Abstract

The three-wave lateral shearing interferometer is an interferometer specially designed for optical testing. It determines three noncollinear phase gradients from one single-fringe pattern. From these quantities, two orthogonal derivatives and the measurement error are estimated, allowing the reconstruction of the aberrated wave front. This new interferometer has several benefits; among them is that its sensitivity and dynamics can be easily adjusted to the analyzed aberrations.

© 1993 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  13. F. Roddier, “Variations on a Hartmann theme,” Opt. Eng. 29, 1239–1242 (1990).
    [CrossRef]

1991

1990

1988

1986

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–9 (1984).
[CrossRef]

1983

1982

1980

1974

J. C. Wyant, “White light extended source shearing interferometer,” Appl. Opt. 13, 200–202 (1974).
[CrossRef] [PubMed]

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1967

Fertin, G.

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

Fontanella, J. C.

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

Freischlad, K. R.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Ichikawa, K.

Koliopoulos, C. L.

Lohmann, A.

Marais, T.

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

Michau, V.

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

Primot, J.

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

Roddier, C.

Roddier, F.

Ronchi, V.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–9 (1984).
[CrossRef]

Takeda, M.

Teague, M. R.

Wyant, J. C.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–9 (1984).
[CrossRef]

Opt. Eng.

F. Roddier, “Variations on a Hartmann theme,” Opt. Eng. 29, 1239–1242 (1990).
[CrossRef]

Other

T. Marais, V. Michau, G. Fertin, J. Primot, J. C. Fontanella, “Deconvolution from wavefront sensing on a 4 m telescope,” in Proceedings of the ESO Conference on High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 589–599.

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Figures (13)

Fig. 1
Fig. 1

Interference pattern observed in the replication plane.

Fig. 2
Fig. 2

Interference pattern observed in a distant plane of the replication plane.

Fig. 3
Fig. 3

Interference pattern observed when the distance between the replication plane and the observation plane increases.

Fig. 4
Fig. 4

Three wave vectors defining a privileged direction of propagation z for which the modulation of the resulting fringe pattern is independent of the longitudinal position of the observation plane.

Fig. 5
Fig. 5

Description of the replication device.

Fig. 6
Fig. 6

Practical implementation of the three-wave lateral shearing interferometer.

Fig. 7
Fig. 7

Geometrical description of the Fourier plane. Localization of the seven harmonics and the associated areas is shown.

Fig. 8
Fig. 8

Block diagram of the algorithm used to reconstruct the wave front from the interferogram.

Fig. 9
Fig. 9

(a) Reference fringe pattern, (b) Fringe pattern with a test default.

Fig. 10
Fig. 10

Harmonic moduli from the test measurement.

Fig. 11
Fig. 11

Three derived phase gradients from the test measurement.

Fig. 12
Fig. 12

(a) Error estimation, sum of the three derivatives; (b) x derivative; (c) y derivative.

Fig. 13
Fig. 13

Reconstructed wave front zoomed on the real support.

Equations (24)

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A ( x ) = [ I ( x ) ] 1 / 2 exp [ i W ( x ) ] ,
A i ( x ) = [ I ( x ) ] 1 / 2 exp { i [ W ( x ) + k i · x ] } , with i = 1 , 3 ,
A P r ( x ) = i = 1 3 A i ( x ) = [ I P r ( x ) ] 1 / 2 exp [ i Φ ( x ) ] ,
I P r ( x ) = I ( x ) M ( x ) ,
M ( x ) = 3 + i , j = 1 i j 3 exp [ i ( k i k j ) · x ] ,
Φ ( x ) = W ( x ) + Ψ ( x ) ,
Ψ ( x ) = phase [ i = 1 3 exp ( i k i · x ) ] .
k I P r z ( x ) = I P r ( x ) · Φ ( x ) + I P r ( x ) 2 Φ ( x ) ,
k i · z = k cos α , i , i = 1 , 3 ,
I P r ( x ) · Ψ ( x ) + I P r ( x ) 2 Ψ ( x ) = 0 .
k I P r z ( x ) = I P r ( x ) · W ( x ) + I P r ( x ) 2 W ( x ) .
I P 0 ( x ) = I P r ( x ) L k [ I P r ( x ) · W ( x ) + I P r ( x ) 2 W ( x ) ] .
I P 0 ( x ) = I ( x ) { 3 f ( x ) + i , j = 1 i j 3 [ f ( x ) i α L W u i j ( x ) ] × exp [ i ( k i k j ) · x ] } ,
f ( x ) = 1 L k I ( x ) [ I ( x ) · W ( x ) + I ( x ) 2 W ( x ) ] ,
( k i k j ) · z = 0 , ( i , j ) i = 1 , 3 , j = 1 , 3 .
I ˜ P 0 ( ν ) = I ˜ ( ν ) * { 3 f ˜ ( ν ) + i , j = 1 i j 3 [ f ˜ ( ν ) i α L W ˜ i j ( ν ) ] * δ ( ν u i j ) } ,
H 00 ( x ) = 3 I ( x ) f ( x ) ,
H i j ( x ) = I ( x ) [ f ( x ) i α L W u i j ( x ) ] .
Re H i j ( x ) = I ( x ) f ( x ) , Im H i j ( x ) = α L I ( x ) W u i j ( x ) .
f ( x ) = 1 L I ( x ) I z ( x ) .
H i j ( x ) = I ( x ) exp [ i ( α L W u i j ) ] .
I ( x ) = H i j ( x ) , W u i j ( x ) = 1 L α phase [ H i j ( x ) ] .
W u i j ( x ) phase ( H i j s H i j w * ) .
σ e 2 = 1 3 [ W u 12 ( x ) + W u 23 ( x ) + W u 31 ( x ) ] 2 ,

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