Abstract

The precision and accuracy of interferometers using quadrature fringe detection are often limited not by the interferometer itself but by the detector system. There are three typical errors: unequal gain in the two channels; quadrature phase shift error; and zero offsets. This paper describes a simple method for determining the quadrature errors from experimental data obtained in the interferometer and correcting for them. A numerical example demonstrating the significant improvement in the precision of interferometer data is given.

© 1981 Optical Society of America

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References

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  1. P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
    [CrossRef]

1977 (1)

P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
[CrossRef]

Heydemann, P. L. M.

P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
[CrossRef]

Hyland, R. W.

P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
[CrossRef]

Tilford, C. R.

P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
[CrossRef]

J. Vac. Sci. Technol. (1)

P. L. M. Heydemann, C. R. Tilford, R. W. Hyland, J. Vac. Sci. Technol. 14, 597 (1977).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Plots of uncorrected (○) and corrected (●) experimental data sets.

Tables (1)

Tables Icon

Table I Quadrature Detector Correction Terms Obtained from Data Plotted in Fig. 1

Equations (20)

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r 1 = 2 cos ω t             r 2 = 2 cos ( ω t + π / 2 ) .
u 1 = R [ cos β L + cos ( 2 ω t - β L ) ] , u 2 = R [ sin β L + sin ( 2 ω t - β L ) ] .
u 1 = R cos β L             u 2 = R sin β L .
F = ½ | u 1 u 1 - u 2 u 2 | + 2 u 1 u 1 · arcsin u 2 π .
Δ L = [ ( η + F ) 1 - ( η + F ) 0 ] λ 4 ,
u 1 d = u 1 + p , u 2 d = 1 r ( u 2 cos α - u 1 sin α ) + q ,
( u 1 + p ) 2 + ( 1 r u 2 cos α - 1 r u 1 sin α + q ) 2 = R 2 ,
cos β L + p R = 0 ,             β L = arcos ( - p R ) + ( m + 1 2 ) π 1             m = 0 1 1 1 ,
sin ( β L + α ) + q r R = 0 ,             β L = arcsin ( - q r R ) - α + n π ,             n = 0 , 1 , .
1 = 1 β [ arccos ( - p R ) - π 2 ] ,
2 = 1 β [ arcsin ( - q r R ) - α ] .
i λ = { 0.004 near β L = π / 2 , 0.036 near β L = 0.
( u 1 d - p ) 2 + [ ( u 2 d - q ) r + ( u 1 d - p ) sin α cos α ] 2 = R 2
A u 1 d 2 + B u 2 d 2 + C u 1 d u 2 d + D u 1 d + E u 2 d = 1 ,
A = ( R 2 cos 2 α - p 2 - r 2 q 2 - 2 r p q sin α ) - 1 , B = A r 2 , C = 2 A r sin α , D = - 2 A ( p + r q sin α ) , E = - 2 A r ( r q + p sin α ) ,
α = arcsin C ( 4 A B ) - 1 / 2
δ α α [ ( 1 2 δ A A ) 2 + ( 1 2 δ B B ) 2 + ( 1 2 δ C C ) 2 ] 1 / 2 , r = ( B A ) 1 / 2 ,
p = 2 B D - E C C 2 - 4 A B ,
q = 2 A E - D C C 2 - 4 A B .
u 1 = u 1 d - p ; u 2 = 1 cos α [ ( u 1 d - p ) sin α + r ( u 2 d - q ) ] .

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