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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 5 — Feb. 10, 2012
  • pp: 644–650

Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays

Jeffrey A. Davis and Don M. Cottrell  »View Author Affiliations

Applied Optics, Vol. 51, Issue 5, pp. 644-650 (2012)

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We reexamine a previously published algorithm for performing a fast Fresnel diffraction calculation that uses two Fourier transform operations and is computationally much faster than the conventional approach. We analyze this technique using a ray matrix analysis and find explicit expressions for the maximum and minimum distances over which this algorithm is accurate. These distances coincide with the experimental distances that are appropriate when patterns are encoded onto liquid crystal displays. We show two examples that confirm our ideas. We expect that these results will be very useful for computational comparison with experimental studies of a variety of diffraction phenomena.

© 2012 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.2730) Geometric optics : Matrix methods in paraxial optics
(070.2025) Fourier optics and signal processing : Discrete optical signal processing

ToC Category:
Fourier Optics and Signal Processing

Original Manuscript: November 11, 2011
Revised Manuscript: November 18, 2011
Manuscript Accepted: November 21, 2011
Published: February 9, 2012

Jeffrey A. Davis and Don M. Cottrell, "Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays," Appl. Opt. 51, 644-650 (2012)

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  1. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  2. J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965). [CrossRef]
  3. F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994).
  4. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).
  5. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).
  6. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. 32, 844–846 (2007). [CrossRef]
  7. J. A. Davis, B. M. L. Pascoquin, C. S. Tuvey, and D. M. Cottrell, “Fourier transform pupil functions for modifying the depth of focus of optical imaging systems,” Appl. Opt. 48, 4893–4898 (2009). [CrossRef]
  8. I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35, 1536–1538 (2010). [CrossRef]
  9. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995). [CrossRef]
  10. D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997). [CrossRef]
  11. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999). [CrossRef]
  12. B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917–927 (2005). [CrossRef]
  13. B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005). [CrossRef]
  14. A. Stern, “Why is the linear canonical transform so little known?,” in Proceedings of 5th International Workshop on Information Optics, G. Cristóbal, B. Javidi, and S. Vallmitjana, eds (Springer, 2006), pp. 225–234.
  15. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006). [CrossRef]
  16. J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008). [CrossRef]
  17. J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010). [CrossRef]
  18. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 8.
  19. D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
  20. J. A. Davis, M. A. Waring, G. W. Bach, R. A. Lilly, and D. M. Cottrell, “Compact optical correlator design,” Appl. Opt. 28, 10–11 (1989). [CrossRef]
  21. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written on programmable spatial light modulators,” Appl. Opt. 29, 2505–2509 (1990). [CrossRef]
  22. J. A. Davis, B. A. Slovick, C. S. Tuvey, and Don M. Cottrell, “High diffraction efficiency from one- and two-dimensional Nyquist frequency binary phase gratings,” Appl. Opt. 47, 2828–2834 (2008).
  23. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chap. 2.
  24. J. A. Davis and R. A. Lilly, “Ray-matrix approach for diffractive optics,” Appl. Opt. 32, 155–159 (1993). [CrossRef]
  25. 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).
  26. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).
  27. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990). [CrossRef]

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