Abstract

It is often of interest to measure the centroid of a light intensity pattern in order to deduce physical properties. Examples are the Hartmann–Shack sensor, which measures the wavefront slopes, and position sensors. We investigate whether amplitude changes of the incoming electromagnetic field can affect the location of the centroid, and we show that the effect is strongly dependent on the relative size of the diffraction pattern in relation to the lenslet size. We show that if the phase varies slowly in space—and the focal spot size relative to the centroid integration area approaches zero—this variation does not affect the centroid. This is a consequence of symmetry properties of the Fresnel operator. We then show that when the focal width is not infinitely small, changes in the field amplitude can exacerbate distortion of the centroid results.

© 2008 Optical Society of America

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References

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  15. E. N. Ribak and E. Gershnik, “Light propagation through multilayer atmospheric turbulence,” Opt. Commun. 142, 99-105 (1997).
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2006 (1)

2004 (1)

2003 (2)

2002 (2)

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann-Shack sensor,” Opt. Commun. 215, 285-288 (2002).
[CrossRef]

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66-76 (2002).

2001 (1)

2000 (2)

1997 (1)

E. N. Ribak and E. Gershnik, “Light propagation through multilayer atmospheric turbulence,” Opt. Commun. 142, 99-105 (1997).
[CrossRef]

1994 (1)

1987 (1)

1986 (1)

1982 (1)

1981 (1)

1980 (1)

1977 (2)

Aragón, J. L.

Arnold, R.

Artal, P.

Bara, S.

Barrett, T.

Berkefeld, T.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66-76 (2002).

Carmon, Y.

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann-Shack sensor,” Opt. Commun. 215, 285-288 (2002).
[CrossRef]

Cuellar, L.

Freischlad, K.

Fried, D. L.

Fukumitsu, O.

Gershnik, E.

E. N. Ribak and E. Gershnik, “Light propagation through multilayer atmospheric turbulence,” Opt. Commun. 142, 99-105 (1997).
[CrossRef]

Goelz, S.

Gray, A.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications in Physics, 2nd ed. (Dover, 1922), p. 43.

Hofer, H. J.

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 357-378 (1977).
[CrossRef]

Johnson, P.

Kanzaki, O.

Koliopoulos, C. L.

Lane, R. G.

Lefebvre, M.

Mathews, G. B.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications in Physics, 2nd ed. (Dover, 1922), p. 43.

Prieto, P. M.

Primot, J.

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81-89 (2003).
[CrossRef]

Rego, A.

Relton, F. E.

F. E. Relton, Applied Bessel Functions (Dover, 1946), p. 55.

Ribak, E. N.

A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23, 288-297 (2006).
[CrossRef]

A. Talmi and E. N. Ribak, “Direct demodulation of Hartman-Shack patterns,” J. Opt. Soc. Am. A 21, 632-639 (2004).
[CrossRef]

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann-Shack sensor,” Opt. Commun. 215, 285-288 (2002).
[CrossRef]

E. N. Ribak and E. Gershnik, “Light propagation through multilayer atmospheric turbulence,” Opt. Commun. 142, 99-105 (1997).
[CrossRef]

Saga, N.

Sandler, D. G.

Singer, B.

Smith, G.

Soltau, D.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66-76 (2002).

Southwell, W. H.

Spivey, B.

Talmi, A.

Tanaka, K.

Taylor, G.

Teague, M. R.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

Van Dam, M. A.

Vargas-Martin, F.

von der Luhe, O.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66-76 (2002).

Williams, D. R.

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (8)

Opt. Commun. (3)

J. Primot, “Theoretical description of Shack-Hartmann wave-front sensor,” Opt. Commun. 222, 81-89 (2003).
[CrossRef]

E. N. Ribak and E. Gershnik, “Light propagation through multilayer atmospheric turbulence,” Opt. Commun. 142, 99-105 (1997).
[CrossRef]

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann-Shack sensor,” Opt. Commun. 215, 285-288 (2002).
[CrossRef]

Proc. SPIE (1)

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66-76 (2002).

Other (3)

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications in Physics, 2nd ed. (Dover, 1922), p. 43.

F. E. Relton, Applied Bessel Functions (Dover, 1946), p. 55.

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Figures (6)

Fig. 1
Fig. 1

Schematic sketch of the aperture plane and the image plane of a single lenslet.

Fig. 2
Fig. 2

(Left) A schematic drawing of the integration range Ξ. (Right) The effective support of the integral, A Ξ ( r ¯ ) A Ξ f φ ( r ¯ ) . In the black area the function has a value of 1 and in gray a value of 1, and the area in white is where the function vanishes. As the duty cycle tends to zero, the contribution of the integrand becomes negligible over these regions until at the limit, the integral vanishes.

Fig. 3
Fig. 3

The full intensity I ( r ¯ ) (bold solid curve) decays more slowly when the perturbed intensity I 2 ( r ¯ ) (dotted-dashed curve) is added to it relative to the case of I 1 ( r ¯ ) , in which it is absent (thin solid curve).

Fig. 4
Fig. 4

Standard deviation ratio std [ I 1 ( r ¯ ) + I 2 ( r ¯ ) ] std [ I 1 ( r ¯ ) ] as a function of the coupling coefficient ε at a fixed duty cycle. As ε is increased, the standard deviation of the intensity increases, making the centroid operator more susceptible to distortion in finite duty cycles.

Fig. 5
Fig. 5

Ratio of the centroid distortion to its signal, Δ ¯ Ξ f φ , as a function of the duty cycle and of the coupling coefficient ε of the perturbation term. The relative distortion increases with ε and with duty cycle. Significantly, sensitivity to change in ε is enhanced as the duty cycle becomes higher.

Fig. 6
Fig. 6

Induced-error-to-signal ratio, Δ ¯ Ξ f φ , for the diffraction pattern centroid as the direction of the amplitude gradient is changed but the amplitude gradient size, phase gradient and duty cycle are kept constant. The solid curve is the error of the x component, and the dashed curve is the error of the y component. It is seen that the direction of the amplitude gradient has a significant effect on the error.

Equations (25)

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U ( x , y ) = γ U ( x 1 , y 1 ) exp [ i α ( x x 1 + y y 1 ) ] d x 1 d y 1 .
U 1 ( r ¯ ) = U 1 * ( r ¯ ) ,
Re { U 1 ( r ¯ ) } = Re { U 1 ( r ¯ ) } ,
Im { U 1 ( r ¯ ) } = Im { U 1 ( r ¯ ) } ,
U ( r ¯ ) = U 1 S ( r ¯ f φ ) + i U 1 A S ( r ¯ f φ ) ,
I ( r ¯ ) = [ U 1 S ( r ¯ f φ ) ] 2 + [ U 1 A S ( r ¯ f φ ) ] 2 ,
r ¯ = f φ + Ξ ( r ¯ f φ ) I d 2 r Ξ I d 2 r = f φ + Δ ¯ Ξ .
Δ ¯ Ξ = ( I 0 Ξ ) 1 Ξ ( r ¯ f φ ) I d 2 r = ( I 0 Ξ ) 1 R 2 A Ξ ( r ¯ f φ ) I d 2 r ,
A Ξ ( r ¯ ) = { 1 , r ¯ Ξ 0 , r ¯ Ξ }
A Ξ = [ A Ξ A ( Ξ f φ ) ] + A ( Ξ f φ ) ,
A Ξ ( r ¯ ) A Ξ f φ ( r ¯ ) = { 1 , r ¯ Ξ \ ( Ξ f φ ) 1 , r ¯ ( Ξ f φ ) \ Ξ 0 , otherwise } .
r ¯ = f A P I ( r ¯ 1 ) φ ( r ¯ 1 ) d 2 r 1 A P I ( r ¯ 1 ) d 2 r 1 .
f φ + lim Ξ R 2 Δ ¯ Ξ = f φ A P I ( r ¯ 1 ) d 2 r 1 A P I ( r ¯ 1 ) d 2 r 1 ,
U 1 ( r ¯ ) = γ circ R ( r ¯ 1 ) ( A 0 + A r ¯ 1 ) exp ( i α r ¯ r ¯ 1 ) d r ¯ 1 .
U 1 S ( r ¯ ) = γ A 0 circ R ( r ¯ 1 ) exp ( i α r ¯ r ¯ 1 ) d r ¯ 1
U 1 A S ( r ¯ ) = γ circ R ( r ¯ 1 ) A r ¯ 1 exp ( i α r ¯ r ¯ 1 ) d r ¯ 1 .
A r ¯ 1 = A r cos ( θ ψ ) = A r [ C cos ( θ ξ ) S sin ( θ ξ ) ] ,
U 1 A S ( r , ξ ) = γ A 0 R r 2 0 2 π [ cos ( θ ξ ) C sin ( θ ξ ) S ] × exp [ i α r r 1 cos ( θ ξ ) ] d r 1 d θ .
J n ( x ) = 1 π i 3 n 0 π cos ( n θ ) exp ( i x cos θ ) d r d θ ,
C 0 2 π cos ( θ ξ ) exp [ i α r r 1 cos ( θ ξ ) ] d θ = i 2 π cos ( ξ ψ ) J 1 ( α r r 1 ) .
S 0 2 π sin ( θ ξ ) exp [ i α r r 1 cos ( θ ξ ) ] d θ
0 2 π sin ( θ ξ ) exp [ i α r r 1 cos ( θ ξ ) ] d θ = 0 .
U 1 A S ( r , ξ ) = i 2 π R 3 γ cos ( ξ ψ ) A J 2 ( α R r ) ( α R r ) .
U ( r ¯ ) = γ π R 2 A 0 { J 1 ( β ) β [ i ( R A A 0 ) β ¯ β ] J 2 ( β ) β } .
I ( r ¯ ) = I 0 [ J 1 2 ( β ) β 2 i ε cos 2 ( ψ ξ ) J 2 2 ( β ) β 2 ] ,

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