Abstract

The Ronchi test with a LCD amplitude sinusoidal grating is used for testing nominally flat surfaces. We prove that it is possible to measure flat surfaces without using a reference element by modifying the common optical Ronchi set up. The Ronchi rulings are computer generated and displayed on the LCD. By displaying various phase-shifted rulings and capturing the corresponding images, the phase is obtained with the conventional phase-shifting algorithms. Theoretical and experimental results are shown.

© 2003 Optical Society of America

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References

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  1. P. Hariharan, �??Improved Oblique-Incidence Interferometer,�?? Opt. Eng. 14, 257-258 (1974).
  2. M.V.R.K. Murty, R.P. Shukla, �??An Oblique Incidence Interferometer,�?? Opt. Eng. 15, 461-463 (1976).
  3. D. Boebel, B. Packro�?, H.J. Tiziani, �??Phase shifting in an oblique incidence interferometer,�?? Opt. Eng. 30, 1910-1914 (1991).
    [CrossRef]
  4. H. Nürge, J. Schwider, �??Testing of cylindrical lenses by grazing incidence interferometry,�?? Optik 111, 545-555 (2000).
  5. Peter de Groot, �??Diffractive grazing-incidence interferometer,�?? Appl. Opt. 39, 1527-1530 (2000).
    [CrossRef]
  6. M. Mora González and N. Alcalá Ochoa, �??The Ronchi test with an LCD grating,�?? Opt. Commun. 191, 203-207 (2001).
    [CrossRef]
  7. M. Mora-González and N. Alcalá Ochoa, �??Sinusoidal liquid crystal display grating in the Ronchi test,�?? Opt. Eng. 42, 1725-1729 (2003).
    [CrossRef]
  8. R. Barakat, �??General Diffraction Theory of Optical Aberration Tests, from the Point of View of Spatial Filtering,�?? J. Opt. Soc. Am. 59, 1432-1439 (1969).
    [CrossRef]
  9. J. E. Greivenkamp, J. H. Bruning, �??Phase Shifting Interferometry,�?? Chapter 14 in Optical Shop Testing, D. Malacara, Ed., pp. 548-551, Wiley, New York, (1992).
  10. K. Hibino, D.I. Farrant, B.K. Ward, and B.F. Oreb, �??Dynamic range of Ronchi test with a phase-shifted sinusoidal grating,�?? Appl. Opt. 36, 6178-6189 (1997).
    [CrossRef]
  11. R.M. Goldstein, H.A. Zebker, and C.L. Werner, �??Satellite radar interferometry: two-dimensional phase unwrapping,�?? Radio Science, 23, 713-720 (1988).
    [CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

M. Mora González and N. Alcalá Ochoa, �??The Ronchi test with an LCD grating,�?? Opt. Commun. 191, 203-207 (2001).
[CrossRef]

Opt. Eng. (4)

M. Mora-González and N. Alcalá Ochoa, �??Sinusoidal liquid crystal display grating in the Ronchi test,�?? Opt. Eng. 42, 1725-1729 (2003).
[CrossRef]

P. Hariharan, �??Improved Oblique-Incidence Interferometer,�?? Opt. Eng. 14, 257-258 (1974).

M.V.R.K. Murty, R.P. Shukla, �??An Oblique Incidence Interferometer,�?? Opt. Eng. 15, 461-463 (1976).

D. Boebel, B. Packro�?, H.J. Tiziani, �??Phase shifting in an oblique incidence interferometer,�?? Opt. Eng. 30, 1910-1914 (1991).
[CrossRef]

Optical Shop Testing (1)

J. E. Greivenkamp, J. H. Bruning, �??Phase Shifting Interferometry,�?? Chapter 14 in Optical Shop Testing, D. Malacara, Ed., pp. 548-551, Wiley, New York, (1992).

Optik (1)

H. Nürge, J. Schwider, �??Testing of cylindrical lenses by grazing incidence interferometry,�?? Optik 111, 545-555 (2000).

Radio Science (1)

R.M. Goldstein, H.A. Zebker, and C.L. Werner, �??Satellite radar interferometry: two-dimensional phase unwrapping,�?? Radio Science, 23, 713-720 (1988).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the Grazing Ronchi Device used for testing flat surfaces.

Fig. 2.
Fig. 2.

Experimental results obtained with a digital vertical sinusoidal grating (a) Ronchigram. (b) Wrapped phase corresponding to the four vertical ronchigrams shifted in phase.

Fig. 3.
Fig. 3.

Surface topography obtained with the Ronchi test after unwrapping the horizontal and vertical wrapped ronchigrams and its integration

Fig. 4.
Fig. 4.

Surface topography obtained with a commercial Fizeau interferometer.

Fig. 5.
Fig. 5.

Interferometric fringes representation of Figs. 3 and 4. (a) From Ronchi results. (b) From Fizeau results.

Equations (10)

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D ( x , y ) w ( x , y ) 2 cos ( θ ) λ = w ( x , y ) 2 λ eqv
λ eqv = λ cos ( θ )
F 0 ( x 0 , y 0 ) = exp ( i 2 π w ( x 0 , y 0 ) )
U ( x r , y r ) = F 0 ( x 0 , y 0 ) · exp ( i 2 π r λ ( x r x 0 + y r y 0 ) ) d x 0 d y 0 ,
G ( x 1 , y 1 ) = U ( x r , y r ) · M ( x r , y r ) · exp ( i 2 π f c λ ( x r x 1 + y r y 1 ) ) d x r d y r
M ( x r , y r ) = [ m R ( x r , y r ) * * rect ( x r a x , y r a y ) ] · comb ( x r Δ x , y r Δ y ) ,
m R ( x r , y r ) = A 2 [ 1 + cos ( π x r p β ) ]
I ( x 1 , y 1 ; β ) = C + 4 V ( x 1 , y 1 ) cos [ φ ( x 1 , y 1 ) β ] + Γ ( x 1 , y 1 ) ,
φ ( x 1 , y 1 ) = π λ s [ w ( x 1 , y 1 ) x 1 cos ( γ ) w ( x 1 , y 1 ) y 1 sin ( γ ) ]
φ ( x 1 , y 1 ) = arctg [ I ( x 1 , y 1 ; 2 β ) I ( x 1 , y 1 ; 0 β ) I ( x 1 , y 1 ; 3 β ) I ( x 1 , y 1 ; 1 β ) ] .

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