Abstract

The optical transfer functions for variable focus error are contained as a single picture representation in the ambiguity function that is associated with the pupil function. This picture representation is shown to be useful for designing pupil functions that increase the depth of focus. We specify a criterion for an optical transfer function with low sensitivity defocus in terms of a nonlinear differential equation for the point spread function. Based on this approach, we design and compare five new spatial filters for achieving high focal depth.

© 1988 Optical Society of America

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References

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  1. M. Mino, Y. Okano, “Improvement in the OTF of a Defocused Optical System Through the Use of Shade Apertures,” Appl. Opt. 10, 2219 (1971).
    [CrossRef] [PubMed]
  2. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458 (1983).
    [CrossRef] [PubMed]
  3. C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. 2, 799 (1985).
    [CrossRef]
  4. G. Indebetow, H. Bai, “Imaging with Fresnel Zone Pupils Masks: Extended Depth of Field,” Appl. Opt. 23, 4299 (1984).
    [CrossRef]
  5. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
    [CrossRef] [PubMed]
  6. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular Apodizer for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487 (1986).
    [CrossRef] [PubMed]
  7. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Strehl Ratio with Low Sensitivity to Spherical Aberration,” J. Opt. Soc. Am. (accepted).
  8. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
    [CrossRef]
  9. A. W. Lohmann, “Teaching how to Invent: Is it possible?” Isr. J. Technol. 18, 214 (1980).
  10. J. Ojeda-Castaneda, “Focus-Error Operator and Related Special Functions,” J. Opt. Soc. Am. 73, 1042 (1983).
    [CrossRef]
  11. H. Bartelt, J. Ojeda-Castaneda, E. E. Sicre, “Misfocus Tolerance Seen by Simple Inspection of the Ambiguity Function,” Appl. Opt. 23, 2693 (1984).
    [CrossRef] [PubMed]
  12. J. P. Mills, B. J. Thompson, “Effect of Aberrations and Apodization on the Performance of Coherent Optical Systems. I. The Amplitude Impulse Response,” J. Opt. Soc. Am. A 3, 694 (1986).
    [CrossRef]
  13. J. P. Mills, B. J. Thompson, “Effect of Aberrations and Apodization on the Performance of Coherent Optical Systems. II: Imaging,” J. Opt. Soc. Am. A 3, 704 (1986).
    [CrossRef]
  14. J. Ojeda-Castaneda, P. Andres, E. Montes, “Phase-Space Representation of the Strehl Ratio: Ambiguity Function,” J. Opt. Soc. Am. A 4, 313 (1987).
    [CrossRef]

1987 (1)

1986 (3)

1985 (2)

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. 2, 799 (1985).
[CrossRef]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
[CrossRef] [PubMed]

1984 (2)

1983 (3)

1980 (1)

A. W. Lohmann, “Teaching how to Invent: Is it possible?” Isr. J. Technol. 18, 214 (1980).

1971 (1)

Andres, P.

Bai, H.

Bartelt, H.

Berriel-Valdos, L. R.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

Diaz, A.

J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular Apodizer for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487 (1986).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, P. Andres, A. Diaz, “Strehl Ratio with Low Sensitivity to Spherical Aberration,” J. Opt. Soc. Am. (accepted).

Indebetouw, G.

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. 2, 799 (1985).
[CrossRef]

Indebetow, G.

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

A. W. Lohmann, “Teaching how to Invent: Is it possible?” Isr. J. Technol. 18, 214 (1980).

Mills, J. P.

Mino, M.

Montes, E.

Ojeda-Castaneda, J.

Okano, Y.

Sicre, E. E.

Thompson, B. J.

Varamit, C.

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. 2, 799 (1985).
[CrossRef]

Appl. Opt. (3)

Isr. J. Technol. (1)

A. W. Lohmann, “Teaching how to Invent: Is it possible?” Isr. J. Technol. 18, 214 (1980).

J. Opt. Soc. Am. (2)

C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. 2, 799 (1985).
[CrossRef]

J. Ojeda-Castaneda, “Focus-Error Operator and Related Special Functions,” J. Opt. Soc. Am. 73, 1042 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “The Ambiguity Function as a Polar Display of the OTF,” Opt. Commun. 44, 323 (1983).
[CrossRef]

Opt. Lett. (3)

Other (1)

J. Ojeda-Castaneda, P. Andres, A. Diaz, “Strehl Ratio with Low Sensitivity to Spherical Aberration,” J. Opt. Soc. Am. (accepted).

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Figures (6)

Fig. 1
Fig. 1

Pupil functions Q0(u) of some apodizers that increase the depth of focus.

Fig. 2
Fig. 2

Irradiance point spread functions q0(x).

Fig. 3
Fig. 3

Column A integrand of Eq. (3). Column B modulus of the ambiguity functions for the pupil functions a, b, c, and d of Fig. 1.

Fig. 4
Fig. 4

Same as in Fig. 3 but for the pupil functions e, f, g, and h in Fig. 1.

Fig. 5
Fig. 5

Images of a spoke pattern that are obtained with the apodizers a, b, c, and d in Fig. 1: Column A, in-focus images; Column B, out-of-focus images with a focus error of 1λ.

Fig. 6
Fig. 6

Same as in Fig. 5 but for the apodizers e, f, g, and h in Fig. 1.

Tables (1)

Tables Icon

Table I Out-Of-Focus Imagery

Equations (33)

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Q 0 ( u ) = F ( u ) rect ( u ) .
A ( y , u ) = Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) exp ( i 2 π y υ ) d υ .
A ( W 20 , u ) = A ( W 20 = 0 , u ) + W 20 A W 20 ( W 20 = 0 , u ) + W 20 2 2 ! 2 A W 20 2 ( W 20 = 0 , u ) + .
n A W 20 n ( W 20 = 0 , u ) = 0 , for n = 1 , 2 .
i 4 π u υ Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ = 0 ,
( i 4 π u ) 2 υ 2 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ = 0 .
Q 0 ( u ) = Q * 0 ( u ) ,
υ 2 n + 1 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ = 0 ,
2 n + 1 W 20 2 n + 1 A ( W 20 = 0 , u ) = 0 ,
q 0 ( x ) = Q 0 ( u ) exp ( i 2 π u x ) d u ,
[ d d x q 0 ( x ) ] 2 = q 0 ( x ) d 2 d x 2 q 0 ( x ) ;
q 0 ( x ) = A exp ( B x ) ,
[ d d x q 0 ( x ) ] 2 q 0 ( x ) d 2 d x 2 q 0 ( x ) = ε 0 .
Q 0 ( u ) = Q 0 ( u ) 0 .
s ( u , υ ) = rect ( υ + u / 2 ) rect ( υ u / 2 ) .
| u | 1 and | υ | ( 1 | u | ) / 2 .
Q 0 ( u ) = rect ( u ) ,
Q 0 ( u ) = ( 1 4 u 2 ) rect ( u ) ,
Q 0 ( u ) = [ sinc ( u ) + ( 2 π ) 2 d 2 d u 2 sinc ( u ) ] rect ( u ) .
Q 0 ( u ) = sinc ( u ) rect ( u ) ,
Q 0 ( u ) = sinc 2 ( 2 u ) rect ( u ) ,
Q 0 ( u ) = sinc 2 ( 4 u ) rect ( u ) .
Q 0 ( u ) = exp ( π u 2 ) ,
Q 0 ( u ) = exp ( 3 π u 2 ) .
d d y q 0 ( y ) = i 2 π u Q 0 ( u ) exp ( i 2 π y u ) d u .
| d d y q 0 ( y ) | 2 = ( 2 π ) u α ( Q 0 ( u ) Q * 0 ( α ) × exp [ i 2 π y ( u α ) ] d u d α .
| d d y q 0 ( y ) | 2 exp ( i 2 π u y ) d y = ( 2 π ) 2 ( u + α ) α Q 0 ( u + α ) × Q * 0 ( α ) d u ,
υ = α + u / 2 ,
| d d y q 0 ( y ) | 2 exp ( i 2 π u y ) d y = ( 2 π ) 2 ( υ 2 u 2 / 4 ) × Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ
| d d y q 0 ( y ) | 2 exp ( i 2 π u y ) d y = ( 2 π ) 2 υ 2 Q 0 ( υ + u / 2 ) × Q * 0 ( υ u / 2 ) d υ ( π u ) 2 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ .
( π u ) 2 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ = ( π u ) 2 | q 0 ( y ) | 2 exp ( i 2 π u y ) d y .
( 2 π ) 2 υ 2 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ = [ | d d y q 0 ( y ) | 2 + ( π u ) 2 | q 0 ( y ) | 2 ] exp ( i 2 π u y ) d y .
0 = ( 2 π ) 2 υ 2 Q 0 ( υ + u / 2 ) Q * 0 ( υ u / 2 ) d υ × exp ( i 2 π x u ) d u = | d d x q 0 ( x ) | 2 1 4 d 2 d x 2 | q 0 ( x ) | 2 .

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