Abstract

The phase-space measurement of micro-optical objects with submillimeter dimensions is reported for the first time to our knowledge. The experimental data were compared with simulated results from interferometric measurements and were found to be in good agreement.

© 1999 Optical Society of America

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References

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  1. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
    [Crossref]
  2. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [Crossref]
  3. R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [Crossref]
  4. M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
    [Crossref] [PubMed]
  5. T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
    [Crossref]
  6. T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
    [Crossref]
  7. R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
    [Crossref]
  8. L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
    [Crossref]
  9. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [Crossref]
  10. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–132 (1949).
    [Crossref]
  11. H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
    [Crossref]
  12. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28, 4996–4998 (1989).
    [Crossref] [PubMed]

1993 (1)

T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
[Crossref]

1992 (1)

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[Crossref]

1989 (1)

1988 (1)

L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
[Crossref]

1986 (1)

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[Crossref]

1985 (1)

1984 (1)

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[Crossref]

1983 (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[Crossref]

1982 (1)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[Crossref]

1980 (1)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

1949 (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–132 (1949).
[Crossref]

Asakura, T.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[Crossref]

Bamler, R.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[Crossref]

Bartelt, H. H.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[Crossref]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Berriel-Valdas, L. R.

L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
[Crossref]

Bescos, J.

L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
[Crossref]

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[Crossref]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Conner, M.

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
[Crossref]

Easton, R. L.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[Crossref]

Glünder, H.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[Crossref]

Gonzalo, C.

L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
[Crossref]

Gupta, A. K.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[Crossref]

Iwai, T.

T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
[Crossref]

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[Crossref]

Kawamura, N.

T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
[Crossref]

Li, Y.

Lohmann, A. W.

A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28, 4996–4998 (1989).
[Crossref] [PubMed]

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[Crossref]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Moyal, J. E.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–132 (1949).
[Crossref]

Takemori, T.

T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
[Crossref]

Ticknor, A. J.

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[Crossref]

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[Crossref]

Appl. Opt. (2)

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[Crossref]

Opt. Acta (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[Crossref]

Opt. Commun. (5)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[Crossref]

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[Crossref]

T. Iwai, N. Kawamura, T. Takemori, “Real-time profiling of a pure phase object using an auto-Wigner distribution function,” Opt. Commun. 95, 199–204 (1993).
[Crossref]

L. R. Berriel-Valdas, C. Gonzalo, J. Bescos, “Computation of the Wigner distribution function by the Hartley transform: application to image restoration,” Opt. Commun. 68, 339–344 (1988).
[Crossref]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Opt. Eng. (1)

R. L. Easton, A. J. Ticknor, H. H. Bartelt, “Application of the Radon transform to optical production of the Wigner distribution function,” Opt. Eng. 23, 738–744 (1984).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–132 (1949).
[Crossref]

Other (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1997), Vol. 37, pp. 1–56.
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Experimental setup for measuring the spectrogram.

Fig. 2
Fig. 2

(a) Selected microlens, (b) its experimentally obtained PS image, (c) its simulated PS image with data obtained from interferometric measurements.

Fig. 3
Fig. 3

(a) Selected GRIN H-rod, (b) its experimentally obtained PS image, (c) its simulated PS image with data obtained from interferometric measurements.

Equations (18)

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Wx, p=ϕ x+x2ϕ*x-x2exp-ikxpdx,
ϕoutxout, yout=expik yout2Mf ϕxin×tαxin+βM yout; Δxexpik xinxoutfdxin,
 ϕxintaxin+βM yout; Δxexpik xinxoutfdxin2= ϕxintxin-x; Δx/αexp-ikxinpdxin2=Sx, p; Δx,
p=xout/fx=youtm,  m=-β/Mα;
Sx, p; Δx=Winy, qWtx-y, p-qdydq,
|ϕxout, yout|2exp-yout-tan ϕxout2Θ2,
Θ=2MFkβα4Δx2+k2Δx24F21/2,  tan ϕ=cot θ MFf.
|ϕxout, yout|2exp-k2Δx24α2p0+xoutf2;
xout2¯= xout-xout¯2|ϕxout, yout|2dxoutdyout |ϕxout, yout|2dxoutdyout,
yout2¯= yout-yout¯2|ϕxout, yout|2dxoutdyout |ϕxout, yout|2dxoutdyout,
ξ¯= ξ|ϕxout, yout|2dxoutdyout |ϕxout, yout|2dxoutdyout,  ξ=xout, yout.
xout2¯=f2k2pin2¯+pt2¯,
yout2¯=1m2xin2¯+xt2¯,
ζin2¯= ζ-ζin¯2Winx, pdxdp Winx, pdxdp,
ζin¯= ζWinx, pdxdp Winx, pdxdp,  ζ=x, p,
δxin2¯=xt2¯,  δpin2¯=pt2¯,
δxout2¯=f2k2pt2¯,  δyout2¯=1m2xt2¯.
1m2xt2¯=f2k2pt2¯.

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