Abstract

Diffraction of monochromatic light by rough apertures is analyzed. Correlation functions are derived for the electric field and for the intensity in the optical transform plane. The expectations calculated are over an ensemble of edges; their roughness is described in terms of a second-order density and associated characteristic function. It is shown that in-plane roughness causes a speckle pattern. The analytical details are markedly different from the more usual case in which speckle is caused by longitudinal phase delay across an extended aperture. Detailed solutions are presented for serrated gaps and edges. Both space and wavelength dependences are included and solutions for cross correlations of electric field and intensity are obtained. These are valid for arbitrary roughness and correlation coefficient. Experiments are described contrasting the optical transforms of serrated and sharp edges. Good qualitative agreement is obtained with the theory. The serration causes a damping of the major spike in the edge transform and it leads to considerable scattering of the radiation.

© 1980 Optical Society of America

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References

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  1. See topical issue on Speckle for types of problems: J. Opt. Soc. Am.,  66, 1145–1312 (1976).
  2. R. A. Shore, B. J. Thompson, and R. W. Whitney, “Diffraction by Apertures Illuminated with Partially Coherent Light,” J. Opt. Soc. Am. 56, 733–738 (1966).
    [Crossref]
  3. E. Wolf and W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953–964 (1978).
    [Crossref]
  4. J. Carl Leader, “Far-zone range criteria for quasihomogeneous partially coherent sources,” J. Opt. Soc. Am. 68, 1332–1338 (1978).
    [Crossref]
  5. P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” Proc. SPIE 107, 63–69 (1977).
    [Crossref]
  6. Yu. M. Polischuk, “Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].
  7. Yu. M. Polischuk, “Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].
  8. Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
    [Crossref]
  9. A. C. Livanos and Nicholas George, “Edge Diffraction of a Convergent Wave,” Appl. Opt. 14, 608–613 (1975).
    [Crossref] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  11. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  14. W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.
  15. G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).
  16. Nicholas George and Atul Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
    [Crossref]
  17. I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [Crossref]
  18. C. L. Mehta, “Coherence and Statistics of Radiation,” in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).
  19. B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
    [Crossref]

1978 (2)

1977 (1)

P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” Proc. SPIE 107, 63–69 (1977).
[Crossref]

1976 (1)

1975 (2)

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
[Crossref]

A. C. Livanos and Nicholas George, “Edge Diffraction of a Convergent Wave,” Appl. Opt. 14, 608–613 (1975).
[Crossref] [PubMed]

1974 (2)

Nicholas George and Atul Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Yu. M. Polischuk, “Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

1966 (1)

1962 (1)

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[Crossref]

Carl Leader, J.

Carter, W. H.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

George, Nicholas

A. C. Livanos and Nicholas George, “Edge Diffraction of a Convergent Wave,” Appl. Opt. 14, 608–613 (1975).
[Crossref] [PubMed]

Nicholas George and Atul Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Heitler, W.

W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.

Jain, Atul

Nicholas George and Atul Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Kratsov, Yu. A.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
[Crossref]

Livanos, A. C.

Mehta, C. L.

C. L. Mehta, “Coherence and Statistics of Radiation,” in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).

Morris, G. M.

G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).

Papoulis, A.

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

Peters, P. J.

P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” Proc. SPIE 107, 63–69 (1977).
[Crossref]

Polischuk, Yu. M.

Yu. M. Polischuk, “Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

Yu. M. Polischuk, “Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].

Reed, I. S.

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[Crossref]

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

Rytov, C. M.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[Crossref]

Shore, R. A.

Tatarskii, V. I.

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
[Crossref]

Thompson, B. J.

Whitney, R. W.

Wolf, E.

Appl. Opt. (1)

Appl. Phys. (1)

Nicholas George and Atul Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[Crossref]

J. Opt. Soc. Am. (4)

Proc. SPIE (1)

P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” Proc. SPIE 107, 63–69 (1977).
[Crossref]

Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] (1)

Yu. M. Polischuk, “Fresnel Diffraction at n Rough Screens in a Medium with Large-Scale Nonhomogeneities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)] 19, 2038–2045 (1974), [Radio Eng. Electron. Phys. (USA) 19, 11–17 (1974)].

Usp. Fiz. Nauk (1)

Yu. A. Kratsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction theory,” Usp. Fiz. Nauk 115, 239–262 (1975), [Sov. Phys.-Usp. 18, 118–130 (1975)].
[Crossref]

Other (9)

C. L. Mehta, “Coherence and Statistics of Radiation,” in Lectures in Theoretical Physics VIIC, edited by W. E. Britten (University of Colorado, Boulder, 1965).

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[Crossref]

Yu. M. Polischuk, “Fresnel Diffraction at n Halfplanes with Statistically Nonuniform Edges. Small Irregularities,” Radiotekh. Elektron. [Radio Eng. Electron. Phys. (USSR)], 675–684, [Radio Eng. Electron. Phys. (USA) 16, 728–736 (1971)].

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), Ch. 6.

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

W. Heitler, The Quantum Theory of Radiation (Oxford University, London, 1954), p. 69.

G. M. Morris, Diffraction By Serrated Apertures (Ph.D. thesis, California Institute of Technology, 1979).

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Figures (8)

FIG. 1
FIG. 1

Coordinate system for the serrated aperture defined by the curve A r .

FIG. 2
FIG. 2

Serrated gap with rough edges yra and yrb and average width w.

FIG. 3
FIG. 3

Serrated edge with roughness yr(ξ).

FIG. 4
FIG. 4

Function T3(η0,η0;κ1) vs x plotted from Eq. (43) for (a) σ = 5 μm and (b) σ = 15 μm and for correlation lengths 0 = 50, 100, and 200 μm. Wavelength λ = 5, 145 Å y1 = 10 mm, z0 = 1 m.

FIG. 5
FIG. 5

Optical transform patterns for (a) straight edge and (b) serrated edge with σ = 15 μm (shown inset). Illumination is by a convergent monochromatic wave with z0 = 200 cm and λ = 5, 145 Å. Note the appreciable decrease in energy in the central spike for the roughened edge. The radius of the illuminated sector is 6 mm so the speckle or resolution scale size is on the order, λz0/ρ, of 200 μm. The pattern subtends an angle ± 1/40 rad.

FIG. 6
FIG. 6

(a) Microscope photograph of serrated edge generated by No. 60 sandpaper and (b) its associated correlation coefficient.

FIG. 7
FIG. 7

Intensity variation u(0,y) along the central spike for σ ≅ 6 and 15 μm. The plot starts at a large value y0 = 12.5 mm which is approximately 125 Airy disc units from y = 0.

FIG. 8
FIG. 8

Intensity measurements of the spread of energy perpendicular to the central spike for edges with (a) σ ~ 6 μm and 0 ~ 300 μm, (b) σ ~ 15 μm and 0 = 160 μm (No. 60 grit sandpaper). Other parameters were: y = 15 mm, z0 = 120 cm, λ = 0.5145 μm and Lx = 1.9 mm.

Tables (1)

Tables Icon

TABLE I Density and characteristic function notation12 and selected moment expansions.17,18,19

Equations (88)

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E ( x , y ) = d ξ d γ A r ( ξ , γ ) t ( ξ , γ ) S ( x , y ; ξ , γ ) ,
A r ( ξ , γ ) = { 1 for points ( ξ , γ ) inside the curve A r 0 otherwise .
S ( x , y ; ξ , γ ) = i exp ( i k R ) λ R z 0 R ( 1 + 1 i k R ) ,
S ( x , y ; ξ , γ ) i exp ( i k z 0 ) λ z 0 exp ( i k 2 z 0 [ ( x ξ ) 2 + ( y γ ) 2 ] ) .
E ( x 1 , y 1 ) E * ( x 2 , y 2 ) = d ξ d ξ d γ d γ × A r ( ξ , γ ) A r * ( ξ , γ ) t ( ξ , γ ) t * ( ξ , γ ) × S ( x 1 , y 1 ; ξ , γ ) S * ( x 2 , y 2 ; ξ , γ ) .
E T ( r , ω ) = T T d t e 1 ( r , t ) exp ( i ω t ) ,
P ( r 1 , r 2 ; ω ) = lim T E T ( r 1 , ω ) E * ( r 2 , ω ) 2 T .
e 1 ( r 1 , t ) e * ( r 2 , t τ ) = P ( r 1 , r 2 ; ω ) exp ( + i ω τ ) d ω / ( 2 π ) .
R u = u ( r 1 , k 1 ) u ( r 2 , k 2 ) ;
R u = E ( x 1 , y 1 ; k 1 ) E * ( x 1 , y 1 ; k 1 ) × E ( x 2 , y 2 ; k 2 ) E * ( x 2 , y 2 ; k 2 ) .
E 1 E 2 * = E ( x 1 , y 1 ; k 1 ) E * ( x 2 , y 2 ; k 2 ) ,
E 1 E 2 * = d ξ d ξ d γ d γ A r ( ξ , γ ) A r * ( ξ , γ ) × t ( ξ , γ ) t * ( ξ , γ ) S ( x 1 , y 1 ; ξ , γ ) × S * ( x 2 , y 2 ; ξ , γ ) .
t ( ξ , γ ) = exp [ i k 1 ( ξ 2 + γ 2 ) / ( 2 z 0 ) ] ,
A 1 = i exp [ i k 1 z 0 i k 1 ( x 1 2 + y 1 2 ) / ( 2 z 0 ) ] / ( λ 1 z 0 ) ,
E 1 E 2 * = A 1 A 2 * d ξ d ξ d γ d γ × A r ( ξ , γ ) A r * ( ξ , γ ) × exp [ i ( κ 1 ξ κ 2 ξ + η 1 γ η 2 γ ) ] ,
κ 1 , 2 = k 1 , 2 x 1 , 2 / z 0
η 1 , 2 = k 1 , 2 y 1 , 2 / z 0 ,
k 1 , 2 = 2 π / λ 1 , 2 .
A 1 A 2 * = exp [ i ( k 2 k 1 ) z 0 ] exp [ i k 2 ( x 2 2 + y 2 2 ) / ( 2 z 0 ) ] i k 1 ( x 1 2 + y 1 2 ) / ( 2 z 0 ) ] / ( λ 1 λ 2 z 0 2 ) .
B = 1 / ( λ 1 λ 2 z 0 2 ) 2 .
ω / 2 + y r a ( ξ ) γ ω / 2 + y r b ( ξ ) .
E ( x 1 , y 1 ; k 1 ) = A 1 i η 1 L L d ξ e i κ 1 ξ { exp [ i η 1 w / 2 + i η 1 y r b ( ξ ) ] exp [ i η 1 w / 2 + i η 1 y r a ( ξ ) ] } ,
E 1 = E 1 E 1
E 2 = E 2 E 2 .
E 1 E 2 * = E 1 E 2 * + E 1 E 2 * .
E 1 E 2 * = A 1 A 2 * ( 2 L ) 2 η 1 η 2 ( sin κ 1 L κ 1 L ) ( sin κ 2 L κ 2 L ) × [ exp ( i η 1 w / 2 ) Ψ b ( η 1 ) exp ( i η 1 w / 2 ) Ψ a ( η 1 ) ] × [ exp ( i η 2 w / 2 ) Ψ b * ( η 2 ) exp ( i η 2 w / 2 ) Ψ a * ( η 2 ) ] .
E 1 E 2 * = A 1 A 2 * η 1 η 2 L L d ξ d ξ × exp [ i ( κ 1 ξ κ 2 ξ ) ] { T 1 + T 2 } ,
T 1 = exp [ i ( η 1 η 2 ) w / 2 ] [ exp ( i η 1 y r b ) Ψ b ( η 1 ) ] × [ exp ( i η 2 y r b ) Ψ b * ( η 2 ) ] + exp [ i ( η 1 η 2 ) w / 2 ] [ exp ( i η 1 y r a ) Ψ a ( η 1 ) ] [ exp ( i η 2 y r a ) Ψ a * ( η 2 ) ] ;
T 2 = exp [ i ( η 1 η 2 ) w / 2 ] [ exp ( i η 1 y r b ) Ψ b ( η 1 ) ] × [ exp ( i η 2 y r a ) Ψ a * ( η 2 ) ] exp [ i ( η 1 η 2 ) w / 2 ] [ exp ( i η 1 y r a ) Ψ a ( η 1 ) ] [ exp ( i η 2 y r b ) Ψ b * ( η 2 ) ] .
ρ ( ξ ξ ) = y r a ( ξ ) ( ξ ) σ 2 = { arbitrary , when | ξ ξ | < 0 0 , when | ξ ξ | 0 .
E 1 E 2 * = A 1 A 2 * η 1 η 2 2 L sin ( κ 1 κ 2 ) L ( κ 1 κ 2 ) L × ( exp [ i ( η 1 η 2 ) w / 2 ] 0 0 d ξ exp ( i κ 2 ξ ) × { Φ b [ η 1 , η 2 ; ρ ( ξ ) ] Ψ b ( η 1 ) Ψ b * ( η 2 ) } + exp [ i ( η 1 η 2 ) w / 2 ] 0 0 d ξ exp ( i κ 2 ξ ) × { Φ a [ η 1 η 2 ; ρ ( ξ ) ] Ψ a ( η 1 ) Ψ a * ( η 2 ) } ) .
Φ a [ η 1 , η 2 ; ρ ( ξ ) ] = exp { σ 2 ( η 1 η 2 ) 2 / 2 σ 2 η 1 η 2 [ 1 ρ ( ξ ) ] }
Ψ a ( η 1 ) = exp ( σ 2 η 1 2 / 2 ) .
T 3 ( η 1 , η 2 ; κ 2 ) = 0 0 d ξ exp ( i κ 2 ξ ) { Φ a [ η 1 , η 2 ; ρ ( ξ ) ] Ψ a ( η 1 ) Ψ a ( η 2 ) } .
T 3 ( η 1 , η 2 ; κ 2 ) = 0 0 d ξ exp ( i κ 2 ξ ) [ exp ( σ 2 ( η 1 η 2 ) 2 2 σ 2 η 1 η 2 [ 1 ρ ( ξ ) ] ) exp ( σ 2 ( η 1 2 + η 2 2 ) 2 ) ] .
ρ ( ξ ) = Λ ( ξ / 0 ) = { 1 | ξ | / 0 , when | ξ | 0 0 , when | ξ | > 0 .
T 3 ( η 1 , η 2 ; κ 2 ) = 2 exp [ σ 2 ( η 1 η 2 ) 2 / 2 ] ( σ 2 η 1 η 2 / 0 ) 2 + κ 2 2 × [ σ 2 η 1 η 2 0 + ( σ 2 η 1 η 2 0 cos κ 2 0 + κ 2 sin κ 2 0 ) exp ( σ 2 η 1 η 2 ) ] 2 exp [ σ 2 ( η 1 2 + η 2 2 ) / 2 ] [ sin ( κ 1 0 ) / κ 2 ] .
E 1 E 2 * = A 1 A 2 * η 1 η 2 2 L sin ( κ 1 κ 2 ) L ( κ 1 κ 2 ) L × 2 T 3 cos [ ( η 1 η 2 ) w / 2 ] .
E ( x 1 , y 1 ; k 1 ) = A 1 d ξ d γ exp [ i ( κ 1 ξ + η 1 γ ) ] × { 1 + sgn [ γ y r ( ξ ) ] } / 2 ,
A 1 = i exp [ i k 1 z 0 i k 1 ( x 1 2 + y 1 2 ) / ( 2 z 0 ) ] / ( λ 1 z 0 ) , κ 1 = k 1 x 1 / z 0 , η 1 = k 1 y 1 / z 0 ,
E ( x 1 , y 1 ; k 1 ) = 2 π 2 A 1 δ ( κ 1 , η 1 ) + ( i A 1 / η 1 ) × d ξ exp { i [ κ 1 ξ + η 1 y r ( ξ ) ] } .
E 1 = 2 π 2 A 1 δ ( κ 1 , η 1 ) + ( i A 1 / η 1 ) × d ξ exp ( i κ 1 ξ ) Ψ ( η 1 )
E 1 = 2 π 2 A 1 δ ( κ 1 , η 1 ) + i 2 π A 1 δ ( κ 1 ) Ψ ( η 1 ) / η 1 .
E 1 = ( i A 1 / η 1 ) d ξ exp ( i κ 1 ξ ) × { exp [ i η 1 y r ( ξ ) ] Ψ ( η 1 ) } .
E 1 E 2 * = A 1 A 2 * / ( η 1 η 2 ) L x L x d ξ d ξ exp ( i κ 1 ξ i κ 2 ξ ) × { exp [ i η 1 y r ( ξ ) ] Ψ ( η 1 ) } { exp [ i η 2 y r ( ξ ) ] Ψ ( η 2 ) } .
E 1 E 2 * = [ A 1 A 2 * / ( η 1 η 2 ) ] T 3 ( η 1 , η 2 ; κ 2 ) × 2 L x sin ( κ 2 κ 1 ) L x / [ ( κ 2 κ 1 ) L x ] ,
E 1 E 1 * = ( A 1 A 1 * / ( η 1 2 ) ) 2 L x T 3 ( η 1 , η 1 ; κ 1 ) ,
T 3 ( η 1 , η 1 ; κ 1 ) = 0 0 d ξ exp ( i κ 1 ξ ) × { Φ [ η 1 , η 1 ; ρ ( ξ ) ] Ψ ( η 1 ) Ψ ( η 1 ) } .
T 3 ( η 1 , η 1 ; κ 1 ) = 0 0 d ξ exp ( i κ 1 ξ ) × [ exp { σ 2 η 1 2 [ 1 ρ ( ξ ) ] } exp ( σ 2 η 1 2 ) ] ,
T 3 ( η 1 , η 1 ; κ 1 ) = 2 ( σ 2 η 1 2 / 0 ) 2 + κ 1 2 × [ σ 2 η 1 2 0 + ( σ 2 η 1 2 0 cos κ 1 0 + κ 1 sin κ 1 0 ) × exp ( σ 2 η 1 2 ) ] 2 exp ( σ 2 η 1 2 ) [ sin ( κ 1 0 ) / κ 1 ] .
E 1 E 1 * E 2 E 2 * = [ B / ( η 1 η 2 ) 2 ] × d ξ d ξ d ξ d ξ × exp [ i ( κ 1 ξ κ 1 ξ + κ 2 ξ κ 2 ξ ) ] [ exp ( i η 1 y r ) Ψ ( η 1 ) ] × [ exp ( i η 1 y r ) Ψ ( η 1 ) ] [ exp ( i η 2 y r ) Ψ ( η 2 ) ] × [ exp ( i η 2 y r ) Ψ ( η 2 ) ] .
z 1 = exp ( i η 1 y r ) Ψ ( η 1 ) ,
z 1 = 1 + i η 1 y r + Ψ ( η 1 ) , z 1 i η 1 y r .
E 1 E 1 * E 2 E 2 * = B d ξ d ξ d ξ d ξ × exp [ i ( κ 1 ξ κ 1 ξ + κ 2 ξ κ 2 ξ ) ] × y r y r y r y r rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] × rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] .
D 2 = σ 4 B d ξ d ξ d ξ d ξ × exp [ i ( κ 1 ξ κ 1 ξ + κ 2 ξ κ 2 ξ ) ] × ρ ( ξ ξ ) ρ ( ξ ξ ) rect [ ξ / ( 2 L x ) ] rect [ ( ξ / 2 L x ) ] × rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] .
D 2 = 64 B ( σ 2 L x 0 ) 2 × [ sin [ ( κ 2 κ 1 ) L x ] ( κ 2 κ 1 ) L x ( sin ( κ 1 0 / 2 ) κ 1 0 ) 2 * 2 L x sin ( κ 1 L x ) κ 1 L x ] κ 1 × [ sin [ ( κ 2 κ 1 ) L x ] ( κ 2 κ 1 ) L x ( sin ( κ 2 0 ) κ 2 0 ) 2 * 2 L x sin ( κ 2 L x ) κ 2 L x ] κ 2 .
E 1 E 1 * E 2 E 2 * = D 1 + D 2 + D 3 ,
D 1 64 B ( σ 2 L x 0 ) 2 ( sin ( κ 1 0 / 2 ) κ 1 0 ) 2 ( sin ( κ 2 0 / 2 ) κ 2 0 ) 2 ,
D 2 64 B ( σ 2 L x 0 ) 2 ( sin [ ( κ 2 κ 1 ) L x ] ( κ 2 κ 1 ) L x ) 2 × ( sin ( κ 1 0 / 2 ) κ 1 0 ) 2 ( sin ( κ 2 0 / 2 ) κ 2 0 ) 2 ,
D 3 64 B ( σ 2 L x 0 ) 2 ( sin [ ( κ 2 + κ 1 ) L x ] ( κ 2 + κ 1 ) L x ) 2 × ( sin ( κ 1 0 / 2 ) κ 1 0 ) 2 ( sin ( κ 2 0 / 2 ) κ 2 0 ) 2 ,
exp { i [ η 1 y r ( ξ ) n 1 y r ( ξ ) + η 2 y r ( ξ ) η 2 y r ( ξ ) ] } .
E 1 E 1 * E 2 E 2 * L 1 + L 2 + L 3 ,
L 1 4 B [ L x / ( η 1 η 2 ) ] 2 T 3 ( η 1 , η 1 ; κ 1 ) T 3 ( η 2 , η 2 ; κ 2 ) ,
L 2 4 B [ L x / ( η 1 η 2 ) ] 2 { sin ( κ 2 κ 1 ) L x / [ ( κ 2 κ 1 ) L x ] } 2 × T 3 ( η 1 , η 2 ; κ 1 ) T 3 ( η 1 , η 2 ; κ 2 ) ,
L 3 4 B [ L x / ( η 1 η 2 ) ] 2 { sin ( κ 1 κ 2 ) L x / [ ( κ 1 κ 2 ) L x ] } 2 × T 3 ( η 1 , η 2 ; κ 1 ) T 3 ( η 1 , η 2 ; κ 2 ) ,
T 3 ( η 1 , η 2 ; κ 2 ) σ 2 η 1 η 2 0 0 d ξ ρ ( ξ ) exp ( i κ 2 ξ ) .
T 3 ( η 1 , η 2 ; κ 2 ) σ 2 η 1 η 2 0 [ sin ( κ 2 0 / 2 ) / ( κ 2 0 / 2 ) ] 2 .
E 1 E 1 * = A 1 A 1 * | 2 π 2 δ ( κ 1 , η 1 ) + ( i 2 L x / η 1 ) [ sin ( κ 1 L x ) / κ 1 L x ] × exp ( σ 2 η 1 2 / 2 ) | 2 + ( A 1 A 1 * / η 1 2 ) 2 L x T 3 ( η 1 , η 1 ; κ 1 )
R u = E 1 E 1 * E 2 E 2 * .
E 1 E 1 * E 2 E 2 * = E 1 E 1 * E 2 E 2 * + terms in ( A 3 ) below
+ b 1 E 1 * E 2 E 2 * b 1 b 1 * E 2 E 2 * b 1 b 2 * E 1 * E 2 + b 1 * E 1 E 2 E 2 * b 1 b 2 E 1 * E 2 * b 1 * b 2 E 1 E 2 * + b 2 E 1 E 1 * E 2 * b 1 * b 2 * E 1 E 2 + b 2 * E 1 E 1 * E 2 b 2 b 2 * E 1 E 1 * + 3 b 1 b 1 * b 2 b 2 * .
+ 2 Re ( b 1 E 1 * E 2 E 2 * + b 2 E 1 E 1 * E 2 * + b 1 b 2 E 1 * E 2 * + b 1 * b 2 E 1 E 2 * ) + b 1 b 1 * E 2 E 2 * + b 2 b 2 * E 1 E 1 * + b 1 b 1 * b 2 b 2 * .
E 1 E 1 * E 2 E 2 * = [ B / ( η 1 η 2 ) 2 ] d ξ d ξ d ξ d ξ
× exp [ i ( κ 1 ξ κ 1 ξ + κ 2 ξ κ 2 ξ ) ] × z 1 ( ξ ) z 1 * ( ξ ) z 2 ( ξ ) z 2 * ( ξ ) × rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] × rect [ ξ / ( 2 L x ) ] ,
z 1 ( ξ ) = exp [ i η 1 y r ( ξ ) ] Ψ ( η 1 ) , etc .
P = z 1 ( ξ ) z 1 * ( ξ ) z 2 ( ξ ) z 2 * ( ξ ) .
P = z 1 ( ξ ) z 1 * ( ξ ) z 2 ( ξ ) z 2 * ( ξ ) = 0 ,
| ξ ξ | > 0 L 1 : | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | > 0 | ξ ξ | > 0 | ξ ξ | > 0 ; | ξ ξ | > 0
L 2 : | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | > 0 ;
L 3 : | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | > 0 | ξ ξ | > 0
L 4 : | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | 0 | ξ ξ | > 0 | ξ ξ | 0 | ξ ξ | > 0 .
E 1 E 1 * E 2 E 2 * = L 1 + L 2 + L 3 + m = 4 37 L m
L 1 = [ B / ( η 1 η 2 ) 2 ] d ξ d ξ d ξ d ξ × exp [ i ( κ 1 ξ κ 1 ξ + κ 2 ξ κ 2 ξ ) ] × z 1 ( ξ ) z 1 * ( ξ ) z 2 ( ξ ) z 2 * ( ξ ) rect [ ξ / ( 2 L x ) ] × rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] rect [ ξ / ( 2 L x ) ] × rect [ ( ξ ξ ) / ( 2 0 ) ] rect [ ξ ξ / ( 2 0 ) ] × ( 1 rect [ ( ξ ξ ) / ( 2 0 ) ] ) ( 1 rect [ ( ξ ξ ) / ( 2 0 ) ] ) × ( 1 rect [ ( ξ ξ ) / ( 2 0 ) ] ) ( 1 rect [ ( ξ ξ ) / ( 2 0 ) ] ) .
z 1 ( ξ ) z 1 * ( ξ ) = Φ ( η 1 , η 1 ; ρ [ ξ ξ ] ψ ( η 1 ) ψ ( n 1 ) ,
z 1 ( ξ ) z 1 * ( ξ ) = exp [ σ 2 η 1 2 ] { exp [ σ 2 η 1 2 ρ ( ξ ξ ) ] 1 } ,
L 1 4 B [ L x / ( η 1 η 2 ) ] 2 ( T 3 ( η 1 , η 1 ; κ 1 ) * 2 L x sin ( κ 1 L x ) κ 1 L x ) κ 1 × ( T 3 ( η 2 , η 2 ; κ 2 ) * 2 L x sin ( κ 2 L x ) κ 2 L x ) κ 2 .
L 2 4 B [ L x / ( η 1 η 2 ) ] 2 × ( sin ( κ 2 κ 1 ) L x ( κ 2 κ 1 ) L x T 3 ( η 1 , η 2 ; κ 2 ) * 2 L x sin ( κ 2 L x ) κ 2 L x ) κ 2 . × ( sin ( κ 2 κ 1 ) L x ( κ 2 κ 1 ) L x T 3 ( η 1 , η 2 ; κ 2 ) * 2 L x sin ( κ 2 L x ) κ 2 L x ) κ 2
L 3 4 B [ L x / ( η 1 η 2 ) ] 2 × ( sin ( κ 1 + κ 2 ) L x ( κ 1 + κ 2 ) L x T 3 ( η 1 , η 2 ; κ 2 ) * 2 L x sin ( κ 2 L x ) κ 2 L x ) κ 2 × ( sin ( κ 1 + κ 2 ) L x ( κ 1 + κ 2 ) L x T 3 ( η 1 , η 2 ; κ 2 ) * 2 L x sin ( κ 2 L x ) κ 2 L x ) κ 2 .