Abstract

The optical properties of an aberration-free defocused optical system used to image incoherently illuminated objects are analyzed. The exact diffraction optical transfer function (OTF) and point spread function (PSF) are compared to the geometrical OTF and PSF, respectively, for both small and large amounts of defocusing. It follows that geometrical optics does not describe diffraction optics very well for any reasonable amount of defocusing. Calculation of the diffraction OTF is complicated and time consuming, even with a large computer. An empirically derived analytic approximation to the diffraction OTF which is much easier to calculate, is formulated. It is also shown, and examples are presented to demonstrate, that the exact OTF and PSF are different in planes at equal distances on the two sides of the in-focus plane.

© 1969 Optical Society of America

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References

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  1. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  2. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968), p. 115.
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 485.
  5. This transfer function is obtained by combining Eqs. (10) and (15) in Ref. 1.
  6. A nice example of this is given in Ref. 3, p. 126.
  7. R. E. Stephens and L. E. Sutton, J. Opt. Soc. Am. 581001 (1968).
    [Crossref]
  8. Reference 4, p. 490.
  9. Reference 3, p. 85.
  10. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, 1950), p. 14.
  11. R. Bracewell, The Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 249.
  12. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957), 3rd ed., p. 303.

1968 (1)

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

1955 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 485.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 249.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968), p. 115.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

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

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957), 3rd ed., p. 303.

Stephens, R. E.

Sutton, L. E.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957), 3rd ed., p. 303.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 485.

J. Opt. Soc. Am. (1)

Proc. Phys. Soc. (London) (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

Other (9)

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968), p. 115.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 485.

This transfer function is obtained by combining Eqs. (10) and (15) in Ref. 1.

A nice example of this is given in Ref. 3, p. 126.

Reference 4, p. 490.

Reference 3, p. 85.

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

R. Bracewell, The Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 249.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw–Hill Book Co., New York, 1957), 3rd ed., p. 303.

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Figures (11)

F. 1
F. 1

The optical system. I is the geometrical image of the point O. The reference spheres in object and image space are S and S′, respectively. E is the entrance pupil, E′ the exit pupil. The extreme ray l′ makes an angle α with the optical axis in image space.

F. 2
F. 2

Optical system with defect of focus. E′ denotes the exit pupil, l′ the extreme ray in image space. The reference sphere S1′ is centered on the axis at the out-of-focus point P. The wave-front S′ converges to the in-focus point I. The amount of defocusing can be specified by the out-of-focus distance z or the optical distance w between S′ and S1′.

F. 3
F. 3

OTF’s of a defocused system plotted as a function of the spatial frequency s for small values of the defect of focus optical path length w. Solid curves give the diffraction OTF’s, dashed curves the geometrical OTF’s. The geometrical OTF is not shown for w = 0.

F. 4
F. 4

Magnitude of OTF’s of a defocused system plotted on a logarithmic scale as a function of s for intermediate values of w. Solid curves give the diffraction OTF’s, dashed curves the geometrical OTF’s. Every other sideband of the OTF’s is negative as indicated by the plus and minus signs on the diffraction OTF curve for w = 2λ. The geometrical OTF is not shown for w = 10λ.

F. 5
F. 5

Magnitude of OTF’s of a defocused system plotted on a logarithmic scale as a function of s for large values of w. Solid curves give the diffraction OTF’s, dashed curves the geometrical OTF’s.

F. 6
F. 6

PSF’s of a defocused f/5 system with λ = 0.5 μm. Left and right columns, the diffraction PSF; center column, the geometrical PSF. (a)w = 0.5λ, γ = 1.65; (b) w = 2λ, γ = 6.6; (c) w = 5λ, γ = 16.5; (d) w = 10λ, γ = 33; (e) w = 50λ, γ = 165. The element length is 4 μm in the right column. For the left and center columns, the element lengths are; (a) 0.2 μm, (b) 0.8 μm, (c) 2 μm, (d) 4 μm, (e) 20 μm.

F. 7
F. 7

Magnitude of diffraction OTF’s (solid curves) and of approximate OTF’s (dashed curves) of a defocused system on a logarithmic scale. Every other sideband of the OTF’s (dashed curves) of a defocused system on a logarithmic scale. Every other sideband of the OTF’s is negative as indicated by the plus and minus signs on the curves for w = 2λ.

F. 8
F. 8

PSF’s of a defocused f/5 system with λ = 0.5 μm as described by diffraction optics (top row) and approximation (bottom row). (a) w = 0.5λ, (b) w = 2λ, (c) w = 5λ.

F. 9
F. 9

Nonsymmetry of a defocused system about the in-focus plane which intersects the optical axis at I. P1 and P2 are points which are z1 and −z1 out of focus, respectively. Their reference spheres are S1′ and S2′, respectively. S′ is the wavefront which converges to the in-focus point I.

F. 10
F. 10

ΔwR, the relative difference in |w| for planes at equal distances on the two sides of the focal plane as a function of the relative out-of-focus distance, rF is the focal length, ×, f/2.5, rF = 10mm; □, f/2.5, rF = 250mm; +, f/5, rF = 50mm; ∇, f/10, rF = 50 mm; ○, f/10, rF = 250 mm.

F. 11
F. 11

PSF’s obtained on both sides of focal plane for various relative out-of-focus distances z′/rF. Top row, negative defocusing; bottom row, positive defocusing. (a) z′/rF = 1%, (b) z′/rF = 2%, (c) z′/rF = 5%, (d) z′/rF = 10%. Pictures are made for an f/5 system with focal length rF = 10 mm and λ = 0.5μm.

Equations (29)

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T ( s ) = f ( x + s 2 , y ) f * ( x s 2 , y ) dxdy | f ( x , y ) | 2 dxdy ,
s = ( λ / n sin α ) f ,
f ( x , y ) = { exp [ i ( 2 π / λ ) w ( x 2 + y 2 ) ] , if x 2 + y 2 1 0 , if x 2 + y 2 > 1 ,
T E ( s ) = 4 π a cos [ 1 2 a s ] × { β J 1 ( a ) + n = 1 ( 1 ) n + 1 sin 2 n β 2 n [ J 2 n 1 ( a ) J 2 n + 1 ( a ) ] } 4 π a sin [ 1 2 a s ] n = 0 ( 1 ) n sin ( 2 n + 1 ) β 2 n + 1 × [ J 2 n ( a ) J 2 ( n + 1 ) ( a ) ] ,
a = ( 4 π / λ ) w s , β = cos 1 ( s / 2 ) ,
T G ( s ) = 2 J 1 ( a ) / a , a = ( 4 π / λ ) w s .
T E ( s ) π 1 { 2 arc cos ( s / 2 ) sin [ 2 arc cos ( s / 2 ) ] } = π 1 ( 2 β sin 2 β ) ,
γ = 3.3 ( w / λ ) .
T A ( s ) = 2 J 1 [ a y ( s , w ) ] / [ a y ( s , w ) ] .
y ( s , w ) = 0.5 a s = ( 2 π / λ ) w s 2 .
T A ( 1 + s ) = T A ( 1 s ) .
T A ( s ) = { 2 ( 1 0.69 s + 0.0076 s 2 + 0.043 s 3 ) × [ J 1 ( a 0.5 a s ) / ( a 0.5 a s ) ] , when | s | < 2 0 , when | s | 2 ,
( A P ) 2 = ( A I ) 2 + ( I P ) 2 2 ( A I ) ( I P ) cos ( π α ) .
( r + z ) 2 = ( r + w ) 2 + z 2 2 ( r + w ) z cos ( π α ) ,
w = r z cos α + ( r 2 + 2 r z + z 2 cos 2 α ) 1 2 .
Δ w R = 2 [ | w ( z ) | | w ( z ) | ] | w ( z ) | + | w ( z ) | ,
Δ w R = 2 ( z / r ) .
w = 1 2 z α 2 [ 1 z / ( r + z ) ]
w = z ( 1 cos α ) .
w = 1 2 z sin 2 α ,
T G ( f ) = const [ J 1 ( 2 π r G f ) / 2 π r G f ] .
A = 2 π r G f = 4 π w s / λ cos α = ( 4 π / λ ) w s = a .
T G ( s ) = 2 J 1 ( a ) / a
d G = 2 z tan α .
d G = 4 w / tan α ,
d G = 8 w ( f / )
( f / ) = ( 2 tan α ) 1 .
d A = 2 · 1.22 · λ · ( f / ) .
γ = d G / d A = 3.3 w / λ ,