Abstract

For extending the depth of field, we analyze the result of superimposing several snapshots, which are taken while changing the amount of focus error, at full pupil aperture. We unveil the use of a varifocal lens for controlling the amount of focus error, without modifying either the lateral magnification or light throughput. After recording a set of snapshots, we use suitable acquisition factors for shaping an optical transfer function, which has reduced sensitivity to focus errors.

© 2013 Optical Society of America

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References

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  1. M. Mino and Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219–2225 (1971).
    [CrossRef]
  2. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. L. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
    [CrossRef]
  3. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef]
  4. J. Ojeda-Castaneda, P. Andrés, and A. Diaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef]
  5. J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef]
  6. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
    [CrossRef]
  7. J. Ojeda-Castaneda and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
    [CrossRef]
  8. J. Ojeda-Castañeda and A. Díaz, “High focal depth by quasi-bifocus,” Appl. Opt. 27, 4163–4165 (1988).
    [CrossRef]
  9. E. R. Dowski and T. W. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1865(1995).
    [CrossRef]
  10. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658–5662 (2001).
    [CrossRef]
  11. S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt. 41, 4611–4615 (2002).
    [CrossRef]
  12. N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A 5, s157–s163 (2003).
    [CrossRef]
  13. A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29, 560–562(2004).
    [CrossRef]
  14. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479(2004).
    [CrossRef]
  15. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend depth of field of incoherent hybrid imaging system,” Appl. Opt. 43, 2709–2721 (2004).
    [CrossRef]
  16. J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30, 1647–1649 (2005).
    [CrossRef]
  17. A. Castro, J. Ojeda-Castaneda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878–7884 (2006).
    [CrossRef]
  18. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A 23, 1058–1062 (2006).
    [CrossRef]
  19. G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15, 9184–9193 (2007).
    [CrossRef]
  20. M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. 46, 216–226 (2007).
    [CrossRef]
  21. Y. Takahashi and S. Komatsu, “Optimized free-form phase mask for extension of depth of field in wavefront-coding imaging,” Opt. Lett. 33, 1515 (2008).
    [CrossRef]
  22. N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. 47, E39–E43 (2008).
    [CrossRef]
  23. J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “The use of conjugate phase plates in the analysis of the frequency response of optical systems designed for an extended depth of field,” Appl. Opt. 47, E99–E105 (2008).
    [CrossRef]
  24. J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Optical processor arrays for controlling focal length or for tuning the depth of field,” Photonics Lett. Pol. 3, 44–46 (2011).
    [CrossRef]
  25. J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
    [CrossRef]
  26. H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1957).
    [CrossRef]
  27. G. Haeusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
    [CrossRef]
  28. R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 (ACM, 2006), pp. 795–804.
  29. L. W. Alvarez, “Two-element variable-power spherical lens,” U. S. Patent 3,305,294 (3December1964).
  30. A. W. Lohmann, “Lente focale variabile,” Italian Patent 727,848 (19June1964).
  31. A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” British patent 998,191 (29May1964).
  32. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
    [CrossRef]
  33. B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, 1991), pp. 182–183.

2011 (1)

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Optical processor arrays for controlling focal length or for tuning the depth of field,” Photonics Lett. Pol. 3, 44–46 (2011).
[CrossRef]

2010 (1)

J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
[CrossRef]

2008 (3)

2007 (2)

2006 (2)

2005 (1)

2004 (3)

2003 (1)

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A 5, s157–s163 (2003).
[CrossRef]

2002 (1)

2001 (1)

1995 (1)

1988 (4)

1986 (1)

1985 (1)

1983 (1)

1972 (1)

G. Haeusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

1971 (1)

1970 (1)

1957 (1)

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1957).
[CrossRef]

Agrawal, A.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 (ACM, 2006), pp. 795–804.

Alvarez, L. W.

L. W. Alvarez, “Two-element variable-power spherical lens,” U. S. Patent 3,305,294 (3December1964).

Andrés, P.

Berriel-Valdos, L. R.

Caron, N.

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 5, s157–s163 (2003).
[CrossRef]

Christensen, M. P.

Diaz, A.

Díaz, A.

Dowski, E. R.

Escamilla, H. M.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, 1991), pp. 182–183.

Gan, F.

García-Almanza, E.

J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
[CrossRef]

George, N.

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A 5, s157–s163 (2003).
[CrossRef]

Ghosh, A.

Gómez-Sarabia, C. M.

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Optical processor arrays for controlling focal length or for tuning the depth of field,” Photonics Lett. Pol. 3, 44–46 (2011).
[CrossRef]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “The use of conjugate phase plates in the analysis of the frequency response of optical systems designed for an extended depth of field,” Appl. Opt. 47, E99–E105 (2008).
[CrossRef]

Haeusler, G.

G. Haeusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Harvey, A. R.

Hopkins, H. H.

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1957).
[CrossRef]

Jaroszewicz, Z.

Kolodziejczyk, A.

Komatsu, S.

Landgrave, J. E. A.

Lohmann, A. W.

A. Castro, J. Ojeda-Castaneda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878–7884 (2006).
[CrossRef]

A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
[CrossRef]

A. W. Lohmann, “Lente focale variabile,” Italian Patent 727,848 (19June1964).

A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” British patent 998,191 (29May1964).

Mezouari, S.

Mikula, G.

Mino, M.

Montes, E.

Montes, E. L.

Muyo, G.

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Optical processor arrays for controlling focal length or for tuning the depth of field,” Photonics Lett. Pol. 3, 44–46 (2011).
[CrossRef]

J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
[CrossRef]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “The use of conjugate phase plates in the analysis of the frequency response of optical systems designed for an extended depth of field,” Appl. Opt. 47, E99–E105 (2008).
[CrossRef]

A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29, 560–562(2004).
[CrossRef]

A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479(2004).
[CrossRef]

J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
[CrossRef]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
[CrossRef]

J. Ojeda-Castañeda and A. Díaz, “High focal depth by quasi-bifocus,” Appl. Opt. 27, 4163–4165 (1988).
[CrossRef]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
[CrossRef]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. L. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
[CrossRef]

Okano, Y.

Petelczyc, K.

Raskar, R.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 (ACM, 2006), pp. 795–804.

Sanyal, S.

Sauceda, A.

Sheng, Y.

Sherif, S. S.

Somayaji, M.

Sypek, M.

Takahashi, Y.

Tumblin, J.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 (ACM, 2006), pp. 795–804.

Wang, H.

Yépez-Vidal, E.

J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
[CrossRef]

Appl. Opt. (14)

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
[CrossRef]

J. Ojeda-Castaneda and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
[CrossRef]

J. Ojeda-Castañeda and A. Díaz, “High focal depth by quasi-bifocus,” Appl. Opt. 27, 4163–4165 (1988).
[CrossRef]

E. R. Dowski and T. W. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1865(1995).
[CrossRef]

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658–5662 (2001).
[CrossRef]

S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt. 41, 4611–4615 (2002).
[CrossRef]

M. Mino and Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219–2225 (1971).
[CrossRef]

A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479(2004).
[CrossRef]

S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend depth of field of incoherent hybrid imaging system,” Appl. Opt. 43, 2709–2721 (2004).
[CrossRef]

A. Castro, J. Ojeda-Castaneda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878–7884 (2006).
[CrossRef]

M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. 46, 216–226 (2007).
[CrossRef]

N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. 47, E39–E43 (2008).
[CrossRef]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “The use of conjugate phase plates in the analysis of the frequency response of optical systems designed for an extended depth of field,” Appl. Opt. 47, E99–E105 (2008).
[CrossRef]

A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
[CrossRef]

J. Opt. A (1)

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A 5, s157–s163 (2003).
[CrossRef]

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

Opt. Commun. (1)

G. Haeusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Photon. Lett. Pol. (1)

J. Ojeda-Castañeda, E. Yépez-Vidal, and E. García-Almanza, “Complex amplitude filters for extended depth of field,” Photon. Lett. Pol. 2, 162–164 (2010).
[CrossRef]

Photonics Lett. Pol. (1)

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Optical processor arrays for controlling focal length or for tuning the depth of field,” Photonics Lett. Pol. 3, 44–46 (2011).
[CrossRef]

Proc. Phys. Soc. B (1)

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1957).
[CrossRef]

Other (5)

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, 1991), pp. 182–183.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 (ACM, 2006), pp. 795–804.

L. W. Alvarez, “Two-element variable-power spherical lens,” U. S. Patent 3,305,294 (3December1964).

A. W. Lohmann, “Lente focale variabile,” Italian Patent 727,848 (19June1964).

A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” British patent 998,191 (29May1964).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of an optical processor.

Fig. 2.
Fig. 2.

MTFs of (a) diffraction limited aperture, (b) focus error W=1, (c) the average MTF, and (d) cubic phase mask and a Gaussian apodizer.

Fig. 3.
Fig. 3.

Varifocal lens for controlling the focus error coefficient.

Fig. 4.
Fig. 4.

Hauesler proposal for extending the depth of field.

Fig. 5.
Fig. 5.

Weighting factors in modulated time photography.

Fig. 6.
Fig. 6.

Averaging 10 snapshots having focus error coefficients with random values.

Fig. 7.
Fig. 7.

MTF versus focus error and average MTF versus maximum integration value.

Tables (2)

Tables Icon

Table 1. Classical Trade-Offs for Extending the Depth of Focus

Tables Icon

Table 2. Impact of Focus Error: (a) Interferograms of the Pupil Function, (b) PSF, (c) Siemen Star Images, and (d) Lena Images

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

P(μ;W2,0)=Q(μ)exp[i2πW(μΩ)2]rect(μ2Ω).
HQ(μ;W)=(2Ω|μ|)2(2Ω|μ|)2Q(ν+μ2)Q*(νμ2)ei2π(2μνΩ2)Wdν.
I(x;W)=I˜0(μ)·HQ(μ;W)ei2πxμdμ+|N(x;W)|2.
I(x)=g(W)I(x;W)dW.
I(x)=I˜0(μ)g(W)HQ(μ;W)dWei2πxμdμ+g(W)|N(x;W)|2dW=I˜0(μ)HQ(μ)ei2πxμdμ+|N(x)|2.
I(x)|N(x)|2=I˜0(μ)HQ(μ)ei2πxμdμ.
I˜0(μ)=[HQ(μ)]1[I(x)|N(x)|2]ei2πμxdx.
HQ(μ)=(2Ω|μ|)2(2Ω|μ|)2Q(ν+μ2)Q*(νμ2)G(2μνΩ2)dν.
G(2μνΩ2)=g(W)ei2π(2μνΩ2)WdW.
T(μ;η)=exp{i2πa[(μ+η2Ω)3(μ+η2Ω)3]}rect(μ2Ω)=exp{i(π2)(ηΩ)3}exp{i2π(3aηΩ)(μΩ)2}rect(μ2Ω).
P(μ;η;W)=Q(μ)T(μ;η)ei2πW(μΩ)2rect(μ2Ω).
P(μ;η;W)=ei(π2)(ηΩ)3Q(μ)ei2π[W3aηΩ](μΩ)2rect(μ2Ω).
η=(Ω3a)W.
E(x,tn;Wn)=M(tn)I(x,Wn).
Wn=f(tn);or equivalentlytn=f1(Wn).
E(x,tn;Wn)=M(f1(Wn))I(x,Wn)=g(Wn)I(x,Wn).
s(W)=I(x;W)I(0;0)=sinc2(Wε2).
R(μ;W)=|HQ(μ;W)||HQ(μ;0)|.
R(μ;W)=|HQ(μ;W)||HQ(μ;0)|=sinc2[8(Wε2)(μ2εΩ)(1|μ2εΩ|)].
sinc2[8(Wε2)(μ2εΩ)(1|μ2εΩ|)]L.
sinc2[W(2ε1)]L.
(2ε1)W=(2ε1)W2,0λ0.9,orW2,09λ10(2ε1).
g(W)=(1M)m=0M1Cmδ(WWm).
G(2μνΩ2)=(1M)m=0M1Cmei2π(2μνΩ2)Wm.
HQ(μ)=(1M)m=0M1CmHQ(μ;Wm).
g(W)=(1Wmax)rect(WWmax).
G(2μνΩ2)=sinc(2μνWmaxΩ2).
HQ(μ)=18π(μ2Ω)WmaxSi(8π(μ2Ω)(1|μ2Ω|)Wmax).
Wmax=18(σ2Ω)(1σ2Ω).
|N(x)|2=g(W)|N(x;W)|2dW.
P(μ)=|N(x)|2ei2πμxdx=g(W)N(x;W)N*(x;W)ei2πμxdxdW.
P(μ)=g(W)N(ν+μ2;W)N*(νμ2;W)dνdW=N(ν+μ2)N*(νμ2)dν.

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