Abstract

The Wigner distribution function is used to describe general properties of the irradiance distribution at the Fresnel diffraction planes of one-dimensional phase gratings. We report a remarkably simple, analytical expression for the irradiance distribution at one quarter of the Talbot length of any phase grating. As illustrative examples, we consider binary and triangular phase profiles.

© 1992 Optical Society of America

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References

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  1. A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).
  2. J. M. Cowley, A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sect. B 76, 378–384 (1960).
    [Crossref]
  3. K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase diffraction grating,” Opt. Appl. 11, 627–631 (1981).
  4. T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
    [Crossref]
  5. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [Crossref] [PubMed]
  6. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  7. M. J. Bastiaans, “Local-frequency description of optical signals and systems,” (Eindhoven University of Technology, Eindhoven, The Netherlands, 1988).
  8. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).
  9. B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,” IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
    [Crossref]

1990 (1)

1988 (3)

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,” IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[Crossref]

1981 (1)

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase diffraction grating,” Opt. Appl. 11, 627–631 (1981).

1980 (1)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

1960 (1)

J. M. Cowley, A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sect. B 76, 378–384 (1960).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “Local-frequency description of optical signals and systems,” (Eindhoven University of Technology, Eindhoven, The Netherlands, 1988).

Boashash, B.

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,” IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sect. B 76, 378–384 (1960).
[Crossref]

Jinhong, T.

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[Crossref] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).

Mecklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sect. B 76, 378–384 (1960).
[Crossref]

Patorski, K.

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase diffraction grating,” Opt. Appl. 11, 627–631 (1981).

Szwaykowski, P.

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase diffraction grating,” Opt. Appl. 11, 627–631 (1981).

Thomas, J. A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,” IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

J. Mod. Opt. (1)

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[Crossref]

Opt. Appl. (1)

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase diffraction grating,” Opt. Appl. 11, 627–631 (1981).

Optik (1)

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).

Philips J. Res. (1)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Proc. Phys. Soc. London Sect. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sect. B 76, 378–384 (1960).
[Crossref]

Other (1)

M. J. Bastiaans, “Local-frequency description of optical signals and systems,” (Eindhoven University of Technology, Eindhoven, The Netherlands, 1988).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the optical setup. The phase grating, t(x), is located at z = 0.

Fig. 2
Fig. 2

Binary phase grating with w/d ≤ 1/2: (a) phase function ϕ(x); (b) function S(x); (c) irradiance distribution at z = Zt/4 for ϕ0 = π/2.

Fig. 3
Fig. 3

Triangular phase grating with w/d = 1/2 and ϕ0 = π/2: (a) phase function ϕ(x); (b) function S(x); (c) cosinusoidal irradiance distribution at z = Zt/4.

Fig. 4
Fig. 4

Numerically evaluated irradiance distributions at Fresnel distances of the triangular phase profile in Fig. 3(a) for (a) 0 ≤ zZt/4; (b) Zt/4 ≤ zZt/2; (c) Zt/2 ≤ z ≤ 3Zt/4; (d) 3Zt/4 ≤ zZt.

Equations (40)

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W 0 ( x , ν ) = - t ( z + y / 2 ) t * ( x - y / 2 ) exp ( - i 2 π ν y ) d y ,
W ( x , ν ; z ) = W 0 ( x - λ z ν , ν ) .
I ( x , z ) = - W ( x , ν ; z ) d ν = - W 0 ( x - λ z ν , ν ) d ν .
t ( x ) = exp [ i ϕ ( x ) ] = m = - a m exp ( i 2 π m x / d ) .
W 0 ( x , ν ) = m = - { n = - a m + n a n * δ [ ν - ( m + 2 n ) / 2 d ] } × exp ( i 2 π m x / d ) .
W ( x , ν ; z ) = m = - { n = - a m + n a n * δ [ ν - ( m + 2 n ) / 2 d ] } × exp ( i 2 π m x / d ) exp [ - i 2 π m ( m + 2 n ) λ z / 2 d 2 ] .
I ( x , z ) = m = - c m ( z ) exp ( i 2 π m x / d ) ,
c m ( z ) = exp [ - i 2 π m 2 ( z / Z t ) ] n = - a m + n a n * × exp [ - i 2 π m n ( 2 z / Z t ) ] .
c m ( Z t / 2 ) = δ m , 0 ,
I ( x , z = Q Z t / 2 ) = 1 ,
c m ( z ) = c - m ( z )
c m ( z ) = c m * ( - z ) .
c m ( z ) = c - m * ( z ) .
c m ( z ) = c - m ( - z )
I ( x , z ) = I ( - x , - z ) .
c m ( z + Z t / 2 ) = exp ( - i π m 2 ) c m ( z ) .
c 2 m ( z + Z t / 2 ) = c 2 m ( z ) ,
c 2 m + 1 ( z + Z t / 2 ) = - c 2 m + 1 ( z ) .
I ( x , z ) = I ( x + d / 2 , z + Z t / 2 ) .
c - m * ( - z , - ϕ ) = c m ( z , ϕ ) .
I ( x , z , ϕ ) = I ( x , - z , - ϕ ) ,
I ( x - d / 2 , Z t / 2 - z , - ϕ ) = I ( x , z , ϕ ) ,
I ( x - d / 2 , Z t / 2 - z , - ϕ ) = m = - c - m * ( Z t / 2 - z , - ϕ ) × exp [ i 2 π ( x - d / 2 ) m / 2 ] .
I ( x - d / 2 , Z t / 2 - z , - ϕ ) = m = - ( - 1 ) m 2 + m c m ( z , ϕ ) × exp ( i 2 π m x / d ) .
I ( x , Z t - z , - ϕ ) = m = - c m ( Z t - z , - ϕ ) exp ( i 2 π m x / d ) = m = - c - m * ( Z t - z , - ϕ ) exp ( i 2 π m x / d ) = m = - c m ( z - Z t , ϕ ) exp ( i 2 π m x / d ) = I ( x , z , ϕ ) .
I ( x , Z t / 4 ) = m = - c m ( Z t / 4 ) exp ( i 2 π m x / d ) = 1 + sin [ S ( x ) ] ,
S ( x ) = ϕ ( x ) - ϕ ( x - d / 2 ) ;
S ( x ) = { π 2 + 2 π x d - d / 2 x < 0 π 2 - 2 π x d 0 x < d / 2 ,
I ( x , z = Z t 4 ) = 1 + cos ( 2 π x d ) ,
t ( x ) 2 = 1 = m = - ( n = - a m + n a n * ) exp ( i 2 π m x / d ) .
n = - a m + n a n * = δ m , 0 ,
t ( x ) = n = - a 2 n exp [ i 2 π ( 2 n ) x / d ] + n = - a 2 n + 1 exp [ i 2 π ( 2 n + 1 ) x / d ] .
t ( x ) + t ( x - d / 2 ) = 2 n = - a 2 n exp [ i 2 π ( 2 n ) x / d ] ,
t ( x ) - t ( x - d / 2 ) = 2 n = - a 2 n + 1 exp [ i 2 π ( 2 n + 1 ) x / d ] .
[ t ( x ) - t ( x - d / 2 ) ] [ t * ( x ) + t * ( x - d / 2 ) ] = 2 i sin [ ϕ ( x ) - ϕ ( x - d / 2 ) ] = 4 m = - [ n = - a 2 ( m + n ) + 1 a 2 n * ] exp [ i 2 π ( 2 m + 1 ) x / d ] .
m = - [ - 2 i n = - a 2 ( m + n ) + 1 a 2 n * ] exp [ i 2 π ( 2 m + 1 ) x / d ] = sin [ ϕ ( x ) - ϕ ( x - d / 2 ) ] .
c m ( Z t / 4 ) = exp ( - i π m 2 / 2 ) n = - a m + n a n * exp ( - i π m n ) ,
c 2 m ( Z t / 4 ) = n = - a m + n a n * = δ m , 0 ,
c 2 m + 1 ( Z t / 4 ) = - i n = - a 2 ( m + n ) + 1 a n * + i n = - a 2 ( m + n + 1 ) a 2 n + 1 * = - 2 i n = - a 2 ( m + n ) + 1 a 2 n * ,
I ( x , Z t / 4 ) = 1 + sin [ ϕ ( x ) - ϕ ( x - d / 2 ) ] ,

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