Abstract

On the basis of a matrix formalism, we analyze the paraxial optical systems composed by generalized lenses and fixed free-space intervals, suitable for orthosymplectic transformations in phase space. Flexible configurations to perform the attractive operations for optical information processing such as image rotation, separable fractional Fourier transformation, and twisting for different parameters are proposed.

© 2006 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).
  2. J. Shamir, "Cylindrical lens described by operator algebra," Appl. Opt. 18, 4195-4202 (1979).
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  4. H. Braunecker, O. Bryngdahl, and B. Schnell, "Optical system for image rotation and magnification," J. Opt. Soc. Am. 70, 137-141 (1980).
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  5. D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transform and their optical implementation," J. Opt. Soc. Am. A 10, 1875-1881 (1993).
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  6. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional order Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  7. G. Nemes and A. G. Kostenbauder, "Optical systems for rotating a beam," in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R.Martinez-Herrero, and A.Gonzales-Urena, eds. (Sociedad Española de Optica, 1993), pp. 99-109.
  8. G. Nemes and A. E. Seigman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994).
    [CrossRef]
  9. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktas, "Anamorphic fractional Fourier transform: optical implementation and applications?" Appl. Opt. 34, 7451-7456 (1995).
    [CrossRef] [PubMed]
  10. M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, "Design of dynamically adjustable anamorphic fractional transformer Fourier," Opt. Commun. 136, 52-60 (1997).
    [CrossRef]
  11. A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters," Appl. Opt. 37, 2130-2141 (1998).
    [CrossRef]
  12. 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]
  13. A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
    [CrossRef]
  14. R. Simon and K. B. Wolf, "Fractional Fourier transforms in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000).
    [CrossRef]
  15. R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000).
    [CrossRef]
  16. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
    [CrossRef]
  17. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
    [CrossRef]
  18. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
  19. T. Alieva and M. Bastiaans, "Alternative representation of the linear canonical integral transform," Opt. Lett. 30, 3302-3304 (2005).
    [CrossRef]

2006 (1)

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

2005 (1)

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

2003 (1)

2000 (2)

1998 (1)

1997 (1)

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, "Design of dynamically adjustable anamorphic fractional transformer Fourier," Opt. Commun. 136, 52-60 (1997).
[CrossRef]

1995 (1)

1994 (1)

1993 (3)

1983 (1)

1980 (1)

1979 (1)

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

Alieva, T.

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Arsenault, H. H.

Bastiaans, M.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Bitran, Y.

Braunecker, H.

Bryngdahl, O.

Crabtree, K.

Davis, J. A.

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, "Design of dynamically adjustable anamorphic fractional transformer Fourier," Opt. Commun. 136, 52-60 (1997).
[CrossRef]

Ferreira, C.

Garcia, J.

Kostenbauder, A. G.

G. Nemes and A. G. Kostenbauder, "Optical systems for rotating a beam," in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R.Martinez-Herrero, and A.Gonzales-Urena, eds. (Sociedad Española de Optica, 1993), pp. 99-109.

Lohmann, A. W.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

Macukow, B.

Malyutin, A. A.

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

Mendlovic, D.

Moreno, I.

Nemes, G.

G. Nemes and A. E. Seigman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994).
[CrossRef]

G. Nemes and A. G. Kostenbauder, "Optical systems for rotating a beam," in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R.Martinez-Herrero, and A.Gonzales-Urena, eds. (Sociedad Española de Optica, 1993), pp. 99-109.

Ozaktas, H. M.

Sahin, A.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters," Appl. Opt. 37, 2130-2141 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, "Design of dynamically adjustable anamorphic fractional transformer Fourier," Opt. Commun. 136, 52-60 (1997).
[CrossRef]

Schnell, B.

Seigman, A. E.

Shamir, J.

Simon, R.

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Wolf, K. B.

Appl. Opt. (4)

J. Opt. A, Pure Appl. Opt. (1)

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, "Design of dynamically adjustable anamorphic fractional transformer Fourier," Opt. Commun. 136, 52-60 (1997).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Opt. Lett. (1)

Quantum Electron. (1)

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

Other (3)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966).

G. Nemes and A. G. Kostenbauder, "Optical systems for rotating a beam," in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R.Martinez-Herrero, and A.Gonzales-Urena, eds. (Sociedad Española de Optica, 1993), pp. 99-109.

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

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Figures (2)

Fig. 1
Fig. 1

Optical setup. P in and P out are the input and output planes, respectively. Generalized lenses L m and free-space intervals z m are also displayed.

Fig. 2
Fig. 2

Optical system associated with L = L 1 L 2 L 3 generalized lens.

Equations (80)

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[ r o q o ] = [ A B C D ] [ r i q i ] = T [ r i q i ] .
Q = [ 0 I I 0 ] ,
AB t = BA t , CD t = DC t , AD t BC t = I ,
A t C = C t A , B t D = D t B , A t D C t B = I ,
T = [ A B C D ] = [ I 0 L I ] [ S 0 0 S 1 ] [ X Y Y X ] ,
S = ( AA t + BB t ) 1 2 = S t ,
X + i Y = ( AA t + BB t ) 1 2 ( A + i B ) ,
L = ( CA t + DB t ) ( AA t + BB t ) 1 = L t .
X f r FT = [ cos γ x 0 0 cos γ y ] , Y f r FT = [ sin γ x 0 0 sin γ y ] ,
X rot = [ cos ϑ sin ϑ sin ϑ cos ϑ ] , Y rot = 0
X gyr = [ cos α 0 0 cos α ] , Y gyr = [ 0 sin α sin α 0 ]
XY t = YX t , X t Y = Y t X , XX t + YY t = I .
L = 1 2 [ i = 1 m p i ( 1 + cos 2 φ i ) i = 1 m p i sin 2 φ i i = 1 m p i sin 2 φ i i = 1 m p i ( 1 cos 2 φ i ) ] ,
T n = [ A n B n C n D n ] = [ I z n + 1 I 0 I ] [ I 0 L n I ] [ I z n I 0 I ] . [ I 0 L 1 I ] [ I z 1 I 0 I ] .
[ A 1 B 1 C 1 D 1 ] = [ I z 2 I 0 I ] [ I 0 L 1 I ] [ I z 1 I 0 I ] = [ L 1 z 2 + I L 1 z 1 z 2 + I ( z 1 + z 2 ) L 1 L 1 z 1 + I ]
B 1 = A 1 z 1 C 1 z 1 z 2 + D 1 z 2 .
B 0 = A 0 z 1 ,
B 1 = A 1 z 1 C 1 z 1 z 2 + D 1 z 2 ,
B 2 = A 2 z 1 C 2 z 1 z 3 + D 2 z 3 + I z 2 ,
B 3 = A 3 z 1 C 3 z 1 z 4 + D 3 z 4 + L 2 z 2 z 3 + I ( z 2 + z 3 ) ,
B 4 = A 4 z 1 C 4 z 1 z 5 + D 4 z 5 + L 2 z 2 ( z 3 + z 4 ) + L 3 z 4 ( z 2 + z 3 ) + L 3 L 2 z 2 z 3 z 4 + I ( z 2 + z 3 + z 4 ) .
B n = A n z 1 C n z 1 z n + 1 + D n z n + 1 + B n 2 e ,
T = M 1 ˜ M 2 ˜ , , M m ˜ M m , , M 2 M 1 = R ̃ R ,
M k = M L k M z k = [ a b c d ] = [ I z k I L k z k L k + I ] ,
M k ˜ = M z k M L k = [ d b c a ] = [ z k L k + I z k I L k I ] ,
M k ˜ 1 = [ d b c a ] 1 = [ a t b t c t d t ] = [ a b c d ] = [ I z k I L k z k L k + I ] .
R = [ A r B r C r D r ] ,
R ̃ = [ D r t B r t C r t A r t ] .
T = R ̃ R = [ D r t A r + B r t C r D r t B r + B r t D r C r t A r + A r t C r C r t B r + A r t D r ] ,
T = [ A B C D ] = [ D r t A r + B r t C r 2 D r t B r 2 C r t A r ( D r t A r + B r t C r ) t ] .
X = X t ,
Y = Y t .
XY = YX , X 2 + Y 2 = I .
X = 2 D r t A r I ,
Y = 2 D r t B r = 2 C r t A r ,
D r t B r + C r t A r = 0 .
A r = L 1 z 2 + I ,
B r = I z 2 ,
C r = L 1 + 1 2 L 2 + 1 2 L 2 L 1 z 2 ,
D r = 1 2 L 2 z 2 + I .
X = z 2 ( 2 L 1 + L 2 + L 1 L 2 z 2 ) + I ,
Y = z 2 ( L 2 z 2 + 2 I ) .
L 1 = ( X + I ) Y 1 1 z 2 I ,
L 2 = 1 z 2 2 ( Y 2 z 2 I ) .
L 1 ( γ x , γ y ) = [ cot ( γ x 2 ) 1 z 2 0 0 cot ( γ y 2 ) 1 z 2 ] ,
L 2 ( γ x , γ y ) = 1 z 2 2 [ sin γ x 2 z 2 0 0 sin γ y 2 z 2 ] .
L 1 ( α ) = [ 1 z 2 cot ( α 2 ) cot ( α 2 ) 1 z 2 ] ,
L 2 ( α ) = 1 z 2 2 [ 2 z 2 sin α sin α 2 z 2 ] .
X = ( L 1 + 1 z 2 I ) Y I .
L 1 ( γ x , π ) = [ cot ( γ x 2 ) 1 z 2 0 0 0 ] ,
L 2 ( γ x , π ) = 1 z 2 2 [ sin γ x 2 z 2 0 0 2 z 2 ] .
L = p sin 2 ω [ sin 2 Ω cos 2 Ω cos 2 Ω sin 2 Ω ] + p 3 I ,
L 1 ( α , α ) = [ cot ( α 2 ) 1 z 2 0 0 cot ( α 2 ) 1 z 2 ] ,
X ref ( θ ) = [ cos θ sin θ sin θ cos θ ] , Y = 0 .
X rot ( θ 1 θ 2 ) = X ref ( θ 2 ) X ref ( θ 1 ) = [ cos θ 2 sin θ 2 sin θ 2 cos θ 2 ] [ cos θ 1 sin θ 1 sin θ 1 cos θ 1 ] = [ cos ( θ 1 θ 2 ) sin ( θ 1 θ 2 ) sin ( θ 1 θ 2 ) cos ( θ 1 θ 2 ) ] .
A 2 = L 1 ( z 2 + z 3 ) + L 2 z 3 + L 2 L 1 z 2 z 3 + I ,
B 2 = L 1 ( z 2 + z 3 ) z 1 + L 2 ( z 2 + z 1 ) z 3 + L 2 L 1 z 2 z 3 z 1 + I ( z 2 + z 3 + z 1 ) ,
C 2 = L 1 + L 2 + L 2 L 1 z 2 ,
D 2 = L 1 z 1 + L 2 ( z 1 + z 2 ) + L 2 L 1 z 2 z 1 + I ,
B 2 = A 2 z 1 C 2 z 1 z 3 + D 2 z 3 + I z 2 .
C 2 z 1 z 3 I z 2 = X ( z 1 + z 3 ) .
T = [ X 0 C 2 X ]
C 2 = z 1 + z 3 z 1 z 3 [ cos θ sin θ sin θ cos θ ] + [ 1 0 0 1 ] z 2 z 1 z 3 = L 1 + L 2 + L 2 L 1 z 2 .
L 1 ( θ ) = z 3 z 1 z 2 [ cos θ sin θ sin θ cos θ ] z 1 + z 2 z 1 z 2 [ 1 0 0 1 ] ,
L 2 ( θ ) = z 1 z 2 z 3 [ cos θ sin θ sin θ cos θ ] z 2 + z 3 z 2 z 3 [ 1 0 0 1 ] ,
L 3 ( θ ) = z 2 z 1 z 3 [ cos θ sin θ sin θ cos θ ] z 1 + z 3 z 1 z 3 [ 1 0 0 1 ] .
A 3 = C 3 z 4 + B 1 e L 1 + L 2 z 3 + I ,
B 3 = A 3 z 1 C 3 z 1 z 4 + D 3 z 4 + B 1 e ,
C 3 = L 3 [ L 1 ( z 2 + z 3 ) + L 2 z 3 + L 2 L 1 z 2 z 3 + I ] + [ L 1 + L 2 + L 2 L 1 z 2 ] ,
D 3 = C 3 z 1 + L 3 B 1 e + L 1 z 1 + L 2 ( z 1 + z 2 ) + L 2 L 1 z 2 z 1 + I ,
B 1 e = L 2 z 3 z 2 + I ( z 2 + z 3 ) ,
X rot = 1 ( z 1 + z 4 ) ( C 3 z 1 z 4 B 1 e ) .
L 2 = [ 0 0 0 p ] .
L 1 = ( B 1 e ) 1 ( I X rot ) 2 z I ,
L 3 = [ z ( B 1 e ) 1 ( I X rot ) + 3 X rot + I + 2 z B 1 e ] [ z X rot + B 1 e ] 1 ,
L 4 = 1 z 2 ( 2 z I + B 1 e X 1 ) .
L 1 = 1 2 z [ 1 + cos θ sin θ sin θ 1 cos θ ] 1 z [ 1 0 0 2 ] ,
L 2 = 4 z [ 0 0 0 1 ] ,
L 3 = 1 2 z [ 1 + cos θ sin θ sin θ 1 cos θ ] 1 z [ 1 0 0 2 ] ,
L 4 = 2 z [ 1 + cos θ sin θ sin θ 1 cos θ ] .

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