Abstract

We investigate the effect of finite bandwidth of the incident radiation on scattering by thin layers that introduce random phase or amplitude variations. In particular, we calculate the scintillation index of the propagating radiation for smoothly varying and fractal phase screens and for random telegraph wave and checkerboard amplitude screens. Increasing the bandwidth of the incident radiation reduces the fluctuations of the scattered intensity over the whole propagation path, except in the case of the smoothly varying phase screen, where geometrical optics features in the pattern persist in the focusing region.

© 2010 Optical Society of America

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  1. E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis, 2006).
    [CrossRef]
  2. D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
    [CrossRef]
  3. C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
    [CrossRef]
  4. R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
    [CrossRef]
  5. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [CrossRef]
  6. E. Jakeman and B. J. Hoenders, “Scattering by a random surface of rectangular grooves,” Opt. Acta 29, 1587–1598(1982).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Springer-Verlag, 1978).
  8. H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
    [CrossRef]
  9. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  10. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [CrossRef]
  11. K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
    [CrossRef]
  12. D. L. Knepp, “Multiple phase-screen calculation of the temporal behaviour of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  13. E. Jakeman, S. M. Watson, and K. D. Ridley, “Intensity-weighted phase-derivative statistics,” J. Opt. Soc. Am. A 18, 2121–2131 (2001).
    [CrossRef]
  14. E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
    [CrossRef]
  15. R. H. Dicke, “Scatter-hole cameras for x-rays and gamma-rays,” Astrophys. J. 153, L101–L106 (1968).
    [CrossRef]

2009 (1)

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

2007 (1)

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

2001 (1)

1997 (1)

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

1983 (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behaviour of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1982 (2)

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

E. Jakeman and B. J. Hoenders, “Scattering by a random surface of rectangular grooves,” Opt. Acta 29, 1587–1598(1982).
[CrossRef]

1979 (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

1978 (1)

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

1968 (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma-rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

1962 (1)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

1950 (1)

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
[CrossRef]

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Booker, H. G.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
[CrossRef]

De Villiers, G.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

de Villiers, G. D.

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma-rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

Dravins, D.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Eismann, M.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Springer-Verlag, 1978).

Gordon, N.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Hoenders, B. J.

E. Jakeman and B. J. Hoenders, “Scattering by a random surface of rectangular grooves,” Opt. Acta 29, 1587–1598(1982).
[CrossRef]

Jakeman, E.

E. Jakeman, S. M. Watson, and K. D. Ridley, “Intensity-weighted phase-derivative statistics,” J. Opt. Soc. Am. A 18, 2121–2131 (2001).
[CrossRef]

E. Jakeman and B. J. Hoenders, “Scattering by a random surface of rectangular grooves,” Opt. Acta 29, 1587–1598(1982).
[CrossRef]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis, 2006).
[CrossRef]

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behaviour of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Lewis, K.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Lindegren, L.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

McDonald, G.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

McNie, M.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Mercier, R. P.

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

Mezey, E.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Parry, G.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Payne, D.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Payne, D. A.

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

Pike, E. R.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Pusey, P. N.

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Ratcliffe, J. A.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
[CrossRef]

Ridley, K.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Ridley, K. D.

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

E. Jakeman, S. M. Watson, and K. D. Ridley, “Intensity-weighted phase-derivative statistics,” J. Opt. Soc. Am. A 18, 2121–2131 (2001).
[CrossRef]

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis, 2006).
[CrossRef]

Shinn, D. H.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
[CrossRef]

Slinger, C.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Slinger, C. W.

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

Strens, M.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Watson, S. M.

Wilson, R.

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Wilson, R. A.

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

Young, A. T.

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Astrophys. J. (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma-rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

Contemp. Phys. (1)

E. Jakeman, G. Parry, E. R. Pike, and P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[CrossRef]

Opt. Acta (1)

E. Jakeman and B. J. Hoenders, “Scattering by a random surface of rectangular grooves,” Opt. Acta 29, 1587–1598(1982).
[CrossRef]

Phil. Trans. R. Soc. A (1)

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. A 242, 579–609 (1950).
[CrossRef]

Proc. Cambridge Philos. Soc. A (1)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

Proc. IEEE (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behaviour of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Proc. SPIE (2)

K. D. Ridley, G. D. de Villiers, D. A. Payne, R. A. Wilson, and C. W. Slinger, “Visible band lens-free imaging using coded aperture techniques,” Proc. SPIE 7468, 746809 (2009).
[CrossRef]

C. Slinger, M. Eismann, N. Gordon, K. Lewis, G. McDonald, M. McNie, D. Payne, K. Ridley, M. Strens, G. De Villiers, and R. Wilson, “An investigation of the potential for the use of a high resolution adaptive coded aperture system in the mid-wave infrared,” Proc. SPIE 6714, 671408 (2007).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

D. Dravins, L. Lindegren, E. Mezey, and A. T. Young, “Atmospheric intensity scintillation of stars: Dependence on optical wavelength,” Publ. Astron. Soc. Pac. 109, 725–737 (1997).
[CrossRef]

Other (3)

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (Taylor and Francis, 2006).
[CrossRef]

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

J. W. Goodman, Statistical Optics (Springer-Verlag, 1978).

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Figures (6)

Fig. 1
Fig. 1

Diagram showing the scattering geometry, with the scattering layer placed at z = 0 .

Fig. 2
Fig. 2

Results for the phase screen with Gaussian correlation function. The solid curves, in order of descending peak, correspond to r = 0 , 0.05, 0.2, 0.5, and 1.0.

Fig. 3
Fig. 3

Results for the fractal model. The top three theory curves have the same parameters as the corresponding simulation results; the lowest curve is for σ = 50 .

Fig. 4
Fig. 4

Results for the telegraph wave model. The lowest theory curve is for σ = 100 .

Fig. 5
Fig. 5

Intensity autocorrelation function for the telegraph wave. This is normalized by the square of the average intensity, so the result for zero distance is the normalized second moment. Here, k 0 = 2 π / λ 0 with λ 0 = 500 nm , R = 10 5 m 1 , and z = 25 mm . The curves are for different spectral bandwidths: monochromatic, σ = 0.05 k 0 , and σ = 0.1 k 0 .

Fig. 6
Fig. 6

Results for the checkerboard model. The approximate theory curve is the result of Eq. (32).

Equations (33)

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E ( r , z ; t ) = i k E k ( z , t ) 2 π z d r A k ( r , t ) exp ( i k 2 z ( r r ) 2 ) .
E ( x , z ; t ) = i k 2 π z E k ( z , t ) d x A k ( x , t ) exp ( i k 2 z ( x x ) 2 ) .
I 0 ( t ) I 0 ( t + τ ) = k , j , l , m a k a j * a l a m * exp { i ( k j ) c t + i ( l m ) c ( t + τ ) i ( k j + l m ) z } = k , j | a k | 2 | a j | 2 { 1 + exp ( i ( j k ) c τ ) } = I 0 2 + | k S k exp ( i k c t ) | 2 .
E ( x , z ; t ) = i k a k k 2 π z exp [ i k ( c t z ) ] d x A k ( x , t ) exp ( i k 2 z ( x x ) 2 ) .
I ( x , z ; t ) = k S k k 2 π z d x 1 d x 2 A k ( x 1 , t ) A k * ( x 2 , t ) exp ( i k 2 z [ ( x x 1 ) 2 ( x x 2 ) 2 ] ) .
I ( x , z ) I ( x , z ) = ( 1 2 π z ) 2 k 1 k 2 S ( k 1 ) S ( k 2 ) d k 1 d k 2 × d x 1 d x 4 A ( k 1 , x 1 ) A * ( k 1 , x 2 ) A ( k 2 , x 3 ) A * ( k 2 , x 4 ) × exp { i k 1 2 z [ ( x x 1 ) 2 ( x x 2 ) 2 ] + i k 2 2 z [ ( x x 3 ) 2 ( x x 4 ) 2 ] } .
E ( x , z ; t ) = i k 2 π z E k ( z , t ) n = A n n Δ ( n + 1 ) Δ d x exp ( i k 2 z ( x x ) 2 ) .
I 2 ( z ) = ( 1 2 π z ) k 1 S ( k 1 ) S ( k 2 ) d k 1 d k 2 d x 1 d x 2 × exp { h 2 k 2 2 [ 1 + p 2 p 2 ρ ( x 1 x 2 ) ρ ( p ( x 1 x 2 ) ) ] } exp { h 2 k 2 2 p [ ρ ( x 1 + p ( x 1 x 2 ) ) + ρ ( x 2 ) ρ ( x 1 ) ρ ( x 2 + p ( x 1 x 2 ) ) ] } exp { i k 1 2 z ( x 1 2 x 2 2 ) + i k 1 p 2 z ( x 1 x 2 ) 2 } .
I 2 ( z ) = S ( k 1 ) S ( k 2 ) d k 1 d k 2 ( k 1 2 π z ) d s d t exp { h 2 k 2 2 [ 1 + p 2 p 2 ρ ( t ) ρ ( p t ) ] } exp { h 2 k 2 2 p [ ρ ( s + 1 2 ( 1 + p ) t ) + ρ ( s 1 2 ( 1 + p ) t ) ρ ( s + 1 2 ( 1 p ) t ) ρ ( s 1 2 ( 1 p ) t ) ] } × exp ( i k 1 s t z ) .
I 2 ( z ) = S ( k 1 ) S ( k 2 ) d k 1 d k 2 ( I 1 + I 2 + I 3 ) .
I 1 = q π exp ( q 2 ) [ ln ( 8 γ k 1 2 h 2 3 ) + π erf i ( q ) R ( q ) ] ,
q = ξ 2 2 z h 6 , erf i ( q ) = 2 π 0 q d x exp ( x 2 ) , R ( q ) = 2 π 0 q d x exp ( x 2 ) erf c ( x ) ,
S ( k ) = 1 σ 2 π exp [ ( k k 0 ) 2 2 σ 2 ] ,
I 2 = q π exp ( q 2 ) [ ln ( 8 γ k 0 2 h 2 3 ) + π erf i ( q ) R ( q ) ] + erf c ( q ) + 2 q π 1 d x exp ( q 2 x 2 ) 1 + r 2 x 2 ,
I 2 ( z ) = S ( k 1 ) S ( k 2 ) d k 1 d k 2 × ( k 1 2 π z ) 0 d s 0 d t exp { k 1 k 2 L [ 1 2 ( 1 + p ) t s ] } × exp { k 1 k 2 L [ | s + 1 2 ( 1 p ) t | + | s 1 2 ( 1 p ) t | | s 1 2 ( 1 + p ) t | ] } cos ( k 1 s t z ) .
S ( k ) = 1 2 σ exp ( | k k 0 | σ ) .
I 2 = k 0 π z σ 2 0 d ε exp ( ε / σ ) ( ε + σ ) ( K 1 + K 2 + K 3 ) .
I 2 = 1 2 [ 1 2 C ( α ) ] 2 2 [ 1 2 S ( α ) ] 2 + J ( β ) where     J ( β ) = 2 π 0 t d t [ ( 1 + β t ) 2 + t 4 ] { 1 + 2 ( 1 + β t ) [ ( 1 + β t ) 2 + t 4 ] } ,
E ( x , z ) = i 2 + i 2 k 0 2 π z d x T ( x ) exp [ i k 2 z ( x x ) 2 ] = i 2 ( 1 + F ) .
I 2 I 2 = 7 4 + 1 2 { [ 1 S ( α T ) C ( α T ) ] cos ( π α T 2 2 ) + [ C ( α T ) S ( α T ) ] sin ( π α T 2 2 ) } 1 2 { [ 1 2 C ( α T ) ] 2 + [ 1 2 S ( α T ) ] 2 } .
I 2 I 2 = ( 1 4 π z ) 2 k 1 k 2 S ( k 1 ) S ( k 2 ) d k 1 d k 2 × d x 1 d x 4 { 1 + T 1 T 2 + T 1 T 3 + T 1 T 4 + T 2 T 3 + T 2 T 4 + T 3 T 4 + T 1 T 2 T 3 T 4 } × exp { i k 1 2 z [ x 1 2 x 2 2 ] + i k 2 2 z [ x 3 2 x 4 2 ] } .
I 2 I 2 = 1 + 1 2 { [ 1 S ( α T ) C ( α T ) ] cos ( π α T 2 2 ) + [ C ( α T ) S ( α T ) ] sin ( π α T 2 2 ) } + 1 ( 16 + β T 4 ) 2 { 128 + β T 2 ( 3 β T 6 8 β T 4 2 + 14 β T 2 40 ) } 1 2 { [ 1 2 C ( α T ) ] 2 + [ 1 2 S ( α T ) ] 2 } + 1 4 J ( β T ) .
g ( ω ) = 1 I 2 d x exp ( i ω χ ) I ( 0 ) I ( χ ) ,
g ( ω ) = 2 π δ ( ω ) + 2 R 4 R 2 + ω 2 ( cos ω 2 z k 0 + 1 [ 1 + ( σ z ω 2 / 2 k 0 2 ) 2 ] ) + 2 R 16 R 2 + ω 2 × { 1 + ( 8 σ z R ω / k 0 2 ) [ 1 + ( σ z R ω / k 0 2 ) ] 2 + σ 3 z 3 R ω 5 / 2 k 0 6 [ 1 + ( 2 σ z R ω / k 0 2 ) 2 + ( σ z ω 2 / 2 k 0 2 ) 2 ] 2 e 4 R z ω / k 0 ω ( 4 R sin ω 2 z k 0 + ω cos ω 2 z k 0 ) } .
f n = i 2 k 2 π z 0 Δ d x A ( x + n Δ ) exp [ i k 2 z ( x x n Δ ) 2 ] .
I I = p q r s ( b p + 1 ) ( b q + 1 ) ( b r + 1 ) ( b s + 1 ) ) f p f q * f r f s * = p q r s ( b p b q b r b s + b p b q + b p b r + b p b s + b q b r + b q b s + b r b s + 1 ) f p f q * f r f s * .
p q r s b p b q b r b s = ( n | f n | 2 ) ( m | f m | 2 ) + | n f n f n | 2 + | n f n f n * | 2 2 n | f n | 2 | f n | 2 .
| E | 2 n | f n | 2 + | E | 2 n | f n | 2 + E * E * n f n f n + c c + E * E n f n f n * + c c + | E E | 2 .
I I = I I + | E E | 2 + | E E * | 2 2 | E E | 2 2 n | f n | 2 | f n | 2 .
var I I 2 = 1 + 4 ( | E 2 | 2 2 | E | 4 2 n | f n | 4 ) .
n f n 2 1 4 λ z Δ d y 0 Δ d x 0 Δ d x exp { i k 2 z [ ( x x y ) 2 + ( x x y ) 2 ] } .
E 2 = E 2 E 0 2 ( 1 + i ) 4 × { C ( Δ / λ z ) + i S ( Δ / λ z ) + i λ z Δ π [ exp ( i k Δ 2 4 z ) 1 ] } .
I 2 I 2 = 7 4 + 1 2 [ C ( a ) + S ( a ) + C ( a ) 2 + S ( a ) 2 ] 1 2 π a { [ 1 + 2 C ( a ) ] sin π a 2 2 + [ 1 + 2 S ( a ) ] ( 1 cos π a 2 2 ) } + 1 π 2 a 2 ( 1 cos π a 2 2 ) 1 π 0 1 d x 1 x x sin π x a 2 2 .

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