Abstract

We show how the focus longitudinal error z can be deduced from the observed image. This deduction requires some digital image processing. Once the focus error z is known, one may compensate for it either by shifting the lens or by deblurring the image digitally. We have studied systems with a circular or with a rectangular aperture. In the former case, the defocus causes some dark rings in the frequency spectrum of the image. With a rectangular aperture, the defocus creates a set of straight, orthogonal dark lines. These dark zones provide a quantitative diagnosis of the focus error. Some experimental evidence supports our concept.

© 1997 Optical Society of America

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References

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  1. R. Dandliker, “Heterodyne holographic interferometry,” Progress in Optics, Vol. 17, E. Wolf, ed. (North-Holland, Amsterdam, 1980).
    [CrossRef]
  2. G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” Appl. Opt. Electr. Eng. 9, 73–110 (1983).
    [CrossRef]
  3. G. E. Hege, H. J. Tiziani, “Speckle techniques for absolute distance measurement,” Tech. Messen 54 (6), 237–242 (1987)(in German)
  4. K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).
  5. M. Forrer, M. Frenz, A. D. Zweig, V. Romano, H. P. Weber, J. Ochsenbein, “Autofocus system for surgical CO2 laser,” Appl. Opt. 30, 1480–1486 (1991).
    [CrossRef] [PubMed]
  6. M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 6.
  8. P. M. Duffieux, L’Integrale de Fourier et ses Applications a L’Optique (Faculte des Sciences, Besancon, 1946), (in French); P. M. Duffieux, The Fourier Transform and its Application to Optics, 2nd. ed. (Wiley, New York, 1983).
  9. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
    [CrossRef] [PubMed]

1993 (1)

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

1991 (1)

1987 (2)

G. E. Hege, H. J. Tiziani, “Speckle techniques for absolute distance measurement,” Tech. Messen 54 (6), 237–242 (1987)(in German)

K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).

1983 (1)

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” Appl. Opt. Electr. Eng. 9, 73–110 (1983).
[CrossRef]

1970 (1)

Bouwhuis, G.

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” Appl. Opt. Electr. Eng. 9, 73–110 (1983).
[CrossRef]

Braat, J. J. M.

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” Appl. Opt. Electr. Eng. 9, 73–110 (1983).
[CrossRef]

Choi, T.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Dandliker, R.

R. Dandliker, “Heterodyne holographic interferometry,” Progress in Optics, Vol. 17, E. Wolf, ed. (North-Holland, Amsterdam, 1980).
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, L’Integrale de Fourier et ses Applications a L’Optique (Faculte des Sciences, Besancon, 1946), (in French); P. M. Duffieux, The Fourier Transform and its Application to Optics, 2nd. ed. (Wiley, New York, 1983).

Forrer, M.

Frenz, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 6.

Hege, G. E.

G. E. Hege, H. J. Tiziani, “Speckle techniques for absolute distance measurement,” Tech. Messen 54 (6), 237–242 (1987)(in German)

Leonhardt, K.

K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).

Lohmann, A. W.

Nikzad, A.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Ochsenbein, J.

Rippert, K. H.

K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).

Romano, V.

Subbarao, M.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Tiziani, H. J.

K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).

G. E. Hege, H. J. Tiziani, “Speckle techniques for absolute distance measurement,” Tech. Messen 54 (6), 237–242 (1987)(in German)

Weber, H. P.

Zweig, A. D.

Appl. Opt. (2)

Appl. Opt. Electr. Eng. (1)

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” Appl. Opt. Electr. Eng. 9, 73–110 (1983).
[CrossRef]

Opt. Eng. (1)

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Tech. Messen (2)

G. E. Hege, H. J. Tiziani, “Speckle techniques for absolute distance measurement,” Tech. Messen 54 (6), 237–242 (1987)(in German)

K. Leonhardt, K. H. Rippert, H. J. Tiziani, “Optical microprofilometry and roughness measurement,” Tech. Messen 54 (6), 243–252 (1987) (in German).

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 6.

P. M. Duffieux, L’Integrale de Fourier et ses Applications a L’Optique (Faculte des Sciences, Besancon, 1946), (in French); P. M. Duffieux, The Fourier Transform and its Application to Optics, 2nd. ed. (Wiley, New York, 1983).

R. Dandliker, “Heterodyne holographic interferometry,” Progress in Optics, Vol. 17, E. Wolf, ed. (North-Holland, Amsterdam, 1980).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Standard optical system with focus error.

Fig. 2
Fig. 2

Telecentric one-to-one imaging system.

Fig. 3
Fig. 3

Description of the integration limits of Eq. (10).

Fig. 4
Fig. 4

Rectangular aperture.

Fig. 5
Fig. 5

Optical transfer function for various amounts of defocusing: vertical, OTF; horizontal, reduced frequency; φ, defocus parameter.

Fig. 6
Fig. 6

First zero-crossing frequency as a function of the defocusing.

Fig. 7
Fig. 7

Zero stripes that appear in the frequency domain if the aperture is rectangular.

Fig. 8
Fig. 8

Evidence of the dark fringes in the frequency domain of the defocused image: (a1) object, (b1) defocused image IB when a circular aperture is used, (c1) OTF of the misfocused object (spectrum of I B), (d1) cross section of c1, (a2)–(d2) same as (a1)–(d1) but for rectangular aperture.

Fig. 9
Fig. 9

Varifocal lens that may be used for the focus action.

Equations (32)

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IBx, y; z=ITx, y*x*yFx, y; z, 
I˜bνx, νy; z=I˜Tνx, νyF˜ νx, νy; z.
Ĩ0νx, νy=ĨTνx, νyF˜0νx, νy.
I˜Bνx, νy; z=I˜0νx, νyF˜νx, νy; zF˜0νx, νy=I˜0·H˜.
P˜x1λf, y1λf; z=circx1, y1; Bexp-iπλzx12+y12λf2.
circx1, y1; B=1x12+y12B/220otherwise
Px, y; z=P˜x1λf, y1λf; zexp-2πix1x+y1yλfdx.
Px, y; z2=Fx, y; z.
F˜νx, νy; z=Fx, y; zexp-2πiνxx+νyydxdy.
F˜νx, νy; z=1NP˜νx+νx2, νy+νy2; z×P˜*νx-νx2, νy-νy2; zdνxdνy.
exp-iπλzνx2+νy2.
-2πλzνxνx+νyνy.
F˜νx, νy; z=1Ncircνx+νx2, νy+νy2, Bλf×circνx-νx2, νy-vy2, Bλf×exp-2πiλzνxνx+νyνydνxdνy.
P˜νx, νy; z=P˜xνx; zP˜yνy; z.
P˜xνx; z=Rνx/Δνexp-iπλzνx2,
F˜νx, νy; z=F˜xνx; zF˜yνy; z.
F˜xνx; z=1ΔνRνx+vx/2ΔνRνx-νx/2Δν×exp-πλzννdν.
R+R-=Rνx/ΔνRνx/Δν-νx.
F˜xνx; z=F˜0νxH˜νx, z,
F˜0νx=Rνx/Δν1-νx/Δν=trianνx/Δν,
H˜νx, z=sincΔν-νxλzνx.
μ=ν/Δν; μ1,
φ=z/δz; δz=4/λΔν2=4λΦ2.
ν0z0+ε<ν0z0,
φ1-μμ=1/4.
μ=1/21-1-1/φ1/21/4φ φ1.
IBx, y; z.
IBx, y; zexp-2πiνxx+νyydxdy=I˜Bνx, νy; z.
I˜Bνx, νy; z.
SBρ, z=02νI˜Bρ, θ; zdθ, SBρ, z=SOBJρSSYSTρ, z.
I˜Bνx, νydνy=SBνx.
I˜0νx, νy=I˜Bνx, νy; zF˜0νx, νyF˜νx, νy; z.

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