Abstract

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

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References

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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]

2010

2009

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]

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).

2008

2007

2006

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]

2005

1999

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]

1997

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]

1995

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

1993

1990

1989

1984

D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

1965

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1836

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Bach, G. W.

Bandres, M. A.

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).

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).

Bernardo, L. M.

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]

Campos, J.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Cottrell, D. M.

Cottrell, Don M.

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).

Davis, J. A.

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]

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).

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]

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).

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).

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]

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).

J. A. Davis and R. A. Lilly, “Ray-matrix approach for diffractive optics,” Appl. Opt. 32, 155–159 (1993).
[CrossRef]

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]

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]

Ferreira, C.

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).

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]

Garcia, J.

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]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 8.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Gori, F.

F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994).

Healy, J. J.

Hedman, T. R.

Hennelly, B. M.

Iemmi, C.

Kelly, D. P.

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]

Konforti, N.

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]

Leger, J. R.

Lilly, R. A.

López-Coronado, O.

Marinho, F.

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]

Mas, D.

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]

Mateos, F.

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).

McAuley, K. P.

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).

Mendlovic, D.

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]

Mitry, M. J.

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).

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).

Moreno, I.

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]

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).

Paek, E. G.

D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Pascoquin, B. M. L.

Psaltis, D.

D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Rhodes, W. T.

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]

Ruiz, I.

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).

Sánchez-López, M. M.

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).

Sheridan, J. T.

Slovick, B. A.

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).

Stern, A.

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.

Swanson, G. J.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Tuvey, C. S.

Venkatesh, AS. S.

D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Waring, M. A.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chap. 2.

Yuan, X.-C.

Yzuel, M. J.

Zalevsky, A.

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]

Zhang, N.

Appl. Opt.

Eur. J. Phys.

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).

J. Mod. Opt.

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]

J. Opt. Soc. Am. A

Math. Comput.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun.

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]

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Opt. Eng.

D. Psaltis, E. G. Paek, and AS. S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

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]

Opt. Express

Opt. Lett.

Philos. Mag.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Other

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chap. 2.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 8.

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.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

F. Gori, “Why is the Fresnel transform so little known?,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

Fresnel lens phase patterns for the Nyquist focal length lens and a lens pattern having a focal length of fN/3.

Fig. 2.
Fig. 2.

Magnitude and phase of the Fourier spectrum for a n=6 accelerating parabolic beam.

Fig. 3.
Fig. 3.

Comparison of computational results for the direct Fresnel diffraction calculation (left column) and the fast Fresnel diffraction calculation (right column) for a n=6 accelerating parabolic beam at distances from the Fourier transform plane of (from top to bottom) z=0, 600, 1200, 1800, 2400 mm.

Fig. 4.
Fig. 4.

Comparison of computational results for the intensity for the direct Fresnel diffraction calculation (left column) and the fast Fresnel diffraction calculation (right column) for an amplitude grating having a period of two pixels at distances from the grating of (from top to bottom) z=0, 2.28, 4.57, 6.85, 9.14, 11.43, 13.71 mm.

Equations (18)

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

G^F(x2,y2)=exp(ikz)iλzg(x1,y1)exp{ik2z[(x2x1)2+(y2y1)2]}dx1dy1.
G^F(x2,y2)=exp(ikz)iλzexp[ik(x22+y222z)]g(x1,y1)exp[ik(x12+y122z)]exp[ik(x1x2z+y1y2z)]dx1dy1.
G^F(x2,y2)=exp(ikz)iλzZ(x2,y2)I[g(x1,y1)Z(x1,y1)].
G^F(x2,y2)=g(x1,y1)Z(x2x1,y2y1)dx1=g(x1,y1)Z(x1,y1).
I[g(x1,y1)Z(x1,y1)]=G(ξ,η)Z*(ξ,η)=G(ξ,η)exp[iπ(ξ2+η2)λz].
G^F(x2,y2)=I1[G(ξ,η)Z*(ξ,η)].
r¯1=(r1r1).
(r2r2)=(ABCD)(r1r1).
T=(1d01),
Z*=(101/f1).
I=(1f01)(101/f1)(1f01)(0fFFT1/fFFT0).
I1=(0fFFT1/fFFT0).
G^F=(1z01).
G^F=(1z01)=ZIZ=(101/f11)(0fFFT1/fFFT0)(101/f11)=(fFFT/f1fFFTfFFT/f121/fFFTfFFT/f1).
G^F=(1z01)=I1Z*I=(0fFFT1/fFFT0)(101/f11)(0fFFT1/fFFT0)=(1fFFT2/f101).
f1=fFFT2/z.
z=fFFT2/f1.
NΔ2/λ=zMAX>z>zMIN=0.

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