Abstract

Methods of imaging phase objects are considered. First the square-root filter is inferred from a definition of fractional-order derivatives given in terms of the integration of a fractional order called the Riemann-Liouville integral. Then we present a comparison of the performance of three frequency-domain real filters: square root, Foucault, and Hoffman. The phase-object imaging method is useful as a phase-shift measurement technique under the condition that the output image intensity is a known function of object phase. For the square-root filter it is the first derivative of the object phase function. The Foucault filter, in spite of its position, gives output image intensities expressed by Hilbert transforms. The output image intensity obtained with the Hoffman filter is not expressed by an analytical formula. The performance of the filters in a 4f imaging system with coherent illumination is simulated by use of VirtualLab 1.0 software.

© 2003 Optical Society of America

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References

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  1. M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. 1, 192–200 (1984).
    [CrossRef]
  2. 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]
  3. J. Ojeda-Castaneda, “A proposal to classify methods employed to detect thin phase structures under coherent illumination,” Opt. Acta 27, 917–929 (1980).
    [CrossRef]
  4. R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
    [CrossRef] [PubMed]
  5. H. Furuhashi, K. Matsuda, C. P. Grover, “Visualization of phase objects by use of a differentiation filter,” Appl. Opt. 42, 218–226 (2003).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988).
    [CrossRef] [PubMed]
  7. C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474–7485 (1995).
    [CrossRef] [PubMed]
  8. J. Glückstad, P. C. Mogensen, “Optimal phase contrast in common path interferometry,” Appl. Opt. 40, 268–282 (2001).
    [CrossRef]
  9. D. Sánchez-de-la-Llave, M. D. Iturbe Castillo, “Influence of illuminating beyond the object support on Zernike type phase contrast filtering,” Appl. Opt. 41, 2607–2612 (2002).
    [CrossRef] [PubMed]
  10. R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
    [CrossRef] [PubMed]
  11. B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
    [CrossRef] [PubMed]
  12. 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]
  13. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
    [CrossRef]
  14. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
    [CrossRef]
  15. L. M. Soroko, Hilbert Optics (Science, Moscow, 1981), pp. 34–94 (in Russian).
  16. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).
  17. H. Kasprzak, “On the possibility of optical performing of non-integer order derivatives,” Opt. Appl. 10, 289–292 (1980).
  18. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [CrossRef] [PubMed]
  19. J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
    [CrossRef]
  20. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).
  21. S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and Their Applications (Science and Technique, Minsk, Russia, 1987; in Russian).
  22. G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, Berlin, 2001).
    [CrossRef]
  23. M. Pluta, Advanced Light Microscopy. Volume 2: Specialized Methods (PWN and Elsevier, Warsaw, 1989).
  24. E. W. S. Hee, “Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol. 7, 75–79 (1975).
    [CrossRef]
  25. J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, J. Campos, “Fractional derivatives—analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
    [CrossRef]
  26. J. A. Davis, M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41, 4835–4839 (2002).
    [CrossRef] [PubMed]

2003 (1)

2002 (2)

2001 (2)

1998 (1)

1997 (2)

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-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997).
[CrossRef]

1995 (1)

1988 (1)

1984 (1)

M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. 1, 192–200 (1984).
[CrossRef]

1982 (1)

1980 (2)

H. Kasprzak, “On the possibility of optical performing of non-integer order derivatives,” Opt. Appl. 10, 289–292 (1980).

J. Ojeda-Castaneda, “A proposal to classify methods employed to detect thin phase structures under coherent illumination,” Opt. Acta 27, 917–929 (1980).
[CrossRef]

1978 (1)

1976 (1)

J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
[CrossRef]

1975 (2)

E. W. S. Hee, “Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol. 7, 75–79 (1975).
[CrossRef]

R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
[CrossRef] [PubMed]

1972 (1)

Anderson, C. S.

Bracewell, R. N.

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

Campos, J.

Climent, V.

Cottrell, D. M.

Davis, J. A.

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.

Furuhashi, H.

Glückstad, J.

Gross, L.

Grover, C. P.

Hee, E. W. S.

E. W. S. Hee, “Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol. 7, 75–79 (1975).
[CrossRef]

Hoffman, R.

Horwitz, B. A.

Iturbe Castillo, M. D.

Kasprzak, H.

H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
[CrossRef] [PubMed]

H. Kasprzak, “On the possibility of optical performing of non-integer order derivatives,” Opt. Appl. 10, 289–292 (1980).

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).

Kowalczyk, M.

M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. 1, 192–200 (1984).
[CrossRef]

Lancis, J.

Lavoie, J. L.

J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
[CrossRef]

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).

Matsuda, K.

McNamara, D. E.

Mogensen, P. C.

Nowak, M. D.

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, “A proposal to classify methods employed to detect thin phase structures under coherent illumination,” Opt. Acta 27, 917–929 (1980).
[CrossRef]

Oldham, K. B.

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

Osler, T. J.

J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
[CrossRef]

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]

Pluta, M.

M. Pluta, Advanced Light Microscopy. Volume 2: Specialized Methods (PWN and Elsevier, Warsaw, 1989).

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).

Sánchez-de-la-Llave, D.

Schwider, J.

Settles, G. S.

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, Berlin, 2001).
[CrossRef]

Smith, D. A.

Soroko, L. M.

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

Spanier, J.

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

Sprague, R. A.

Streibl, N.

Szoplik, T.

Tajahuerce, E.

Thomas, J.

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]

Tremblay, R.

J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
[CrossRef]

Appl. Opt. (13)

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

H. Furuhashi, K. Matsuda, C. P. Grover, “Visualization of phase objects by use of a differentiation filter,” Appl. Opt. 42, 218–226 (2003).
[CrossRef] [PubMed]

A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988).
[CrossRef] [PubMed]

C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474–7485 (1995).
[CrossRef] [PubMed]

J. Glückstad, P. C. Mogensen, “Optimal phase contrast in common path interferometry,” Appl. Opt. 40, 268–282 (2001).
[CrossRef]

D. Sánchez-de-la-Llave, M. D. Iturbe Castillo, “Influence of illuminating beyond the object support on Zernike type phase contrast filtering,” Appl. Opt. 41, 2607–2612 (2002).
[CrossRef] [PubMed]

R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
[CrossRef] [PubMed]

B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
[CrossRef] [PubMed]

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]

T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
[CrossRef]

H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
[CrossRef] [PubMed]

J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, J. Campos, “Fractional derivatives—analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
[CrossRef]

J. A. Davis, M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41, 4835–4839 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. 1, 192–200 (1984).
[CrossRef]

Opt. Acta (1)

J. Ojeda-Castaneda, “A proposal to classify methods employed to detect thin phase structures under coherent illumination,” Opt. Acta 27, 917–929 (1980).
[CrossRef]

Opt. Appl. (1)

H. Kasprzak, “On the possibility of optical performing of non-integer order derivatives,” Opt. Appl. 10, 289–292 (1980).

Opt. Laser Technol. (1)

E. W. S. Hee, “Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol. 7, 75–79 (1975).
[CrossRef]

Pure Appl. Opt. (1)

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

SIAM Rev. (1)

J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976).
[CrossRef]

Other (7)

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).

G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, Berlin, 2001).
[CrossRef]

M. Pluta, Advanced Light Microscopy. Volume 2: Specialized Methods (PWN and Elsevier, Warsaw, 1989).

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, 1981), pp. 34–94 (in Russian).

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

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Figures (4)

Fig. 1
Fig. 1

Amplitude transmittances of (a) the normalized square-root real filter, (b) the Foucault filter, and (c) the three-level Hoffman filter.

Fig. 2
Fig. 2

Profile of a cylindrical lens with variable f-number used as the phase object in simulations.

Fig. 3
Fig. 3

Output image intensity distributions for small phase gradients obtained with (a) Foucault, (b) Hoffman, and (c) square root filters and corresponding intensity profiles (d), (e), and (f), respectively, calculated for three different focal-length values.

Fig. 4
Fig. 4

Output image intensity distributions for large phase gradients obtained with (a) Foucault, (b) Hoffman, and (c) square-root filters and corresponding intensity profiles (d), (e), and (f), respectively, calculated for three different focal-length values.

Tables (2)

Tables Icon

Table 1 Values of Fitting Parameters a and b (mm/rad) for Output Image Intensity Distributions Obtained for Three Filters in the Case of a Thin Phase Lens

Tables Icon

Table 2 Values of Fitting Parameters a and b (mm/rad) for Output Image Intensity Distributions Obtained for the Square-Root Filter in the Case of a Thick Phase Lens

Equations (21)

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dqfξdξq=FT-1{2πiuqFTfξ},
dqfξdξ-aq= 1Γ-qaξξ-τ-q-1fτdτ,
d1/2fξdξ-a1/2= ddξd-1/2fξdξ-a-1/2= 1πddξaξξ-τ-1/2fτdτ.
FT-1{2πiu1/2 FTfξ} = FT-1{2πiu-1/22πiuFTfξ}=FT-12πiu-1/2FT-1{2πiuFTfξ}=FT-12πiu-1/2 dfξdξ= ddξFT-112πiufξ.
FT-112πiu=1/πξξ>00ξ0,
FT-112πiufξ= 1π-ξ-τ-1/2fτdτξ-τ>00ξ-τ0 = 1π-ξξ-τ-1/2fτdτ,
FT-1{2πiu1/2 FTfξ}= 1πddξ-ξξ-τ-1/2fτdτ.
dqfξgξdξq=j=0qjdq-jfξdξq-jdjgξdξj,
Ix= 12+ 14πu0dϕxdx.
F-u, v=-1+iϕξ, ηexp-i2πξu+ηvdξdη=δu, v+iΦu, v,
I1x, y1+ 12HTxϕx, y21+HTxϕx, y,
HTxϕx, y= m2cos2πxL,
ϕx, yx= m22πLcos2πxL.
HTxϕx, y= L2πϕx, yx,
I1x, y1+ L2πϕx, yx=1+ m2cos2πxL,
Ix, y=a+b dϕx, ydx,
I2x, y12+ 12HTxϕx, y2 14+ 12HTxϕx, y.
I2x, y 14+ L4πϕx, yx.
I3x, y ϕ2x, y4+ HTxϕx, y24 HTxϕx, y24.
ϕξ, η= 1-16ξ6η+5s2ϕmax|ξ|< |6η+5s|160otherwise,
Ix=a+b dϕxdx.

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