Abstract

A new optical implementation of the Radon–Wigner display for one-dimensional objects is presented, making use of the fractional Fourier transform approach. The proposed setup makes use of only two conventional refractive elements: a cylindrical lens and a varifocal lens. Although the exact magnifications cannot be achieved simultaneously for all the fractional transforms, an optimum design can be obtained through balancing the conflicting magnification requirements. Experimental results are obtained with a commercially available progressive addition lens. For comparison, computer simulations are also provided.

© 1997 Optical Society of America

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References

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  1. J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).
  2. J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon-Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
    [CrossRef]
  3. S. Kay, G. F. Boudreaux-Bartels, “On optimality of the Wigner distribution for detection,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).
  4. J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
    [CrossRef]
  5. D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, C. Ferreira, “Optical illustration of a varied fractional Fourier transform order and the Radon-Wigner display,” Appl. Opt. 35, 3925–3929 (1996).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  7. 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]
  8. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  9. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  10. 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]
  11. T. Alieva, F. Agulló-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
    [CrossRef]

1996 (2)

1995 (2)

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (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]

1994 (4)

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]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon-Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

1993 (1)

1992 (1)

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

1985 (1)

S. Kay, G. F. Boudreaux-Bartels, “On optimality of the Wigner distribution for detection,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

Agulló-Lopez, F.

T. Alieva, F. Agulló-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

Alieva, T.

T. Alieva, F. Agulló-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

Barry, D. T.

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon-Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Boudreaux-Bartels, G. F.

S. Kay, G. F. Boudreaux-Bartels, “On optimality of the Wigner distribution for detection,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

Dorsch, R. G.

Ferreira, C.

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]

Kay, S.

S. Kay, G. F. Boudreaux-Bartels, “On optimality of the Wigner distribution for detection,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

Lohmann, A. W.

Mendlovic, D.

Ozaktas, H. M.

Pellat-Finet, P.

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]

Soffer, B. H.

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]

Wood, J. C.

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon-Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Zalevsky, Z.

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon-Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

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]

T. Alieva, F. Agulló-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE Int. Conf. Acoust. Speech Signal Process. (2)

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

S. Kay, G. F. Boudreaux-Bartels, “On optimality of the Wigner distribution for detection,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

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Figures (8)

Fig. 1
Fig. 1

Two equivalent setups to obtain the FRT of order p: (a) Lohmann’s setup of type II, (b) free-space propagation setup. The coordinates at the input and the output planes are (x 0, y 0) and (x, y), respectively. The correspondence between the distances in (a) and (b) are given through Eqs. (3) and (4).

Fig. 2
Fig. 2

Optical setup to obtain the Radon–Wigner display of 1-D inputs. The cylindrical lens L C provides the illuminating wavefront that converges to a line that crosses the optical axis at the point S. The free-space propagation provides the FRT’s at distances R′ from the input. Each one of such is imaged simultaneously by the varifocal lens L at the output plane.

Fig. 3
Fig. 3

Magnification provided by the lens L in Fig. 2 for the values z = 426 mm, l = 646 mm, and a′ = 831 mm, computed with Eq. (10) (approximated solution) in a solid curve and with Eq. (11) (exact solution) in a dotted curve.

Fig. 4
Fig. 4

Focal length (solid curve) and optical power (dotted curve) of the designed varifocal lens for values of z, l, and a′ the same as in Fig. 3.

Fig. 5
Fig. 5

Radon–Wigner display of a Ronchi grating of 3 lines/mm: (a) exact numerical simulation, (b) numerical simulation with the approximation discussed in Section 2, (c) experimental result.

Fig. 6
Fig. 6

Radon-Wigner display of a binary grating with a linearly increasing spatial frequency (chirp signal): (top) numerical simulation with the approximation discussed in Section 2, (bottom) experimental result.

Fig. 7
Fig. 7

Vertical profiles (p constant) obtained from Fig. 6 for three values of the fractional order p: (a) p = 0.2, (b) p = 0.45, (c) p = 0.5. The figures on the left-hand side correspond to the numerical simulation [Fig. 6 (top)], and those on the right-hand side correspond to the experimental data [Fig. 6 (bottom)].

Fig. 8
Fig. 8

Experimental results for the Radon-Wigner display: (top) single slit of 2.2 mm, (middle) double slit of 0.23 mm separated by 1.05 mm, (bottom) Ronchi grating of 7 lines/mm.

Equations (17)

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ptx0, y0=expik2zpx2+y2Ux, y; Rp,
ptx0, y0=expiπλf1tanp π2x2+y2×tx0, y0expiπλf1tanpπ2×x02+y02exp-2iπλf1sinpπ2×xx0+yy0dx0dy0,
zp=-f=-f1tanpπ4,
Rp=d=f1 sinpπ2.
1Rp=1zp-1zq+1Rp,
Mp=RpRp.
Rp=f1 tanpπ21-f1zqtanpπ2,
Mp=1+tanpπ2tanpπ41-f1zqtanpπ2.
fp=apaap+a
ap=l-Rp,
fp=al-f1a tanpπ21+lza-l-f1 tanpπ2a+l+zz.
MLp=-aap=aRp-l.
MLp=a-1+f1ztanpπ2l1-tanpπ2f1z+f1l.
MLp=-1Mp=-1+f1ztanpπ21+tanpπ2tanpπ4
Jl, a, z=01ML-MLML2dp=π21+lf12a2z+π41a2l+f12+l2f12z2+1a22l-2f1-lf1a-2lf1z-l2f1az,
a=l12+π4,
z=-lf1l+f1.

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