Abstract

General optical setups that implement the fractional Fourier transforms are proposed by use of the impulse response theory. These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.

© 1995 Optical Society of America

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References

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  1. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  2. A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]
  3. D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am A 10, 1875 (1993).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  5. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  6. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994).
    [CrossRef]
  7. P. Pellet-Finet, Opt. Lett. 19, 1388 (1994).
    [CrossRef]
  8. G. S. Agarwal, R. Simon, Opt. Commun. 110, 23 (1994).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, Opt. Lett. 19, 1678 (1994).
    [CrossRef] [PubMed]
  10. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), Chap. 3, p. 117.

1994 (4)

1993 (3)

1987 (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, R. Simon, Opt. Commun. 110, 23 (1994).
[CrossRef]

Barshan, B.

Kerr, F. H.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Pellet-Finet, P.

Simon, R.

G. S. Agarwal, R. Simon, Opt. Commun. 110, 23 (1994).
[CrossRef]

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), Chap. 3, p. 117.

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Opt. Soc. Am A (1)

D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am A 10, 1875 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

G. S. Agarwal, R. Simon, Opt. Commun. 110, 23 (1994).
[CrossRef]

Opt. Lett. (2)

Other (1)

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), Chap. 3, p. 117.

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Figures (3)

Fig. 1
Fig. 1

Optical setups for the fractional FT proposed by Lohmann5: (a) single lens, (b) double lenses. R = sin(ϕα), Q = tan(ϕα/2).

Fig. 2
Fig. 2

General optical setup for implementations of the fractional FT.

Fig. 3
Fig. 3

Dynamic range of the normalized standard focal length F1α/f at different orders.

Equations (17)

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F α { f ( x ) } = c - f ( x ) × exp { i π λ F 1 α [ x 2 cot ( ϕ α ) - 2 x x f csc ( ϕ α ) + x f 2 cot ( ϕ α ) ] } d x ,
f ( r ) = d - f ( x ) ψ ( x - r ; d ) d x .
g ( r ) = c x u v f ( x ) ψ ( x - u ; d 1 ) ψ * ( u , k 1 ) ψ × ( u - v , d 2 ) ψ * ( v , k 2 ) ψ ( v - r , d 3 ) d x d u d v ,
g ( r ) = c - f ( x ) ψ ( x ; d 1 - n d 1 2 m n - d 2 2 ) × ψ ( r ; d 3 - m d 3 2 m n - d 2 2 ) × exp ( - i 2 π λ d 1 d 2 d 3 m n - d 2 2 x r ) d x ,
d 1 - n d 1 2 m n - d 2 2 = d 3 - m d 3 2 m n - d 2 2 ,
d 1 - n d 1 2 m n - d 2 2 = cot ( ϕ α ) F 1 α ,
d 1 d 2 d 3 m n - d 2 2 = 1 F 1 α sin ( ϕ α ) .
D 2 = ξ [ cos ( ϕ α ) - 1 ] + ( ξ + 1 ) N N - 1 f ,
F 1 α = ξ ( 1 - N ) - ( ξ - M ) cos ( ϕ α ) sin ( ϕ α ) f ,
M = N - ( 1 - ξ ) { N ( ξ + 1 ) - ξ [ 1 - cos ( ϕ α ) ] } 1 + ξ cos ( ϕ α ) .
( ξ - 1 ) [ 1 - cos ( ϕ α ) ] ξ + cos ( ϕ α ) N ξ ξ + 1 [ 1 - cos ( ϕ α ) ]             if             ξ 1 ,
0 N ξ ξ + 1 [ 1 - cos ( ϕ α ) ]             if             0 < ξ < 1 ,
N = M = ξ ξ + 1 [ 1 - cos ( ϕ α ) ] [ 1 - cos ( ϕ α ) ] ,
F 1 α = ξ ξ + 1 sin ( ϕ α ) f sin ( ϕ α ) f .
D 2 = 1 - 2 N - cos ( ϕ α ) 1 - N f ,
F 1 α = ( 1 - N ) tan ( ϕ α / 2 ) f .
N = 1 - sin ( ϕ α ) sin ( ϕ β ) 1 - cos ( ϕ α ) f β f α .

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