Abstract

In many scientific and medical applications wavefront-sensorless adaptive optics (AO) systems are used to correct the wavefront aberration by optimizing a certain target parameter, which is nonlinear with respect to the control signal to the deformable mirror (DM). Hysteresis is the most common nonlinearity of DMs, which can be corrected if the information about the hysteresis behavior is present. We report a general approach to extract hysteresis from the nonlinear behavior of the adaptive optical system, with the illustration of a Foucault knife test, where the voltage–intensity relationship consists of both hysteresis and some memoryless nonlinearity. The hysteresis extracted here can be used for modeling and linearization of the AO system.

© 2008 Optical Society of America

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References

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  1. Q. Yang, C. Ftaclas, M. Chun, and D. Toomey, J. Opt. Soc. Am. A 22, 142 (2005).
    [CrossRef]
  2. A. Dubra, J. Massa, and C. Paterson, Opt. Express 13, 9062 (2005).
    [CrossRef] [PubMed]
  3. I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications (Elsevier, 2003).
  4. K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
    [CrossRef]
  5. R. K. Tyson, Adaptive Optics Engineering Handbook (Dekker, 2000).
  6. G. Vdovin, Proc. SPIE 3353, 902 (1998).
    [CrossRef]
  7. M. A. A. Neil, M. J. Booth, and T. Wilson, Opt. Lett. 25, 1083 (2000).
    [CrossRef]
  8. P. Marsh, D. Burns, and J. Girkin, Opt. Express 11, 1123 (2003).
    [CrossRef] [PubMed]
  9. L. Foucault, Annales de l'Observatoire imperial de Paris 5, 197 (1859).
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  11. M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
    [CrossRef]
  12. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

2005 (2)

2004 (1)

K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
[CrossRef]

2003 (1)

2001 (1)

M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
[CrossRef]

2000 (1)

1998 (1)

G. Vdovin, Proc. SPIE 3353, 902 (1998).
[CrossRef]

1859 (1)

L. Foucault, Annales de l'Observatoire imperial de Paris 5, 197 (1859).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Booth, M. J.

Burns, D.

Chun, M.

Dubra, A.

Foucault, L.

L. Foucault, Annales de l'Observatoire imperial de Paris 5, 197 (1859).

Fraanje, R.

K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
[CrossRef]

Ftaclas, C.

Girkin, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hinnen, K. J. G.

K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
[CrossRef]

Loktev, M.

M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
[CrossRef]

Marsh, P.

Massa, J.

Mayergoyz, I. D.

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications (Elsevier, 2003).

Monteiroa, D. W. D. L.

M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
[CrossRef]

Neil, M. A. A.

Paterson, C.

Toomey, D.

Tyson, R. K.

R. K. Tyson, Adaptive Optics Engineering Handbook (Dekker, 2000).

Vdovin, G.

M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
[CrossRef]

G. Vdovin, Proc. SPIE 3353, 902 (1998).
[CrossRef]

Verhaegen, M.

K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
[CrossRef]

Wilson, T.

Yang, Q.

Annales de l'Observatoire imperial de Paris (1)

L. Foucault, Annales de l'Observatoire imperial de Paris 5, 197 (1859).

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

Opt. Commun. (1)

M. Loktev, D. W. D. L. Monteiroa, and G. Vdovin, Opt. Commun. 192, 91 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. (1)

K. J. G. Hinnen, R. Fraanje, and M. Verhaegen, Proc. Inst. Mech. Eng. Part I - J. Systems Control Eng. 218, 503 (2004).
[CrossRef]

Proc. SPIE (1)

G. Vdovin, Proc. SPIE 3353, 902 (1998).
[CrossRef]

Other (4)

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications (Elsevier, 2003).

R. K. Tyson, Adaptive Optics Engineering Handbook (Dekker, 2000).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (5)

Fig. 1
Fig. 1

Left, schematic of the experimental setup (input, control voltage V i ; output, light intensity measurement V o ). Right, distribution of the actuators in the DM.

Fig. 2
Fig. 2

Transfer from V i to V o . Two hysteresis branches join at x = V i . Virtual output V o l = f ( x 0 ) = f ( V i 0 ) can be approximated by [ f ( x 1 ) + f ( x 2 ) ] 2 with error ϵ.

Fig. 3
Fig. 3

Left, ( V i , V o ) curves for actuator 1 when the razor is at its initial position, tuned by 0.45, 0.52, 0.60, and 0.80 mm (top to bottom). Right, ( V i , V o ) curves for actuator 4 when the razor is at its initial position, tuned by 0.37, 0.50, 0.59, and 0.80 mm (top to bottom).

Fig. 4
Fig. 4

( V i , x ̂ ) curves (solid curve) extracted from A 1 , A 2 , B 1 , and B 2 , with polynomial order M = 8 . The hysteresis curve measured by a position sensor (dashed curve) is used for comparison.

Fig. 5
Fig. 5

( V i , V o ) curves corresponding to actuator 1 (left) and 4 (right) after hysteresis compensation. The razor is at the same positions as in Fig. 3.

Tables (1)

Tables Icon

Table 1 VAF of x ̂ Extracted from A 1 , A 2 , B 1 , and B 2 , with Polynomial Order M = 8 , 12 , 15

Equations (13)

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V o = A Σ + exp [ j 2 π λ ( ϕ i ( ξ , η ) + 2 ϕ m ( ξ , η ) ) ] × exp [ j 2 π λ f ( u ξ + v η ) ] d ξ d η 2 d u d v ,
ϕ m ( z ) = k = 1 K P k W ( z , ζ k ) 16 π D + w 0 + w 1 z x + w 2 z y ,
P k = { k e ( h ( V i ) ϕ m ( ζ k ) ) if actuator k is excited k e ϕ m ( ζ k ) otherwise } ,
k = 1 K P k = 0 , k = 1 K P k ζ k , x = 0 , k = 1 K P k ζ k , y = 0 ,
V o = f ( x ) , x = H s ( V i ) ,
x = f 1 ( V o ) .
V o l = f ( V i ) .
V ̂ o l = 1 2 ( f ( x 1 ) + f ( x 2 ) ) = f ( x 0 ) + ε ,
ε = 1 2 d f ( x ) d x x = x 0 ( x 1 + x 2 2 x 0 ) + 1 2 d 2 f ( x ) 2 ! d x 2 x = x 0 ( ( x 1 x 0 ) 2 + ( x 2 x 0 ) 2 ) + ,
V ̂ o l f ( V i ) .
min a m V i V ̂ i 2 2 , with V ̂ i = f ̂ 1 ( V ̂ o l ) = m = 0 M a m V ̂ o l m ,
x ̂ = f ̂ n 1 ( V o ) , n = 1 , 2 N .
VAF = ( 1 var ( x ̂ x measure ) var ( x measure ) ) × 100 % .

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