Abstract

A method for phase visualization and edge enhancement by spatial self-filtering by use of a polarizer sheet in the Fourier plane of an optical processor is described. Light absorbed by the polarizer sheet induces a thermal lens, which, in turn, produces selective action on certain spatial frequencies of the image to be processed. Some experiments that demonstrate the self-filtering action of the proposed system are presented.

© 2005 Optical Society of America

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References

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    [CrossRef]
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  10. S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1988).
  12. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3.
  13. E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

1998 (2)

E. U. Wagemann, H.-J. Tiziani, “Spatial self-filtering using photorefractive and liquid crystals,” J. Mod. Opt. 45, 1885–1897 (1998).
[CrossRef]

E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

1997 (2)

1996 (1)

1995 (1)

1994 (1)

1993 (1)

1990 (1)

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

1985 (1)

1980 (1)

Dovichi, N. J.

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

Dultz, W.

E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

Egami, C.

Feinberg, J.

Ferrari, J. A.

E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

Frins, E. M.

E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

Goodman, J. W.

Hesselink, L.

Huang, T.

Jürgensen, F.

Kato, J.

Ochoa, E.

Okamoto, N.

Okamoto, T.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3.

Schröer, W.

Sugihara, O.

Suzuki, Y.

Tanaka, H.

Tiziani, H.-J.

E. U. Wagemann, H.-J. Tiziani, “Spatial self-filtering using photorefractive and liquid crystals,” J. Mod. Opt. 45, 1885–1897 (1998).
[CrossRef]

Uemori, T.

Uhrich, C.

Wagemann, E. U.

E. U. Wagemann, H.-J. Tiziani, “Spatial self-filtering using photorefractive and liquid crystals,” J. Mod. Opt. 45, 1885–1897 (1998).
[CrossRef]

Wagner, K. H.

Wu, S.

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

Yamagata, K.

Yamaguchi, I.

Appl. Opt. (3)

J. Appl. Phys. (1)

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

J. Eur. Opt. Soc. A (1)

E. M. Frins, W. Dultz, J. A. Ferrari, “Polarization shifting method for step interferometry,” J. Eur. Opt. Soc. A 7, 53–60 (1998).

J. Mod. Opt. (1)

E. U. Wagemann, H.-J. Tiziani, “Spatial self-filtering using photorefractive and liquid crystals,” J. Mod. Opt. 45, 1885–1897 (1998).
[CrossRef]

Opt. Lett. (5)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1988).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 3.

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Figures (8)

Fig. 1
Fig. 1

Schematic layout of a 4f optical processor with a polarizer sheet (P) in the Fourier plane. SF1, spatial filter; L1, collimating lens; L2,3, Fourier lenses; HW, half-wave plate; PO, test object; C, digital camera.

Fig. 2
Fig. 2

Interferometric setup utilized to retrieve the refractive-index profile of the induced thermal lens. SF1,2, spatial filters; L3, collimating lens; L4, lens that images the TL on output plane C; QW, quarter-wave plate; P1, rotable polarizer; PBSs, polarizing beam splitters; M1,2, mirrors; P, polarizer; HW, half-wave plate; PO, test object; L1, collimating lens; L2, Fourier lens.

Fig. 3
Fig. 3

(a) Interferogram of the TL obtained for θ ≈ 45°, (b) its reconstructed phase-delay profile.

Fig. 4
Fig. 4

(a) Interferogram of the TL obtained for θ ≈ 90°, (b) its reconstructed phase-delay profile.

Fig. 5
Fig. 5

(a), (b) Radial dependence of the phase profiles for θ ≈ 45° and θ ≈ 90°, respectively. Circles, experimental data; continuous curves, fitting with polynomials of fourth order.

Fig. 6
Fig. 6

Piece of transparent film with an eagle’s head impressed on its surface used as test object (PO; Figs. 1 and 2). (a) The TL is not in the Fourier plane of the processor, and the eagle’s head of the transparent piece of film is almost invisible. (b), (c) The TL is in the Fourier plane of the processor. The images were acquired for θ ≈ 45° and θ ≈ 90°, respectively. In (b) the first derivatives of the edges of the transparent film are enhanced; in (c) the second derivatives are enhanced.

Fig. 7
Fig. 7

Column of butane gas flowing from a tube used as a test object (PO in Figs. 1 and 2) in the processor with the TL in the Fourier plane (a) for θ ≈ 45° and (b) for θ ≈ 90°.

Fig. 8
Fig. 8

Spring utilized as a test object in the processor with the TL in the Fourier plane. The double white outline surrounding the edges is characteristic of the enhancement of the second derivatives of the original image.

Equations (12)

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Δ T ( r , t ) - Δ T ( 0 , t ) = ( P H / 4 π κ ) [ Ei ( - 2 r 2 ω H 2 ) - Ei ( - 2 r 2 ω H 2 τ ) + ln ( τ ) ] ,
n ( r , t ) - n ( 0 , t ) = ( n / T ) [ Δ T ( r , t ) - Δ T ( 0 , t ) ] = A [ Ei ( - 2 r 2 ω H 2 ) - Ei ( - 2 r 2 ω H 2 τ ) + ln ( τ ) ] ,
E ˜ ( x , y ) = B - - T { E 0 } ( x , y ) exp { i k l [ n ( r , ) - n ( 0 , ) ] } exp [ - i ( k / f ) ( x x + y y ) ] d x d y ,
T { E 0 } ( x , y ) ( 1 / 2 π ) - - E 0 ( x , y ) exp [ - i ( k / f ) ] × ( x x + y y ) ] d x d y .
n ( r , ) - n ( 0 , ) p 0 + p 1 ( r / ω H ) + p 2 ( r / ω H ) 2 + p 3 ( r / ω H ) 3 + ,
E ˜ ( x , y ) B 0 E 0 ( - x , - y ) + B 1 T { r T { E 0 } ( x , y ) } ( x , y ) + B 2 T { r 2 T { E 0 } ( x , y ) } ( x , y ) + ,
( i k x / f ) n T { E 0 ( x , y ) } ( x , y ) = T { n ( x y ) x n } ( x , y ) ,
( i k y / f ) n T { E 0 ( x , y ) } ( x , y ) = T { n E 0 ( x , y ) x n } ( x , y ) ;
r T { E 0 } ( x , y ) = - ( i f / k ) T { E 0 ( x , y ) } ( x , y ) ,
r 2 T { E 0 } = - ( f 2 / k 2 ) T { 2 E 0 ( x , y ) } ( x , y ) ,
E ˜ ( x , y ) B 0 E 0 ( - x , - y ) - ( i f / k ) B 1 T { T { E 0 ( x , y ) } ( x , y ) - ( f 2 / k 2 ) B 2 2 E 0 ( - x , - y ) + .
T { T { E 0 ( x , y ) } } ( x , y ) = E 0 ( - x ) x .

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