Abstract

We show that an arbitrary paraxial optical system, compounded with its reflection in an appropriately warped mirror, is a pure fractional Fourier transformer between coincident input and output planes. The geometric action of reflection on optical systems is introduced axiomatically and is developed in the paraxial regime. The correction of aberrations by warp of the mirror is briefly addressed.

© 2002 Optical Society of America

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  1. D. Mendlovic, H. M. Ozaktas, “Fourier transforms of fractional order and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).
  3. 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]
  4. K. B. Wolf, G. Krötzsch, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).
  5. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  6. R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
    [CrossRef]
  7. M. Moshinsky, “Canonical transformations in quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).
    [CrossRef]
  8. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1985).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  10. 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]
  11. R. Simon, “Peres–Hodorecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
    [CrossRef] [PubMed]
  12. A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  13. S. Steinberg, “Lie series, Lie transformations, and their applications” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 3, pp. 45–102.
  14. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.
  15. O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.
  16. A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
    [CrossRef]
  17. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  18. K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).
  19. K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
    [CrossRef]
  20. See, e.g., H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  21. K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Non-linear Phenomena, Proceedings of the International School on Applied Mathematics, Centro Internacional de Fı́sica, Paipa, Colombia, D. Levi, P. Winternitz eds., (World Scientific, Singapore, 1988), pp. 376–429.
  22. K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
    [CrossRef]
  23. K. B. Wolf, G. Krötzsch, “mexLIE 2, A set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS-UNAM No. 10 (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México, 1995).
  24. K. B. Wolf, G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
    [CrossRef]

2001 (1)

K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).

2000 (3)

1999 (1)

1997 (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]

1995 (1)

1993 (1)

1991 (2)

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

K. B. Wolf, G. Krötzsch, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).

1988 (1)

1987 (1)

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

1986 (1)

1985 (2)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1985).
[CrossRef]

1982 (1)

1973 (1)

M. Moshinsky, “Canonical transformations in quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).
[CrossRef]

Alper Kutay, M.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

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. A.

See, e.g., H. A. 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, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.

Dragt, A. J.

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.

Krötzsch, G.

K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).

K. B. Wolf, G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
[CrossRef]

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, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

K. B. Wolf, G. Krötzsch, “mexLIE 2, A set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS-UNAM No. 10 (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México, 1995).

López Moreno, E.

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, “Canonical transformations in quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).
[CrossRef]

Mukunda, N.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Navarro-Saad, M.

Ozaktas, H. M.

Simon, R.

R. Simon, “Peres–Hodorecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Steinberg, S.

S. Steinberg, “Lie series, Lie transformations, and their applications” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 3, pp. 45–102.

Sudarshan, E. C. G.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Wolf, K. B.

K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

K. B. Wolf, G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
[CrossRef]

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, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (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]

K. B. Wolf, G. Krötzsch, “mexLIE 2, A set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS-UNAM No. 10 (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México, 1995).

K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Non-linear Phenomena, Proceedings of the International School on Applied Mathematics, Centro Internacional de Fı́sica, Paipa, Colombia, D. Levi, P. Winternitz eds., (World Scientific, Singapore, 1988), pp. 376–429.

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

J. Opt. Soc. Am. (1)

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

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]

J. Symb. Comput. (1)

K. B. Wolf, G. Krötzsch, “Group-classified polynomials of phase space in higher-order aberration expansions,” J. Symb. Comput. 12, 673–695 (1991).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

R. Simon, “Peres–Hodorecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

Rev. Mex. Fi´s. (2)

K. B. Wolf, G. Krötzsch, “El problema de las tres lentes,” Rev. Mex. Fı́s. 47, 291–298 (2001).

K. B. Wolf, G. Krötzsch, “La transformación raı́z de superficies refractantes y espejos,” Rev. Mex. Fı́s. 37, 540–554 (1991).

SIAM J. Appl. Math. (1)

M. Moshinsky, “Canonical transformations in quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).
[CrossRef]

Other (7)

K. B. Wolf, G. Krötzsch, “mexLIE 2, A set of symbolic computation functions for geometric aberration optics,” Manuales IIMAS-UNAM No. 10 (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México, 1995).

See, e.g., H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

K. B. Wolf, “Nonlinearity in aberration optics,” in Symmetries and Non-linear Phenomena, Proceedings of the International School on Applied Mathematics, Centro Internacional de Fı́sica, Paipa, Colombia, D. Levi, P. Winternitz eds., (World Scientific, Singapore, 1988), pp. 376–429.

S. Steinberg, “Lie series, Lie transformations, and their applications” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 3, pp. 45–102.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 4, pp. 105–158.

O. Castaños, E. López Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics, (Springer Verlag, Heidelberg, Germany1986), Chap. 5, pp. 159–182.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

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Figures (5)

Fig. 1
Fig. 1

Reflection of free propagation in a mirror on the q plane, seen as free propagation into the mirror world.

Fig. 2
Fig. 2

Reflection of a refracting surface z=ζ(q) between media m and n is, in the mirror world, a refracting surface z=-ζ(q) between media n and m.

Fig. 3
Fig. 3

Reflection is an antihomomorphism: The reflected concatenation of two systems G1 and G2 is the product of the reflected systems, G1¯ and G2¯, in reversed order.

Fig. 4
Fig. 4

Example of a pure fractional Fourier transformer composed of two media, m and n, separated by a spherical interface. A movable, warpable mirror is indicated by dashed lines. The distance z0 is fixed, while for the mirror z=nχ>0 is adjustable.

Fig. 5
Fig. 5

(a) Mirror warp coefficient Z(χ) as a function of χ=23z>0 in the Fourier transformer of the Fig. 4. Bullets mark χ=2, where the one-pass system is an impure Fourier transformer, and χ=3 where it is an impure inverting imager. (b) Locus of the one-pass systems in the (a, c) plane of matrix elements. (c) Fourier–Iwasawa angle θ counted modulo 4π. The heavy curve shows the range of fractional Fourier transform angles of these systems; the range includes the second metaplectic unit 1 but not the identity 1.

Equations (33)

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

Fn,z¯=Fn,z.
Smn;ζ¯=Sn;-ζ.
G==G,
G1G2¯=G2¯G1¯.
Smn;ζ=Rm;ζRn;ζ-1,
Smn;ζ¯=Rm;ζRn;ζ-1¯=Rn;ζ-1¯Rm;ζ¯Snm;-ζ=Rn;-ζRm;-ζ-1.
Rm;ζ¯=Rm;-ζ-1.
GIIGG¯.
GIIGζIIGRn;ζGRn;ζ¯=GMn;ζG¯
=GζGζ¯,GζGRn;ζ,
Mn;ζRn;ζRn;ζ¯=Rn;ζRn;-ζ-1,
MJMT=J,J=01-10=J-1.
M=ABCD,
ABT,ATC,BTD,CDT symmetric,ADT-BCT=1.
M-1=JMTJ=DT-BT-CTAT.
10z/n11¯=10z/n11.
1G01¯=1G01,GT=G.
MM¯=ABCD¯=DTBTCTAT.
M¯=KM-1K,K=-1001=K-1.
MII=MM¯=ADT+BCT2ABT2CDT(ADT+BCT)T,
M=ABCD=Re UIm U-Im URe U E00ET-1 1G01.
2nZ=-GMRm,Z=Re UIm U-Im URe U E00ET-1.
Mn;ZII=MRn;ZRn;Z¯M¯=Re UUTIm UUT-Im UUTRe UUT,
M=abcd=cos θsin θ-sin θcos θ E00E-1 1G01.
M=10-61 11201 10-χ1=1-12 χ12-6+2χ-2.
θ=arg(a-ic)=arg[(1+6i)-(12+2i)χ],
E=+(a2+c2)1/2=+(37-25χ+174χ2)1/2> 0,
G=ab+cda2+c2=50-17χ148-100χ+17χ2=2nζ2=3ζ2.
G{A;M}(k)  G{aberrationpart;paraxialpart},
=exp{Ak,}××exp{A3,}×exp{A2,}× G(M),
exp{Aj,}exp{Aj,}¯=exp{A¯j,},
A¯j(|p|2, pq, |q|2)=Aj(|p|2, -pq, |q|2),
G{A; M}¯=G{0; K}G{-A; M}-1G{0; K}.

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