Abstract

The efficient Fresnel-transform algorithm (EFTA) is a computational tool that facilitates diffracted field analysis. The computational improvement that is achieved with the EFTA relative to the conventional double fast-Fourier-transform (FFT) algorithm results from the properties of fractional Fresnel diffraction. Some programming simplicities are shown to appear at numerous locations along the propagation axis. The number of those locations is determined by the fractional order. Some computational aspects of the method are presented and compared with those of the FFT algorithm.

© 1995 Optical Society of America

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References

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  1. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  2. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
    [CrossRef]
  3. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  4. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 19, 41–45 (1987).
  5. H. Hamam, J. L. de Bougrenet, “Fractional Talbot 4-level phase only holograms using ferroelectric liquid crystal spatial light modulators,” Opt. Lett. 19, 1654–1656 (1994).
    [CrossRef] [PubMed]
  6. H. Hamam, J. L. de Bougrenet, “Programmable joint fractional Talbot computer generated holograms,” J. Opt. Soc. Am. A 12, 314–324 (1995).
    [CrossRef]
  7. P. Szwaykowski, “The lateral shift of periodic objects under coherent plane wave illumination,” Opt. Acta 31, 563–566 (1984).
    [CrossRef]
  8. A. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  9. J. L. de Bougrenet, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
    [CrossRef]
  10. A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation for complex Fourier series,” Math. Computa. 29, 297–301 (1965).
    [CrossRef]

1995

H. Hamam, J. L. de Bougrenet, “Programmable joint fractional Talbot computer generated holograms,” J. Opt. Soc. Am. A 12, 314–324 (1995).
[CrossRef]

J. L. de Bougrenet, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
[CrossRef]

1994

1990

1987

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 19, 41–45 (1987).

1984

P. Szwaykowski, “The lateral shift of periodic objects under coherent plane wave illumination,” Opt. Acta 31, 563–566 (1984).
[CrossRef]

1978

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

1967

1965

J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. 55, 373–381 (1965).
[CrossRef]

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation for complex Fourier series,” Math. Computa. 29, 297–301 (1965).
[CrossRef]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation for complex Fourier series,” Math. Computa. 29, 297–301 (1965).
[CrossRef]

de Bougrenet, J. L.

Goodman, J. W.

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

Hamam, H.

Lohmann, A.

Lohmann, A. W.

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 19, 41–45 (1987).

Montgomery, W. D.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

Smirnov, A. P.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

Szwaykowski, P.

P. Szwaykowski, “The lateral shift of periodic objects under coherent plane wave illumination,” Opt. Acta 31, 563–566 (1984).
[CrossRef]

Thomas, J. A.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation for complex Fourier series,” Math. Computa. 29, 297–301 (1965).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Appl. Opt.

J. Opt. (Paris)

J. L. de Bougrenet, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Computa.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation for complex Fourier series,” Math. Computa. 29, 297–301 (1965).
[CrossRef]

Opt. Acta

P. Szwaykowski, “The lateral shift of periodic objects under coherent plane wave illumination,” Opt. Acta 31, 563–566 (1984).
[CrossRef]

Opt. Lett.

Opt. Spectrosc. (USSR)

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

Optik

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 19, 41–45 (1987).

Other

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

Diffraction of a periodic object.

Figure 2
Figure 2

Destructive and constructive interference phenomena during propagation.

Fig. 3
Fig. 3

Limits of pattern overlap as a function of the period d, the fractional order q, and the pattern extent Δf.

Fig. 4
Fig. 4

Flow graph for computing four output values: F(0), F(32), F(64), and F(96).

Fig. 5
Fig. 5

Flow graph for computing four output values: F(17), F(49), F(81), and F(113).

Fig. 6
Fig. 6

Computing effort per sample as a function of the fractional order (dashed line) compared with the corresponding FFT (horizontal lines).

Tables (2)

Tables Icon

Table 1 Talbot Coefficients of First Fractional Orders

Tables Icon

Table 2 Computational Effort As a Function of the Fractional Order

Equations (89)

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Δ ( u , z ) = 1 d m = - + exp ( - i π m 2 λ z d 2 ) exp ( i 2 π m u d ) .
F d ( u ) = f ( u ) Δ ( u , z ) ,
f d ( u ) = f ( u ) comb ( x , d ) ,
comb ( u , d ) = j = - + δ ( u - j d ) ,
Δ ( u , p q Z T ) = 1 d m = - + exp ( - 2 i π m 2 q p ) × exp ( i 2 π m u d ) .
Δ d ( u , p q Z T ) = a = 0 q / 2 - 1 T ( a , p , q ) δ ( u - d 2 + 2 a d q ) , T ( a , p , q ) = 2 q b = 0 q / 2 - 1 t ( a , b , p , q ) , t ( a , b , p , q ) = exp [ - i π ( 2 b 2 q p + b ) ] exp ( i π 4 b a q ) ,
Δ d ( u , p q Z T ) = a = 0 q - 1 T ( a , p , q ) δ ( u - a d q ) , T ( a , p , q ) = 1 q b = 0 q - 1 t ( a , b , p , q ) , t ( a , b , p , q ) = exp [ - i 2 π ( b 2 q p ) ] exp ( i 2 π b a q ) ,
F d ( u ) = f d ( u ) Δ d ( u , p q Z T ) .
q = 2 k ,             k an interger > 3.
t ( a , b + q / 2 , p , q ) = t ( a , b , p , q ) ,
t ( a , b + q / 4 , p , q ) = ( - 1 ) a + b t ( a , b , p , q ) ,
t ( a + q / 4 , b , p , q ) = ( - 1 ) b t ( a , b , p , q ) ,
t ( q / 2 - a , b , p , q ) = t ( a , q / 2 - b , p , q ) .
T ( a , p , q ) = 2 q b = 0 q / 4 - 1 t ( a , 2 b , p , q ) ,
T ( a , p , q ) = 2 q b = 0 q / 4 - 1 t ( a , 2 b + 1 , p , q ) .
T ( q / 2 - a , p , q ) = T ( - a , p , q ) = T ( a , p , q )
T ( q / 4 + a , p , q ) = ( - 1 ) a T ( a , p , q ) .
T ( q / 4 - a , p , q ) = ( - 1 ) a T ( a , p , q ) .
T ( a , p + q , q ) = T ( a , p , q ) ,
T ( a , q - p , q ) = T ( a , p , q ) * ,
T ( a , p + q / 2 , q ) = ( - 1 ) a T ( a , p , q ) ,
T ( a , q / 2 - p , q ) = ( - 1 ) a T ( a , p , q ) * ,
T ( 2 a , p + q / 4 , q ) = T ( 2 a , p , q ) ,
T ( 2 a + 1 , p + q / 4 , q ) = - i T ( 2 a + 1 , p , q ) ,
T ( 2 a , q / 4 - p , q ) = T ( 2 a , p , q ) * ,
T ( 2 a + 1 , q / 4 - p , q ) = - i T ( 2 a + 1 , p , q ) * .
T ( 2 a , p , 4 q ) = ( - 1 ) a 2 T ( a , p , q ) ,
T ( 4 a + p , p , 4 q ) = - 1 2 exp ( i π 8 a + p 2 q ) × T ( 2 a , p , q ) ,
T ( 4 a + p + 2 , 4 q ) = 1 2 exp ( i π 8 a + p + 4 2 q ) × T ( 2 a + 1 , p , q ) .
F ( x ) = 1 i λ z + - exp [ i π λ z ( x - u ) 2 ] f ( u ) d u .
Δ f < d ,
F d ( x ) = 1 i λ z + - exp [ i π λ z ( x - u ) 2 ] f d ( u ) d u .
F d ( x ) = a = 0 q / 2 - 1 T ( a , p , q ) f d ( x - d 2 + 2 a d q ) ,
F ( x ) = lim d a = 0 q / 2 - 1 T ( a , p , q ) f d ( x - d 2 + 2 a d q ) .
1 i λ z + - exp [ i π λ z ( x - u ) 2 ] f ( u ) d u - a = 0 q / 2 - 1 T ( a , p , q ) f d ( x - d 2 + 2 a d q ) d 2 < .
a = 0 2 q - 1 T ( a , p , q ) f 2 d ( u - d + 4 a d 4 q ) - a = 0 q / 2 - 1 T ( a , p , q ) f d ( x - d 2 + 2 a d q ) d 2 < .
2 d / q < Δ f .
z 3 π 4 λ Δ f 4 ,
( p q ) 3 Z T 3 π 4 λ Δ f 4 .
q p ( 32 α π λ 2 d 6 Δ f 4 ) 1 / 3 .
q ( 32 α π λ 2 d 6 Δ f 4 ) 1 / 3
q 2 β d Δ f .
q q ( + ) = ( 32 R 6 M 2 α π ) 1 / 3 , q q ( - ) = 2 β R ,
1 p q ( + ) q ( - ) .
F ( x ) = F 0 ( x ) + F ( x ) ,
F 0 ( x ) = a = 0 q / 4 - 1 T ( 2 a , p , q ) f d ( x - d 2 + 2 2 a d q ) , F 1 ( x ) = a = 0 q / 4 - 1 T ( 2 a + 1 , p , q ) f d [ x - d 2 + 2 ( 2 a + 1 ) d q ] ,
F ( x + d / 2 ) = a = 0 q / 4 - 1 T ( 2 a , p , q ) f d ( x + d 2 - d 2 + 2 2 a d q ) + a = 0 q / 4 - 1 T ( 2 a + 1 , p , q ) × f d [ x + d 2 - d 2 + 2 ( 2 a + 1 ) d q ] .
F ( x + d / 2 ) = a = 0 q / 4 - 1 T ( 2 a + q / 4 , p , q ) × f d [ x - d / 2 + 2 ( 2 a + q / 4 ) d q ] - a = 0 q / 4 - 1 T ( 2 a + 1 + q / 4 , p , q ) × f d [ x - d / 2 + 2 ( 2 a + 1 + q / 4 ) d q ] .
T ( a , p , q ) = T ( a + q / 2 , p , q )
F ( x + d / 2 ) = F 0 ( x ) - F 1 ( x ) .
F ( x ) = F 0 ( x ) + F 1 ( x ) , F ( x + d / 2 ) = F 0 ( x ) - F 1 ( x ) .
F 0 ( x ) = a = 0 q / 4 - 1 T ( 2 a , p , q ) f d ( x - d 2 + 2 2 a d q ) = a = 0 q / 8 - 1 T ( 2 a , p , q ) f d ( x - d 2 + 2 2 a d q ) + a = 0 q / 8 - 1 T ( 2 a + q 4 , p , q ) × f d [ x - d 2 + 2 ( 2 a + q / 4 ) d q ] .
F 0 ( x ) = a = 0 q / 8 - 1 T ( 2 a , p , q ) { f d ( x - d 2 + 2 2 a d q ) + f d [ x - d 2 + 2 ( 2 a + q / 4 ) d q ] } ,
F 1 ( x ) = a = 0 q / 8 - 1 T ( 2 a + 1 , p , q ) { f d [ x - d 2 + 2 2 ( a + 1 ) d q ] - f d [ x - d 2 + 2 ( 2 a + 1 + q / 4 ) d q ] } .
T ( a + q / s , p , q ) = ( - 1 ) 4 a / s T ( a , p , q ) ,             q > s 2 / 2 ,             s a power of 2.
T ( q / s - a , p , q ) = ( - 1 ) - 4 a / s T ( a , p , q ) ,             q > s 2 / 2 ,             s a power of 2.
F ( x + w 2 d s ) = k = 0 s / 2 - 1 ( - 1 ) - ( 4 w k / s ) F k ( x ) ,
F k ( x ) = a = 0 q / s - 1 T ( s 2 a + k , p , q ) × f d [ x - d 2 + 2 ( s 2 a + k ) d q , ] ,             w { 0 , s / 2 - 1 } .
F k ( x ) = a = 0 2 q / s 2 - 1 T ( s 2 a + k , p , q ) r = 0 s / 2 - 1 ( - 1 ) 4 r k / s × f d [ x - d 2 + 2 ( s 2 a + k + r q s ) d q ] .
( - 1 ) 4 ( r + s / 8 ) k / s = ( - 1 ) k / 2 ( - 1 ) 4 r k / s ,
F k ( x ) = a = 0 2 q / s 2 - 1 T ( s 2 a + k , p , q ) r = 0 s / 8 - 1 ( - 1 ) 4 r k / s × ( f d [ x - d 2 + 2 ( s 2 a + k + r q s ) d q ] + ( - 1 ) k / 2 f d { x - d 2 + 2 [ s 2 a + k + ( r + s 8 ) q s ] d q } + ( - 1 ) k f d { x - d 2 + 2 [ s 2 a + k + ( r + s 4 ) q s ] d q } + ( - 1 ) 3 k / 2 f d { x - d 2 + 2 [ s 2 a + k + ( r + 3 s 8 ) q s ] d q } ) .
F [ x + ( w + v s 8 ) 2 d 8 ] = k = 0 s / 8 - 1 ( - 1 ) - ( 4 w k / s ) ( - 1 ) - ( v / 2 ) k × [ F k ( x ) + ( - 1 ) w / 2 F k + s / 8 ( x ) × ( - 1 ) w F k + s / 4 ( x ) + ( - 1 ) 3 w / 2 F k + 3 s / 8 ( x ) ] ,
w { 0 , s / 8 - 1 } ,             v { 0 , , 3 } ,             s > 16.
C ( N , q , s ) = 2 N s q s = N q 32 ,             s = 8.
C ( N , q , s ) = 2 N s ( q 8 + A ) ,
A = s 4 ( s 4 - log 2 s + 1 ) ,
Δ d x , d y ( u , v , p x q x Z T x , p y q y Z T y ) = a y q y / 2 - 1 a x q x / 2 - 1 T ( a y , p y , q y ) T ( a x , p x , q x ) × δ ( u - d x 2 + 2 a x d x q x , v - d y 2 + 2 a y d y q y ) ,
Δ d x , d y ( u , v , p x q x Z T x , p y q y Z T y ) = a y q y / 2 - 1 T ( a y , p y , q y ) a x q x / 2 - 1 T ( a x , p x , q x ) × δ ( u - d x 2 + 2 a x d x q x , v - d y 2 + 2 a y d y q y ) .
T ( a , p , q ) = 2 q b = 0 q / 4 - 1 t ( a , 2 b , p , q ) + b = 0 q / 4 - 1 t ( a , 2 b + 1 , p , q ) .
T ( a , p , q ) = 2 q b = 0 q / 4 - 1 t ( a , 2 b , p , q ) + b = 0 q / 8 - 1 t ( a , 2 b + 1 , p , q ) + t ( a , 2 b + 1 + q / 4 , p , q ) .
T ( a , p , q ) = 2 q b = 0 q / 4 - 1 t ( a , 2 b , p , q ) + b = 0 q / 8 - 1 t ( a , 2 b + 1 , p , q ) - t ( a , 2 b + 1 , p , q ) .
T ( 2 a , p , 4 q ) = 1 2 q b = 0 q - 1 t ( 2 a , 2 b , p , 4 q ) , T ( 2 a , p , 4 q ) = 1 2 q b = 0 q - 1 exp [ - i π ( 2 b 2 q p ) ] exp ( i π 4 b a q ) .
T ( 2 a , p , 4 q ) = 2 2 q b = 0 q / 2 - 1 exp [ - i π ( 2 b 2 q p ) ] × exp ( i π 4 b a q )
T ( 2 a , p , 4 q ) = ( - 1 ) a 2 T ( a , p , q ) .
T ( 2 a + p , p , 4 q ) = 1 2 q b = 0 q - 1 t ( 2 a + p , 2 b + 1 , p , q ) , T ( 2 a + p , p , 4 q ) = 1 2 q b = 0 q - 1 exp [ - i π ( 2 4 b 2 + 4 b + 1 4 q p + 2 b + 1 ) ] × exp [ 4 i π ( 2 b + 1 ) ( 2 a + p ) 4 q ] .
T ( 2 a + p , p , 4 q ) = - 1 2 q exp ( i π 4 a + p 2 q ) × b = 0 q - 1 exp [ - i π ( 2 b 2 q p ) ] exp ( i π 4 b a q ) ,
T ( 2 a + p , p , 4 q ) = { - 1 2 exp ( i π 4 a + p 2 q ) T ( a , p , q ) a even 1 2 exp ( i π 4 a + p 2 q ) T ( a , p , q ) a odd .
a = k s 2 + j ,             j < s 2 ,             s a power of 2.
T ( a , p , q ) = b = 0 q / s - 1 t ( a , b s 2 + j , p , q ) .
T ( k s 2 + j , p , q ) = b = 0 q / s - 1 t ( k s 2 + j , b s 2 + j , p , q )
T ( k s 2 + j , p , q ) = T ( 2 k s 2 + j , p , q ) = b = 0 q / s - 1 t ( 2 k s 2 + j , b s 2 + j , p , q ) = b = 0 q / 2 s - 1 t ( k s 2 + j , 2 b s 2 + j , p , q ) + b = 0 q / 2 s - 1 t [ k s 2 + j , ( 2 b + 1 ) s 2 + j , p , q ] .
t ( a , b + q / s , p , q ) = ( - 1 ) ( 2 q / s 2 ) p - q / s ( - 1 ) [ ( 4 b + s b ) / s ] p × ( - 1 ) 4 a s t ( a , b , p , q ) ,
t [ k s 2 + j , ( 2 b + 1 ) s 2 + q s + j , p , q ] = ( - 1 ) ( 2 b + 1 ) p t [ k s 2 + j , ( 2 b + 1 ) s 2 + j , p , q ] = - t [ k s 2 + j , ( 2 b + 1 ) s 2 + j , p , q ] .
t [ k s 2 + j , ( 2 b + 1 ) s 2 + q s + j , p , q ] = t { k s 2 + j , [ 2 ( b + q s 2 ) + 1 ] s 2 + j , p , q } .
T ( a + q / s , p , q ) = b = 0 q / s - 1 t ( a + q s , b s 2 + j , p , q ) , a = k s 2 + j .
T ( a + q / s , p , q ) = b = 0 q / s - 1 t ( a , b s 2 + j , p , q ) × exp [ i π 4 ( k s 2 + j ) ( b s 2 + j ) q ] = exp ( i π 4 j s ) b = 0 q / s - 1 t ( a , b s 2 + j , p , q ) ,
T ( a + q / s , p , q ) = ( - 1 ) 4 a / s T ( a , p , q ) .
A = 4 [ s 16 ( s 8 - 1 ) + s 32 ( s 8 - 2 ) + s 64 ( s 8 - 4 ) + 2 ( s 8 - s 32 ) + ( s 8 - s 16 ) + 0 ] ,
A = s 4 ( s 4 - log 2 s + 1 ) .

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