Abstract

We study the relation between optical lens systems that perform a fractional Fourier transform (FRFT) with the geometrical cardinal planes. We demonstrate that lens systems symmetrical with respect to the central plane provide an exact FRFT link between the input and output planes. Moreover, we show that the fractional order of the transform has real values between 0 and 2 when light propagation is produced between principal planes and antiprincipal planes, respectively. Finally, we use this new point of view to design an optical lens system that provides FRFTs with variable fractional order in the range (0,2) without moving the input and output planes.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. A. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  4. K. B. Wolf, "Construction of optical systems," Geometric Optics in Phase Space (Springer-Verlag, 2004), Chap. 10.
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    [CrossRef] [PubMed]
  6. C. Sheppard and K. G. Larkin, "Similarity theorems for fractional Fourier transforms and fractional Hankel transforms," Opt. Commun. 154, 173-178 (1998).
    [CrossRef]
  7. K. K. Sharma and S. D. Joshi, "On scaling properties of the fractional Fourier transform and its relation with other transforms," Opt. Commun. 257, 27-38 (2006).
    [CrossRef]
  8. L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 24, 249-252 (2002).
  9. A. W. Lohmann, "A fake zoom lens for fractional Fourier experiments," Opt. Commun. 115, 437-443 (1995).
    [CrossRef]
  10. I. Moreno, J. A. Davis, and K. Crabtree, "Fractional Fourier transform optical system with programmable diffractive lenses," Appl. Opt. 42, 6544-6548 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 742-740 (1996).
    [CrossRef]
  17. I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
    [CrossRef]
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  19. C.-C. Shih, "Optical interpretation of a complex-order Fourier transform," Opt. Lett. 20, 1178-1180 (1995).
    [CrossRef]
  20. L. M. Bernardo and O. D. D. Soares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
    [CrossRef]
  21. A. Torre, "The fractional Fourier transform and some of its applications to optics," Prog. Opt. 43, 531-596 (2002).
    [CrossRef]
  22. A. Yariv, "Imaging of coherent fields through lenslike systems," Opt. Lett. 19, 1607-1608 (1994).
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    [CrossRef]

2006 (1)

K. K. Sharma and S. D. Joshi, "On scaling properties of the fractional Fourier transform and its relation with other transforms," Opt. Commun. 257, 27-38 (2006).
[CrossRef]

2005 (1)

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

2004 (1)

K. B. Wolf, "Construction of optical systems," Geometric Optics in Phase Space (Springer-Verlag, 2004), Chap. 10.

2003 (1)

2002 (2)

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 24, 249-252 (2002).

A. Torre, "The fractional Fourier transform and some of its applications to optics," Prog. Opt. 43, 531-596 (2002).
[CrossRef]

1998 (1)

C. Sheppard and K. G. Larkin, "Similarity theorems for fractional Fourier transforms and fractional Hankel transforms," Opt. Commun. 154, 173-178 (1998).
[CrossRef]

1997 (2)

1996 (2)

1995 (2)

A. W. Lohmann, "A fake zoom lens for fractional Fourier experiments," Opt. Commun. 115, 437-443 (1995).
[CrossRef]

C.-C. Shih, "Optical interpretation of a complex-order Fourier transform," Opt. Lett. 20, 1178-1180 (1995).
[CrossRef]

1994 (3)

1993 (2)

1983 (1)

1975 (1)

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).

1971 (1)

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

1970 (1)

1937 (1)

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. U.S.A. 23, 158-164 (1937).
[CrossRef] [PubMed]

Arsenault, H. H.

Bernardo, L. M.

L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 742-740 (1996).
[CrossRef]

L. M. Bernardo and O. D. D. Soares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
[CrossRef]

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).

Cai, L. Z.

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 24, 249-252 (2002).

Casperson, L. W.

Caulfield, J. C.

Collins, S. A.

Condon, E. U.

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. U.S.A. 23, 158-164 (1937).
[CrossRef] [PubMed]

Crabtree, K.

Davis, J. A.

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

I. Moreno, J. A. Davis, and K. Crabtree, "Fractional Fourier transform optical system with programmable diffractive lenses," Appl. Opt. 42, 6544-6548 (2003).
[CrossRef] [PubMed]

Dorsch, R. G.

Ferreira, C.

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, "Optical illustration of a varied fractional Fourier transform order and the Radon-Wigner display," Appl. Opt. 35, 3925-3929 (1996).
[CrossRef] [PubMed]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).

Joshi, S. D.

K. K. Sharma and S. D. Joshi, "On scaling properties of the fractional Fourier transform and its relation with other transforms," Opt. Commun. 257, 27-38 (2006).
[CrossRef]

Larkin, K. G.

C. Sheppard and K. G. Larkin, "Similarity theorems for fractional Fourier transforms and fractional Hankel transforms," Opt. Commun. 154, 173-178 (1998).
[CrossRef]

Liu, S.

Lohmann, A.

Lohmann, A. W.

Macukow, B.

Mateos, F.

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

Mendlovic, D.

Moreno, I.

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

I. Moreno, J. A. Davis, and K. Crabtree, "Fractional Fourier transform optical system with programmable diffractive lenses," Appl. Opt. 42, 6544-6548 (2003).
[CrossRef] [PubMed]

Moshinski, M.

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

Ozaktas, H. M.

Quesne, C.

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

Ren, H.

Sánchez-López, M. M.

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

Sharma, K. K.

K. K. Sharma and S. D. Joshi, "On scaling properties of the fractional Fourier transform and its relation with other transforms," Opt. Commun. 257, 27-38 (2006).
[CrossRef]

Sheppard, C.

C. Sheppard and K. G. Larkin, "Similarity theorems for fractional Fourier transforms and fractional Hankel transforms," Opt. Commun. 154, 173-178 (1998).
[CrossRef]

Shih, C.-C.

Soares, O. D. D.

Torre, A.

A. Torre, "The fractional Fourier transform and some of its applications to optics," Prog. Opt. 43, 531-596 (2002).
[CrossRef]

Tovar, A. A.

Wang, Y. Q.

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 24, 249-252 (2002).

Wolf, K. B.

K. B. Wolf, "Construction of optical systems," Geometric Optics in Phase Space (Springer-Verlag, 2004), Chap. 10.

Yariv, A.

Zalevsky, Z.

Zhang, J.

Zhang, X.

Appl. Opt. (4)

Eur. J. Phys. (1)

I. Moreno, M. M. Sánchez-López, C. Ferreira, J. A. Davis, and F. Mateos, "Teaching Fourier optics through ray matrices," Eur. J. Phys. 26, 261-271 (2005).
[CrossRef]

J. Math. Phys. (1)

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1780 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

A. W. Lohmann, "A fake zoom lens for fractional Fourier experiments," Opt. Commun. 115, 437-443 (1995).
[CrossRef]

C. Sheppard and K. G. Larkin, "Similarity theorems for fractional Fourier transforms and fractional Hankel transforms," Opt. Commun. 154, 173-178 (1998).
[CrossRef]

K. K. Sharma and S. D. Joshi, "On scaling properties of the fractional Fourier transform and its relation with other transforms," Opt. Commun. 257, 27-38 (2006).
[CrossRef]

Opt. Eng. (Bellingham) (1)

L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 742-740 (1996).
[CrossRef]

Opt. Laser Technol. (1)

L. Z. Cai and Y. Q. Wang, "Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system," Opt. Laser Technol. 24, 249-252 (2002).

Opt. Lett. (2)

Proc. Natl. Acad. Sci. U.S.A. (1)

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. U.S.A. 23, 158-164 (1937).
[CrossRef] [PubMed]

Prog. Opt. (1)

A. Torre, "The fractional Fourier transform and some of its applications to optics," Prog. Opt. 43, 531-596 (2002).
[CrossRef]

Other (2)

K. B. Wolf, "Construction of optical systems," Geometric Optics in Phase Space (Springer-Verlag, 2004), Chap. 10.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1975).

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Figures (7)

Fig. 1
Fig. 1

Symmetrical lens system

Fig. 2
Fig. 2

Lohmann type I FRFT lens system.

Fig. 3
Fig. 3

FRFT order p and normalized scaling factor b f as a function of the ratio d f in the Lohmann type I system.

Fig. 4
Fig. 4

FRFT order p and normalized scaling factor b f as a function of the ratio d f in the Lohmann type II system.

Fig. 5
Fig. 5

Lohmann type II FRFT lens system and its cardinal planes for different propagation distances. F 1 is the focal plane of the first lens (with focal length f ). S is the central symmetry plane. I, O are the input and output (FRFT) planes. H, H are the principal planes. F, F are the focal planes. A, A are the antiprincipal planes. Distances d O , d H , d F , and d A are measured with origin at S. f S is the focal length of the optical system and ξ is the normalized Newtonian distance.

Fig. 6
Fig. 6

FRFT lens system with fixed input and output planes. FRFT order p is changed in the range [0,2] when shifting the lateral lenses distance x in the range [ 2 f , 4 f ] from the central symmetry plane S. Other parameters are as in Fig. 5.

Fig. 7
Fig. 7

FRFT order p and normalized scaling factor b f in the optical system in Fig. 6 versus the ratio d f , d being the distance from the lateral lenses to the central lens and taking values in the range ( 2 f , 4 f ) .

Equations (21)

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M = [ A B C D ] ,
g 2 ( r 2 ) = 1 i λ B g 1 ( r 1 ) exp [ i π λ B ( A r 1 2 + D r 2 3 2 r 1 r 2 ) ] d r 1 ,
M FRFT ( p , b ) = [ cos ( ϕ ) b sin ( ϕ ) 1 b sin ( ϕ ) cos ( ϕ ) ] ,
g 2 ( r 2 ) = exp [ i π r 2 2 λ b tan ( ϕ ) ] i λ b sin ( ϕ ) g 1 ( r 1 ) exp [ i π r 1 2 λ b tan ( ϕ ) ] exp ( i 2 π r 1 r 2 λ b sin ( ϕ ) ) d r 1 ,
M R ( I O ) = R M 0 1 R M 0 ,
R = [ + 1 0 0 1 ] ,
M R ( I O ) = [ A D + B C 2 B D 2 A C A D + B C ] .
cos ( p π 2 ) = A D + B C ,
b = B D A C .
M = [ 1 d 0 1 ] [ 1 0 1 f 1 ] [ 1 d 0 1 ] = [ 1 d f d ( 2 d f ) 1 f 1 d f ] ,
cos ( p π 2 ) = 1 d f ,
b f = d f ( 2 d f ) .
ξ = ξ ,
cos ( p π 2 ) = ξ .
ξ = ξ ( 1 , + 1 ) .
M = [ 1 0 1 f 1 ] [ 1 d 0 1 ] [ 1 0 1 f 1 ] = [ 1 d f d ( 2 d f ) f 1 d f ] ,
b f = 1 2 ( d f ) 1 .
M IO = [ 1 4 f d 0 1 ] [ 1 0 1 f 1 ] [ 1 d 0 1 ] [ 1 0 1 2 f 1 ] [ 1 d 0 1 ] [ 1 0 1 f 1 ] [ 1 4 f d 0 1 ] ,
M IO = [ 9 + 12 x 9 2 x 2 + 1 2 x 3 f ( 32 + 48 x 26 x 2 + 6 x 3 1 2 x 4 ) 1 d ( 5 2 x + 3 x 2 1 2 x 3 ) 9 + 12 x 9 2 x 2 + 1 2 x 3 ] ,
cos ( p π 2 ) = 9 + 12 x 9 2 x 2 + 1 2 x 3 ,
b f = 32 + 48 x 26 x 2 + 6 x 3 1 2 x 4 81 216 x + 225 x 2 117 x 3 + 129 4 x 4 9 2 x 5 + 1 4 x 6 .

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