Abstract

We study fractional Fourier transformation in the metaxial regime of geometric optics. Two commonly used optical arrangements that perform fractional Fourier transformation are a symmetric thick lens and a length of graded-index waveguide. By means of Lie methods in phase space, we can correct some of their aberrations: for the first, through deforming the lens surfaces to a polynomial shape, and for the second, by warping the output screen at the end of the waveguide. We correct the planar cases to third, fifth, and seventh aberration orders; checks are provided on the convergence of aberration series in phase space. We add some comments on the usefulness of these corrected devices for fractional transformers in scalar wave optics.

© 1999 Optical Society of America

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References

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  1. K. B. Wolf, “The Fourier transform in metaxial geometric optics,” J. Opt. Soc. Am. A 8, 1399–1403 (1991).
    [CrossRef]
  2. V. I. Man’ko, K. B. Wolf, “The map between Heisenberg–Weyl and Euclidean optics is comatic,” in Lie Methods in Optics, Vol. 352 of Lecture Notes in Physics, J. Sánchez-Mondragón, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1989), Chap. 7, pp. 163–197.
  3. K. B. Wolf, G. Krötzsch, “mexLIE 2, a set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS–UNAM (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México) No. 10 (June, 1995).
  4. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
    [CrossRef]
  5. A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. (N.Y.) 17, 2215–2227 (1976);A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982); S. Steinberg, “Factored product expansions of nonlinear differential equations,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 108–115 (1984); S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
    [CrossRef]
  6. K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
    [CrossRef]
  7. H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  8. M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
    [CrossRef]
  9. E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997).
    [CrossRef]
  10. K. B. Wolf, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991).
    [CrossRef]
  11. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transforms,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
    [CrossRef]
  12. M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the XVth Solvay Conference in Physics, E. Progogine, ed. (Gordon & Breach, New York, 1974);C. Quesne, M. Moshinsky, “Linear canonical transformations and their unitary representations,” J. Math. Phys. (N.Y.) 12, 1772–1780 (1971); M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. (N.Y.) 12, 1780–1783 (1971).
    [CrossRef]
  13. K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. (N.Y.) 15, 1295–1301 (1974);K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
    [CrossRef]
  14. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  15. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–158 (1980).
    [CrossRef]
  16. O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 5, pp. 159–182.
  17. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their implementations. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–750 (1995); D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  18. M. A. Alonso, G. W. Forbes, “Uniform asymptotic expansion for wave propagators via fractional transformations,” J. Opt. Soc. Am. A 14, 1279–1292 (1997).
    [CrossRef]
  19. B.-Zh. Dong, Y. Zhang, B.-Y. Gu, G.-Zh. Yang, “Numerical investigation of phase retrieval of a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997);Y. Zhang, B.-Zh. Dong, B.-Y. Gu, G.-Zh. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
    [CrossRef]
  20. K. B. Wolf, “The Heisenberg–Weyl ring in quantum mechanics,” in Group Theory and Its Applications, III, E. M. Loebl, ed. (Academic, New York, 1975), pp. 189–247.
  21. M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
    [CrossRef]
  22. A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
    [CrossRef]

1997 (4)

1993 (1)

1991 (2)

1988 (1)

1986 (2)

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
[CrossRef]

1980 (2)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–158 (1980).
[CrossRef]

1976 (1)

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. (N.Y.) 17, 2215–2227 (1976);A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982); S. Steinberg, “Factored product expansions of nonlinear differential equations,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 108–115 (1984); S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

1974 (1)

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. (N.Y.) 15, 1295–1301 (1974);K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
[CrossRef]

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transforms,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef]

Alonso, M. A.

Atakishiyev, N. M.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Atzema, E. J.

E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997).
[CrossRef]

Buchdahl, H.

H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Castaños, O.

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 5, pp. 159–182.

Chumakov, S. M.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transforms,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef]

Dong, B.-Zh.

Dragt, A. J.

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. (N.Y.) 17, 2215–2227 (1976);A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982); S. Steinberg, “Factored product expansions of nonlinear differential equations,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 108–115 (1984); S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Finn, J.

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. (N.Y.) 17, 2215–2227 (1976);A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982); S. Steinberg, “Factored product expansions of nonlinear differential equations,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 108–115 (1984); S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

Forbes, G. W.

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Garci´a-Bullé, M.

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
[CrossRef]

Gu, B.-Y.

Krötzsch, G.

E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997).
[CrossRef]

K. B. Wolf, G. Krötzsch, “mexLIE 2, a set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS–UNAM (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México) No. 10 (June, 1995).

Lassner, W.

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
[CrossRef]

López Moreno, E.

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 5, pp. 159–182.

Man’ko, V. I.

V. I. Man’ko, K. B. Wolf, “The map between Heisenberg–Weyl and Euclidean optics is comatic,” in Lie Methods in Optics, Vol. 352 of Lecture Notes in Physics, J. Sánchez-Mondragón, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1989), Chap. 7, pp. 163–197.

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the XVth Solvay Conference in Physics, E. Progogine, ed. (Gordon & Breach, New York, 1974);C. Quesne, M. Moshinsky, “Linear canonical transformations and their unitary representations,” J. Math. Phys. (N.Y.) 12, 1772–1780 (1971); M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. (N.Y.) 12, 1780–1783 (1971).
[CrossRef]

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Navarro-Saad, M.

Nazarathy, M.

Ozaktas, H. M.

Quesne, C.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the XVth Solvay Conference in Physics, E. Progogine, ed. (Gordon & Breach, New York, 1974);C. Quesne, M. Moshinsky, “Linear canonical transformations and their unitary representations,” J. Math. Phys. (N.Y.) 12, 1772–1780 (1971); M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. (N.Y.) 12, 1780–1783 (1971).
[CrossRef]

Rivera, A. L.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Shamir, J.

Wolf, K. B.

E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

K. B. Wolf, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991).
[CrossRef]

K. B. Wolf, “The Fourier transform in metaxial geometric optics,” J. Opt. Soc. Am. A 8, 1399–1403 (1991).
[CrossRef]

K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
[CrossRef]

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. (N.Y.) 15, 1295–1301 (1974);K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

V. I. Man’ko, K. B. Wolf, “The map between Heisenberg–Weyl and Euclidean optics is comatic,” in Lie Methods in Optics, Vol. 352 of Lecture Notes in Physics, J. Sánchez-Mondragón, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1989), Chap. 7, pp. 163–197.

K. B. Wolf, G. Krötzsch, “mexLIE 2, a set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS–UNAM (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México) No. 10 (June, 1995).

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 5, pp. 159–182.

K. B. Wolf, “The Heisenberg–Weyl ring in quantum mechanics,” in Group Theory and Its Applications, III, E. M. Loebl, ed. (Academic, New York, 1975), pp. 189–247.

Yang, G.-Zh.

Zhang, Y.

J. Inst. Math. Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Math. Phys. (N.Y.) (3)

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math. Phys. (N.Y.) 17, 2215–2227 (1976);A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982); S. Steinberg, “Factored product expansions of nonlinear differential equations,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 108–115 (1984); S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. (N.Y.) 27, 29–36 (1986).
[CrossRef]

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. (N.Y.) 15, 1295–1301 (1974);K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

K. B. Wolf, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their implementations. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–750 (1995); D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

M. A. Alonso, G. W. Forbes, “Uniform asymptotic expansion for wave propagators via fractional transformations,” J. Opt. Soc. Am. A 14, 1279–1292 (1997).
[CrossRef]

B.-Zh. Dong, Y. Zhang, B.-Y. Gu, G.-Zh. Yang, “Numerical investigation of phase retrieval of a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997);Y. Zhang, B.-Zh. Dong, B.-Y. Gu, G.-Zh. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

K. B. Wolf, “The Fourier transform in metaxial geometric optics,” J. Opt. Soc. Am. A 8, 1399–1403 (1991).
[CrossRef]

J. Phys. A (1)

E. J. Atzema, G. Krötzsch, K. B. Wolf, “Canonical transformations to warped surfaces: correction of aberrated optical images,” J. Phys. A 30, 5793–5803 (1997).
[CrossRef]

Phys. Rev. A (1)

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transforms,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef]

Other (7)

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the XVth Solvay Conference in Physics, E. Progogine, ed. (Gordon & Breach, New York, 1974);C. Quesne, M. Moshinsky, “Linear canonical transformations and their unitary representations,” J. Math. Phys. (N.Y.) 12, 1772–1780 (1971); M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. (N.Y.) 12, 1780–1783 (1971).
[CrossRef]

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 5, pp. 159–182.

K. B. Wolf, “The Heisenberg–Weyl ring in quantum mechanics,” in Group Theory and Its Applications, III, E. M. Loebl, ed. (Academic, New York, 1975), pp. 189–247.

H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

V. I. Man’ko, K. B. Wolf, “The map between Heisenberg–Weyl and Euclidean optics is comatic,” in Lie Methods in Optics, Vol. 352 of Lecture Notes in Physics, J. Sánchez-Mondragón, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1989), Chap. 7, pp. 163–197.

K. B. Wolf, G. Krötzsch, “mexLIE 2, a set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS–UNAM (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México) No. 10 (June, 1995).

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 4, pp. 105–158.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Paraxial fractional Fourier transformations are rigid rotations of the phase plane around its origin, the optical center corresponding to the design ray, shown here for α=0, 14π, 12π (the Fourier transform), and π (a perfect, inverted imager).

Fig. 2
Fig. 2

Fractional Fourier transformer built as a symmetric optical arrangement with a thick lens (optical length t) between the object and screen planes (at distances z). The (circular) surfaces of the lens will be corrected to a polynomial shape ζ(q).

Fig. 3
Fig. 3

Phase-space maps (in the interaction picture) of Fourier transformers (α=12π) built with a symmetric lens. Top row: uncorrected system with circular refracting line (r=1) computed to third, fifth, and seventh aberration orders; bottom row: corrected systems with polynomial-line lenses to the same aberration orders.

Fig. 4
Fig. 4

Phase-space maps of an uncorrected lens fractional Fourier transformer (with circular refracting lines) for α=15°, 30°, 45°,,180°, computed to seventh aberration order.

Fig. 5
Fig. 5

Corrected fractional Fourier transform map of phase space (with the polynomial lens coefficients of Table 1) for α=15°, 30°, 45°,, 180°, to seventh aberration order.

Fig. 6
Fig. 6

A length of a graded-index medium (a waveguide) acts as an (aberrating) fractional Fourier transformer. The correction tactic is applied to warp the output screen to a polynomial shape ζ(q).

Fig. 7
Fig. 7

Phase-space maps (in the interaction picture) of a (an inverse) Fourier transformer (α=-12π) built with an elliptic-index-profile waveguide. Top row: computation to third, fifth, and seventh aberration orders; Bottom row: corrected map to the same aberration orders.

Fig. 8
Fig. 8

Uncorrected waveguide arrangement for fractional Fourier transformation (in the interaction picture) for α=15°, 30°, 45°,, 180°, to seventh aberration order.

Fig. 9
Fig. 9

Waveguide arrangement for fractional Fourier transformation corrected by warping the exit sensor line so that there is spherical aberration (in the interaction picture) to seventh aberration order for transform angles of α=15°, 30°, 45°,, 180°.

Tables (2)

Tables Icon

Table 1 Coefficients of the Polynomial Refracting Line That Correct the Lens Arrangement for Fractional Fourier Transformation of Angle α in Steps of 15° to Third, Fifth, and Seventh Aberration Orders

Tables Icon

Table 2 Coefficients of the Polynomial Exit Sensor Line That Correct the Waveguide Arrangements for Fractional Fourier Transformation of Angle α in Steps of 15° to Third, Fifth, and Seventh Aberration Ordersa

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

{F(p, q), G(p, q)}=F(p, q)qG(p, q)p-F(p, q)pG(p, q)q,
{q, p}=1,{q, 1}=0,{p, 1}=0.
Φ(α)=cos α-sin αsin αcos α.
G{0; Φ(α)}pq=[Φ(α)]-1pq=pαqα=p cos α+q sin α-p sin α+q cos α.
F(α)=G{A; Φ(α)}=G{A; 1}G{0; Φ(α)},
G{A; 1}=×exp{A3, °}exp{A5/2, °}exp{A2, °}×exp{A3/2, °},
exp{Ak, °}=1+{Ak, °}+12!{Ak, {Ak, °}}+,
{Ak, °}=Ak(p, q)qp-Ak(p, q)pq,
Ak(p, q)=m=-kkAmkpk+mqk-m.
G{A; M}G{B; N}=G{A  D(M)B; MN},
[A  D(M)B]k=Ak+Dk(M)Bk+multilineartermsofhigherranks.
A2(p, q)=A22p4+A12p3q+A02p2q2+A-12pq3+A-22q4.
G{A; 1}pq(1+{A2, °})pq
=p+A12p3+2A02p2q+3A-12pq2+4A-22q3q-4A22p3-3A12p2q-2A02pq2-A-12q3=pA[3](p, q)qA[3](p, q),
pA[3](p, 0)=p+A12p3,pA[3](0, q)=4A-22q3,
qA[3](p, 0)=-4A22p3,qA[3](0, q)=q-A-12q3.
ΔvA(p)=1+3A12p2-12A22p2Δp,
ΔwA(q)=12A-22q21-3A-12q2Δq.
ΔvA·ΔwA=|ΔvA||ΔwA|cos ω=[12(A-22q2-A22p2)+36(A12A-22+A22A-12)p2q2]ΔpΔq.
A22=A-22,
pA[5](p, q)=(1+{A3, °})1+{A2, °}+12!{A2, {A2, °}}p,
pA[7](p, q)=(1+{A4, °})(1+{A3, °})1+{A2, °}+12!{A2, {A2, °}}+13!{A2, {A2, {A2, °}}}p,
(p=n sin θ, q)(p, q+z tan θ)=(p, q+zp/n2-p2).
Z(z, n)=exp(z{n2-p2, °})=exp-z16n5{p6, °}exp-z8n3{p4, °}×exp-z2n{p2, °}
=G{, Z3, Z2; Z}
Z(z)=10-z/n1
ζ(q)=ζ2q2+ζ4q4+ζ6q6+ζ8q8+,
S(n1, n2; ζ)=G{, S4, S3, S2; S},
S=1g01
S2(p, q)=0p4+0p3q+121n2-1n1ζ2p2q2-2n2-n1n2ζ22pq3+(n2-n1)×2n2-n1n2ζ23-ζ4q4.
A(z, n; α)=Z(z, 1)S(1, n; ζ)Z(τ, n)S(n, 1;-ζ)Z(z, 1).
A(z, n; α)=10-z1 1g01 10-t1 1g01 10-z1=1-gt-2gz+g2ztg(2-gt)-2z-t+2z2g+2gzt-g2z2t1-gt-2gz+g2zt,
z=1g+cotα2,τn=t=sin α+2gg2.
ζr(q)=r-r2-q2q22r+q48r3+q616r5+5q8128r7+,
ne(q)=(ν2-μ2q2)1/2=ν-μ22νq2-μ48ν3q4-μ616ν5q6-
E(ν, μ; z)=exp[z{(ν2-p2-μ2q2)1/2, °}]=×exp-z8ν3{(p2+μ2q2)2, °}×exp-z2ν{p2+μ2q2, °}
=G{, E2; E(z)},
E(z)=cosμzνμ sinμzν-1μsinμzνcosμzν
E2(p, q; z)=-(z/8ν3)p4-(zμ2/4ν3)p2q2-(zμ4/8ν3)q4,
E(ν; z)pq=cos κz-sin κzsin κzcos κzpq=P(p, q; z)Q(p, q; z),
κ(p, q)=1[ν2-(p2+q2)]1/2=1ν+12ν3(p2+q2)+38ν5(p2+q2)2+.
q¯=R(ne; ζ),q=Q(p, q; ζ(q¯)),
p¯=R(ne; ζ),p=P(p, q; ζ(q¯))+[ν2-(p2+q2)]1/2dζ(q¯)dq¯.
R(ne; ζ)=G{, R3, R2; R},
R=1-2νζ201
R2(p, q)=-(ζ2/2ν)p2q2+(νζ4-ζ2/2ν)q4,
A(z, ne; α)=E(ne; z)R(ne; ζ).
A2=E2+D(2)(Φ(α))R2,
A22=-(α/8ν2)+νζ4 sin4 α,
A02=-(α/4ν2)+6νζ4 sin2 α cos2 α,
A-22=-(α/8ν2)+νζ4 cos4 α,

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