Abstract

The concept of an extended fractional Fourier transform (FRT) is suggested. Previous FRT’s and complex FRT’s are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [Crossref]
  2. A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [Crossref]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]
  4. C.-C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
    [Crossref]
  5. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transform and their optical implementation. I” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [Crossref]
  6. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [Crossref]
  7. A. W. Lohmann, D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt. 33, 7661–7664 (1994).
    [Crossref] [PubMed]
  8. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [Crossref] [PubMed]
  9. C.-C. Shih, “Optical interpretation of a complex-order Fourier transform,” Opt. Lett. 20, 1178–1180 (1995).
    [Crossref] [PubMed]
  10. L. Bernardo, O. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
    [Crossref] [PubMed]
  11. S. Liu, J. Xu, Y. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
    [Crossref] [PubMed]
  12. G. Dorsch, “Fractional Fourier transforms of variable order based on a modular lens system,” Appl. Opt. 34, 6016–6020 (1995).
    [Crossref] [PubMed]
  13. L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
    [Crossref]
  14. H. M. Ozaktas, B. Barshan, D. Mendlovic, “Fractional Fourier transforms as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [Crossref] [PubMed]
  15. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens-design problem,” Appl. Opt. 34, 4111–4112 (1995).
    [Crossref] [PubMed]
  16. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [Crossref] [PubMed]
  17. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [Crossref] [PubMed]
  18. H. M. Ozktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [Crossref]
  19. H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139 Part III, 285–288 (1994).
  20. S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
    [Crossref]
  21. S. A. Collings, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  22. S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
    [Crossref]

1996 (1)

1995 (7)

1994 (8)

1993 (3)

1987 (1)

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[Crossref]

1970 (1)

Abe, S.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[Crossref]

Barshan, B.

Bernardo, L.

Bernardo, L. M.

Bitran, Y.

Chen, L.

Collings, S. A.

Dorsch, G.

Granieri, S.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Li, C.

Liu, S.

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[Crossref]

Onural, L.

Ozaktas, H. M.

Ozktas, H. M.

Pellat-Finet, P.

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[Crossref]

Shih, C.-C.

C.-C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
[Crossref]

C.-C. Shih, “Optical interpretation of a complex-order Fourier transform,” Opt. Lett. 20, 1178–1180 (1995).
[Crossref] [PubMed]

Sicre, E. E.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Soares, O. D.

Trabocchi, O.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

Urey, H.

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139 Part III, 285–288 (1994).

Xu, J.

Zhang, Y.

Appl. Opt. (6)

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Inst. Phys. Conf. Ser. (1)

H. M. Ozaktas, B. Barshan, D. Mendlovic, H. Urey, “Space-variant filtering in fractional Fourier domains,” Inst. Phys. Conf. Ser. 139 Part III, 285–288 (1994).

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

C.-C. Shih, “Fractionalization of Fourier transform,” Opt. Commun. 118, 495–498 (1995).
[Crossref]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[Crossref]

S. Abe, J. T. Sheridan, “Almost-Fourier and almost-Fresnel transformations,” Opt. Commun. 113, 385–388 (1995).
[Crossref]

Opt. Lett. (4)

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Geometry of the lens transform.

Fig. 2
Fig. 2

Domain maps of the parameters a2, b2, and cos ϕ. Map (a) presents the domain of a2 and b2. When (l/f, l/f) falls into the striped region, a2 and b2 are both real (R); otherwise, they are both imaginary (I). Map (b) is for ϕ. Within the striped region, ϕ is real.

Fig. 3
Fig. 3

Equivalent cascading procedure of the extended FRT’s.

Fig. 4
Fig. 4

Equivalent function of the lens-cascaded system.

Equations (42)

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

Fαf(x)=exp[i(π/4-α/2)](2π sin α)1/2-+f(x)×exp-i x2+y22 tan α+i xysin αdx,
exp[i(π/4-α/2)]/(2π sin α)1/2
FαFβf(x)=Fα+βf(x).
up(x)=Fp[u0(x0)]=-+u0(x0)×expiπλfs tan ϕ(x02+x2)×exp-iπ2x0xλfs sin ϕdx0,
u(y)=u[v(x)|a, ϕ, b|y]=-+v(x)×expiπ a2x2+b2y2tan ϕ-i2π absin ϕxydx
u[v(x)|a, ϕ, b|y]=-+v(x0/a)expiπtan ϕ(x02+y02)-i2πsin ϕx0y0dx0 1a.
u[v(x)|a, ϕ, b|y]=u[v(x/a)|1, ϕ, 1|y0]=u[v(x/a)|1, ϕ, 1|by].
a2=1λf-lf-l1[f2-(f-l)(f-l)]1/2,
ϕ=arccosf-lf-lf,
b2=1λf-lf-l1[f2-(f-l)(f-l)]1/2,
u(y)=-+v(x)×expi πλ(f-l)x2+(f-l)y2-2fxylf+lf-lldx.
a2=b2=1/λf-3,
ϕ=arccos(-2)=π-1.1317i,
a2=-i/22λf,b2=i2/λf,cos ϕ=i,
u[v(xa0)|a0/λfs, ϕ, 1/λfs|y]
=u[v(x)|1/λfs, ϕ, 1/λfs|y].
a2b2=(1/λf)2/sin2 ϕ,
a2(f-l)=b2(f-l).
a1=1/λfs,
b1=a2,
b2=1/λfs,
ϕ1+ϕ2=ϕ.
f1 sin ϕ1=f2 sin ϕ2.
ϕ1=arccot[f1/(f2 sin ϕ)+cot ϕ],
ϕ2=arccot[f2/(f1 sin ϕ)+cot ϕ].
l1=f11-b1a1cos ϕ1,
l1=f11-a1b1cos ϕ1,
l2=f21-b2a2cos ϕ2,
l2=f21-a2b2cos ϕ2,
f=1λab sin ϕ,f-l=1λa2 tan ϕ,
f-l=1λb2 tan ϕ.
u{u[v(x)|1, ϕ1, 1|y]|1, ϕ2, 1|z}
=u[v(x)|1, ϕ1+ϕ2, 1|z].
u{u[v(x)|a1, ϕ1, b1|y]|a2, ϕ2, b2|z}
=u{u[v(x/a1)|1, ϕ1, 1|yb1]|a2, ϕ2, b2|z}=u{u[v(x/a1)|1, ϕ1, 1|yb1/a2]|1, ϕ2, 1|zb2}.
u{u[v(x)|a1, ϕ1, b1|y]|a2, ϕ2, b2|z}
=u{u[v(x/a1)|1, ϕ1+ϕ2, 1|zb2}=u[v(x)|a1, ϕ1+ϕ2, b2|z].
u(z)=-+-+v(x)×expiπ a12x2+b12y2tan ϕ1-i2π a1b1sin ϕ1xydx×expiπ a22y2+b22z2tan ϕ2-i2π a2b2sin ϕ2yzdy=-+dx v(x)expiπ a12x2tan ϕ1+b22z2tan ϕ2×-+dy expiπyb12tan ϕ1+a22tan ϕ21/2-a1b1xsin ϕ1+a2b2zsin ϕ2b12tan ϕ1+a22tan ϕ21/22-iπa1b1xsin ϕ1+a2b2zsin ϕ2b12tan ϕ1+a22tan ϕ21/22=-+dx v(x)expiπ a12x2tan ϕ1+b22z2tan ϕ2×exp-iπa1b1xsin ϕ1+a2b2zsin ϕ2b12tan ϕ1+a22tan ϕ21/22×-+dy expiπyb12tan ϕ1+a22tan ϕ21/2-a1b1xsin ϕ1+a2b2zsin ϕ2b12tan ϕ1+a22tan ϕ21/22.
u(z)=-+dx v(x)×expiπa12tan ϕ1-a12b12sin2 ϕ1b12tan ϕ1+a22tan ϕ2x2+iπb22tan ϕ2-a22b22sin2 ϕ2b12tan ϕ1+a22tan ϕ2z2-i2π a1b1a2b2(sin ϕ1 sin ϕ2)b12tan ϕ1+a22tan ϕ2xz.
a12tan ϕ1-a12sin2 ϕ1b12tan ϕ1+a22tan ϕ2=a2tan ϕ,
b22tan ϕ2-a22b22sin2 ϕ2b12tan ϕ1+a22tan ϕ2=b2tan ϕ,
a1b1a2b2(sin ϕ1 sin ϕ2)b12tan ϕ1+a22tan ϕ2=absin ϕ.

Metrics