Abstract

The diffraction focus of optical systems with circular and annular exit pupils affected with primary, secondary and tertiary spherical aberration is calculated for some values of aberration coefficients and obscuration ratios. The depth of focus of these systems is also given.

© 1990 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 461.
  2. W. B. Wetherell, “The Calculation of Image Quality,” in Applied Optics and Optical Engineering, Vol. 8, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980), p. 193.
  3. Ref. 1, p. 438.
  4. C. A. Taylor, B. J. Thompson, “Attempt to Investigate Experimentally the Intensity Distribution near the Focus in the Error-Free Diffraction Patterns of Circular and Annular Apertures,” J. Opt. Soc. Am. 48, 844–850 (1958).
    [CrossRef]
  5. T. Brender Andersen, “Evaluating rms Spot Radii by Ray Tracing,” Appl. Opt. 21, 1241–1248 (1982).
    [CrossRef]
  6. V. N. Mahajan, “Zernike Annular Polynomials for Imaging Systems with Annular Pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  7. Ref. 2, p. 199.

1982 (1)

1981 (1)

1958 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (4)

Ref. 2, p. 199.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 461.

W. B. Wetherell, “The Calculation of Image Quality,” in Applied Optics and Optical Engineering, Vol. 8, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1980), p. 193.

Ref. 1, p. 438.

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Figures (1)

Fig. 1
Fig. 1

Qualitative behavior of the intensity distribution along the optical axis: curve A, diffraction-limited systems; curve B, systems with a small amount of spherical aberration.

Tables (9)

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Table I Diffraction Focus Position and Intensity for Optical Systems with Primary Spherical Aberration Function: W(r) = W40r4 (W40 in Wavelength Units)

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Table II Diffraction Focus Position and Intensity for Optical Systems With Primary Plus Secondary Spherical Aberration Function: W(r) = W40r4 + W60r6 (W40 = W60 in Wavelength Units)

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Table III Diffraction Focus Position and Intensity for Optical Systems With Primary Plus Secondary and Tertiary Spherical Aberration Function: W(r) = W40r4 + W60r6 + W80r8 (W40 = W60 = W80 in Wavelength Units)

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Table IV Diffraction Focus Position and Positions With Intensity at 80% of the Diffraction Focus Intensity for Optical Systems With Aberration Function as in Table I (for Each Value of the Depth of Focus is the Difference Between the Third and First Columns)

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Table V Diffraction Focus Position and Positions With Intensity at 80% of the Diffraction Focus Intensity for Optical Systems With Aberration Function as in Table II (for Each Value of the Depth of Focus is the Difference Between the Third and First Columns)

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Table VI Diffraction Focus Position and Positions With Intensity at 80 % of the Diffraction Focus Intensity for Optical Systems with Aberration Function as in Table III (for Each Value of the Depth of Focus is the Difference Between the Third and First Columns)

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Table VII Ratio of the Diffraction Focus Intensity to the Focal Plane Intensity for Optical Systems With Aberration Function as in Table I

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Table VIII Ratio of the Diffraction Focus Intensity to the Focal Plane Intensity for Optical Systems With Aberration Function as in Table II

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Table IX Ratio of the Diffraction Focus intensity to the Focal Plane Intensity for Optical Systems With Aberration Function as in Table III

Equations (13)

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I ( u , v , ϕ ) = C 2 + S 2 ) π 2 ,
C ( u , v , ϕ ) S ( u , v , ϕ ) = 1 0 2 π T ( r , θ ) cos sin { K W ( r , θ ) - [ v r cos ( θ - ϕ ) + ½ u r 2 ] } r d r d θ .
u = π 2 λ N 2 Z             v = π R λ N
W ( r , θ ) = W 20 r 2 + W 40 r 4 + W 60 r 6 + W 80 r 8 + W 22 r 2 cos 2 θ + W 42 r 4 cos 2 θ + W 31 r 3 cos θ + W 51 r 5 cos θ + W 33 r 3 cos 3 θ + W 11 r cos θ .
I ( u , v ) = 4 ( C 2 + S 2 ) ,
C ( u , v ) S ( u , v ) = 1 T ( r ) cos sin [ K W ( r ) - 1 2 u r 2 ] J 0 ( v r ) r d r ,
I ( 0 , v ) = 4 [ J 1 ( v ) v - J 1 ( v ) v ] 2 ,
I ( u , 0 ) = [ sin 1 4 u ( 1 - 2 ) 1 4 u ] 2 .
d I d u = 4 1 cos [ K W ( r ) - 1 2 u r 2 ] r d r × 1 sin [ K W ( r ) - 1 2 u r 2 ] r 3 d r - 4 1 sin [ K W ( r ) - 1 2 u r 2 ] r d r × 1 cos [ K W ( r ) - 1 2 u r 2 ] r 3 d r .
Z DF = 8 ( 1 + 2 ) N 2 W 40 .
u DF = 4 π ( 1 + 2 ) W 40
Δ Z = ± 2 , 08 λ N 2 1 - 2 .
Δ u = ± 2 , 08 π 2 ( 1 - 2 ) .

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