Abstract

The linear transform kernel for fractional Fourier transforms is derived. The spatial resolution and the space–bandwidth product for propagation in graded-index media are discussed in direct relation to fractional Fourier transforms, and numerical examples are presented. It is shown how fractional Fourier transforms can be made the basis of generalized spatial filtering systems: Several filters are interleaved between several fractional transform stages, thereby increasing the number of degrees of freedom available in filter synthesis.

© 1993 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  4. A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967).
  5. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New. York, 1989), Chap. 2, pp. 18–26; Chap. 6, pp. 106–127.
  6. J. W. Goodman, A. R. Dias, L. M. Woody, “Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms,” Opt. Lett. 2, 1–3 (1978).
    [CrossRef] [PubMed]
  7. S. K. Case, P. R. Haugen, O. J. Løberge, “Multifacet holographic optical elements for wave front transformations,” Appl. Opt. 20, 2670–2675 (1981).
    [CrossRef] [PubMed]
  8. D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformations and optical-interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
    [CrossRef] [PubMed]
  9. H. I. Jeon, M. A. G. Abushagur, A.A. Sawchuk, B. K. Jenkins, “Digital optical processor based on symbolic substitution using holographic matched filtering,” Appl. Opt. 29, 2113–2125 (1990).
    [CrossRef] [PubMed]
  10. M. W. Haney, J. J. Levy, “Optically efficient free-space folded perfect shuffle network,” Appl. Opt. 30, 2833–2840 (1991).
    [CrossRef] [PubMed]
  11. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
    [CrossRef]
  12. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” submitted to Appl. Opt.
  13. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 712–760.

1993 (4)

1991 (1)

1990 (1)

1982 (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

1981 (1)

1978 (1)

Abushagur, M. A. G.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 712–760.

Case, S. K.

Dias, A. R.

Dickinson, B. W.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Goodman, J. W.

Haney, M. W.

Haugen, P. R.

Jenkins, B. K.

Jeon, H. I.

Levy, J. J.

Løberge, O. J.

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” submitted to Appl. Opt.

Mendlovic, D.

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformations and optical-interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” submitted to Appl. Opt.

Ozaktas, H. M.

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformations and optical-interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” submitted to Appl. Opt.

Sawchuk, A.A.

Steiglitz, K.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Woody, L. M.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New. York, 1989), Chap. 2, pp. 18–26; Chap. 6, pp. 106–127.

Appl. Opt. (4)

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

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Opt. Lett. (1)

Other (4)

A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New. York, 1989), Chap. 2, pp. 18–26; Chap. 6, pp. 106–127.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” submitted to Appl. Opt.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 712–760.

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Figures (13)

Fig. 1
Fig. 1

Hermite–Gaussian (HG) functions of different orders n. (a) Solid curve, n = 0; dashed curve, n = 1; dashed–dotted curve, n = 2; (b) solid curve, n = 0; dashed curve, n = 40. The one-dimensional version of Eq. (3) has been plotted normalized so as to have unit energy.

Fig. 2
Fig. 2

Solid lines, input signal; dashed curve, magnitude of its first-order Fourier transform.

Fig. 3
Fig. 3

Phase of the first-order Fourier transform of the input signal in Fig. 2.

Fig. 4
Fig. 4

Energy distribution among the first 100 HG orders.

Fig. 5
Fig. 5

Fractional Fourier transform of order a = 0.25. (a) Magnitude, (b) phase.

Fig. 6
Fig. 6

Fractional Fourier transform of order a = 0.5. (a) Magnitude, (b) phase.

Fig. 7
Fig. 7

Fractional Fourier transform of order a = 0.75. (a) Magnitude, (b) phase.

Fig. 8
Fig. 8

Solid line, input signal (delta function); dashed curve, its second-order Fourier transform.

Fig. 9
Fig. 9

Energy distribution among the first 100 HG orders for the delta function.

Fig. 10
Fig. 10

Magnitude of the a = 0.1th-order fractional Fourier transform of the delta function.

Fig. 11
Fig. 11

Magnitude of the a = 0.4th-order fractional Fourier transform of the delta function.

Fig. 12
Fig. 12

Magnitude of the first-order Fourier transform of the delta function.

Fig. 13
Fig. 13

System for performing generalized spatial filtering consists of several fractional Fourier transform stages with spatial filters inserted in between.

Equations (58)

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( F 1 f ) ( x , y ) = - + - + f ( x , y ) exp [ - i 2 π ( x x + y y ) s 2 ] d x d y ,
n 2 ( r ) = n 1 2 [ 1 - ( n 2 / n 1 ) r 2 ] ,
Ψ l m ( x , y ) = H l ( 2 x ω ) H m ( 2 y ω ) exp ( - x 2 + y 2 ω 2 ) ,
β l m = k [ 1 - 2 k ( n 2 n 1 ) 1 / 2 ( l + m + 1 ) ] 1 / 2 k - ( n 2 n 1 ) 1 / 2 ( l + m + 1 ) ,
f ( x , y ) = l m A l m Ψ l m ( x , y ) ,
A l m = - - f ( x , y ) Ψ l m ( x , y ) h l m d x d y ,
F a [ f ( x , y ) ] = l m A l m Ψ l m ( x , y ) exp ( i β l m a L ) .
A l = - f ( x ) Ψ l ( x ) h l d x ,
F a [ f ( x ) ] = l A l Ψ l ( x ) exp ( i β l α L ) ,
β l = k [ 1 - 2 k ( n 2 n 1 ) 1 / 2 ( l + 1 2 ) ] 1 / 2
k - ( n 2 n 1 ) 1 / 2 ( l + 1 2 )
= k - π 2 1 L ( l + 1 2 ) ,
F a [ f ( x ) ] = - f ( x ) B a ( x , x ) d x ,
B a ( x , x ) = l = 0 exp ( i β l a L ) h l Ψ l ( x ) Ψ l ( x ) .
B a ( x , x ) = 2 ω 1 π exp [ ikaL - i ( π 4 ) a ] × exp [ - ( x 2 + x 2 ) ω 2 ] × l exp [ - ial ( π / 2 ) ] 2 l l ! × H l ( 2 x ω ) H l ( 2 x ω ) .
B a ( x , y ; x , y ) = 2 ω 2 1 π exp [ - ( x 2 + x 2 ) ω 2 ] × exp [ - ( y 2 + y 2 ) ω 2 ] × exp [ ikaL - i ( π 2 ) a ] × l m exp [ - ial ( π / 2 ) ] 2 l l ! exp [ - iam ( π / 2 ) ] 2 m m ! × H l ( 2 x ω ) H l ( 2 x ω ) H m ( 2 y ω ) × H m ( 2 y ω ) .
( n 2 / n 1 ) ( D / 2 ) 2 1 ,
N = N x N y = ( Δ x Δ ν x ) ( Δ y Δ ν y ) = ( D 2 / s 2 ) 2 .
N = ( D 2 π ω 2 ) 2
x ( 2 E l / k ho ) 1 / 2 ( 2 ω ho l / k ho ) 1 / 2 .
2 / ω = ( k ho / ω ho ) 1 / 2 .
x 2 2 l ω 2 / 2 = ω 2 l .
l = D 2 / 4 ω 2 .
2 k n 2 n 1 ( l + m + 1 ) 2 k ( n 2 n 1 ) 1 / 2 ( l + m ) < 2 k ( n 2 n 1 ) 1 / 2 2 D 2 4 ω 2 = n 2 n 1 D 2 2 1 ,
f = [ f 1 f 2 f N ] T .
f ( z 2 ) = B ( Δ z 12 ) f ( z 1 ) ,
B ( Δ z 23 ) B ( Δ z 12 ) = B ( Δ z 23 + Δ z 12 ) = B ( Δ z 13 ) .
i B ( Δ z i , i + 1 ) = B ( i Δ z i , i + 1 ) .
f i = l = 1 N A ˜ l Ψ ˜ i l ,
f = Ψ A ,
A = Ψ - 1 f .
B ( Δ z ) Ψ l = exp ( i β l Δ z ) Ψ l ,
f ( z 1 ) = Ψ A ( z 1 ) ,
A ( z 1 ) = Ψ - 1 f ( z 1 ) .
A ˜ l ( z 2 ) = exp ( i β l Δ z 12 ) A ˜ l ( z 1 ) ,
A ( z 2 ) = β ( Δ z 12 ) A ( z 1 ) ,
f ( z 2 ) = Ψ A ( z 2 ) .
f ( z 2 ) = Ψ β ( Δ z 12 ) Ψ - 1 f ( z 1 ) ,
B ( Δ z 12 ) = Ψ β ( Δ z 12 ) Ψ - 1 .
Ψ - 1 B ( Δ z 12 ) Ψ = β ( Δ z 12 ) .
Ψ l ( x ) Ψ k ( x ) d x = δ l k h l ,
Ψ ˜ l ( x ) Ψ ˜ k ( x ) d x = δ l k .
Ψ l T Ψ k = i = 1 N Ψ ˜ i l Ψ ˜ i k
f ( z 1 - ) = B ( Δ z 01 ) f ( z 0 ) ,
f ( z 1 + = H 1 f ( z 1 - ) ,
f ( z j - ) = B ( Δ z j - 1 , j ) f ( z j - 1 + ) ,
f ( z j + ) = H j f ( z j - ) ,
f ( z out ) = f ( z M + 1 ) = Tf ( z 0 ) = Tf ( z in ) ,
T = B ( Δ z M , M + 1 ) H M B ( Δ z M - 1 , M ) H M - 1 B ( Δ z 1 , 2 ) H 1 B ( Δ z 0 , 1 ) ,
T = B ( Δ z ) H M B ( Δ z ) H M - 1 B ( Δ z ) H 1 B ( Δ z ) .
A ( z out ) = Ψ - 1 T Ψ A ( z in ) ,
T = Ψ β ( Δ z M , M + 1 ) Ψ - 1 H M Ψ β ( Δ z 1 , 2 ) Ψ - 1 × H 1 Ψ β ( Δ z 0 , 1 ) Ψ - 1 ,
S = β ( Δ z M , M + 1 ) G M β ( Δ z 1 , 2 ) G 1 β ( Δ z 0 , 1 ) ,
n 2 ( r ) = n 1 2 [ 1 - ( n p / n 1 ) r p ] ,
f ( x ) = l = 0 A l Ψ l ( x ) ,
A l = f ( x ) Ψ l ( x ) h l d x .
f ( x ) l = 0 N - 1 C l Ψ l ( x ) .
[ f ( x ) - l = 0 N - 1 C l Ψ l ( x ) ] 2 exp ( - x 2 ω 2 ) d x

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