Abstract

A chopper wheel, a bicell photodetector, and an electronic phase detector are used in a mechanization of the Foucault optical test to obtain a linear focus error voltage for an isolated point image. Geometrical and diffraction analyses of the technique are included which show the sensitivity of the technique to aberrations of higher order than focus.

© 1983 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  2. D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Appendix 2.
  3. E. H. Linfoot, Recent Advances in Optics (Oxford U.P., London, 1958), pp. 128–175.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Oxford U.P., London, 1958), pp. 128–175.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978), Appendix 2.

E. H. Linfoot, Recent Advances in Optics (Oxford U.P., London, 1958), pp. 128–175.

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Figures (6)

Fig. 1
Fig. 1

Foucault knife-edge test with the knife-edge (a) in front and (b) behind focus, (c) a mechanization of the Foucault test for focus sensing.

Fig. 2
Fig. 2

Geometry for analysis of the focus sensor.

Fig. 3
Fig. 3

Relative values of the signals X+X and Y+Y for the Zernike aberrations up to n = m = 8 as given by the geometrical analysis.

Fig. 4
Fig. 4

Relative values of the signals X+X and Y+Y or the Zernike aberrations up to n = m = 8 as given by the diffraction analysis.

Fig. 5
Fig. 5

Output voltage vs focus discriminant obtained for an f/30 He–Ne beam using algorithm 1 described in the text.

Fig. 6
Fig. 6

Output voltage vs focus discriminant obtained for an f/30 He–Ne beam using algorithm 2 described in the text.

Equations (22)

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I ( x , y ) = 1 2 I r ( x , y ) - b I r ( x , y ) ϕ ( x , y ) x ,
X + = 1 2 x > 0 I r ( x , y ) d x d y - x > 0 I r ( x , y ) ϕ ( x , y ) x d x d y ,
X - = 1 2 x < 0 I r ( x , y ) d x d y - x < 0 I r ( x , y ) ϕ ( x , y ) x d x d y .
X + - X - = - x > 0 I r ( x , y ) ϕ ( x , y ) x d x d y + x < 0 I r ( x , y ) ϕ ( x , y ) x d x d y ,
ϕ ( x , y ) = i , j 0 α i j x i y j ,
( 2 π λ α i j ) 2 1 ,
I ( x , y ) = I 0 ( x , y ) + { 4 π 2 / λ 0 } 1 r i j 0 α i j x i - r y j ( - 1 ) r ( i r ) × ( x + y ) r - ( x - y ) r r
= I 0 ( x , y ) + Δ I ( x , y ) ,
X + - X - = x > 0 ( x 2 + y 2 ) 1 Δ I ( x , y ) d x d y - x < 0 ( x 2 + y 2 ) 1 Δ I ( x , y ) d x d y .
L ( t ) · C ( F , t ) - ½ L ( t ) = L ( t ) [ C ( F , t ) - ½ ] .
ϕ ( x , y ) = U n m ( ρ , θ ) = R n m ( ρ ) cos m θ ,
x ϕ ( x , y ) = x U n m ( ρ , θ ) = ρ U n m ( ρ , θ ) ρ x + θ U n m ( ρ , θ ) θ x .
ρ x = sin θ             and            θ x = 1 ρ cos θ .
x U n m ( ρ , θ ) = R n m ( ρ ) cos m θ sin θ - m ρ R n m ( ρ ) sin m θ cos θ .
X + - X - = - x > 0 x U n m ( ρ , θ ) d a + x < 0 x U n m ( ρ , θ ) d a = - 0 1 ρ d ρ 0 π d θ [ R n m ( ρ ) cos m θ sin θ - m ρ R n m ( ρ ) sin m θ cos θ ] + 0 1 ρ d ρ - π 0 d θ [ R n m ( ρ ) cos m θ sin θ - m ρ R n m ( ρ ) sin m θ cos θ ] .
X + - X - = - 0 1 ρ d ρ 0 π d θ ( 4 ρ · 1 · sin θ - 0 ) + 0 1 ρ d ρ - π 0 d θ ( 4 ρ · 1 · sin θ - 0 ) = - 16 3 .
ϕ ( x , y ) = U 2 0 ( ρ , θ ) = 2 ρ 2 - 1 = 2 ( x 2 + y 2 ) - 1 ,
α 00 = - 1 ,             α 20 = 2 ,     α 02 = 2.
Δ I ( x , y ) = { 4 π 2 / λ 0 } { 2 · x · 1 · ( - 1 ) 1 ( 2 1 ) ( x + y ) 1 - ( x - y ) 1 1 + 2 · 1 · 1 · ( - 1 ) 2 ( 2 2 ) ( x + y ) 2 - ( x - y ) 2 2 } = { 4 π 2 / λ 0 } { - 4 x y } .
X + = - 1 1 0 1 - y 2 4 π 2 λ ( - 4 x y ) d x d y = 4 π 2 λ ( - 3 π 4 ) .
X - = 4 π 2 λ ( + 3 π 4 )
X + - X - = 4 π 3 λ ( - 3 2 ) .

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