Abstract

We demonstrate that an off-axis illuminated hemispherical-rod microlens acts as a fractional Fourier transformer with a continuously varying degree of fractionality. A complete theoretical treatment of the device as well as experimental results are presented.

© 1999 Optical Society of America

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References

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  1. D. Dragoman, K.-H. Brenner, M. Dragoman, J. Bähr, U. Krackhardt, “Hemispherical-rod microlens as a variant fractional Fourier transformer,” Opt. Lett. 23, 1499–1501 (1998).
    [CrossRef]
  2. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  3. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  5. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
    [CrossRef] [PubMed]
  6. D. Mendlovic, Z. Zalevsky, N. Konforti, R. G. Dorsch, A. W. Lohmann, “Incoherent fractional Fourier transform and its optical implementation,” Appl. Opt. 34, 7615–7620 (1995).
    [CrossRef] [PubMed]
  7. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the 2-D fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
    [CrossRef]
  8. D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, C. Ferreira, “Optical illustration of a varied fractional Fourier order and the Radon–Wigner display,” Appl. Opt. 35, 3925–3929 (1996).
    [CrossRef] [PubMed]
  9. M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
    [CrossRef]
  10. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  11. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformations,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef]
  12. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index media, Wigner distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  13. V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]

1998

1997

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

1996

1995

1994

1993

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

1980

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

Abe, S.

Agarwal, G. S.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Bähr, J.

Bitran, Y.

Brenner, K.-H.

Dorsch, R. G.

Dragoman, D.

Dragoman, M.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Ferreira, C.

Garcia, J.

James, D. F. V.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Konforti, N.

Krackhardt, U.

Lohmann, A. W.

Mendlovic, D.

Namias, V.

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

Ozaktas, H. M.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the 2-D fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index media, Wigner distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Sahin, A.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the 2-D fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

Sheridan, J. T.

Soffer, B. H.

Zalevsky, Z.

Appl. Opt.

J. Inst. Math. Its Appl.

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

J. Opt. Soc. Am. A

Opt. Commun.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the 2-D fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractionalFourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Opt. Lett.

Prog. Opt.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) H-rod microlens and the propagation of rays in the meridional and the sagital planes between the input and the output planes. (b) Rays in the meridional plane incident under different angles propagate over different z distances.

Fig. 2
Fig. 2

(a) Calculated and (b) measured output light intensities for an incident Gaussian light beam with a width ω = 50 µm, i = -400 µm, xi = 0.

Fig. 3
Fig. 3

Calculated (curve) dependence on θ xi of the ratio between the distances from the center of the output light distribution and the H-rod ends (L - )/. Points, experimental values.

Equations (11)

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

nx, y=n01-A2/2x2+y2,  x0,
xopxoyopyo=cosAzsinAzn0A00-n0A sinAzcosAz0000cosAzsinAzn0A00-n0A sinAzcosAzxipxiyipyi=Azz00CzDz0000Azz00CzDzxipxiyipyi,
z¯=1A arctan-n0Ax¯ip¯xi,  p¯xo=-n0Ax¯i sinAz¯+p¯xi cosAz¯,  y¯o=cosAz¯y¯i+sinAz¯n0A p¯yi,  p¯yo=-noAy¯i sinAz¯+p¯yi cosAz¯.
zA x¯isin Az¯pxoyopyo=Az¯z¯00Cz¯Dz¯0000Az¯z¯00Cz¯Dz¯=xipxiyipyi.
fxo, yo; z= fxi, yiexpik2zAzxi2+yi2-2xixo+yiyo+Dzxo2+yo2dxidyi
f0, yo; z= fxi-x¯i, yi-y¯iexpikn0A cosAzxi22 sinAz×expikn0A2 sinAzcosAzyi2+yo2-2yiyodxidyi=gz, yo; x¯i, y¯i.
Fαu, vf=K  fxi, yiexp  ikn0A2 sinα π2cosα π2 ×u2+xi2+v2+yi2-2uxi+vyidxidyi,
K=expi1-απ22π sinα π2.
gz, yo; x¯i, y¯i=1K F2Az/πx¯i, yofxi, yi-y¯i×expikn0Ax¯icosAz+1sinAz.
|gz, yo; x¯i, z¯|2= fxi, yiexpiknoAxix¯icosAz¯sinAz¯×expi kn0A2xi2cosAz¯sinAz¯ -2 xix¯iAzsin2(Az¯)×expi kn0A2yi2cosAz+z¯sinAz+z¯-2 yiyosinAz+z¯dxidyi21|K|2F2Az¯/πx¯iAzsinAz¯, yofxi, yi×expikn0Axix¯icosAz¯sinAz¯2.
|gz, y0; x¯i, z¯|2=πω221+kn0Aω22 tanAz¯2 exp -kω2×p¯xi+Azx¯in0Asin Az¯22+y02n0Asin Az+z¯221+kn0Aω22 tanAz¯2  .

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