Abstract

A new approach for obtaining line-spread functions (LSF’s), which vary slowly in out-of-focus planes, is described. Based on this approach, we report a LSF that shows relatively less sensitivity to focus errors than that shown by both the diffraction-limited LSF and a LSF previously reported.

© 1983 Optical Society of America

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References

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  1. J. Ojeda-Castaneda, “Focus-error operator and related special functions,” J. Opt. Soc. Am. 73, 1042–1047 (1983).
    [CrossRef]
  2. M. Mino, Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shadow apertures,” Appl. Opt. 10, 2219 (1971).
    [CrossRef] [PubMed]
  3. H. H. Hopkins, “Canonical coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 4, Suppl. 1, 31–35 (1965).
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  5. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950) p. 14.

1983 (1)

1971 (1)

1965 (1)

H. H. Hopkins, “Canonical coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 4, Suppl. 1, 31–35 (1965).

Goodman, J. W.

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

Hopkins, H. H.

H. H. Hopkins, “Canonical coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 4, Suppl. 1, 31–35 (1965).

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950) p. 14.

Mino, M.

Ojeda-Castaneda, J.

Okano, Y.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

H. H. Hopkins, “Canonical coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 4, Suppl. 1, 31–35 (1965).

Other (2)

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

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950) p. 14.

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Figures (4)

Fig. 1
Fig. 1

In-focus LSF irradiance distributions for a diffraction-limited pupil, LSF0; for the filter after Mino and Okano, LSF1; and for the proposed filter, LSF2.

Fig. 2
Fig. 2

Out-of-focus irradiance distributions for the LSF’s shown in Fig. 1, with W20 = 0.5λ. The continuous line is the graph for the in-focus LSF irradiance for a diffraction-limited pupil.

Fig. 3
Fig. 3

Out-of-focus irradiance distributions for the LSF’s shown in Fig. 1, with W20 = 1.0λ.

Fig. 4
Fig. 4

On-axis maximum irradiance versus the defocus coefficient W20 for the LSF’s indicated in Fig. 1.

Equations (20)

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a ( x ) = A ( u ) exp ( j 2 π u x ) d u ,
P ( u ) = exp ( j k W 20 u 2 ) Q ( u ) ,
A ( u ) = m = 0 b m u 2 m Q ( u ) A 0 ( u ) .
a ( x ) = m = 0 b m ( j / 2 π ) m × ( j 2 π u ) 2 m Q ( u ) A 0 ( u ) exp ( j 2 π u x ) d u
a ( x ) = m = 0 c m d 2 m d x 2 m q ( x ) ,
q ( x ) = Q ( u ) A 0 ( u ) exp ( j 2 π u x ) d u .
a ( x ) = m = 0 c m d 2 m d x 2 m q ( x ) = q ( x ) .
q ( x ) = constant ,
Q ( u ) = δ ( u ) ,
q ( x ) = α 0 + α 1 x + α 2 x 2 .
q ( x ) = ( α 0 + α 1 x + α 2 x 2 ) rect ( x / x 0 ) ,
rect ( x / x 0 ) = { 1 , | x | | x 0 | 0 , elsewhere .
q ( x ) = [ ( α 0 + α 1 x + α 2 x 2 ) rect ( x / x 0 ) ] * sin c ( 2 π x ) ,
α 1 = 0 , α 2 < 0 , | α 2 | < α 0 .
α 0 | α 2 | x 2 | x = x 0 = 0 ,
α 0 / | α 2 | = x 0 .
α 0 = 1 ;
q ( x ) = { [ 1 ( x / x 0 ) 2 ] rect ( x / x 0 ) } * sin c ( 2 π x )
Q ( u ) = 2 x 0 [ sin c 2 π ux o + ( 1 2 π x 0 ) 2 d 2 d u 2 sin c 2 π ux o ] rect ( u ) .
Q ( u ) = ( 1 u 2 ) rect ( u ) .

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