 exp [ ( n π f 2 ) 2 ] .
I z ( x ) = exp ( 2 x 2 / w env 2 ) [ exp ( 2 x 2 / w spot 2 ) n δ ( x + 2 n d ) ] ,
w env = 8 N d π [ 1 + 1 / σ 2 ( a / d ) 2 + 1 f 2 ] 1 / 2 ,
w spot = a [ 1 + ( 2 a g / a ) 2 ] 1 / 2 .
I z ( x ) = A 0 ( 0 , 0 ) n m a n a m * × exp [ ( π a ) 2 2 ( Δ ν n m d ) 2 ] × exp { 1 2 ( λ z a ) 2 ( 1 + 1 σ 2 ) [ Δ ν z 1 ( n m ) z d ] 2 } × exp { i 2 π Δ ν [ x λ z 2 ( n + m ) 2 d ] } d Δ ν = A 0 ( 0 , 0 ) n m a n a m * exp [ A ( Δ ν B ) 2 ] × exp [ C ( Δ ν D ) 2 ] exp [ i 2 π Δ ν ( x E ) ] d Δ ν ,
A = ( π a ) 2 / 2 ,
B = ( n m ) / d ,
C = ½ ( λ z / a ) 2 ( 1 + 1 / σ 2 ) ,
D = [ z 1 ( n m ) ] / z d ,
E = [ λ z 2 ( n + m ) ] / 2 d .
( π A + C ) 1 / 2 exp [ A C ( B D ) 2 A + C ] exp { [ π ( x E ) ] 2 A + C } × exp [ i 2 π ( x E ) A B + C D A + C ] .
2 π 1 w z exp { 2 w z 2 [ z 2 z T ( n m ) d ] 2 ( 1 + 1 σ 2 ) } × exp { 2 w z 2 [ x z 2 z T ( n + m ) d ] 2 } × exp { i 2 π x n m d ( R z z 2 R z ) } × exp [ i 2 π ( n 2 m 2 ) z 2 z T ( R z z 2 R z ) ] ,
R z M z T R z = N M .
M N + 1 ,
z 0 = π a 2 / λ ,
C = ( 1 + 1 σ 2 ) 1 / 2 ,
R z = z + 1 z ( z 0 C ) 2 .
R z = z 1 + M z T + 1 z 1 + M z T ( z 0 C ) 2 .
( M N ) C 2 z 1 2 + M ( M 2 N ) C 2 z T z 1 + [ ( M N ) z 0 2 C 2 M 2 N z T 2 ] = 0 ,
z 1 = M ( M 2 N ) 2 ( M N ) z T ± { ( M 2 C z T ) 2 [ 2 ( M N ) z 0 ] 2 } 1 / 2 2 ( M N ) C .
M 2 N 0 ,
M 2 M N C z T 2 z 0 1 .
N + 1 M 2 N .
M = 2 N ,
z 1 = 2 N z T [ 1 ( z 0 2 N C z T ) 2 ] 1 / 2 .
I z ( x ) = exp ( 2 x 2 w z 2 ) m exp ( α m 2 ) × exp ( β m ) exp ( i 2 π x m 2 d ) × n exp ( α n 2 ) exp [ ( β + γ m ) n ] exp ( i 2 π x n 2 d ) ,
α = ( π f 2 ) 2 + γ ( σ 2 + 1 2 ) ,
β = 8 N d w z 2 x ,
γ = ( 4 N d w z σ ) 2 .
t ( x ) = A ( x ) n δ ( x n d )
= n a n exp ( i 2 π n x d ) ,
a n = 1 d d / 2 d / 2 A ( x ) exp ( i 2 π n x d ) d x 1 d A ( x ) exp ( i 2 π n x d ) d x ,
n = exp ( β 2 4 α ) exp ( β γ 2 α m ) exp ( γ 2 4 α m 2 ) × ( n { exp [ x 2 α ( 2 d / π ) 2 ] exp ( i 2 π x β + γ m 4 α d ) } δ ( x + 2 n d ) ) ,
i z ( x ) = exp ( 2 x 2 w z 2 ) exp ( β 2 4 α ) exp ( i 2 π x β 4 α d ) × n exp [ ( x + 2 n d ) 2 α ( 2 d / π ) 2 ] exp ( i 2 π β 2 α n ) × m exp [ ( α γ 2 4 α ) m 2 ] exp [ β ( 1 + γ 2 α ) m ] × exp { i 2 π [ ( 1 γ 2 α ) x γ d α n ] m 2 d } .
n = exp [ β 2 ( 2 α + γ ) 4 α ( 2 α γ ) ] × ( m { exp [ π 2 α d 2 ( 2 α + γ ) ( 2 α γ ) × ( 2 α γ 2 α x γ d α n ) 2 ] × exp [ i 2 π ( 2 α γ 2 α x γ d α n ) β 2 d ( 2 α γ ) ] } δ [ ( 2 α γ 2 α x γ d α n ) 2 m d ] ) ,
i z ( x ) = exp ( β 2 2 α γ ) exp ( 2 x 2 w z 2 ) × n m exp { 1 ( d / π ) 2 ( 2 α + γ ) [ x + ( n m ) d ] 2 } × exp [ π 2 2 α γ ( n + m ) 2 ] exp [ i 2 π β 2 α γ ( n + m ) ] .
i z ( x ) = exp ( 2 x 2 w env 2 ) × n m exp { 2 w spot 2 [ x + ( n m ) d ] 2 } × exp [ 2 f 2 + ( a / d 2 ) σ 2 / ( 1 + σ 2 ) ( n + m ) 2 ] × exp { i 2 π ( n + m ) x 4 N d 1 [ f / ( a / d ] 2 ( 1 + 1 / σ 2 ) + 1 } ,

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