Abstract

We propose to use a differential operator for representing the influence of phase-only filters on the defocused modulation transfer function of the clear pupil aperture. We present a phase-only filter that implements optically Taylor's theorem in phase space. We show numerical simulations of the modulation transfer functions and the images that can be obtained by using the proposed filter.

© 2006 Optical Society of America

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References

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  1. G. Häusler, "A method to increase the depth of focus by two step image processing," Opt. Commun. 6, 38-42 (1972).
    [Crossref]
  2. M. Minski, "Double focusing stage scanning microscope," U.S. patent 3,0130467 (19 December 1961).
  3. T. Wilson, Confocal Microscopy (Academic, 1990).
  4. J. Ojeda-Castañeda and A. Noyola-Isgleas, "High focal depth by apodization and digital restoration," Appl. Opt. 27, 2583-2586 (1988).
    [Crossref] [PubMed]
  5. T. W. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
    [Crossref] [PubMed]
  6. E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
    [Crossref]
  7. E. B. Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with extended depth of field," Appl. Opt. 44, 2792-2798 (2005).
    [Crossref]
  8. E. R. Dowski and T. W. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1865 (1995).
    [Crossref] [PubMed]
  9. S. Mezouari and A. A. Harvey, "Phase pupil functions reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
    [Crossref] [PubMed]
  10. N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003).
    [Crossref]
  11. A. Sauceda and J. Ojeda-Castañeda, "High focal depth with fractional-power wave fronts," Opt. Lett. 29, 560-562 (2004).
    [Crossref] [PubMed]
  12. A. Castro and J. Ojeda-Castañeda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
    [Crossref] [PubMed]
  13. S. Sanyal and A. Gosh, "High focal depth with a quasi-focus birefringent lens," Appl. Opt. 27, 4163-4165 (2000).
  14. H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-5662 (2001).
    [Crossref]
  15. S. S. Sherif, W. T. Cathey, and E. R. Dowski, "Phase plate to extend the depth of field of incoherent hybrid imaging system," Appl. Opt. 43, 2709-2721 (2004).
    [Crossref] [PubMed]
  16. J. Ojeda-Castañeda, J. E. A. Landgrave, and H. M. Escamilla, "Annular phase-only mask for high focal depth," Opt. Lett. 30, 1647-1649 (2005).
    [Crossref] [PubMed]
  17. K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
    [Crossref]
  18. J. Ojeda-Castañeda and L. R. Berriel-Valdos, "Ambiguity function as a design tool for high focal depth," Appl. Opt. 27, 790-795 (1988).
    [Crossref] [PubMed]
  19. E. D. Rainville, Elementary Differential Equations, 3rd ed. (Macmillan, 1964) pp. 234-237.
  20. F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (Dover, 1970), pp. 358-361.
  21. R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002) pp. 262-266.

2005 (2)

2004 (3)

2003 (3)

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

S. Mezouari and A. A. Harvey, "Phase pupil functions reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
[Crossref] [PubMed]

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003).
[Crossref]

2002 (2)

T. W. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
[Crossref] [PubMed]

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002) pp. 262-266.

2001 (1)

2000 (1)

1995 (1)

1988 (2)

1983 (1)

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[Crossref]

1972 (1)

G. Häusler, "A method to increase the depth of focus by two step image processing," Opt. Commun. 6, 38-42 (1972).
[Crossref]

1970 (1)

F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (Dover, 1970), pp. 358-361.

1964 (1)

E. D. Rainville, Elementary Differential Equations, 3rd ed. (Macmillan, 1964) pp. 234-237.

1961 (1)

M. Minski, "Double focusing stage scanning microscope," U.S. patent 3,0130467 (19 December 1961).

Berriel-Valdos, L. R.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[Crossref]

Castro, A.

Cathey, T. W.

Cathey, W. T.

Chi, W.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003).
[Crossref]

Dowski, E. R.

Eliezer, E. B.

E. B. Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with extended depth of field," Appl. Opt. 44, 2792-2798 (2005).
[Crossref]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

Escamilla, H. M.

Gan, F.

George, N.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003).
[Crossref]

Gonzalez, R.

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002) pp. 262-266.

Gosh, A.

Harvey, A. A.

Häusler, G.

G. Häusler, "A method to increase the depth of focus by two step image processing," Opt. Commun. 6, 38-42 (1972).
[Crossref]

Konforti, N.

E. B. Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with extended depth of field," Appl. Opt. 44, 2792-2798 (2005).
[Crossref]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

Landgrave, J. E. A.

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[Crossref]

Marom, E.

E. B. Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with extended depth of field," Appl. Opt. 44, 2792-2798 (2005).
[Crossref]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

Mezouari, S.

Minski, M.

M. Minski, "Double focusing stage scanning microscope," U.S. patent 3,0130467 (19 December 1961).

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

Olver, F. W. J.

F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (Dover, 1970), pp. 358-361.

Rainville, E. D.

E. D. Rainville, Elementary Differential Equations, 3rd ed. (Macmillan, 1964) pp. 234-237.

Sanyal, S.

Sauceda, A.

Sherif, S. S.

Wang, H.

Wilson, T.

T. Wilson, Confocal Microscopy (Academic, 1990).

Woods, R.

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002) pp. 262-266.

Zalevsky, Z.

E. B. Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with extended depth of field," Appl. Opt. 44, 2792-2798 (2005).
[Crossref]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

Appl. Opt. (9)

J. Opt. A: Pure Appl. Opt. (2)

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003).
[Crossref]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S163 (2003).
[Crossref]

Opt. Commun. (2)

G. Häusler, "A method to increase the depth of focus by two step image processing," Opt. Commun. 6, 38-42 (1972).
[Crossref]

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[Crossref]

Opt. Lett. (3)

Other (5)

E. D. Rainville, Elementary Differential Equations, 3rd ed. (Macmillan, 1964) pp. 234-237.

F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (Dover, 1970), pp. 358-361.

R. Gonzalez and R. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002) pp. 262-266.

M. Minski, "Double focusing stage scanning microscope," U.S. patent 3,0130467 (19 December 1961).

T. Wilson, Confocal Microscopy (Academic, 1990).

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Figures (7)

Fig. 1
Fig. 1

Schematics of the optical setup. The phase-only mask is located at the exit pupil plane with spatial frequency coordinates ( ν , μ ) .

Fig. 2
Fig. 2

Phase profiles: (a) sinusoidal phase filter and (b) the asymmetric phase-only filters in Ref. 12.

Fig. 3
Fig. 3

First quadrant of the modulus of the ambiguity function for the sinusoidal phase-only filter, with 2 π ( a / λ ) = 30 .

Fig. 4
Fig. 4

MTFs associated with the ambiguity function in Fig. 3.

Fig. 5
Fig. 5

Numerical simulations of the images that can be obtained by using the proposed phase profile.

Fig. 6
Fig. 6

Digital image restoration of an image for various amount of defocus. Column (a), image with additive Gaussian noise ( SNR = 50 ) ; column (b), image on (a) digitally restored using an inverse filter; and column (c), image on (a) digitally restored using a Wiener filter.

Fig. 7
Fig. 7

Digital image restoration of an image for various amount of defocus. Column (a), image with additive Gaussian noise ( SNR = 16 ) ; column (b), image on (a) digitally restored using an inverse filter; and column (c), image on (a) digitally restored using a Wiener filter.

Equations (16)

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P ( ν , μ ; W 20 ) = P ( ν ; W 20 ) P ( μ ; W 20 ) = Q ( ν ) exp [ i 2 π ( W 20 / λ ) ( ν / Ω ) 2 ] × rect ( ν / 2 Ω ) Q ( μ ) × exp [ i 2 π ( W 20 / λ ) ( μ / Ω ) 2 ] × rect ( μ / 2 Ω ) .
p ( x , y ) = - - P ( ν , μ ; W 20 ) exp [ i2π ( x ν + y μ ) ] d ν dμ= - Q ( ν ) rect ( ν ∕2 Ω ) exp [ i2π ( W 20 ∕λ ) ( ν Ω ) 2 ] × exp ( i2π x ν ) d ν - Q ( μ ) rect ( μ∕2 Ω ) × exp [ i2π ( W 20 ∕λ ) ( μ∕ Ω ) 2 ] exp ( i2π y μ ) = q ( x , W 20 ) q ( y , W 20 ) .
H ( μ , W 20 ) = - | q ( y , W 20 ) | 2 exp ( - i 2 πμ y ) d y - | q ( y , 0 ) | 2 d y = - Q ( ζ + μ / 2 ) Q ( ζ - μ / 2 ) × rect [ ( ζ + μ / 2 ) / 2 Ω ] rect [ ( ζ - μ / 2 ) / 2 Ω ] × exp ( i 2 π ( 2 W 20 μ / λΩ 2 ) ζ ) d ζ - | Q ( ζ ) | 2 rect ( ζ / 2 Ω ) d ζ .
A ( μ , s ) = - Q ( ζ + μ / 2 ) Q ( ζ - μ / 2 ) × rect [ ( ζ + μ / 2 ) / 2 Ω ] rect [ ( ζ - μ / 2 ) / 2 Ω ] × exp ( i 2 π s ζ ) d ζ .
| H ( μ , W 20 ) | = | A ( μ , s = 2 W 20 μ / λΩ 2 ) | / 2 Ω .
Q ( μ ) = exp [ - i 2 πφ ( μ ) ] .
A ( μ , s ) = - exp { - i 2 π [ φ ( ζ + μ / 2 ) - φ ( ζ - μ / 2 ) ] } × rect [ ( ζ + μ / 2 ) / 2 Ω ] rect [ ( ζ - μ / 2 ) / 2 Ω ] × exp ( i 2 π s ζ ) d ζ .
( i 2 π ) - 1 ( / s ) A 0 ( μ , s ) = - ζ rect [ ( ζ + μ / 2 ) / 2 Ω ] × rect [ ( ζ - μ / 2 ) / 2 Ω ] × exp ( i 2 π s ζ ) d ζ .
φ [ ( i 2 π ) - 1 ( / s ) ] A 0 ( μ , s ) = - φ ( ζ ) rect [ ( ζ + μ / 2 ) / 2 Ω ] × rect [ ( ζ - μ / 2 ) / 2 Ω ] × exp ( i 2 π s ζ ) d ζ .
A ( μ , s ) = exp ( i 2 π { φ [ ( i 2 π ) - 1  ∂ / s + μ / 2 ] - φ [ ( i 2 π ) - 1  ∂ / s - μ / 2 ] } ) × - rect [ ( ζ + μ / 2 ) / 2 Ω ] × rect [ ( ζ - μ / 2 ) / 2 Ω ] exp ( i 2 π s ζ ) d ζ ,
A ( μ , s ) = exp ( i 2 π { φ [ ( i 2 π ) - 1  ∂ / s + μ / 2 ) - φ [ ( i 2 π ) - 1  ∂ / s - μ / 2 ] } ) A 0 ( μ , s ) .
φ ( μ ) = 2 π ( a / λ ) sin ( πμ / 2 Ω ) .
exp { - i 2 π [ φ ( ζ + μ / 2 ) - φ ( ζ - μ / 2 ) ] } = exp { - i [ 4 π ( a / λ ) sin ( πμ / 4 Ω ) ] cos ( πζ / 2 Ω ) } = m = - ( - i ) m J m [ 4 π ( a / λ ) sin ( πμ / 4 Ω ) ] × exp [ ( - m ζ / 2 Ω ) ] .
exp ( i 2 π { φ [ ( i 2 π ) - 1  ∂ / s + μ / 2 ] - φ [ ( i 2 π ) - 1  ∂ / s μ / 2 ] } ) = m = - ( - i ) m J m [ 4 π ( a / λ ) sin ( πμ / 4 Ω ) ] × exp [ ( - m / 4 Ω ) / s ] .
A ( μ , s ) = m = - ( - i ) m J m [ 4 π ( a / λ ) sin ( πμ / 4 Ω ) ] × exp [ ( - m / 4 Ω ) / s ] A 0 ( μ , s ) = m = - ( - i ) m J m [ 4 π ( a / λ ) sin ( πμ / 4 Ω ) ] × A 0 ( μ , s - m / 4 Ω ) .
A ( s , μ = Ω ) = Ω m = - ( - i ) m J m [ 2 π ( a / λ ) 2 ] × sinc [ ( s Ω - m / 4 ) ] ,

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