Abstract

A general graphical convolution method for fringe analysis of testing methods that use a spatially incoherent light source is presented. Results are shown for the grating-slit test using optical narrow and cross slits. The convolution method shows that phase shifting grating-slit test can be done with the cross slit at the expense of a reduced fringe contrast ratio. A phase shifting measurement with a cross-slit modulator yields high measurement precision and validates our fringe contrast theory.

© 2010 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).
  2. V. Ronchi, Appl. Opt. 3, 437 (1964).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. C. W. Liang and J. Sasian, Opt. Express 15, 1738 (2007).
    [CrossRef] [PubMed]
  5. C.-W. Liang and C.-F. Ou, Proc. SPIE 7068, 70680I (2008).
    [CrossRef]
  6. J. L. Rayces, J. Mod. Opt. 11, 85 (1964).
  7. F. Roddier and C. Roddier, Appl. Opt. 30, 1325 (1991).
    [CrossRef] [PubMed]

2008

C.-W. Liang and C.-F. Ou, Proc. SPIE 7068, 70680I (2008).
[CrossRef]

2007

1991

1986

1964

V. Ronchi, Appl. Opt. 3, 437 (1964).
[CrossRef]

J. L. Rayces, J. Mod. Opt. 11, 85 (1964).

Cornejo Rodriguez, A.

Liang, C. W.

Liang, C. -W.

C.-W. Liang and C.-F. Ou, Proc. SPIE 7068, 70680I (2008).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).

Ou, C. -F.

C.-W. Liang and C.-F. Ou, Proc. SPIE 7068, 70680I (2008).
[CrossRef]

Patorski, K.

Rayces, J. L.

J. L. Rayces, J. Mod. Opt. 11, 85 (1964).

Roddier, C.

Roddier, F.

Ronchi, V.

Sasian, J.

Appl. Opt.

J. Mod. Opt.

J. L. Rayces, J. Mod. Opt. 11, 85 (1964).

Opt. Express

Proc. SPIE

C.-W. Liang and C.-F. Ou, Proc. SPIE 7068, 70680I (2008).
[CrossRef]

Other

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992).

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Figures (4)

Fig. 1
Fig. 1

Illustration of incoherent optical testing method: black rays represent object imaging optical path by tested optics and gray (blue online) rays represent pupil imaging optical path by field lens.

Fig. 2
Fig. 2

The imaged sinusoidal grating is laterally shifted by the amount of the transverse ray aberration within the paraxial region.

Fig. 3
Fig. 3

Cross slit and its equivalent three subaperture combination, after equality from left to right are vertical slit, horizontal slit, and slit crossover area.

Fig. 4
Fig. 4

(a) Experimental setup of the Grating-slit test. (b) Measured surface deformation by the phase shifting grating-slit test with a cross slit (units of waves, 632 nm)

Equations (10)

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W ( x , y ) x = ε x r W ,
W ( x , y ) y = ε y r W .
I ( x p , y p ) = K ε x   min ε x   max ε y   min ε y   max Image [ x a ε x ( x p , y p ) , y a ε y ( x p , y p ) ] Aperture ( x a , y a ) d x a d y a ,
L ( x a , y a ) = 1 + γ   cos [ 2 π x a p + δ ] ,
I ( x P , y P ) = K { 1 + γ   cos [ 2 π p [ x a ε x ( x p , y p ) ] + δ ] } [ Rect ( x a d ) Rect ( y a h y ) ] d x a d y a .
I ( x P , y P ) = K d h y { 1 + γ   sinc ( π d p ) cos [ 2 π ε x ( x p , y p ) p + δ ] } .
V = γ   sinc ( π d p ) .
I CS ( x P , y P ) = I VS ( x P , y P ) + I HS ( x P , y P ) I OA ( x P , y P ) ,
I VS ( x P , y P ) I OA ( x P , y P ) = K d ( h y d ) { 1 + sinc ( π d p ) cos [ 2 π ε x ( x p , y p ) p + δ x ] } .
I HS ( x P , y P ) = K d h x { 1 + sinc ( π h x p ) cos [ 2 π ε x ( x p , y p ) p + δ x ] } .

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