Abstract

The influence of fifth-order spherical aberration in the measurement error in an interferometer system is analyzed. The criterion of the aberration correction and the optimum correction are proposed. The influence of the focusing error and the wave tilt as the adjustment errors are determined. As a practical example, the optimum wave spherical aberration, optimum longitudinal spherical aberration and the error the distribution in the observation field are given.

© 1990 Optical Society of America

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References

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  1. W. G. A. Taylor, “Spherical Aberration in the Fizeau Interferometer,” J. Sci. Instrum. 34, 399–402 (1957).
    [CrossRef]
  2. P. R. Yoder, W. W. Hollis, “Design of a Compact Wide Aperture Fizeau Interferometer,” J. Opt. Soc. Am. 47, 858–861 (1957).
    [CrossRef]
  3. S. A. Rodionov, I. P. Agurok, “The Influence of the Optical System Defects of the Interferometer on the Measurement Accuracy of Surface Shapes,” Opt. Meck. Promst. Nr 8, 3–5 (1988) (in Russian).
  4. R. Jozwicki, “Propagation of an Aberrated Wave with Nonuniform Amplitude Distribution and its Influence upon the Interferometric Measurement Accuracy,” Opt. Applicata20, Nr 3 (1990) (in press).
  5. H. H. Hopkins, Wave Theory of Aberrations(Clarendon, Oxford, 1950), Chap. 4.
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford1980), Chap. 5.
  7. D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Chap. 1.

1988 (1)

S. A. Rodionov, I. P. Agurok, “The Influence of the Optical System Defects of the Interferometer on the Measurement Accuracy of Surface Shapes,” Opt. Meck. Promst. Nr 8, 3–5 (1988) (in Russian).

1957 (2)

W. G. A. Taylor, “Spherical Aberration in the Fizeau Interferometer,” J. Sci. Instrum. 34, 399–402 (1957).
[CrossRef]

P. R. Yoder, W. W. Hollis, “Design of a Compact Wide Aperture Fizeau Interferometer,” J. Opt. Soc. Am. 47, 858–861 (1957).
[CrossRef]

Agurok, I. P.

S. A. Rodionov, I. P. Agurok, “The Influence of the Optical System Defects of the Interferometer on the Measurement Accuracy of Surface Shapes,” Opt. Meck. Promst. Nr 8, 3–5 (1988) (in Russian).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford1980), Chap. 5.

Hollis, W. W.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations(Clarendon, Oxford, 1950), Chap. 4.

Jozwicki, R.

R. Jozwicki, “Propagation of an Aberrated Wave with Nonuniform Amplitude Distribution and its Influence upon the Interferometric Measurement Accuracy,” Opt. Applicata20, Nr 3 (1990) (in press).

Rodionov, S. A.

S. A. Rodionov, I. P. Agurok, “The Influence of the Optical System Defects of the Interferometer on the Measurement Accuracy of Surface Shapes,” Opt. Meck. Promst. Nr 8, 3–5 (1988) (in Russian).

Taylor, W. G. A.

W. G. A. Taylor, “Spherical Aberration in the Fizeau Interferometer,” J. Sci. Instrum. 34, 399–402 (1957).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford1980), Chap. 5.

Yoder, P. R.

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

W. G. A. Taylor, “Spherical Aberration in the Fizeau Interferometer,” J. Sci. Instrum. 34, 399–402 (1957).
[CrossRef]

Opt. Meck. Promst. Nr (1)

S. A. Rodionov, I. P. Agurok, “The Influence of the Optical System Defects of the Interferometer on the Measurement Accuracy of Surface Shapes,” Opt. Meck. Promst. Nr 8, 3–5 (1988) (in Russian).

Other (4)

R. Jozwicki, “Propagation of an Aberrated Wave with Nonuniform Amplitude Distribution and its Influence upon the Interferometric Measurement Accuracy,” Opt. Applicata20, Nr 3 (1990) (in press).

H. H. Hopkins, Wave Theory of Aberrations(Clarendon, Oxford, 1950), Chap. 4.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford1980), Chap. 5.

D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Chap. 1.

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Figures (7)

Fig. 1
Fig. 1

Basic arrangement of objective U, standard S, and tested element T in a Fizeau interferometer.

Fig. 2
Fig. 2

Evolution of the wave propagation in interferometers with reflection (Fig. 1) into one-direction wave propagation.

Fig. 3
Fig. 3

Functions W sf ( a n ) / B for primary spherical aberration (dashed lines) and fifth-order (solid lines) without defocusing (O) and with optimum defocusing (OD). All curves concern the same maximum measurement error ΔWm and the positive sign of the primary spherical aberration (W40 > 0).

Fig. 4
Fig. 4

Relationship between the aberration Δr and longitudinal aberration δs′.

Fig. 5
Fig. 5

Wave aberrations for primary spherical aberration (dashed lines) and fifth-order included (solid lines) without defocusing (O) and with optimum defocusing (OD). All curves concern the same maximum measurement error ΔWm = 0.001, the positive sign of the primary spherical aberration (W40 > 0), and zm = 4a0 (B = 0.859). The wave aberrations with the fifth-order included are described by Eqs. (32a) and (33a).

Fig. 6
Fig. 6

Longitudinal spherical aberration δ s ( a n ) related to the wave aberration shown in Fig. 5 for focal length f′ = 1000 mm of objective U for flatness measurement. The solid lines concern the spherical aberration with the fifth-order included and the dashed lines are the primary spherical aberration; O indicates no defocusing, OD with optimum defocusing. The aberrations with the fifth-order included are described by Eqs. (32b) and (33b) (BD = 0.1087). Plane δs′ = 0 is the focusing plane.

Fig. 7
Fig. 7

Measurement error distribution Δ W ( a n ) in the observation field for two principal sections: θ = 0 and θ = π(1) and θ = π/2, and θ = 3π/2 (2); for optimum defocusing with fifth-order spherical aberration. The error is described by Eq. (4) (A = −5.054 10−5, W11 = 5).

Equations (43)

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z = 2 t 1 + t z 0 ,
Δ W ( a ¯ n ) = z λ 2 a 0 2 ( 1 - z z 0 ) [ ( W a x n ) 2 + ( W a y n ) 2 ] ,
W ( a ¯ n ) = W 11 a x n + W 20 a n 2 + W 40 a n 4 + W 60 a n 6 .
a n 2 = a x n 2 + a y n 2 , a n a x n = a x n a n , a n a y n = a y n a n and a x n = a n cos θ ,
Δ W ( a n , θ , z ) = A ( z ) { W sf ( a n ) [ W sf ( a n ) + W 11 cos θ ] } ,
W sf ( a n ) = a n ( W 20 + 2 W 40 a n 2 + 3 W 60 a n 4 )
A ( z ) = - 2 z λ ( 1 - z z 0 ) a 0 2
A m = A ( z max ) ,
W a = W 11 max ,
W sf ( W sf + W a ) Δ W m A m ,
W sf B ,
B 2 + B W a - Δ W m / A m = 0.
W 11 = a 0 ϕ λ .
B = Δ W m a 0 2 z m ϕ .
W sf = W 40 R ,
R = a n ( R 24 + 2 a n 2 ) ,
R 24 = W 20 W 40 .
W 40 B R m ,
W 40 0.5 B .
W 40 2 B .
W sf = W 60 R ;
R = a n ( R 26 + 2 R 46 a n 2 + 3 a n 4 ) ,
R 26 = W 20 W 60 ,
R 46 = W 40 W 60 .
W 60 B R m ,
R = a n 3 ( 3 a n 2 + 2 R 46 ) .
W 60 2.5465 B .
W 60 5.3333 B
R = a n Δ W 20 W 60 + 0.1875 ,
R m = Δ W 20 W 60 + 0.1875 ,
W 60 = W 60 - 5.3333 Δ W 20 .
Δ W 20 < B .
Δ r = z 0 λ W a
δ s = z 0 2 λ a W a .
W ( a n ) = W 20 a n 2 + W 40 a n 4 + W 60 a n 6 ,
δ s = D ( W 20 + 2 W 40 a n 2 + 3 W 60 a n 4 ) .
D = 2 λ u 0 2
W ( a n ) = 2.5465 B ( - 1.304 a n 4 + a n 6 )
δ s = 2.5465 B D ( - 2.608 a n 2 + 3 a n 4 )
W ( a n ) = 5.3333 B ( 0.9375 a n 2 - 1.875 a n 4 + a n 6 )
δ s = 5.3333 B D ( 0.9375 - 3.75 a n 2 + 3 a n 4 ) .
Δ W ( a ) = - i 2 ( a ) z 2 λ ( 1 - z z 0 ) ,
i 2 ( a ¯ ) = i a 2 ( a ¯ ) + 2 i a ( a ¯ ) ϕ + ϕ 2 ,

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