Abstract

The propagation of interfering waves through the whole optical system of the Fizeau interferometer is analyzed. A general formula is found that describes the measurement errors introduced by optical elements of the system. In particular, we studied the influence of the position of the beam splitter and the objective of the imaging system of the interferometer.

© 1991 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. Mekh. Prom. 8, 3–5 (1988).
  4. R. JóŸwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl. 20, 229–253 (1990).
  5. R. JóŸwicki, “Correction of the spherical aberration in an interferometric sytstem with nonzero difference between optical paths and fringeless observation field,” Appl. Opt. 29, 3575–3582 (1990).
    [CrossRef]
  6. R. JóŸwicki, “Influence of spherical aberration of an interferometric system on the measurement error for a fringe observation field,” Appl. Opt. (to be published).
  7. R. JóŸwicki, “Imaging synthesis in the Fresnel approximations,” Opt. Acta 31, 169–180 (1984).
    [CrossRef]
  8. R. JóŸwicki, J. Wójciak, “Influence of the field truncation by the aperture stop of interferometers upon the intensity distribution in the observation field,” Opt. Appl. 19, 439–450 (1989).
  9. R. JóŸwicki, “Telecentricity of the interferometric imaging system and its importance in the measurement accuracy,” Opt. Appl. 19, 470–475 (1989).
  10. R. JóŸwicki, “Transformation of reference spheres by an aberration free and infinitely large optical system in the Fresnel approximation,” Opt. Acta 29, 1383–1383 (1982).
    [CrossRef]
  11. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), Chap. 4.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 4.6.2.

1990

R. JóŸwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl. 20, 229–253 (1990).

R. JóŸwicki, “Correction of the spherical aberration in an interferometric sytstem with nonzero difference between optical paths and fringeless observation field,” Appl. Opt. 29, 3575–3582 (1990).
[CrossRef]

1989

R. JóŸwicki, J. Wójciak, “Influence of the field truncation by the aperture stop of interferometers upon the intensity distribution in the observation field,” Opt. Appl. 19, 439–450 (1989).

R. JóŸwicki, “Telecentricity of the interferometric imaging system and its importance in the measurement accuracy,” Opt. Appl. 19, 470–475 (1989).

1988

S. A. Rodionov, I. P. Agurok, “The influence of the optical system defects of the interferometer on the measurement accuracy of surface shapes,” Opt. Mekh. Prom. 8, 3–5 (1988).

1984

R. JóŸwicki, “Imaging synthesis in the Fresnel approximations,” Opt. Acta 31, 169–180 (1984).
[CrossRef]

1982

R. JóŸwicki, “Transformation of reference spheres by an aberration free and infinitely large optical system in the Fresnel approximation,” Opt. Acta 29, 1383–1383 (1982).
[CrossRef]

1957

P. R. Yoder, W. W. Hollis, “Design of a compact wide aperture Fizeau interferometer,” J. Opt. Soc. Am. 47, 858–861 (1957).
[CrossRef]

W. G. A. Taylor, “Spherical aberration in the Fizeau interferometer,” J. Sci. Instrum. 34, 399–402 (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. Mekh. Prom. 8, 3–5 (1988).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 4.6.2.

Hollis, W. W.

Hopkins, H. H.

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

JóŸwicki, R.

R. JóŸwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl. 20, 229–253 (1990).

R. JóŸwicki, “Correction of the spherical aberration in an interferometric sytstem with nonzero difference between optical paths and fringeless observation field,” Appl. Opt. 29, 3575–3582 (1990).
[CrossRef]

R. JóŸwicki, “Telecentricity of the interferometric imaging system and its importance in the measurement accuracy,” Opt. Appl. 19, 470–475 (1989).

R. JóŸwicki, J. Wójciak, “Influence of the field truncation by the aperture stop of interferometers upon the intensity distribution in the observation field,” Opt. Appl. 19, 439–450 (1989).

R. JóŸwicki, “Imaging synthesis in the Fresnel approximations,” Opt. Acta 31, 169–180 (1984).
[CrossRef]

R. JóŸwicki, “Transformation of reference spheres by an aberration free and infinitely large optical system in the Fresnel approximation,” Opt. Acta 29, 1383–1383 (1982).
[CrossRef]

R. JóŸwicki, “Influence of spherical aberration of an interferometric system on the measurement error for a fringe observation field,” Appl. Opt. (to be published).

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. Mekh. Prom. 8, 3–5 (1988).

Taylor, W. G. A.

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

Wójciak, J.

R. JóŸwicki, J. Wójciak, “Influence of the field truncation by the aperture stop of interferometers upon the intensity distribution in the observation field,” Opt. Appl. 19, 439–450 (1989).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 4.6.2.

Yoder, P. R.

Appl. Opt.

J. Opt. Soc. Am.

J. Sci. Instrum.

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

Opt. Acta

R. JóŸwicki, “Imaging synthesis in the Fresnel approximations,” Opt. Acta 31, 169–180 (1984).
[CrossRef]

R. JóŸwicki, “Transformation of reference spheres by an aberration free and infinitely large optical system in the Fresnel approximation,” Opt. Acta 29, 1383–1383 (1982).
[CrossRef]

Opt. Appl.

R. JóŸwicki, J. Wójciak, “Influence of the field truncation by the aperture stop of interferometers upon the intensity distribution in the observation field,” Opt. Appl. 19, 439–450 (1989).

R. JóŸwicki, “Telecentricity of the interferometric imaging system and its importance in the measurement accuracy,” Opt. Appl. 19, 470–475 (1989).

R. JóŸwicki, “Propagation of an aberrated wave with nonuniform amplitude distribution and its influence upon the interferometric measurement accuracy,” Opt. Appl. 20, 229–253 (1990).

Opt. Mekh. Prom.

S. A. Rodionov, I. P. Agurok, “The influence of the optical system defects of the interferometer on the measurement accuracy of surface shapes,” Opt. Mekh. Prom. 8, 3–5 (1988).

Other

R. JóŸwicki, “Influence of spherical aberration of an interferometric system on the measurement error for a fringe observation field,” Appl. Opt. (to be published).

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 4.6.2.

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Figures (7)

Fig. 1
Fig. 1

Optical system of the Fizeau interferometer.

Fig. 2
Fig. 2

Transition of the optical system of the interferometer to the substitutional system of distorters.

Fig. 3
Fig. 3

Substitutional system of distorters for the wave reflected from the standard plane.

Fig. 4
Fig. 4

Beam splitter with a spherical reflecting surface as an element of the imaging system of the interferometer.

Fig. 5
Fig. 5

Transfer of the beam splitter distortion to the object space of the focusing objective Uo2.

Fig. 6
Fig. 6

Exemplary position of the vectors ān and ρ ¯ e with respect to vector N ¯ (the vector perpendicular to the fringe lines).

Fig. 7
Fig. 7

Elimination of the influence of the aberration of objective Ui by inserting aperture lens AL.

Equations (48)

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1 f o + f i - 1 z i = 1 f o ,
A 2 s ( a ¯ n ) = A 1 ( a ¯ n ) + C s grad 2 A 1 ( a ¯ n ) ,
C s = - λ z 2 a o 2 = - λ s a o 2 ,
a ¯ n = a ¯ a o
A 2 s ( a ¯ n ) = A 1 ( a ¯ n - ρ ¯ s n ) + C s grad 2 A 1 ( a ¯ n - ρ ¯ s n ) + 2 α 0 λ α ¯ s α ¯ n ,
ρ ¯ s n = 2 z s α ¯ s α o .
A 2 s ( a ¯ n ) = W 1 ( a ¯ n - ρ ¯ s n ) + C s grad 2 W 1 ( a ¯ n - ρ ¯ s n ) + 2 a 0 λ α ¯ s a ¯ n + W 1 ( a n ) .
A 2 t ( a ¯ n ) = W 1 ( a ¯ n - ρ ¯ t n ) + C t grad 2 W 1 ( a ¯ n - ρ ¯ t n ) + 2 α 0 λ α ¯ t a ¯ n + W 1 ( a n ) ,
C t = λ z t a o 2 ,
ρ ¯ t n = 2 z t α ¯ t a o .
A B r ( a ¯ n ) = A 2 r ( a ¯ n ) + C B grad 2 A 2 r ( a ¯ n ) + W B ( a ¯ n ) ,             r = s , t ,
A i r ( a ¯ n ) = A B r ( a ¯ n ) + C i B grad 2 A B r ( a ¯ n ) + W B ( a ¯ n ) ,             r = s , t ,
C B = - λ z B 2 a o 2 ,
C i B = - λ ( z i - z B ) 2 a o 2 .
C p i = - λ z p - z i 2 a o 2 ,
A d r ( a ¯ n ) = A i r ( a ¯ n ) + C p i grad 2 A i r ( a ¯ n ) ,             r = s , t .
A d r ( a ¯ n ) = A 2 r + W B + W i + C p grad 2 A 2 r + C p B grad 2 W B + C p grad 2 W i + 2 C p B grad A 2 r grad W B + 2 C p i grad A 2 r grad W i + 2 C p i grad W B grad W i ,             r = s , t .
Δ A ( a ¯ n ) = A d t - A d s ,
Δ A ( a ¯ n ) = A 2 t - A 2 s + C p ( grad 2 A 2 t - grad 2 A 2 s ) + 2 C p B grad W B × ( grad A 2 t - grad A 2 s ) + 2 C p i grad W i ( grad A 2 t - grad A 2 s ) ,
A 2 r ( a ¯ n ) = 2 W 1 + 2 a o λ α ¯ r a ¯ n ,             r = s , t ,
grad A 2 r ( a ¯ n ) = 2 grad W 1 + 2 a o λ α ¯ r ,             r = s , t .
Δ A ( a ¯ n ) = W 1 ( a ¯ n - ρ ¯ t n ) - W 1 ( a ¯ n - ρ ¯ s n ) + ( C t - C s ) grad 2 W 1 ( a n ) + 0.5 N ¯ a ¯ n + ( 4 a o λ ) 2 C p ( α t 2 - α s 2 ) + 2 C p N ¯ grad W 1 ( a n ) + C p B N ¯ grad W B ( a ¯ n ) + C p i N ¯ grad W i ( a ¯ n ) ,
N ¯ = 4 a o λ ( α ¯ t - α ¯ s ) .
( 4 a o λ ) 2 C p ( α t 2 - α s 2 )
E ( a ¯ n ) = E d ( a ¯ n ) + E 1 ( a ¯ n ) + E B ( a ¯ n ) + E i ( a ¯ n ) ,
E d ( a ¯ n ) = W 1 ( a ¯ n - ρ ¯ t n ) - W 1 ( a ¯ n - ρ ¯ s n ) + ( C t - C s ) grad 2 W 1 ( a n ) ,
E 1 ( a ¯ n ) = 2 C P N ¯ grad W 1 ( a n ) ,
E B ( a ¯ n ) = C P B N ¯ grad W B ( a ¯ n ) ,
E i ( a ¯ n ) = C P i N ¯ grad W i ( a ¯ n ) .
E r ( a ¯ n ) = C p r N ¯ grad [ W r ( a ¯ n ) ] ,
C p r = - λ ( z p - z r ) 2 a o 2 .
W B ( a ¯ n ) = W 22 ( a ¯ n a ¯ s o ) 2 + W 20 a n 2 ,
E B ( a ¯ n ) = 2 a n N C p B ( W 22 a ¯ n o a ¯ s o N ¯ n o a ¯ s o + W 20 a ¯ n o N ¯ o ) .
E B ( a ¯ n ) = 2 a n N C P B [ W 22 ( N ¯ o a ¯ s o ) 2 + W 20 ] .
cos 2 i t t + cos 2 i t = cos i + cos i R ;
1 t s + 1 t = cos i + cos i R ,
Δ t = t t - t s 2 t 2 R S i ,
z B = z r B f o f o - z r B = ( f o - t ) f 0 t ,
W 22 = Δ h e λ = S i t 2 R λ ( a o f o ) 2 .
E B ( a ¯ n ) = S i N a n f o R λ H ( t ) ( N ¯ o a ¯ s o ) 2 ,
H ( t ) = t f o [ ( z p f o + 1 ) t f o - 1 ] .
W i ( a ¯ n ) = W 20 ( a ¯ n - ρ ¯ e n ) 2 + W 40 ( a ¯ n - ρ ¯ e n ) 4 + W 60 ( a ¯ n - ρ ¯ e n ) 6 ,
ρ ¯ e n = ρ ¯ e a o .
W i ( a ¯ n ) W i o ( a n ) - 2 a ¯ n o ρ ¯ e n W s f ( a n ) ,
W s f ( a n ) = a n ( W 20 + 2 a n 2 W 40 + 3 a n 4 W 60 ) .
E i ( a ¯ n ) = 2 C p i N ( b o + b e ) ,
b o = W s f ( a n ) N ¯ o a ¯ n o ,
b e = - ρ e n [ W s f ( a n ) a n ρ ¯ e n o N ¯ o + 4 a n 2 ( W 40 + 3 a n 2 W 60 ) ρ ¯ e n o a ¯ n o N ¯ o a ¯ n o ] .

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