Abstract

Two-dimensional sinusoid fitting and Fourier transform methods of analyzing fringes to determine the wave-front topography are described. The methods are easy to apply because they do not require finding fringe centers and fringe orders. Also, they are accurate. For an active optics experiment in which we have used these techniques, experimental noise exceeds the error resulting from analysis of noise-free theoretical fringe patterns.

© 1983 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  2. L. Mertz, Appl. Opt. 22, 1535 (1983).
    [CrossRef] [PubMed]
  3. M. Malin, W. Macy, G. Ferguson, Proc. Soc. Photo-Opt. Instrum. Eng.365, in press.
  4. Y. Ichioka, M. Inuiya, Appl. Opt. 11, 1507 (1972).
    [CrossRef] [PubMed]

1983 (1)

1982 (1)

1972 (1)

Ferguson, G.

M. Malin, W. Macy, G. Ferguson, Proc. Soc. Photo-Opt. Instrum. Eng.365, in press.

Ichioka, Y.

Ina, H.

Inuiya, M.

Kobayashi, S.

Macy, W.

M. Malin, W. Macy, G. Ferguson, Proc. Soc. Photo-Opt. Instrum. Eng.365, in press.

Malin, M.

M. Malin, W. Macy, G. Ferguson, Proc. Soc. Photo-Opt. Instrum. Eng.365, in press.

Mertz, L.

Takeda, M.

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Figures (4)

Fig. 1
Fig. 1

(a) Contour plot of a test phase pattern 2π(x2y2). (b)–(d), which are to be compared with (a), are derived from a test interferogram having 3 pixels/fringe computed with Eq. (20). Methods used to analyze the test interferogram are (b) Fourier transforms, (c) sinusoid fitting, and (d) smoothing (c). Contours represent 1/10 wave variations.

Fig. 2
Fig. 2

Dependence of the standard deviation of wave fronts derived from test interferograms [Figs. 1(b)–(d) for 3 pixels/fringe] with the true wave front [Fig. 1(a)] on the number of pixels/fringe. Wave fronts are derived using Fourier transforms (solid line), sinusoid fitting (dashed line), and smoothing of sinusoid-fitting wave fronts (dash-dot line).

Fig. 3
Fig. 3

(a) and (b) contour plots of the phase derived from test interferograms having 5 pixels/fringe. Wave fronts are derived using (a) sinusoid fitting and (b) smoothing of (a). Contours represent 1/10 wave variations.

Fig. 4
Fig. 4

(a)–(c), contour plots of a wave front derived from interferograms obtained from an actuated mirror experiment. Methods used for analysis are (a) Fourier transforms, (b) sinusoid fitting, and (c) Zernike polynomial fitting with a Zygo phase measuring system. Contours represent 1/10 wave variations. (d) is a 3-D plot of (c).

Equations (20)

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g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ] ,
g ( x , y ) = a ( x , y ) + c ( x , y ) exp ( 2 π i f 0 x ) c * ( x , y ) × exp ( 2 π i f 0 x ) ,
c ( x , y ) = ( ½ ) b ( x , y ) exp [ i ϕ ( x , y ) ] ,
G ( f , y ) = A ( f , y ) + C ( f f 0 , y ) + C * ( f + f 0 , y ) .
ϕ ( x , y ) = tan 1 { Re [ c ( x , y ) ] / Im [ c ( x , y ) ] } ,
g ( x , y ) = d 0 + d 1 cos ( 2 π x / 3 ) + e 1 sin ( 2 π x / 3 ) .
ϕ ( x , y ) = tan 1 ( e 1 / d 1 ) .
d 1 = ( ½ ) g ( x j 1 , y ) + g ( x j , y ) ( ½ ) g ( x j + 1 + y ) ,
e 1 = ( 3 / 2 ) g ( x j 1 , y ) + ( 3 / 2 ) g ( x j + 1 , y ) ,
ϕ ( x j , y ) = tan 1 ( e 1 / d 1 ) 2 π j / 3 .
ϕ ( x j , y ) = ϕ ( x j , y ) + 2 n π ,
π < ϕ ( x j , y ) ϕ ( x j 1 , y ) < π .
0 < ϕ ( x j , y ) ϕ ( x j 1 , y ) + ( 2 n + 1 ) π < 2 π .
ϕ ( x j , y ) = AMOD [ ϕ ( x j , y ) ϕ ( x j 1 , y ) + 201 π , 2 π ] + ϕ ( x j 1 , y ) π .
Φ ( x c , y k ) = ϕ ( x c , y k ) 2 π n k ,
π < Φ ( x c , y k ) Φ ( x c , y k 1 ) < π .
0 < [ ϕ ( x c , y k ) + ϕ ( x c , y k 1 ) ] / 2 π + 0.5 n k + n k 1 < 1 ,
n k = INT [ ϕ ( x c , y k ) ϕ ( x c , y k 1 ) ] / 2 π + 0.5 ) + n k 1 .
Φ ( x j , y k ) = ϕ ( x j , y k ) 2 π n k .
I = sin [ 2 π T + Φ ( x , y ) ] ,

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