Abstract

The use of thin phase diffusers in coherent imaging systems is analyzed. A condition is derived that must be fulfilled to ensure that no part of incident light is specularly transmitted. This condition is expressed in terms of first-order statistics of the phase of light emerging from a diffuser. It is shown that random diffusers whose amplitude-transmittance argument (phase) is uniformly distributed in the (−π, π) interval do not pass the specular light while their rms phase corresponds to only 0.29 of a wavelength. Such diffusers will be called uniform ones. A method for forming a uniform diffuser is proposed. It is based on a recording of normal speckle patterns in phase-photosensitive materials with consideration of the exposure characteristics. Autocorrelation functions and power spectra of diffuser transmittance are evaluated for two types of exposure characteristics. For both these cases the image contrast of the uniformly illuminated diffuser is calculated.

© 1984 Optical Society of America

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References

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  1. M. J. Lahart, A. S. Marathay, “Image speckle patterns of weak diffusers,” J. Opt. Soc. Am. 65, 769–778 (1975).
    [CrossRef]
  2. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968).
    [CrossRef] [PubMed]
  3. J. S. Chandler, J. J. DePalma, “High brightness projection screens with high ambient light rejection,” J. Soc. Motion Pict. Telev. Eng. 77, 1012–1020 (1968).
  4. J. O. Porteus, “Relation between the height distribution of a rough surface and the reflection at normal incidence,” J. Opt. Soc. Am. 53, 1394–1402 (1963).
    [CrossRef]
  5. A. Iwamoto, “Artificial diffuser for Fourier transform hologram recording,” Appl. Opt. 19, 215–221 (1980).
    [CrossRef] [PubMed]
  6. C. N. Kurtz, “Transmittance characteristics of surface diffusers and design of nearly band-limited binary diffusers,” J. Opt. Soc. Am. 62, 982–989 (1972).
    [CrossRef]
  7. B. J. Thompson, “Optical data processing,” in Optical Transforms, H. S. Lipson, ed. (Academic, London, 1972), pp. 267–298.
  8. C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. 63, 1080–1092 (1973).
    [CrossRef]
  9. Y. Nakayama, M. Kato, “Linear recording of Fourier transform holograms using a pseudorandom diffuser,” Appl. Opt. 21, 1410–1418 (1982).
    [CrossRef] [PubMed]
  10. The authors of Ref. 8 do not state explicitly that the rms phase is large, but this results from the discussion of their approach to validity. That is, the approach presented concerns, in substance, the continuous part of the spectrum, whereas the zero-order term is eliminated because of the rms phase increase.
  11. See the comment on this definition in Ref. 16.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 49.
  13. R. C. Waag, K. T. Knox, “Power-spectrum analysis of exponential diffusers,” J. Opt. Soc. Am. 62, 877–881 (1972).
    [CrossRef]
  14. See the approximation made in Eq. (A7) to Ref. 5, where 1-D formulas analogous to our Eqs. (5) and (6) are derived.
  15. B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–256 (1983).
    [CrossRef]
  16. M. G. Miller, A. M. Schneiderman, P. F. Kellen, “Second-order statistics of laser speckle patterns,” J. Opt. Soc. Am. 65, 779–785 (1975).
    [CrossRef]
  17. R. A. Bartolini, “Photoresists,” in Holographic Recording Materials, H. M. Smith, ed. (Springer-Verlag, New York, 1977).
    [CrossRef]
  18. F. Oberhettinger, Fourier Transforms of Distributions and Their Inverses (Academic, New York, 1973), Table I.
  19. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. 14, E. Wolf, ed. (North-Holland, New York, 1975).
  20. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  21. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 162.
  22. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
    [CrossRef]
  23. A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, p. 189.
  24. This is an assumption indeed since this does not follow from the fact that the first-order density is Gaussian. Also, the central-limit theorem may not be applied here since we assume that successively recorded patterns are not correlated. Thus they may be statistically dependent.
  25. Ref. 23, p. 195 .
  26. B. R. Levin, Teoretitcheskie Osnovy Statistitcheskoj Radiotekhniki (Sovetskoe Radio, Moscow, 1974), Vol. 1, p. 183.
  27. Ref. 21, p. 201 .
  28. Ref. 21, p. 315 .
  29. Ref. 21, p. 178 .
  30. J. W. Goodman, “Role of coherence concept in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).
  31. Ref. 21, p. 337 .

1983

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–256 (1983).
[CrossRef]

1982

1980

1979

J. W. Goodman, “Role of coherence concept in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

1978

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

1975

1973

1972

1968

H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968).
[CrossRef] [PubMed]

J. S. Chandler, J. J. DePalma, “High brightness projection screens with high ambient light rejection,” J. Soc. Motion Pict. Telev. Eng. 77, 1012–1020 (1968).

1963

Bartolini, R. A.

R. A. Bartolini, “Photoresists,” in Holographic Recording Materials, H. M. Smith, ed. (Springer-Verlag, New York, 1977).
[CrossRef]

Chandler, J. S.

J. S. Chandler, J. J. DePalma, “High brightness projection screens with high ambient light rejection,” J. Soc. Motion Pict. Telev. Eng. 77, 1012–1020 (1968).

Dainty, J. C.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–256 (1983).
[CrossRef]

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. 14, E. Wolf, ed. (North-Holland, New York, 1975).

DePalma, J. J.

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am. 63, 1080–1092 (1973).
[CrossRef]

J. S. Chandler, J. J. DePalma, “High brightness projection screens with high ambient light rejection,” J. Soc. Motion Pict. Telev. Eng. 77, 1012–1020 (1968).

Erdélyi, A.

A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, p. 189.

Gerritsen, H. J.

Goodman, J. W.

J. W. Goodman, “Role of coherence concept in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 49.

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Hannan, W. J.

Hoadley, H. O.

Iwamoto, A.

Kato, M.

Kellen, P. F.

Knox, K. T.

Kurtz, C. N.

Lahart, M. J.

Levin, B. R.

B. R. Levin, Teoretitcheskie Osnovy Statistitcheskoj Radiotekhniki (Sovetskoe Radio, Moscow, 1974), Vol. 1, p. 183.

Levine, B. M.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–256 (1983).
[CrossRef]

Marathay, A. S.

Miller, M. G.

Nakayama, Y.

Oberhettinger, F.

F. Oberhettinger, Fourier Transforms of Distributions and Their Inverses (Academic, New York, 1973), Table I.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 162.

Porteus, J. O.

Ramberg, E. G.

Schneiderman, A. M.

Thompson, B. J.

B. J. Thompson, “Optical data processing,” in Optical Transforms, H. S. Lipson, ed. (Academic, London, 1972), pp. 267–298.

Waag, R. C.

Appl. Opt.

J. Opt. Soc. Am.

J. Soc. Motion Pict. Telev. Eng.

J. S. Chandler, J. J. DePalma, “High brightness projection screens with high ambient light rejection,” J. Soc. Motion Pict. Telev. Eng. 77, 1012–1020 (1968).

Opt. Acta

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Opt. Commun.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–256 (1983).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

J. W. Goodman, “Role of coherence concept in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

Other

Ref. 21, p. 337 .

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 162.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
[CrossRef]

A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, p. 189.

This is an assumption indeed since this does not follow from the fact that the first-order density is Gaussian. Also, the central-limit theorem may not be applied here since we assume that successively recorded patterns are not correlated. Thus they may be statistically dependent.

Ref. 23, p. 195 .

B. R. Levin, Teoretitcheskie Osnovy Statistitcheskoj Radiotekhniki (Sovetskoe Radio, Moscow, 1974), Vol. 1, p. 183.

Ref. 21, p. 201 .

Ref. 21, p. 315 .

Ref. 21, p. 178 .

The authors of Ref. 8 do not state explicitly that the rms phase is large, but this results from the discussion of their approach to validity. That is, the approach presented concerns, in substance, the continuous part of the spectrum, whereas the zero-order term is eliminated because of the rms phase increase.

See the comment on this definition in Ref. 16.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 49.

R. A. Bartolini, “Photoresists,” in Holographic Recording Materials, H. M. Smith, ed. (Springer-Verlag, New York, 1977).
[CrossRef]

F. Oberhettinger, Fourier Transforms of Distributions and Their Inverses (Academic, New York, 1973), Table I.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, Vol. 14, E. Wolf, ed. (North-Holland, New York, 1975).

See the approximation made in Eq. (A7) to Ref. 5, where 1-D formulas analogous to our Eqs. (5) and (6) are derived.

B. J. Thompson, “Optical data processing,” in Optical Transforms, H. S. Lipson, ed. (Academic, London, 1972), pp. 267–298.

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Figures (4)

Fig. 1
Fig. 1

Imaging system with diffraction noise reduction or holographic imaging system with a smoothed-out object beam in the spatial-frequency plane. Recording and reconstructing systems are presented jointly. Elements described in parentheses appear in the holographic system only. Diffraction noise reduction also takes place in the holographic system as an additional effect.

Fig. 2
Fig. 2

System for forming uniform diffusers. Speckle pattern that appears in the (x′, y′) plane is recorded in the phase-photosensitive material. In order to ensure stationarity of the uniform diffuser transmittance td(x, y), the central part of the speckle pattern, where the average irradiance distribution 〈Ie(x,′ y′)〉 is nearly constant, should be utilized only. Circularly symmetric illumination of the strong diffuser (dotted area) is required to ensure isotropy of the random field Ie(x,′ y′) and consequently the isotropy of td(x, y).

Fig. 3
Fig. 3

Characteristics of phase-recording materials suitable for forming uniform diffusers.

Fig. 4
Fig. 4

Average contrast of the noise pattern in the image plane.

Equations (67)

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S t ( ω x , ω y ) circ ( ω x 2 + ω y 2 ) 1 / 2 ρ d = circ ρ ρ d ,
φ ( x , y ) = 2 π λ [ n d ( x , y ) - 1 ] h ( x , y ) ,
R t ( Δ x , Δ y ) t d ( x 1 , y 1 ) t d * ( x 2 , y 2 ) ,
S t ( ω x , ω y ) | - - t d ( x , y ) exp [ 2 π i ( x ω x + y ω y ) ] d x d y | 2 .
I ( u , v ; ρ s ) 1 - 1 π 2 ρ s 2 { [ φ ( u , v ) u ] 2 + [ φ ( u , v ) v ] 2 } + 1 4 π 4 ρ s 4 ( { [ φ ( u , v ) u ] 2 + [ φ ( u , v ) v ] 2 } 2 + [ 2 φ ( u , v ) u 2 + 2 φ ( u , v ) v 2 ] 2 ) ,
I ( u , v ; b ) 1 - λ b 2 π [ 2 φ ( u , v ) u 2 + 2 φ ( u , v ) v 2 ] + ( λ b 4 π ) 2 ( [ 2 φ ( u , v ) u 2 + 2 φ ( u , v ) v 2 ] 2 + { [ φ ( u , v ) u ] 2 + [ φ ( u , v ) v ] 2 } 2 ) .
I f ( ξ , η ) { T s ( ω x , ω y ) 2 [ α δ ( ω x , ω y ) + β S t c ( ω x , ω y ) ] } ω x = ξ / λ f w y = η / λ f = [ α T s ( ω x , ω y ) 2 + β T s ( ω x , ω y ) 2 S t c ( ω x , ω y ) ] ω x = ξ / λ f , ω y = η / λ f
I ( u , v ) t s 2 ( u , v ) + 1 - 2 t s ( u , v ) cos ϕ ( u , v ) .
I ( u , v ) 1 - 1 π 2 ρ s 2 { [ φ ( u , v ) u ] 2 + [ φ ( u , v ) v ] 2 } 1 - D 2 ( u , v ) π 2 ρ s 2 ,
R t ( ) = lim τ - - exp [ i ( φ 1 - φ 2 ) ] × W ( φ 1 , φ 2 ; τ ) d φ 1 d φ 2 ,
R t ( ) = - - exp [ i ( φ 1 - φ 2 ) ] × lim τ W ( φ 1 , φ 2 ; τ ) d φ 1 d φ 2 = - exp ( i φ 1 ) W ( φ 1 ) d φ 1 × - exp ( - i φ 2 ) W ( φ 2 ) d φ 2 = Φ ( 1 ) 2 ,
Φ ( 1 ) 2 = 0.
W ( φ ) = { ( 2 π ) - 1 φ < π 0 otherwise .
φ e ( I e ) = π [ 1 - 2 exp ( - I e I e ) ] ,
φ g ( E g ) = π erf ( E g - E g 2 σ E g ) ,
erf s = 2 π 0 s exp ( - t 2 ) d t
R t e ( τ ) = 0 0 exp { i [ φ e ( I e 1 ) - φ e ( I e 2 ) ] } × W ( I e 1 , I e 2 ; τ ) d I e 1 d I e 2 .
W ( I e 1 , I e 2 ; τ ) = exp [ - ( I e 1 + I e 2 ) I e ( 1 - μ e ( τ ) 2 ) ] I e 2 ( 1 - μ e ( τ ) 2 ) × I 0 ( 2 I e 1 I e 2 μ e ( τ ) I e ( 1 - μ e ( τ ) 2 ) ) ,
μ e ( τ ) = 0 P ( l ) 2 J 0 ( 2 π τ l λ z ) l d l 0 P ( l ) 2 l d l ,
1 1 - s exp ( - t 1 + t 2 1 - s ) I 0 [ 2 ( t 1 t 2 s ) 1 / 2 1 - s ] = n = 0 L n ( t 1 ) L n ( t 2 ) s n ,
L n ( t ) = exp ( t ) n ! d n d t n [ t n exp ( - t ) ] ,
R t e ( τ ) = n = 0 c n 2 μ e ( τ ) 2 n ,
c n = 0 1 L n ( - ln t ) exp ( - 2 π i t ) d t .
R t e ( τ ) = n = 0 c n 2 r I e n ( τ ) .
R t g ( τ ) = m = 0 d m 2 m ! ( r E g ( τ ) 2 ) m ,
d m = 1 π - H m ( t ) exp ( - t 2 + i π erf t ) d t ,
H m ( τ ) = ( - 1 ) m exp ( t 2 ) d m d t m [ exp ( - t 2 ) ] .
W ( E g 1 , E g 2 ; τ ) = [ 1 - r E g 2 ( τ ) ] - 1 / 2 2 π σ E g 2 exp [ - ( E g 1 - E g ) 2 - 2 r E g ( τ ) ( E g 1 - E g ) ( E g 2 - E g ) + ( E g 1 - E g ) 2 2 σ E g 2 [ 1 - r E g 2 ( τ ) ] ] .
1 1 - s 2 exp [ 2 t 1 t 2 s - ( t 1 2 + t 2 2 ) s 2 2 ( 1 - s 2 ) ] = k = 0 H k ( t 1 2 ) H k ( t 2 2 ) k ! ( s 2 ) k .
d 0 = 1 π - exp [ - t 2 + i π erf t ] d t = 1 π 0 d [ sin ( π erf t ) ] = 0 .
φ ( x , y ) x = D ( x , y ) cos γ ( x , y ) ,
ζ e ( φ e 1 , φ e 2 ; Δ x ) φ e 2 + φ e 1 2 = φ e ( x + Δ x , y ) + φ e ( x , y ) 2 ,
ψ e ( φ e 1 , φ e 2 ; Δ x ) φ e 2 - φ e 1 Δ x = φ e ( x + Δ x , y ) - φ e ( x , y ) Δ x .
lim Δ x 0 ζ e ( φ e 1 , φ e 2 ; Δ x ) = φ e ,
lim Δ x 0 ψ e ( φ e 1 , φ e 2 ; Δ x ) = φ e / x .
lim Δ x 0 W ( ζ e , ψ e ; Δ x ) = W ( φ e , φ e x ) .
φ e i = π [ 1 - 2 exp ( - I e i I e ) ] ,             i = 1 , 2.
I e i = - I e ln ( 1 2 - φ e i 2 π ) ,
( φ e 1 , φ e 2 ) ( I e 1 , I e 2 ) = 4 π 2 I e 2 ( 1 2 - φ e 1 2 π ) ( 1 2 - φ e 2 2 π ) .
W ( φ e 1 , φ e 2 ; τ ) = [ ( 1 2 - φ e 1 2 π ) ( 1 2 - φ e 2 2 π ) ] μ e ( τ ) 2 1 - μ e ( τ ) 2 4 π 2 ( 1 - μ e ( τ ) 2 ) × I 0 ( 2 [ ln ( 1 2 - φ e 1 2 π ) ln ( 1 2 - φ e 2 2 π ) ] 1 / 2 μ e ( τ ) 1 - μ e ( τ ) 2 ) ,             φ e i < π .
W ( ζ e , ψ e ; Δ x ) = Δ x [ ( 1 2 - ζ e 2 π ) 2 - ( ψ e Δ x 4 π ) 2 ] μ e ( Δ x ) 2 1 - μ e ( Δ x ) 2 4 π 2 [ 1 - μ e ( Δ x ) 2 ] × I 0 ( 2 [ ln ( 1 2 - ζ e 2 π + ψ e Δ x 4 π ) ln ( 1 2 - ζ e 2 π - ψ e Δ x 4 π ) ] 1 / 2 μ e ( Δ x ) 1 - μ e ( Δ x ) 2 ) ,             ζ e < π ,             ψ e Δ x < 2 π .
W ( φ e , φ e x ) = exp ( - ( φ e x ) 2 2 { ( π - φ e ) [ r ¨ I e ( 0 ) ln ( 1 2 - φ e 2 π ) ] 1 / 2 } 2 ) 2 π 2 π ( π - φ e ) [ r ¨ I e ( 0 ) ln ( 1 2 - φ e 2 π ) ] 1 / 2 ,             φ e < π ,             | φ e x | < ,
r ¨ I e ( 0 ) = ( 2 r I e ( Δ x , 0 ) ( Δ x ) 2 ) Δ x = 0 = ( 2 r I e ( τ ) τ 2 ) τ = 0 .
r I e ( τ ) τ τ 0 0
r I e ( Δ x , 0 ) - r I e ( 0 , 0 ) ( Δ x ) 2 Δ x 0 1 2 [ 2 r I e ( Δ x , 0 ) ( Δ x ) 2 ] Δ x = 0 .
W ( φ e x | φ e ) = W ( φ e , φ e x ) W ( φ e ) = exp ( - ( φ e x ) 2 2 { ( π - φ e ) [ r ¨ I e ( 0 ) ln ( 1 2 - φ e 2 π ) ] 1 / 2 } 2 ) 2 π ( π - φ e ) [ r ¨ I e ( 0 ) ln ( 1 2 - φ e 2 π ) ] 1 / 2 .
σ φ ˙ e 2 ( φ e ) = ( π - φ e ) 2 r ¨ I e ( 0 ) ln ( 1 2 - φ e 2 π ) .
W ( φ g 1 , φ g 2 ; τ ) = [ 1 - r I e 2 ( τ ) ] - 1 / 2 4 π 2 exp [ - r I e 2 ( τ ) Θ 2 ( φ g 1 π ) - 2 r I e ( τ ) Θ ( φ g 1 π ) Θ ( φ g 2 π ) + r I e 2 ( τ ) Θ 2 ( φ g 2 π ) 1 - r I e 2 ( τ ) ] ,             φ g i < π ,
W ( φ g , φ g x ) = exp [ Θ 2 ( φ g π ) ] 4 π 2 - r ¨ I e ( 0 ) exp [ - ( φ g x ) 2 exp [ 2 Θ 2 ( φ g π ) ] 4 π [ - r ¨ I e ( 0 ) ] ] ,             φ g < π , | φ g x | < ,
W ( φ g x | φ g ) = 2 π W ( φ g , φ g x ) ,
σ φ ˙ g 2 ( φ g ) = - 2 π r ¨ I e ( 0 ) exp [ - 2 Θ 2 ( φ g π ) ] .
C = [ I ( u , v ) - I ( u , v ) ] 2 1 / 2 I ( u , v ) .
C = 2 [ 2 3 ( φ u ) 4 - ( φ u ) 2 2 ] 1 / 2 π 2 ρ s 2 - 2 ( φ u ) 2 .
X Y = X ,
( φ u ) 2 = - ( φ u | φ W ( φ ) d φ = 1 2 π - π π σ φ ˙ 2 ( φ ) d φ ,
( φ u ) 4 = - ( φ u ) 4 | φ W ( φ ) d φ = 3 2 π - π π σ φ ˙ 4 ( φ ) d φ .
C e = 8 ( 199 405 ) 1 / 2 ( - r ¨ I e ( 0 ) ρ s 2 ) 5 - 40 9 ( - r ¨ I e ( 0 ) ρ s 2 ) ,
C g = 4 ( 2 5 - 1 3 ) 1 / 2 ( - r ¨ I e ( 0 ) ρ s 2 ) π - 4 3 ( - r ¨ I e ( 0 ) ρ s 2 ) .
φ ( E ) = 2 π [ Z ( E ) - 0.5 ] ,
I f ( ξ , η ) [ T s ( ω x , ω y ) 2 S t ( ω x , ω y ) ] ω x = ξ / λ f ω y = η / λ f .
I f ( ω x , ω y ) F [ t s ( x , y ) t d ( x , y ) ] 2 ,
F [ t s ( x , y ) t d ( x , y ) ] 2 = T s ( ω x , ω y ) 2 S t ( ω x , ω y ) .
F [ t s ( x , y ) t d ( x , y ) ] 2 = - - t s ( x 1 ) t s * ( x 2 ) t d ( x 1 ) t d * ( x 2 ) × exp [ 2 π i ω · ( x 1 - x 2 ) ] d x 1 d x 2 = - - t s ( x 1 ) t s * ( x 2 ) R t ( x 1 - x 2 ) × exp [ 2 π i ω · ( x 1 - x 2 ) ] d x 1 d x 2 = - t s ( x 1 ) d x 1 - t s * ( x 2 ) R t ( x 1 - x 2 ) × exp [ 2 π i ω · ( x 1 - x 2 ) ] d x 2 = - - t s ( x 1 ) t s * ( x 1 - x 2 ) R t ( x 2 ) × exp [ 2 π i ω · x 2 ] d x 1 d x 2 = - R t ( x 2 ) exp ( 2 π i ω · x 2 ) d x 2 × - t s ( x 1 ) t s * ( x 1 - x 2 ) d x 1 = - R t ( x 2 ) [ t s ( x 2 ) t s ( x 2 ) ] exp ( 2 π i ω · x 2 ) d x 2 = S t ( ω ) T s ( ω ) 2 ,
R I e ( Δ x , Δ y ; n ) = n 2 I e 2 + n I e 2 r I e ( Δ x , Δ y ) .
R I e ( Δ x , Δ y ; 1 ) = I e 2 + I e 2 r I e ( Δ x , Δ y ) ;
R X + Y ( Δ x , Δ y ) = R X ( Δ x , Δ y ) + R Y ( Δ x , Δ y ) + 2 X Y ,
R I e ( Δ x , Δ y ; n + 1 ) = R I e ( Δ x , Δ y ; n ) + R I e ( Δ x , Δ y ; 1 ) + 2 n I e 2 = ( n + 1 ) 2 I e 2 + ( n + 1 ) I e 2 r I e ( Δ x , Δ y ) .

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