Abstract

A method of visualization of phase changes in two-dimensional pure-phase objects by use of two orthogonal Fourier plane filters that realize the half-order differentiation is presented. Real semiderivative filters used in two dimensions and in sequence yield output-image intensity signals proportional to the first derivatives of the input-object phase that appear on a constant background. This nonlinear filtration of spatial frequencies permits the alleviation of the consequences of square-law detection and makes phase changes visible. Phase changes in gradient-index phosphate glass are calculated experimentally. We discuss the accuracy of the proposed method.

© 1998 Optical Society of America

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References

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  1. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).
  2. S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and their Applications (Science and Technique, Minsk, Russia, 1987) (in Russian).
  3. M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).
  4. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
    [CrossRef]
  5. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
    [CrossRef]
  6. V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.
  7. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
    [CrossRef]
  8. L. M. Soroko, Hilbert Optics (Science, Moscow, Russia, 1981), pp. 34–94 (in Russian).
  9. K. Gniadek, Optical Information Processing (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1992), pp. 257–269, 306–311 (in Polish).
  10. A. W. Lohmann, D. P. Paris, “Computer generated spatial filters for coherent optical data processing,” Appl. Opt. 7, 651–655 (1968).
    [CrossRef] [PubMed]
  11. R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
    [CrossRef] [PubMed]
  12. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).
  13. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [CrossRef] [PubMed]
  14. B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
    [CrossRef] [PubMed]
  15. R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1990).
  16. M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
    [CrossRef]
  17. I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
    [CrossRef]
  18. M. Wychowaniec, “Phosphate glass for gradient-index lenses,” Opt. Eng. 36, 1622–1624 (1997).
    [CrossRef]

1997 (3)

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

M. Wychowaniec, “Phosphate glass for gradient-index lenses,” Opt. Eng. 36, 1622–1624 (1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

1995 (2)

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

1993 (1)

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

1982 (1)

1978 (1)

1972 (1)

1968 (1)

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).

Campos, J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Cichocki, T.

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

Climent, V.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
[CrossRef]

Fernández-Alonso, M.

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

Gniadek, K.

K. Gniadek, Optical Information Processing (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1992), pp. 257–269, 306–311 (in Polish).

Gorecki, C.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Horwitz, B. A.

Kasprzak, H.

Kilbas, A. A.

S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and their Applications (Science and Technique, Minsk, Russia, 1987) (in Russian).

Kober, V.

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

Kowalczyk, M.

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

Lancis, J.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

Lohmann, A. W.

Maritchev, O. I.

S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and their Applications (Science and Technique, Minsk, Russia, 1987) (in Russian).

Martinez-Corral, M.

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

Michalski, M. W.

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

Moreno, I.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Oldham, K. B.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

Paris, D. P.

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
[CrossRef]

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
[CrossRef]

Samko, S. G.

S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and their Applications (Science and Technique, Minsk, Russia, 1987) (in Russian).

Soroko, L. M.

L. M. Soroko, Hilbert Optics (Science, Moscow, Russia, 1981), pp. 34–94 (in Russian).

Spanier, J.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

Sprague, R. A.

Szoplik, T.

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

Tajahuerce, E.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

Thompson, B. J.

R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
[CrossRef] [PubMed]

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
[CrossRef]

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1990).

Wychowaniec, M.

M. Wychowaniec, “Phosphate glass for gradient-index lenses,” Opt. Eng. 36, 1622–1624 (1997).
[CrossRef]

Yzuel, M. J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Appl. Opt. (5)

Diss. Math. (1)

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

Opt. Eng. (2)

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

M. Wychowaniec, “Phosphate glass for gradient-index lenses,” Opt. Eng. 36, 1622–1624 (1997).
[CrossRef]

Pure Appl. Opt. (2)

M. Kowalczyk, T. Cichocki, M. Martinez-Corral, V. Kober, “Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures,” Pure Appl. Opt. 4, 553–570 (1995).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-objet fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

Other (8)

V. Climent, M. Fernández-Alonso, J. Lancis, E. Tajahuerce, T. Szoplik, “Visualization of phase changes using semiderivative Fourier plane filter,” in Diffractive Optics, EOS Topical Meeting Digest Series Vol. 12 (European Optical Society, Orsay, France, 1997), pp. 272–273.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502.
[CrossRef]

L. M. Soroko, Hilbert Optics (Science, Moscow, Russia, 1981), pp. 34–94 (in Russian).

K. Gniadek, Optical Information Processing (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1992), pp. 257–269, 306–311 (in Polish).

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and their Applications (Science and Technique, Minsk, Russia, 1987) (in Russian).

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1990).

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Figures (4)

Fig. 1
Fig. 1

Gray-level representations of the output intensity measured for the radial-gradient-index rod lens by use of the 1-D real semiderivative filter in two orthogonal directions: (a) horizontal and (b) vertical.

Fig. 2
Fig. 2

Lines of equal intensity obtained by slicing of the intensity distributions of Fig. 1 into 16 equally separated levels. Shown are the fractional derivative in (a) the horizontal and (b) the vertical directions.

Fig. 3
Fig. 3

Lines of equal value of the modulus of the gradient vector field calculated by use of Eq. (13).

Fig. 4
Fig. 4

Representation of the phase distribution Φ(x, y) in the input object with isophase lines.

Equations (14)

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- r 1 f x ,   y x r 1 exp - 2 π i ux + vy d x d y = 2 π iu r 1 F u ,   v ,
- r 2 f x ,   y y r 2 exp - 2 π i ux + vy d x d y = 2 π iv r 2 F u ,   v .
r 1 x r 1   f x ,   y = - - 2 π iu r 1 F u ,   v × exp 2 π i xu + yv d u d v ,
r 2 y r 2   f x ,   y = - - 2 π iv r 2 F u ,   v × exp 2 π i xu + yv d u d v .
r 1 , r 2 f x ,   y = r 1 f x ,   y x r 1 i + r 2 f x ,   y y r 2 j ,
r 1 x r 1 exp 2 π ixu 0 f x ,   y = exp 2 π ixu 0 - - 2 π i α + u 0 r 1 F α ,   ν × exp 2 π i x α + y ν d α d ν ,
r 2 y r 2 exp 2 π iyv 0 f x ,   y = exp 2 π iyv 0 - - 2 π i β + v 0 r 2 F u ,   β × exp 2 π i xu + y β d u d β ,
r fg x r = j = - Γ r + 1 Γ r - j + 1 Γ j + 1 d r - j f d x r - j j g x j .
r exp 2 π iu 0 x exp i Φ x ,   y x r = Γ r + 1 Γ r + 1 Γ 1 2 π iu 0 r exp 2 π iu 0 x exp i Φ x ,   y + i Γ r + 1 Γ r Γ 2 2 π iu 0 r - 1 × exp 2 π iu 0 x exp i Φ x ,   y Φ x ,   y x + Γ r + 1 Γ r - 1 Γ 3 2 π iu 0 r - 2   exp 2 π iu 0 x × exp i Φ x ,   y i 2 Φ x ,   y x 2 - Φ x ,   y x 2 + .
I out = 1 / 2 exp 2 π iu 0 x exp i Φ x ,   y x 1 / 2 2 = 2 π u 0 + 2 Γ 3 / 2 Γ 1 / 2 Γ 2 Φ x ,   y x .
I x = 1 2 + 1 4 π u 0 Φ x ,   y x ,
I y = 1 2 + 1 4 π u 0 Φ x ,   y y .
Φ x ,   y = Φ x ,   y x i + Φ x ,   y y j ,
Δ x min = l f s x N x ,     Δ y min = l f s y N y .

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