Abstract

A visibility-dependent depth of focus is developed for incoherent sinusoidal sources. This analysis is based on a derived relationship between the visibility and the optical transfer function (OTF). The formalism developed is general for any aperture topology and arbitrary orientation of sinusoidal sources. To illustrate the application of the method the cases of an annular aperture and a Gaussian aperture are analyzed. It is found that as the level of defect of defocus increases, the maximum visibility for which a particular spatial frequency can be resolved decreases.

© 1992 Optical Society of America

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References

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  1. H. H. Hopkins, “Frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [CrossRef]
  2. E. L. O’Neill, “Transfer function of an annular aperture,” J. Opt. Soc. Am. 46, 285–288 (1956).
    [CrossRef]
  3. A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).
  4. H. F. A. Tschunko, “Imaging performance of annular apertures,” Appl. Opt. 13, 1820–1823 (1974).
    [CrossRef] [PubMed]
  5. R. J. Pieper, J. Park, T.-C. Poon, “Resolution-dependent depth of focus for an incoherent imaging system,” Appl. Opt. 27, 2040–2047 (1988).
    [CrossRef] [PubMed]
  6. J. W. Coltman, “The specification of imaging properties by response to a sine wave input,” J. Opt. Soc. Am. 44, 468–471 (1954).
    [CrossRef]
  7. R. J. Pieper, T.-C. Poon, “Optical transfer functions for defocused two-pupil systems involving annular geometries,” J. Mod. Opt. 37, 2055–2072 (1990).
    [CrossRef]
  8. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1976), Chap. 9, pp. 309–310.
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 112–119.
  10. A. Papoulis, Systems and Transforms with Applications in Optics (KriegerPublishing, Melbourne, Fla., 1968), Chap. 3, pp. 55–56.
  11. M. Born, E. Wolf, Principles of Optics (McGraw-Hill, New York, 1968), Chap. 7, p. 267.
  12. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 521–526 (1985).
    [CrossRef]
  13. T.-C. Poon, M. Motamedi, “Optical/digital incoherent image processing for extended depth of field,” Appl. Opt. 26, 4612–4615 (1987).
    [CrossRef] [PubMed]

1990

R. J. Pieper, T.-C. Poon, “Optical transfer functions for defocused two-pupil systems involving annular geometries,” J. Mod. Opt. 37, 2055–2072 (1990).
[CrossRef]

1988

1987

1985

1974

A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).

H. F. A. Tschunko, “Imaging performance of annular apertures,” Appl. Opt. 13, 1820–1823 (1974).
[CrossRef] [PubMed]

1956

1955

H. H. Hopkins, “Frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

1954

Bhatnagar, G. S.

A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).

Born, M.

M. Born, E. Wolf, Principles of Optics (McGraw-Hill, New York, 1968), Chap. 7, p. 267.

Coltman, J. W.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 112–119.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1976), Chap. 9, pp. 309–310.

Hopkins, H. H.

H. H. Hopkins, “Frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Kavathekar, A. K.

A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).

Motamedi, M.

O’Neill, E. L.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (KriegerPublishing, Melbourne, Fla., 1968), Chap. 3, pp. 55–56.

Park, J.

Pieper, R. J.

R. J. Pieper, T.-C. Poon, “Optical transfer functions for defocused two-pupil systems involving annular geometries,” J. Mod. Opt. 37, 2055–2072 (1990).
[CrossRef]

R. J. Pieper, J. Park, T.-C. Poon, “Resolution-dependent depth of focus for an incoherent imaging system,” Appl. Opt. 27, 2040–2047 (1988).
[CrossRef] [PubMed]

Poon, T.-C.

Singh, K.

A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).

Tschunko, H. F. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (McGraw-Hill, New York, 1968), Chap. 7, p. 267.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1976), Chap. 9, pp. 309–310.

Appl. Opt.

Indian J. Pure Appl. Phys.

A. K. Kavathekar, G. S. Bhatnagar, K. Singh, “Diffraction images of truncated periodic objects formed by a diffraction-limited imaging system with an annular aperture,” Indian J. Pure Appl. Phys. 12, 138–142 (1974).

J. Mod. Opt.

R. J. Pieper, T.-C. Poon, “Optical transfer functions for defocused two-pupil systems involving annular geometries,” J. Mod. Opt. 37, 2055–2072 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. R. Soc. London Ser. A

H. H. Hopkins, “Frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Other

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1976), Chap. 9, pp. 309–310.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, pp. 112–119.

A. Papoulis, Systems and Transforms with Applications in Optics (KriegerPublishing, Melbourne, Fla., 1968), Chap. 3, pp. 55–56.

M. Born, E. Wolf, Principles of Optics (McGraw-Hill, New York, 1968), Chap. 7, p. 267.

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Figures (8)

Fig. 1
Fig. 1

Two impulse responses with defining parameters for the Rayleigh criterion.

Fig. 2
Fig. 2

(a) Response to sinusoidal input. (b) Rayleigh criterion fails with the addition of a small amount of dc.

Fig. 3
Fig. 3

Optical configuration examined for the determination of the impulse response.

Fig. 4
Fig. 4

(a) Relationship between spatial coordinates u and υ and rotated spatial coordinates ũ, υ ˜; the line source is oriented along the ũ axis and at an angle θ with the u axis. (b) Relationship between scaled spatial coordinates α and ω and rotated scaled spatial coordinates α ˜ and ω ˜

Fig. 5
Fig. 5

(a) Gaussian aperture. (b) Annular aperture.

Fig. 6
Fig. 6

Depth-of-focus curves for Gaussian aperture.

Fig. 7
Fig. 7

OTF curve for annular aperture; obscuration ratio 0.5 (after Ref. 7).

Fig. 8
Fig. 8

(a) Depth-of-focus curves for the annular aperture with obscuration ratios of (a) 0.0; (b) 0.5; (c) 0.9.

Tables (1)

Tables Icon

Table I Maximum Deviation Δd′ for Various Visibilities

Equations (77)

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l W ,
h c ( u , υ ) = { h d ( x , y ) exp [ j k 2 F ( x 2 + y 2 ) ] P ( x , y ) } * h d ( x , y ) ,
r * s = s ( x , y ) r ( u x , υ y ) d x d y ,
h d ( x , y ) = c 1 exp [ j k 2 d ( x 2 + y 2 ) ] ,
h d ( x , y ) = c 2 exp [ j k 2 d ( x 2 + y 2 ) ] ,
= 1 F 1 d 1 d ,
h c ( u , υ ) = exp [ j k 2 d ( u 2 + υ 2 ) ] F x y { exp [ j k 2 ( x 2 + y 2 ) ] P ( x , y ) } ,
F x y [ t ( x , y ) ] = t ( x , y ) exp [ j 2 π ( f x x + f y y ) ] d x d y T ( f x , f y ) ,
h i ( u , υ ) = | h c ( u , υ ) | 2 .
x = x / R ,
y = y / R ,
β = R k π ,
α = k u R / π d ,
ω = k υ R / π d ,
p ( x , y ) = P ( R x , R y ) P ( x , y ) .
h ˆ c ( α , ω ) = A ( P ) A ( p ) F x y { exp [ j π 2 β 2 ( x 2 + y 2 ) ] p ( x , y ) } | f x ¯ α/ 2 , f y = ω / 2 ,
A ( p ) A ( P ) R 2 ,
A ( P ) = P ( x , y ) d x d y .
A ( p ) = p ( x , y ) d x d y .
h ˆ i ( α , ω ) = | h ˆ c ( α , ω ) | 2 .
OTF = H i ( f u , f υ ) H i ( 0 , 0 ) ,
H i ( f u , f υ ) = F u υ [ h i ( u , υ ) ] ,
OTF ( f u , f υ ) = H ( f u , f υ ) H ( f u , f υ ) | H ( f u , f υ ) | 2 d f u d f υ ,
r s = r ( x , y ) s * ( x u , y υ ) d x d y ;
H ( f u , f υ ) = F u υ [ h c ( u , υ ) ]
H ˆ ( f u , f υ ) F αω [ h ˆ c ( α , ω ) ] .
H ( f α , f ω ) = ( π d k R ) 2 H ˆ ( f α , f ω ) ,
f α = π d k R f u ,
f ω = π d k R f υ ,
OTF ( f α , f ω ) = H ˆ H ˆ i ( f α , f ω ) H ˆ i ̂ ( 0 , 0 ) ,
H ˆ i ( f α , f ω ) = H ˆ ( f α , f ω ) H ˆ ( f α , f ω ) .
q θ ( u , υ ) = h i ( u u , υ υ ) δ ( υ cos θ u sin θ ) d u d υ .
u = u ˜ cos θ υ ˜ sin θ,
υ = u ˜ sin θ + υ ˜ cos θ .
q θ ( υ ˜ ) = h i ( g ˜ cos θ υ ˜ sin θ , g ˜ sin θ + υ ˜ cos θ ) d g ˜ ,
q ˆ θ ( ω ˜ ) = π d k R h ˆ i ( α ˜ cos θ ω ˜ sin θ , α ˜ sin θ + ω ˜ cos θ ) d α ˜ ,
Q θ ( f ) = q θ ( υ ˜ ) exp ( j 2 π f υ ˜ ) d υ ˜ .
Q ˆ θ ( f n ) = ( π d k R ) q ˆ θ ( ω ˜ ) exp ( j 2 π f n ω ˜ ) d ω ˜ ,
Q ˆ θ ( f n ) = ( π d k R ) F ω ˜ { q ˆ θ ( ω ˜ ) } ,
f n = π d k R f .
k R π d = 2 R λ d = 2 f c ,
V = I max I min I max + I min ,
I in = I o [ 1 + m cos ( 2 π f o υ ) ] ,
I out = I o ( Q ( 0 ) + m | Q ( f o ) | cos { 2 π f o υ + Arg [ Q ( f o ) ] } ) ,
Q ( f o ) = | Q ( f o ) | exp { j Arg [ Q ( f o ) ] } ,
I max = | Q ( 0 ) | + m | Q ( f o ) | ,
I min = | Q ( 0 ) | m | Q ( f o ) | ,
V = m | Q ( f o ) | | Q ( 0 ) | .
V = m | Q θ ( f ) | | Q θ ( 0 ) | .
V = m | Q ˆ θ ( f n o ) | | Q ˆ θ ( 0 ) | ,
V = m OTF ( f n cos θ , f n sin θ ) .
P ( x , y ) = exp ( r 2 R 2 ) .
p ( x , y ) = exp ( r 2 ) .
A ( p ) = 2 π 0 exp ( r 2 ) r d r = π .
h ˆ i ( α , ω ) = π 2 R 4 [ 1 + ( π 2 β 2 ) 2 ] exp [ 2 π 2 ( α 2 + ω 2 ) F ( β ) ] ,
F ( β ) = 4 + ( πβ 2 ) 2 .
q ˆ ( ω ) = 2 2 π 5 d R 3 k F ( β ) exp [ 2 π 2 ω 2 F ( β ) ] .
Q ˆ ( f n ) = 2 π 3 d R 3 k 2 exp [ f n 2 F ( β ) 2 ] .
V = m | 2 Q ˆ ( f n ) Q ˆ ( 0 ) | = m exp [ f n 2 F ( β ) 2 ] .
f n = { ln m V 2 [ 1 + ( π 2 β 2 ) 2 ] } 1 / 2 .
P A = P A ( f α , f ω ) exp [ j 2 πβ 2 ( f α 2 + f ω 2 ) ] ,
P A ( f α , f ω ) = circ [ ( f α 2 + f ω 2 ) 1 / 2 2 a ˆ ] ;
a ˆ = a R
P = P A P B ,
= 1 π ( a 2 b 2 ) [ P A P A + P B P B 2 Re ( P B P A ) ] ,
Q θ ( f ) = h i ( g ˜ cos θ υ ˜ sin θ , g ˜ sin θ + υ ˜ cos θ ) × exp [ j 2 π f ( υ ˜ cos θ g ˜ sin θ ) ] d υ ˜ d g ˜ .
Q θ ( f ) = h i ( u , υ ) × exp [ j 2 π f ( υ cos θ u sin θ ) ] d u d υ .
Q θ ( f ) = H i ( f u , f υ ) ,
f u = f sin θ,
f υ = f cos θ .
OTF ( f sin θ , f cos θ ) = Q θ ( f ) Q θ ( 0 ) .
OTF ( f n sin θ , f n cos θ ) = Q ˆ θ ( f n ) Q ˆ θ ( 0 ) ,
F = 10 cm , d = 20 cm , λ = 0 . 6 × 10 4 cm , R = R = 4 . 0 cm .
f n = λ d 2 R f .
= ( β R ) 2 λ 2 .
= d F d d F d Δ d d F d Δ d d F 2 .
Δ d = ( β d F R ) 2 λ 2 .

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