## Metaxial correction of fractional Fourier transformers

JOSA A, Vol. 16, Issue 4, pp. 821-830 (1999)

http://dx.doi.org/10.1364/JOSAA.16.000821

Enhanced HTML Acrobat PDF (388 KB)

### 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

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(080.1010) Geometric optics : Aberrations (global)

(080.2720) Geometric optics : Mathematical methods (general)

**History**

Original Manuscript: June 29, 1998

Revised Manuscript: October 8, 1998

Manuscript Accepted: November 13, 1998

Published: April 1, 1999

**Citation**

Kurt Bernardo Wolf and Guillermo Krötzsch, "Metaxial correction of fractional Fourier transformers," J. Opt. Soc. Am. A **16**, 821-830 (1999)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-4-821

Sort: Year | Journal | Reset

### References

- K. B. Wolf, “The Fourier transform in metaxial geometric optics,” J. Opt. Soc. Am. A 8, 1399–1403 (1991). [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).
- 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]
- 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]
- K. B. Wolf, “Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988). [CrossRef]
- H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
- 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]
- 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, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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.
- 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, “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.
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.