Abstract

Digital wave-front measuring interferometry is a well-established technique but only few investigations of systematic error sources have been carried out so far. In this work three especially serious error sources are discussed in some detail: inaccuracies of the reference phase values needed for this type of evaluation technique; disturbances due to extraneous fringes; and spatially high frequency noise on the wave fronts caused by dust particles, inhomogeneities, etc. For the first two error sources formulas of the resulting phase deviation are derived and compensation possibilities discussed and experimentally verified. To study the occurrence of wave-front irregularities caused by dust particles a model has been developed and countermeasures derived which assure sufficient regularity of contour line plots. The repeatability of the present experimental setup was better than λ/200 within the 3σ limits.

© 1983 Optical Society of America

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References

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  1. R. Mahany, M. Buzawa, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 50 (1979).
  2. R. Moore, F. Slaymaker, Proc. Soc. Photo-Opt. Instrum. Eng. 220, 75 (1980).
  3. K. Stumpf, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 42 (1978).
  4. B. Dörband, Optik 60, 161 (1982).
  5. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, Appl. Opt. 13, 2693 (1974).
    [Crossref] [PubMed]
  6. R. P. Grosso, R. Crane, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 65 (1979).
  7. B. Baule, Die Mathematik des Naturforschers und Ingenieurs, Vol. 2 (S. Hirzel, Leipzig, 1956), p. 60.
  8. R. Jones, P. Kadakia, Appl. Opt. 7, 1477 (1968).
    [Crossref] [PubMed]
  9. M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [Crossref]
  10. A. Guenther, D. Liebenberg, “Optical Interferograms-Reduction and Interpretation,” ASTM Special Technical Publication 666, Philadelphia (1978).
  11. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

1982 (2)

1980 (1)

R. Moore, F. Slaymaker, Proc. Soc. Photo-Opt. Instrum. Eng. 220, 75 (1980).

1979 (2)

R. P. Grosso, R. Crane, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 65 (1979).

R. Mahany, M. Buzawa, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 50 (1979).

1978 (1)

K. Stumpf, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 42 (1978).

1974 (1)

1968 (1)

Baule, B.

B. Baule, Die Mathematik des Naturforschers und Ingenieurs, Vol. 2 (S. Hirzel, Leipzig, 1956), p. 60.

Brangaccio, D. J.

Bruning, J. H.

Buzawa, M.

R. Mahany, M. Buzawa, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 50 (1979).

Crane, R.

R. P. Grosso, R. Crane, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 65 (1979).

Dörband, B.

B. Dörband, Optik 60, 161 (1982).

Gallagher, J. E.

Grosso, R. P.

R. P. Grosso, R. Crane, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 65 (1979).

Guenther, A.

A. Guenther, D. Liebenberg, “Optical Interferograms-Reduction and Interpretation,” ASTM Special Technical Publication 666, Philadelphia (1978).

Herriott, D. R.

Ina, H.

Jones, R.

Kadakia, P.

Kobayashi, S.

Liebenberg, D.

A. Guenther, D. Liebenberg, “Optical Interferograms-Reduction and Interpretation,” ASTM Special Technical Publication 666, Philadelphia (1978).

Mahany, R.

R. Mahany, M. Buzawa, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 50 (1979).

Malacara, D.

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

Moore, R.

R. Moore, F. Slaymaker, Proc. Soc. Photo-Opt. Instrum. Eng. 220, 75 (1980).

Rosenfeld, D. P.

Slaymaker, F.

R. Moore, F. Slaymaker, Proc. Soc. Photo-Opt. Instrum. Eng. 220, 75 (1980).

Stumpf, K.

K. Stumpf, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 42 (1978).

Takeda, M.

White, A. D.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Optik (1)

B. Dörband, Optik 60, 161 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

R. P. Grosso, R. Crane, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 65 (1979).

R. Mahany, M. Buzawa, Proc. Soc. Photo-Opt. Instrum. Eng. 192, 50 (1979).

R. Moore, F. Slaymaker, Proc. Soc. Photo-Opt. Instrum. Eng. 220, 75 (1980).

K. Stumpf, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 42 (1978).

Other (3)

B. Baule, Die Mathematik des Naturforschers und Ingenieurs, Vol. 2 (S. Hirzel, Leipzig, 1956), p. 60.

A. Guenther, D. Liebenberg, “Optical Interferograms-Reduction and Interpretation,” ASTM Special Technical Publication 666, Philadelphia (1978).

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

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Figures (23)

Fig. 1
Fig. 1 Scheme of the Twyman-Green interferometer with the phase measuring technique.

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Fig. 2
Fig. 2 Contour lines and pseudo-3-D plot of the wave aberrations of a microobjective of N.A. = 0.32, where the contour line distance is λ/32.

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Fig. 3
Fig. 3 Demonstration of the consequences of reference-phase errors. (a) Difference of two consecutive runs with arbitrary reference-phase deviations. First run: φ1 = 0, φ2 = π/2 + π/16; φ3 = π + π/8, φ4 = (3/2)π + (3/16)π. Second run: φ1 = π/2, φ2 = π + (π/16), φ3 = (3/2)π + π/8, φ4 = 2π + (3/16)π. The scale value on the left-hand side corresponds to a surface deviation of λ/256. (b) Intensity distribution with fringe adjustment. Note that the number of fringes is half of the number of maxima in (a).

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Fig. 4
Fig. 4 Averaging effect in dependence of the phase step number. For the purpose of display an alifu representation has been chosen. The alifu program adds a linear function to the deviation data. From contour line to contour line a λ/64 increase has been introduced artificially (on the left). The perspective plot shows the same with a λ/128 increase from section to section (on the right). Top: Difference of two runs, where the first run had a phase error; i.e., one value of 16 had a deviation of π/8, Below: same as above but with only four phase steps per run and the same phase error for 1 value of 4.

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Fig. 5
Fig. 5 Demonstration of reference-phase error cancellation. (a) Two runs superimposed, second run with π/2 reference-phase offset, 3 fringes/diameter, one division = λ/128 surface deviation. (b) A posteriori averaging of the two runs of (a). (c) Averaging of the numerator and the denominator of another phase profile with λ/256 sensitivity two runs (on the left) in comparison with error-free parallel adjustment evaluation (on the right).

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Fig. 6
Fig. 6 Phase error due to extraneous fringes. (a) Phase error due to glass reflection introduced into the object arm of the interferometer, scale value: λ/64 surface deviation. Extraneous fringes can be perceived by slightly inclining the figure. (b) Average of two runs, where the second was made with a π offset of the object wave front, same sensitivity as in (a).

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Fig. 7
Fig. 7 Average of two runs where the second was carried out with a tilt of λ from left to right. Scale value: λ/64 surface deviation. The beat frequency of the extraneous fringes is identical to the frequency of the fringes to be evaluated.

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Fig. 8
Fig. 8 Intensity distribution overlaid with extraneous fringes. (a) Fringe pattern occurring if a glass plate has been introduced in one arm of the interferometer. (b) Section through the fringe pattern from top to bottom of (a). Extraneous fringes modulate the useful interference pattern.

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Fig. 9
Fig. 9 Phase distribution associated with the intensity distribution of Fig. 8; scale value: λ/128 surface deviation.

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Fig. 10
Fig. 10 Two runs superimposed on each other. The second run was made with a phase offset of c ·(2/3)π.

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Fig. 11
Fig. 11 Total cancellation of the phase disturbances due to the algorithm of Eq. (19).

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Fig. 12
Fig. 12 Averaging of the two runs of Fig. 11 a posteriori. Cancellation of extraneous fringes is only poor because of the phase mismatch of the object phase between the two runs.

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Fig. 13
Fig. 13 Demonstration of the cancellation efficiency of the algorithm due to Eq. (19). The second run was carried out with an additional tilt (compare Fig. 7), which means that Φ has been changed by the additional phase χ exceeding the required limits. Note the phase jumps in the neighborhood of χ = 2nπ. The cancellation works sufficiently well for the greater part of the phase interval which can be outlined from the jump-corrected second curve.

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Fig. 14
Fig. 14 Demonstration of the (η − Φ) dependence of ΔΦ. (a) Fringepattern adjusted in such a manner that along a perpendicular section η is constant (b) Phase disturbances due to a Φ variation (scale value: λ/128 surface deviation).

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Fig. 15
Fig. 15 Series of pictures demonstrating the influence of Φ on the resulting erroneous deviation. (a) Φ = const: left, phase distribution; right, inference pattern. (b) sgnη = sgnΦ: left, phase distribution; right, interference pattern. (c) sgnη ≠ sgnΦ: left, phase distribution; right, interference pattern (scale value: λ/128 surface deviation).

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Fig. 16
Fig. 16 Schematic illustration of the relation of the coherent noise amplitude to the background signal used in the phase measurement: ub = background wave amplitude; un = noise amplitude; ΔΦ = maximum deviation from background phase Φ.

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Fig. 17
Fig. 17 Scan through the intensity distribution in the detector plane. No filtering operations administered. Parallel adjustment of the interference pattern.

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Fig. 18
Fig. 18 Same as Fig. 17 but with a moving scatterer at the pinhole position of the illuminating system. Note the smoothing effect of the moving scatterer.

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Fig. 19
Fig. 19 Phase distribution for a wedge adjustment of c. 10 fringes. Scale value: λ/256 surface deviation.

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Fig. 20
Fig. 20 Same phase distribution as in Fig. 19 but taken in parallel adjustment; same scale value as in Fig. 19.

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Fig. 21
Fig. 21 Same as Fig. 20 but with spatial filter inserted.

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Fig. 22
Fig. 22 Rotating scatterer inserted into the ray path at the position of an intermediate image of the surface to be tested: on the left, scatterer defocused; on the right, scatterer sharply imaged onto the detector array, spatial filter removed. Scale value: λ/256 surface deviation.

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Fig. 23
Fig. 23 Phase difference of two consecutive runs made in parallel adjustment. For the display of the difference we used the alifu representation. alifu means the addition of a linear function to the phase distribution. Here (λ/64)/diameter were added: on the left, contour line plot, on the right, pseudo-3-D plot, because of the number of sections two consecutive lines are approximately λ/128 apart.

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Equations (51)

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tan Φ ( x , y ) = ∑ r = 1 R I r ( x , y ) sin φ r ∑ r = 1 R I r ( x , y ) cos φ r ,
I r ′ ( x , y ) = I 0 ( x , y ) { 1 + V ( x , y ) cos [ Φ ( x , y ) − ψ r ] } ,
ɛ r = φ r − ψ r .
I r ′ ( x , y ) = I 0 { 1 + V cos [ Φ − φ r + ɛ r ] } .
Δ Φ = Φ ′ − Φ .
tan Φ ′ = N D = ∑ r I r ′ sin φ r ∑ r I r ′ cos φ r ,
Δ Φ = arctan ( N D ) − arctan ( tan Φ ) .
Δ Φ = arctan N − D tan Φ D + N tan Φ ,
Δ Φ = arctan ∑ r = 1 R ɛ r − C cos 2 Φ − S sin 2 Φ R − C sin 2 Φ + S cos 2 Φ ,
C = ∑ r = 1 R ɛ r cos 2 φ r , S = ∑ r = 1 R ɛ r sin 2 φ r .
1 R ∑ r = 1 R ɛ r
tan Φ = I 2 − I 4 I 1 − I 3 .
I 0 = a + b cos Φ I 1 = a + b cos ( Φ + φ ) , I 3 = a + b cos ( Φ + 3 φ ) , I 4 = a + b cos ( Φ + 4 φ ) ,
φ = arccos 1 2 I 4 − I 0 I 3 − I 1 .
tan Φ ′ = R sin Φ + ∑ r = 1 R ɛ r cos Φ − C cos Φ − S sin Φ R cos Φ − ∑ r = 1 R ɛ r sin Φ − C sin Φ + S cos Φ = N D .
tan Φ ∼ = N 1 + N 2 D 1 + D 2 ,
tan Φ ∼ = sin Φ ( 1 + S ′ ) + cos Φ ( C ′ + ɛ ′ ) cos Φ ( 1 − S ′ ) + sin Φ ( C ′ − ɛ ′ ) ,
ɛ ′ = 1 2 R ∑ r = 1 R [ ɛ r ( 1 ) + ɛ r ( 2 ) ] , C ′ = 1 2 R ∑ r = 1 R [ ɛ r ( 2 ) − ɛ r ( 1 ) ] cos 2 φ r , S ′ = 1 2 R ∑ r = 1 R [ ɛ r ( 2 ) − ɛ r ( 1 ) ] sin 2 φ r ,
Φ ∼ = Φ + arctan ɛ ′ .
ɛ r = ∑ k = 0 ∞ ζ k r k ;
ɛ r ( 2 ) = ∑ k = 0 ∞ ζ k ( r + a ) k ,
C ′ = ∑ r = 1 R ∑ k = 0 ∞ ζ k [ ( r + a ) k − r k ] cos 2 φ r , S ′ = ∑ r = 1 R ∑ k = 0 ∞ ζ k [ ( r + a ) k − r k ] sin 2 φ r .
∑ r = 1 R cos 2 φ r = ∑ r = 1 R sin 2 φ r = 0 ,
Δ Φ ≈ ( ɛ ¯ − C ¯ cos 2 Φ − S ¯ sin 2 Φ ) ( 1 + C ¯ sin 2 Φ − S ¯ cos 2 Φ + Q 2 ) ,
ɛ ¯ = 1 R ∑ r = 1 R ɛ r , C ¯ = 1 R C , S ¯ = 1 R S ,
Q 2 = ( C ¯ sin 2 Φ − S ¯ cos 2 Φ ) 2 .
δ Φ = ½ ( Δ Φ 1 + Δ Φ 2 ) , δ Φ = ɛ ¯ − ½ C ¯ 2 sin 4 Φ + C S ¯ cos 4 Φ + ½ S ¯ 2 sin 4 Φ + ɛ ¯ Q 2 .
I r = 2 + q 2 + 2 cos ( Φ − φ r ) + 2 q cos ( η − φ r ) + 2 q cos ( Φ − η ) .
Φ ′ = arctan sin Φ + q sin η cos Φ + q cos η .
Δ Φ = arctan q sin ( η − Φ ) 1 + q cos ( η − Φ ) .
δ Φ = 1 2 arctan q 2 sin ( 2 η − 2 Φ ) q 2 cos ( 2 η − 2 Φ ) − 1 .
Φ ∼ = arctan N 1 − N 2 D 1 − D 2 .
Φ ∼ = arctan sin Φ − sin ( Φ + χ ) cos Φ − cos ( Φ + χ ) , Φ ∼ = − arctan 1 tan ( Φ + χ / 2 ) .
arctan x + arctan 1 x = { π 2 x > 0 , − π 2 x < 0 ,
Φ ∼ = Φ + χ − 2 + π 2 .
I = I b + I n + 2 I b I n cos φ n ,
I max − I min I b = 4 | u n u b | ≈ 4 Δ Φ .
Φ = Φ 0 + Δ Φ cos 2 π ν m x .
δ Φ = Δ Φ 2 π ν m Δ x sin 2 π ν m x .
δ Φ p υ = 4 π ν m Δ Φ Δ x .
δ Φ p υ = 2 π ν m N s λ P Δ Φ .
ψ r ( 1 ) = φ r − ɛ r ( 1 ) , ψ r ( 2 ) = φ r + π 2 − ɛ r ( 2 ) ,
tan Φ ∼ = ∑ r = 1 R I r ( 1 ) sin φ r + ∑ r = 1 R I r ( 2 ) sin ( φ r + π 2 ) ∑ r = 1 R I r ( 1 ) cos φ r + ∑ r = 1 R I r ( 2 ) cos ( φ r + π 2 ) ,
I r ( 1 ) = I 0 { 1 + V cos [ Φ − φ r + ɛ r ( 1 ) ] } , I r ( 2 ) = I 0 { 1 + V cos [ Φ − π 2 − φ r + ɛ r ( 2 ) ] } .
tan Φ ∼ = ∑ r = 1 R cos [ Φ − φ r + ɛ r ( 1 ) ] sin φ r + ∑ r = 1 R sin [ Φ − φ r + ɛ r ( 2 ) ] cos φ r ∑ r = 1 R cos [ Φ − φ r + ɛ r ( 1 ) ] cos φ r − ∑ r = 1 R sin [ Φ − φ r + ɛ r ( 2 ) ] sin φ r .
tan Φ ∼ = 2 R sin Φ + ∑ r = 1 R [ ɛ r ( 1 ) + ɛ r ( 2 ) ] cos Φ + ( C 2 − C 1 ) cos Φ + ( S 2 − S 1 ) sin Φ 2 R cos Φ − ∑ r = 1 R [ ɛ r ( 1 ) + ɛ r ( 2 ) ] sin Φ − ( S 2 − S 1 ) cos Φ + ( C 2 − C 1 ) sin Φ ,
C i = ∑ r = 1 R ɛ r ( i ) cos 2 φ r and S i = ∑ r = 1 R ɛ r ( i ) sin 2 φ r ( i = 1 , 2 ) .
ζ 1 = ɛ R / R .
ɛ ′ = 1 2 R ∑ r = 1 R ζ 1 ( r − 2 + a )
arctan ɛ R 2 [ 5 4 − 1 R ] .
tan ϕ ∼ = 2 I 2 − 2 I 4 I 1 − 2 I 3 + I 5 .

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