Abstract

A modal phase-reconstruction method for wave-front analysis in lateral shearing interferometry is presented. Pseudo-Zernike polynomial functions describe the differential wave fronts and are related to a Zernike polynomial description of the original wave front. We show that this reconstruction is robust for shear ratios in the range 0.15–0.50. The error propagation properties of this differential Zernike polynomial matrix-inversion method are discussed on the basis of both analysis and simulation. It is concluded that the method allows wave-front analysis with an absolute inaccuracy of 2 mλ rms for diffraction-limited wave fronts and with 1% relative inaccuracy for more strongly aberrated wave fronts.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 3, 531–534 (1964).
    [CrossRef]
  2. P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
    [CrossRef]
  3. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  4. G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wave-front analysis with high accuracy by the use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef] [PubMed]
  5. H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969).
    [CrossRef]
  6. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
    [PubMed]
  7. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  8. R. H. Hudgin, “Wave-front reconstruction for compensating imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  9. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  10. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  11. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  12. R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979).
    [CrossRef] [PubMed]
  13. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  14. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  15. J. Herrmann, “Cross coupling and aliasing in modal wavefront estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  16. D. Korwan, “Lateral shearing interferogram analysis,” in Precision Surface Metrology, J. C. Wyant, ed., Proc. SPIE429, 194–198 (1983).
  17. 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]
  18. F. Zernike, “Beugungstheorie des Schneidenverfahrans und Seiner Verbesserten Form, der Phasekontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  20. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 470–472.
  21. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

1996 (1)

1986 (1)

1981 (1)

1980 (2)

1979 (3)

1978 (1)

1977 (2)

1975 (1)

1974 (3)

1969 (1)

H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969).
[CrossRef]

1964 (1)

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrans und Seiner Verbesserten Form, der Phasekontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Baxter, B. S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Brangaccio, D. J.

Bruning, J. H.

Cubalchini, R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Freischlad, K. R.

Fried, D. L.

Frost, R. L.

Gallagher, J. E.

Harbers, G.

Hariharan, P.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Herriott, D. R.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Koliopoulos, C. L.

Korwan, D.

D. Korwan, “Lateral shearing interferogram analysis,” in Precision Surface Metrology, J. C. Wyant, ed., Proc. SPIE429, 194–198 (1983).

Kunst, P. J.

Leibbrandt, G. W. R.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 470–472.

Murty, M. V. R. K.

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Rimmer, M. P.

Rosenfeld, D. P.

Rushforth, C. K.

Southwell, W. H.

Steel, W. H.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Sumita, H.

H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

White, A. D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Wyant, J. C.

M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
[PubMed]

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenverfahrans und Seiner Verbesserten Form, der Phasekontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969).
[CrossRef]

Opt. Commun. (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrans und Seiner Verbesserten Form, der Phasekontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 470–472.

D. Korwan, “Lateral shearing interferogram analysis,” in Precision Surface Metrology, J. C. Wyant, ed., Proc. SPIE429, 194–198 (1983).

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes, The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Overlap area between two laterally sheared beams.

Fig. 2
Fig. 2

Shearing interferograms: (a) a 20 = 1λ, s = 0.2; (b) a 2+2 = 1λ, s = 0.2; (c) a 2−2 = 1λ, s = 0.2; (d) a 40 = 1λ, s = 0.1; (e) a 40 = 1λ, s = 0.2; (f) a 40 = 1λ, s = 0.5; (g) a 3+1 = 1λ, s = 0.2; (h) a 3−1 = 1λ, s = 0.2; (i) a 60 = 1λ, s = 0.2.

Fig. 3
Fig. 3

Interference patterns obtained by shearing (a) front 1, (b) front 2. The shear ratio is s = 0.2.

Fig. 4
Fig. 4

Difference between Zernike coefficients resulting from shear analysis with varying matrix dimension D and Twyman–Green analysis performed on front 1 and front 2. The shear ratio is s = 0.2.

Fig. 5
Fig. 5

Remaining aberrations (noise) in a shear analysis of wave fronts 1 and 2 after the Zernike polynomials are subtracted, as a function of matrix dimension D. The dashed line indicates the noise remaining after a direct fit of Zernike polynomials (nm max) to the wave fronts.

Fig. 6
Fig. 6

(a) Condition number of reduced shear matrix, (b) the reciprocal condition number as functions of the shear ratio s.

Fig. 7
Fig. 7

Standard deviations in the calculated Zernike coefficients anm at a standard deviation of the measured shear coefficients rjk of 0.001λ.

Fig. 8
Fig. 8

Relative errors in the calculated Zernike coefficients at a shear ratio error Δs of 0.002.

Tables (5)

Tables Icon

Table 1 Zernike Polynomials in Cylindrical and Cartesian Coordinates up to the Sixth Order

Tables Icon

Table 2 Shear Polynomials in Cartesian Coordinates a

Tables Icon

Table 3 Nonzero Cross Products between x-Difference and Elliptical Zernike Polynomials up to Order n + |m| ≤ 6

Tables Icon

Table 4 Nonzero Cross Products between y-Difference and Elliptical Zernike polynomials up to order n + |m| ≤ 6

Tables Icon

Table 5 Zernike rms Aberrations as Determined in a Twyman–Green Interferometer of the Two Input Fronts used in the Simulations

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

n 0 V ( x , y ) = n , m a n m X n m | m | n m + n even ,
Z n , n 2 j ( ρ , θ ) = [ i = 0 j ( 1 ) i ( n i ) ! i ! ( j i ) ! ( n j i ) ! ρ n 2 i ] × { sin cos } ( n 2 j ) θ ,
a n m = N n m x 2 + y 2 1 V ( x , y ) X n m d x d y ,
N n m = { ( n + 1 ) / π if m = 0 2 ( n + 1 ) / π otherwise .
s = Δ R 2 R ,
x 2 ( 1 s ) 2 + y 2 ( 1 s ) = 1 ,
Δ V x ( x , y ; s ) = V ( x + s , y ) V ( x s , y ) = a n m X n m ( x + s , y ) a n m X n m × ( x s , y ) = a n m Δ X x n m ( x , y ; s ) ,
Δ X x n m ( x , y ; s ) = X n m ( x + s , y ) X n m ( x s , y ) .
x = x / ( 1 s ) ,
y = y / 1 s ,
r x , j k = N j k x 2 + y 2 1 Δ V x [ x ( 1 s ) , × y 1 s ; s ] X j k ( x , y ) d x d y .
r x , j k = N j k x 2 + y 2 1 n m a n m Δ X x n m [ x ( 1 s ) , × y 1 s ; s ] X j k ( x , y ) d x d y = N j k n m a n m x 2 + y 2 1 Δ X x n m [ x ( 1 s ) , × y 1 s ; s ] X j k ( x , y ) d x d y ,
r x = x a ,
r x = ( r x , j k ) , a = ( a n m ) , x = [ X x , j k n m ] ,
X x , j k n m = N j k x 2 + y 2 1 Δ X x n m [ x ( 1 s ) , × y 1 s ; s ] X j k ( x , y ) d x d y .
r y , j k = N j k n m a n m x 2 + y 2 1 Δ X y n m [ x 1 s , × y ( 1 s ) ; s ] X j k ( x , y ) d x d y ,
r y = y a .
[ r x , j k r y , j k ] = [ x y ] [ a n m ] ,
r = a ,
a = 1 r .
= UWV T .
1 = VW 1 U T .
I ( x , y ) = I 0 + I 0 γ cos [ 2 π n , m a n m Δ X x n m ( x , y ) ] ,
a n m = j k j k 1 n m ( s ) r j k
( Δ a n m ) 2 = [ j k d j k 1 n m ( s ) d s r j k ] 2 × ( Δ s ) 2 + j k [ j k 1 n m ( s ) ] 2 ( Δ r j k ) 2 .
4 1 s s
6 ( 1 s s 1 s s 2 )
6 ( 1 s s 1 s s 2 )
4 ( 1 s s 6 1 s s 2 + 6 1 s s 3 )
4 ( 2 1 s s 3 1 s s 2 + 1 s s 3 )
4 ( 2 1 s s 5 1 s s 2 + 3 1 s s 3 )
4 1 s s
6 [ ( 1 s s ) + 1 s s 2 ]
6 ( 1 s s 1 s s 2 )
4 ( 1 s s 6 1 s s 2 + 6 1 s s 3 )
4 ( 2 1 s s + 5 1 s s 2 3 1 s s 3 )
4 ( 2 1 s s 3 1 s s 2 + 1 1 s s 3 )
( r x , 11 r x , 33 r y , 1 1 r y , 3 3 ) = [ X x , 11 20 X x , 11 22 X x , 11 40 X x , 33 40 X y , 1 1 40 X y , 3 3 40 ] ( a 20 a 22 a 40 ) ,
r = ( D = 4 ) a .
r x , 11 = 0.051 λ , r y , 1 1 = 0.051 λ , r x , 31 = 1.741 λ , r y , 3 1 = 1.741 λ , r x , 33 = 0.307 λ , r y , 3 3 = 0.307 λ .
( a 20 a 40 ) = ( D = 4 ) 1 ( r x , 11 r y , 3 3 ) ,
r x , 11 = 0.246 λ , r y , 1 1 = 0.246 λ , r x , 31 = 0.553 λ , r y , 3 1 = 0.553 λ , r x , 33 = 0.020 λ , r y , 3 3 = 0.020 λ , r x , 51 = 1.178 λ , r y , 5 1 = 1.178 λ ,
a 20 = 0.204 λ , a 40 = 0.317 λ ,
( a 20 a 60 ) = ( D = 6 ) 1 ( r x , 11 r y , 5 1 ) ,
a 20 = 0.0 λ , a 40 = 0.0 λ , a 60 = 1.0 λ .

Metrics