Abstract

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.

© 2012 Optical Society of America

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  1. H. M. Ozaktas, S. Ö. Arık, and T. Coşkun, Opt. Lett. 36, 2524 (2011).
    [CrossRef]
  2. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).
  3. H. M. Ozaktas and D. MendlovicJ. Opt. Soc. Am. A 12, 743 (1995).
    [CrossRef]
  4. H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
    [CrossRef]
  5. H. M. Ozaktas, A. Koc, I. Sari, and M. A. Kutay, Opt. Lett. 31, 35 (2006).
    [CrossRef]
  6. A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
    [CrossRef]
  7. F. S. Oktem and H. M. Ozaktas, IEEE Signal Process. Lett. 16, 727 (2009).
    [CrossRef]
  8. F. S. Oktem and H. M. Ozaktas, J. Opt. Soc. Am. A 27, 1885 (2010).
    [CrossRef]
  9. L. OnuralAppl. Opt. 39, 5929 (2000).
    [CrossRef]
  10. D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
    [CrossRef]
  11. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
    [CrossRef]
  12. B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A 22, 917 (2005).
    [CrossRef]
  13. B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A 22, 928 (2005).
    [CrossRef]
  14. J. J. Healy, B. M. Hennelly, and J. T. Sheridan, Opt. Lett. 33, 2599 (2008).
    [CrossRef]
  15. J. J. Healy and J. T. Sheridan, J. Opt. Soc. Am. A 27, 21 (2010).
    [CrossRef]
  16. J. J. Healy and J. T. Sheridan, Opt. Lett. 35, 947 (2010).
    [CrossRef]
  17. J. J. Healy and J. T. Sheridan, J. Opt. Soc. Am. A 28, 786 (2011).
    [CrossRef]
  18. H. M. Ozaktas, Int. J. Elect. Eng. Edu. 31, 152 (1994).

2011

2010

2009

F. S. Oktem and H. M. Ozaktas, IEEE Signal Process. Lett. 16, 727 (2009).
[CrossRef]

2008

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, Opt. Lett. 33, 2599 (2008).
[CrossRef]

2006

2005

2000

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

1997

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

1995

1994

H. M. Ozaktas, Int. J. Elect. Eng. Edu. 31, 152 (1994).

Arik, S. Ö.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Candan, C.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Coskun, T.

Erden, M. F.

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Healy, J. J.

Hennelly, B. M.

Koc, A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

H. M. Ozaktas, A. Koc, I. Sari, and M. A. Kutay, Opt. Lett. 31, 35 (2006).
[CrossRef]

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Kutay, M. A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

H. M. Ozaktas, A. Koc, I. Sari, and M. A. Kutay, Opt. Lett. 31, 35 (2006).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

H. M. Ozaktas and D. MendlovicJ. Opt. Soc. Am. A 12, 743 (1995).
[CrossRef]

Oktem, F. S.

F. S. Oktem and H. M. Ozaktas, J. Opt. Soc. Am. A 27, 1885 (2010).
[CrossRef]

F. S. Oktem and H. M. Ozaktas, IEEE Signal Process. Lett. 16, 727 (2009).
[CrossRef]

Onural, L.

Ozaktas, H. M.

H. M. Ozaktas, S. Ö. Arık, and T. Coşkun, Opt. Lett. 36, 2524 (2011).
[CrossRef]

F. S. Oktem and H. M. Ozaktas, J. Opt. Soc. Am. A 27, 1885 (2010).
[CrossRef]

F. S. Oktem and H. M. Ozaktas, IEEE Signal Process. Lett. 16, 727 (2009).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

H. M. Ozaktas, A. Koc, I. Sari, and M. A. Kutay, Opt. Lett. 31, 35 (2006).
[CrossRef]

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

H. M. Ozaktas and D. MendlovicJ. Opt. Soc. Am. A 12, 743 (1995).
[CrossRef]

H. M. Ozaktas, Int. J. Elect. Eng. Edu. 31, 152 (1994).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Sari, I.

Sheridan, J. T.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Appl. Opt.

IEEE Signal Process. Lett.

F. S. Oktem and H. M. Ozaktas, IEEE Signal Process. Lett. 16, 727 (2009).
[CrossRef]

IEEE Trans. Signal Process.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, IEEE Trans. Signal Process. 56, 2383 (2008).
[CrossRef]

Int. J. Elect. Eng. Edu.

H. M. Ozaktas, Int. J. Elect. Eng. Edu. 31, 152 (1994).

J. Mod. Opt.

D. Mendlovic, Z. Zalevsky, and N. Konforti, J. Mod. Opt. 44, 407 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

H. M. Ozaktas and M. F. Erden, Opt. Commun. 143, 75 (1997).
[CrossRef]

Opt. Lett.

Other

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

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Figures (2)

Fig. 1.
Fig. 1.

Truncation of G^(σx,σz) in the σz-σx plane.

Fig. 2.
Fig. 2.

Structure grid for Fresnel diffraction.

Equations (7)

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

G^(σx,σz)=F^(σx)δ(σz(1/λλσx2/2)).
g^(x,z)=ei2πz/λeiaπ/41sMexp(iπx2λR)fa(xsM),
a=2πarctanλzs2,M=1+λ2z2s4,R=s4+λ2z2λ2z.
Δx=MΔx,Δσx=Δσx/M,
δz=8M2λΔσx2.
δa=16π1ΔxΔσx,
g^(x,z)=ei2πz/λF(σx)eiπσx2λzei2πσxxdσx.

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