Abstract

Methods proposed by the authors to establish a flatness standard without using a liquid mirror are proved in practice and extended. The extension is performed by a development of methods for the determination and compensation of random and systematic measuring errors by means of condition equations which must be satisfied by the measured sums of deviations from absolute planeness. Linear errors of these sums of deviations which can lead to ambiguities and errors of planeness deviations can be discovered and completely eliminated. Also nonlinear errors, for example, as a result of temperature differences or of mechanical stress, can be recognized without repeating the interference photography procedure. The deviations from absolute planeness of three fused silica plates were determined along seven diameters (angular distance 2π/14) with an accuracy of λ/500 (mean square error). This was performed by evaluating two sets of four different interference photographs, each with contour plane distances of λ/50 (from fringe to fringe).

© 1971 Optical Society of America

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References

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  1. G. Schulz, J. Schwider, Appl. Opt. 6, 1077 (1967).
    [CrossRef] [PubMed]
  2. G. Schulz, Opt. Acta 14, 375 (1967).
    [CrossRef]
  3. J. Schwider, Opt. Acta 14, 389 (1967).
    [CrossRef]
  4. J. Schwider, Opt. Acta 15, 351 (1968).
    [CrossRef]
  5. Rayleigh, Nature 48, 212 (1893).
  6. H. Barrell, R. Marriner, Brit. Sci. News 2, 130 (1949).
  7. E. Einsporn, Feingerätetechnik 4, 539 (1955);Feingerätetechnik 10, 67 (1961).
  8. R. Bünnagel, Z. Angew. Phys. 8, 342 (1956); Opt. Acta 3, 81 (1956); Z. Instrumentenk. 73, 214 (1965).
  9. O. Schönrock, Z. Instrumentenk. 25, 148 (1905); Z. Instrumentenk. 28, 180 (1908).
  10. O. Schönrock, Z. Instrumentenk. 59, 31 (1939).
  11. E. Emerson, J. Res. Nat. Bur. Stand. 49, 336 (1952).
  12. J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
    [CrossRef]
  13. J. Schwider, thesis (Humboldt-University of Berlin, 1966).
  14. G. Dew, J. Sci. Instrum. 43, 809 (1966).
    [CrossRef]
  15. T. Sakurai, K. Shishido, Sci. Rep. Res. Inst. Tohoku Univ. AI, No. 1 (1949).
  16. J. B. Saunders, J. Res. Nat. Bur. Stand. 47, 148 (1951).
    [CrossRef]
  17. D. R. Herriott, J. Opt. Soc. Amer. 51, 1142 (1961).
    [CrossRef]
  18. As an approximation this plane is to contain the three reference points fixed by steps 1 and 2. However, this is not exactly so, since the measured values in some of these points need not be faultless. Therefore small deviations may arise between the reference points of the steps 1 and 2 and the discussed reference plane.

1968 (1)

J. Schwider, Opt. Acta 15, 351 (1968).
[CrossRef]

1967 (3)

G. Schulz, J. Schwider, Appl. Opt. 6, 1077 (1967).
[CrossRef] [PubMed]

G. Schulz, Opt. Acta 14, 375 (1967).
[CrossRef]

J. Schwider, Opt. Acta 14, 389 (1967).
[CrossRef]

1966 (2)

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

G. Dew, J. Sci. Instrum. 43, 809 (1966).
[CrossRef]

1961 (1)

D. R. Herriott, J. Opt. Soc. Amer. 51, 1142 (1961).
[CrossRef]

1956 (1)

R. Bünnagel, Z. Angew. Phys. 8, 342 (1956); Opt. Acta 3, 81 (1956); Z. Instrumentenk. 73, 214 (1965).

1955 (1)

E. Einsporn, Feingerätetechnik 4, 539 (1955);Feingerätetechnik 10, 67 (1961).

1952 (1)

E. Emerson, J. Res. Nat. Bur. Stand. 49, 336 (1952).

1951 (1)

J. B. Saunders, J. Res. Nat. Bur. Stand. 47, 148 (1951).
[CrossRef]

1949 (2)

T. Sakurai, K. Shishido, Sci. Rep. Res. Inst. Tohoku Univ. AI, No. 1 (1949).

H. Barrell, R. Marriner, Brit. Sci. News 2, 130 (1949).

1939 (1)

O. Schönrock, Z. Instrumentenk. 59, 31 (1939).

1905 (1)

O. Schönrock, Z. Instrumentenk. 25, 148 (1905); Z. Instrumentenk. 28, 180 (1908).

1893 (1)

Rayleigh, Nature 48, 212 (1893).

Barrell, H.

H. Barrell, R. Marriner, Brit. Sci. News 2, 130 (1949).

Bünnagel, R.

R. Bünnagel, Z. Angew. Phys. 8, 342 (1956); Opt. Acta 3, 81 (1956); Z. Instrumentenk. 73, 214 (1965).

Dew, G.

G. Dew, J. Sci. Instrum. 43, 809 (1966).
[CrossRef]

Einsporn, E.

E. Einsporn, Feingerätetechnik 4, 539 (1955);Feingerätetechnik 10, 67 (1961).

Emerson, E.

E. Emerson, J. Res. Nat. Bur. Stand. 49, 336 (1952).

Herriott, D. R.

D. R. Herriott, J. Opt. Soc. Amer. 51, 1142 (1961).
[CrossRef]

Marriner, R.

H. Barrell, R. Marriner, Brit. Sci. News 2, 130 (1949).

Minkwitz, G.

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

Rayleigh,

Rayleigh, Nature 48, 212 (1893).

Riekher, R.

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

Sakurai, T.

T. Sakurai, K. Shishido, Sci. Rep. Res. Inst. Tohoku Univ. AI, No. 1 (1949).

Saunders, J. B.

J. B. Saunders, J. Res. Nat. Bur. Stand. 47, 148 (1951).
[CrossRef]

Schönrock, O.

O. Schönrock, Z. Instrumentenk. 59, 31 (1939).

O. Schönrock, Z. Instrumentenk. 25, 148 (1905); Z. Instrumentenk. 28, 180 (1908).

Schulz, G.

G. Schulz, Opt. Acta 14, 375 (1967).
[CrossRef]

G. Schulz, J. Schwider, Appl. Opt. 6, 1077 (1967).
[CrossRef] [PubMed]

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

Schwider, J.

J. Schwider, Opt. Acta 15, 351 (1968).
[CrossRef]

J. Schwider, Opt. Acta 14, 389 (1967).
[CrossRef]

G. Schulz, J. Schwider, Appl. Opt. 6, 1077 (1967).
[CrossRef] [PubMed]

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

J. Schwider, thesis (Humboldt-University of Berlin, 1966).

Shishido, K.

T. Sakurai, K. Shishido, Sci. Rep. Res. Inst. Tohoku Univ. AI, No. 1 (1949).

Appl. Opt. (1)

Brit. Sci. News (1)

H. Barrell, R. Marriner, Brit. Sci. News 2, 130 (1949).

Feingerätetechnik (1)

E. Einsporn, Feingerätetechnik 4, 539 (1955);Feingerätetechnik 10, 67 (1961).

J. Opt. Soc. Amer. (1)

D. R. Herriott, J. Opt. Soc. Amer. 51, 1142 (1961).
[CrossRef]

J. Res. Nat. Bur. Stand. (2)

J. B. Saunders, J. Res. Nat. Bur. Stand. 47, 148 (1951).
[CrossRef]

E. Emerson, J. Res. Nat. Bur. Stand. 49, 336 (1952).

J. Sci. Instrum. (1)

G. Dew, J. Sci. Instrum. 43, 809 (1966).
[CrossRef]

Nature (1)

Rayleigh, Nature 48, 212 (1893).

Opt. Acta (4)

J. Schwider, G. Schulz, R. Riekher, G. Minkwitz, Opt. Acta 13, 103 (1966).
[CrossRef]

G. Schulz, Opt. Acta 14, 375 (1967).
[CrossRef]

J. Schwider, Opt. Acta 14, 389 (1967).
[CrossRef]

J. Schwider, Opt. Acta 15, 351 (1968).
[CrossRef]

Sci. Rep. Res. Inst. Tohoku Univ. (1)

T. Sakurai, K. Shishido, Sci. Rep. Res. Inst. Tohoku Univ. AI, No. 1 (1949).

Z. Angew. Phys. (1)

R. Bünnagel, Z. Angew. Phys. 8, 342 (1956); Opt. Acta 3, 81 (1956); Z. Instrumentenk. 73, 214 (1965).

Z. Instrumentenk. (2)

O. Schönrock, Z. Instrumentenk. 25, 148 (1905); Z. Instrumentenk. 28, 180 (1908).

O. Schönrock, Z. Instrumentenk. 59, 31 (1939).

Other (2)

J. Schwider, thesis (Humboldt-University of Berlin, 1966).

As an approximation this plane is to contain the three reference points fixed by steps 1 and 2. However, this is not exactly so, since the measured values in some of these points need not be faultless. Therefore small deviations may arise between the reference points of the steps 1 and 2 and the discussed reference plane.

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Figures (9)

Fig. 1
Fig. 1

Optical setup. IF, interference filter (bandwidth: 8 nm); HQE 50, mercury spectrum lamp; Pl, photographic plate. Other symbols are explained in the text. The wedge interferometer is equipped with piezodrivers to render possible very sensitive adjustments of distance and wedge angle.

Fig. 2
Fig. 2

Over-all view of the interferometer.

Fig. 3
Fig. 3

(a) Interference fringes in white light indicating correct adjustment. (b) Interference fringes in light filtered with IF and, in addition, light of the green mercury line indicating the subdivision 1:25. (c) Fringes used for the evaluation.

Fig. 4
Fig. 4

Equidensities of the interference pattern of Fig. 3 (c).

Fig. 5
Fig. 5

Two views of the first positional combination (A,B). (a) Top view of the combination. The heavy points fix the reference planes.1 (b) Cut along ν = 0 (a). EA, EB: reference planes; x, y: deviations from ideal planeness; Dd: directly measured sum of deviations; Dd = x + y if the measurement is free from errors.

Fig. 6
Fig. 6

Error control. Assemblage of points vν(r).

Fig. 7
Fig. 7

Error control. Assemblage of points uν(r). Linear errors were taken into consideration.

Fig. 8
Fig. 8

Error control. Assemblage of points uu(r). Linear errors were taken into consideration.

Fig. 9
Fig. 9

Image of the deviations from absolute planeness of plate B. From contour line to contour line the deviations increase by 0.3 × λ/100, λ = 546 nm, plate diameter: 60 mm.

Equations (11)

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D - d a ¯ ν ( r ) ( first positional combination ) , b ¯ ν ( r ) ( second positional combination ) , c ¯ ν ( r ) ( third positional combination ) , a ¯ ν ( r ) ( fourth positional combination ) . }
[ a ¯ ν ( r ) + b ¯ ν ( r ) + c ¯ ν ( r ) ] - [ a ¯ - ν ( r ) + b ¯ - ν ( r ) + c ¯ - ν ( r ) ] = u ν ( r ) , [ b ¯ 1 + ν ( r ) + c ¯ - 1 + ν ( r ) + a ¯ 1 + ν ( r ) ] - [ b ¯ 1 - ν ( r ) + c ¯ - 1 - ν ( r ) + a ¯ 1 - ν ( r ) ] = v ν ( r ) , ν = - 3 + 3 a ¯ ν ( r ) - v = - 3 + 3 a ¯ ν ( r ) = w ( r ) , }
u 1 ( r ) , u 2 ( r ) , u 3 ( r ) , v 1 ( r ) , v 2 ( r ) , v 3 ( r ) , w ( r ) ,
a ¯ ν ( r ) = a ν ( r ) + α ν ( r ) + A ν ( r ) , b ¯ ν ( r ) = b ν ( r ) + β ν ( r ) + B ν ( r ) , c ¯ ν ( r ) = c ν ( r ) + γ ν ( r ) + C ν ( r ) , a ¯ ν ( r ) = a ν ( r ) + α ν ( r ) + A ν ( r ) , }
a ¯ ν ( r ) = a ν ( r ) + α ν ( r ) + U r sin [ ν ( 2 π / 7 ) ] , b ¯ ν ( r ) = b ν ( r ) + β ν ( r ) , c ¯ ν ( r ) = c ν ( r ) + γ ν ( r ) , a ¯ ν ( r ) = a ν ( r ) + α ν ( r ) + V r sin [ ( ν - 1 ) ( 2 π / 7 ) ] - W . }
U = u ν ( r ) - φ ν ( r ) 2 r sin ( ν × 2 π / 7 ) , V = v ν ( r ) - χ ν ( r ) 2 r sin ( ν × 2 π / 7 ) , W = w ( r ) - ψ ( r ) 7 }
φ ν ( r ) = [ α ν ( r ) + β ν ( r ) + γ ν ( r ) ] - [ α - ν ( r ) + β - ν ( r ) + γ - ν ( r ) ]
a ˜ ν ( r ) = a ˜ ν ( r ) - U r sin ( ν · 2 π 7 ) = a ν ( r ) + α ν ( r ) , b ¯ ν ( r ) c ¯ ν ( r ) = b ν ( r ) + β ν ( r ) , = c ν ( r ) + γ ν ( r ) , } ( as before ) a ˜ ν ( r ) = a ¯ ν ( r ) - V r sin [ ( ν - 1 ) 2 π 7 ] + W = a ν ( r ) + α ν ( r ) . }
x 0 = 1 2 ( a ˜ 0 - b ¯ 0 + c ¯ 0 ) , y 2 = a ˜ 0 + x 0 , x - 2 = a ˜ - 2 - y 2 , y - 3 = a ˜ - 2 - x - 2 ; y 0 = 1 2 ( a ˜ 0 + b ¯ 0 - c ¯ 0 ) , x 2 = a ˜ 2 - y 0 , y - 2 = a ˜ 2 - x 2 , x - 3 = a ˜ - 3 - y - 2 ; x 1 = 1 2 ( - b ¯ 1 + c ¯ - 1 + a ˜ 1 ) , y - 1 = a ˜ 1 - x 1 , x 3 = a ˜ 3 - y - 1 , y - 3 = a ˜ 3 - x 3 ; y 1 = 1 2 ( b ¯ 1 - c ¯ - 1 + a ˜ 1 ) , x - 1 = a ˜ - 1 - y 1 , y 3 = a ˜ - 1 - x - 1 , x - 3 = a ¯ - 3 - y 3 ; z ν = c ¯ ν - x - ν ( ν = 0 , ± 1 , ± 2 , ± 3 ) , z ν = b ¯ - ν - y - ν ( ν = 1 , ± 2 , ± 3 ) . }
Mean square error of : x 0 , y 0 , x 1 y , 1 : 1 2 ( 3 ) 1 2 μ , Mean square error of : x - 1 , y - 1 , x 2 , y 2 : 1 2 ( 7 ) 1 2 μ , Mean square error of : x - 2 , y - 2 , x 3 , y 3 : 1 2 ( 11 ) 1 2 μ , Mean square error of : x - 3 , y - 3 : 1 2 ( 15 2 ) 1 2 μ . }
μ = 1.1 × λ / 1000.

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