Abstract

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to λ/100. The photodiode array samples intensities over a 32 × 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

© 1974 Optical Society of America

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References

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  1. J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.
  2. J. H. Bruning, D. R. Herriott, Appl. Opt. 9, 2180 (1970).
    [CrossRef] [PubMed]
  3. A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wavefront Testing Interferometers,” Paper ThG19, Fall OSA Meeting (October 1973), Rochester, N.Y.

1970

Bruning, J. H.

J. H. Bruning, D. R. Herriott, Appl. Opt. 9, 2180 (1970).
[CrossRef] [PubMed]

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

Gallagher, J. E.

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

Herriott, D. R.

J. H. Bruning, D. R. Herriott, Appl. Opt. 9, 2180 (1970).
[CrossRef] [PubMed]

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

Jensen, A. E.

A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wavefront Testing Interferometers,” Paper ThG19, Fall OSA Meeting (October 1973), Rochester, N.Y.

Nenninger, C. L.

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

Rosenfeld, D. P.

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

White, A. D.

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

Appl. Opt.

Other

J. H. Bruning, J. E. Gallagher, D. R. Herriott, C. L. Nenninger, D. P. Rosenfeld, A. D. White, “Real Time Wavefront Measurement System for Rapid Display of Surface Quality and Lens Performance,” Paper WE16, OSA Meeting (October 1972), San Francisco, Calif.

A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wavefront Testing Interferometers,” Paper ThG19, Fall OSA Meeting (October 1973), Rochester, N.Y.

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Figures (12)

Fig. 1
Fig. 1

Twyman-Green interferometer with piezoelectric pathlength control in reference arm.

Fig. 2
Fig. 2

Twyman-Green digital wavefront measuring interferometer.

Fig. 3
Fig. 3

Detailed view of interferometer.

Fig. 4
Fig. 4

PDP8/I computer system.

Fig. 5
Fig. 5

Computer interface and peripherals.

Fig. 6
Fig. 6

Program flow for measurement of a lens surface: (a) appearance of digitized fringe pattern as seen on monitor oscilloscope; (b) output of raw phases after data accumulation showing λ/2 discontinuities in the wavefront; (c) wavefront after aperture determination and discontinuity removal; (d) wavefront after focus and tilt errors are removed; (e) stored wavefront representing interferometer errors; (f) corrected wavefront display in λ/20 contours on the surface under test.

Fig. 7
Fig. 7

Wavefronts processed for absolute calibration of interferometer. W1, W2 and W3 are input wavefronts, W5 represents the test surface in the absence of interferometer errors (see text).

Fig. 8
Fig. 8

Setup for measuring radii of curvature with distance measuring interferometer and wavefront correction.

Fig. 9
Fig. 9

Wavefront and distance interferometer for radius measurement.

Fig. 10
Fig. 10

Lens testing setup (a) for acquiring wavefront of system errors and (b) lens errors plus system errors.

Fig. 11
Fig. 11

Performance of a 10× reduction lens measured at 6328 Å and 0.7 of full field position, (a) Wavefront in entrance pupil of lens, (b) Computed point spread function, (c) Modulus and (d) phase of the optical transfer function. The MTF in the two principal planes is plotted in (e).

Fig. 12
Fig. 12

Lens measurement support. The lens is mounted in the large cylindrical barrel. Object and image planes lie along the knife edges. The interferometer is seen in the background.

Equations (26)

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w 1 = a exp ( 2 i k l ) , k = 2 π / λ , w 2 = b exp [ 2 i k w ( x , y ) ] ,
I ( x , y , l ) = ( w 1 + w 2 ) ( w 1 + w 2 ) * = a 2 + b 2 + 2 a b cos 2 k [ w ( x , y ) l ] .
I ( x , y , l ) a 0 + a 1 cos 2 k l + b 1 sin 2 k l .
a 0 = 1 n p i = 1 n p I ( x , y , l i ) = a 2 + b 2 ,
a 1 = 2 n p i = 1 n p I ( x , y , l i ) cos 2 k l i = 2 a b cos 2 k w ( x , y ) ,
b 1 = 2 n p i = 1 n p I ( x , y , l i ) sin 2 k l i = 2 a b sin 2 k w ( x , y ) ;
2 k w ( x , y ) = tan 1 ( b 1 / a 1 ) mod 2 π . l = l i = i n λ 2 i = 1,2 , , n p .
w 1 = exp ( i ρ ) form reference arm ; w 2 = exp ( i τ ) form test arm ; w 3 = exp ( i η ) extraneous .
I ( x , y ) = α + β cos ( τ ϕ ) ,
ϕ = tan 1 [ ( sin η ) / ( 1 + cos η ) ] .
I TEST = I 0 sin ϕ i , I REF = I 0 cos ϕ i .
R I TEST cos ϕ A = I REF sin ϕ A ,
R I 0 sin ϕ i cos ϕ A = I 0 cos ϕ i sin ϕ A ,
ϕ A = tan 1 ( R tan ϕ i ) .
ϕ i ( cos ϕ i sin ϕ A ) = ϕ i [ R sin ϕ i ( 1 + R 2 tan 2 ϕ i ) 1 / 2 ] = 0 ,
ϕ i = tan 1 ( 1 / R ) 1 / 2 ,
ϕ A = tan 1 ( R ) 1 / 2 .
I max = I 0 [ R / ( 1 + R ) ] .
2 a b / ( a + b ) 2 = ( a 1 2 + b 1 2 ) 1 / 2 / [ a 0 + ( a 1 2 + b 1 2 ) 1 / 2 ]
w ( x i , y j ) = w 0 ( x i , y j ) + A + B x i + c y j + D ( x i 2 + y j 2 ) .
W 1 = W R 0 + W T 0 + W S 0 , W 2 = W R 0 + W T 0 + W S π , W 3 = W R 0 + 1 2 ( W T 0 + W T π ) ,
W 4 = 1 2 ( W 1 + W ¯ 2 ) = 1 2 ( W R 0 + W R π + W T 0 + W T π ) + W S 0 , W 5 = 1 2 ( W 3 + W ¯ 3 ) = 1 2 ( W R 0 + W R π + W T 0 + W T π ) .
W S 0 = W 4 W 5 .
R = R COUNTER + δ 2 δ 1 ; δ i = 2 i / ( N A i ) 2 ; i = focus error determined form wavefront measurement ; N A i = numerical aperture of collimator or surface .
I ( x , y ) = A ( x , y ) A * ( x , y ) , A ( x , y ) = F [ w ( x , y ) ] = ϕ w ( x , y ) exp [ 2 π i ( x x + y y ) ] d x d y .
O ( f x , f y ) = F 1 [ I ( x , y ) ] .

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