Abstract

We present methods used to determine the linear or nonlinear static response and the linear dynamic response of an adaptive optics (AO) system. This AO system consists of a nonlinear microelectromechanical systems deformable mirror (DM), a linear tip–tilt mirror (TTM), a control computer, and a Shack–Hartmann wavefront sensor. The system is modeled using a single-input–single-output structure to determine the one-dimensional transfer function of the dynamic response of the chain of system hardware. An AO system has been shown to be able to characterize its own response without additional instrumentation. Experimentally determined models are given for a TTM and a DM.

© 2008 Optical Society of America

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References

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2007 (1)

2006 (2)

2005 (1)

D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
[CrossRef]

2004 (2)

1998 (1)

Appl. Opt. (4)

Automatica (1)

D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
[CrossRef]

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

Opt. Express (1)

Other (4)

D. S. Watkins, Fundamentals of Matrix Computations (Wiley, 2002).
[CrossRef]

K. Ogata, Discrete-Time Control Systems, 2nd ed. (Prentice Hall, 1995).

S. Haykin, Adaptive Filter Theory, 4th ed. (Prentice Hall, 2002).

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, "Control of a woofer-tweeter system of deformable mirrors," Proc. SPIE 6274, 62741Z (2006).

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Figures (9)

Fig. 1
Fig. 1

Layout of a simple AO system.

Fig. 2
Fig. 2

Block model of the static DM, WFS, and reconstructor.

Fig. 3
Fig. 3

Inclusion of linearization block, L.

Fig. 4
Fig. 4

(Left) Nonlinear response of the MEMS DM; (right) linear response when applying Eq. (21) to output signal.

Fig. 5
Fig. 5

Concept for identifying an AO system's dynamics.

Fig. 6
Fig. 6

Covariance, p ( j ) , for the TTM x axis.

Fig. 7
Fig. 7

(Color online) Estimation error between measurement and the modeled response.

Fig. 8
Fig. 8

(Color online) Covariance versus sample offset for a MEMS DM at 261   Hz sample rate.

Fig. 9
Fig. 9

(Color online) AO system as a SISO system with respect to signals Θ in ( z ) and Θ out ( z ) .

Tables (2)

Tables Icon

Table 1 Experimentally Determined Models for TTM with 66 Hz Sample Rate a

Tables Icon

Table 2 Experimentally Determined Models for MEMS DM a

Equations (50)

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φ Cor ( b ) = N b ,
φ Err ( b , t ) = φ Atm ( t ) + φ Cor ( b ) ,
d ( b , t ) = d 0 + P φ Err ( b , t ) + q wfs ( t ) = d 0 + P N b + P φ Atm ( t ) + q wfs ( t ) .
Δ d Δ b i = P N I i .
Δ d ^ Δ b i = d ( h I i , t 2 i ) d ( h I i , t 2 i + 1 ) 2 h = P N I i + P φ Atm ( t 2 i ) P φ Atm ( t 2 i + 1 ) + q wfs ( t 2 i ) q wfs ( t 2 i + 1 ) 2 h P N I i ,
D ^ = [ Δ d ^ Δ b 1 Δ d ^ Δ b 2 Δ d ^ Δ b n 1 Δ d ^ Δ b n ] .
U Σ V T = D ^ ,
u i T u j = { 1 for   i = j 0 for   i j ,
v i T v j = { 1 for   i = j 0 for   i j ,
σ i , j = { nonnegative for   i = j 0 for   i j ,
D ^ = i = 1 n u i σ i , i v i T .
σ i , i u i = D ^ v i P N v i = P N ( Δ b ) .
i = 1 n m v i u i T σ i , i = V Σ + U T = D ^ + ,
D ^ + D ^ v i = { v i for   1 i n m 0 elsewhere .
M = D ^ + ,
e = M d ,
M b P N ( Δ b ) = Δ e ,
M b P N v b , i = { v b , i for   1 i n m 0 elsewhere ,
e i ( b i , t ) = I i T M b d ( b i , t ) ,
e i ( b i , t ) = I i T M b ( P N I i b i + P φ Atm ( t ) + q wfs ( t ) + d 0 ) ,
e i ( b i ) I i T M b ( P N I i b i + d 0 ) ,
a i = { c 2 , i b i 2 + c 1 , i b i + c 0 , i for   b i > j c 5 , i b i 2 + c 4 , i b i + c 3 , i for   b i j } = e i ( b i ) | max ( e i ) min ( e i ) | .
0 = { c 2 , i b i 2 + c 1 , i b i a i for   a i > 0 c 5 , i b i 2 + c 4 , i b i a i for   a i 0 ,
b i = { c 1 , i + sgn ( c 1 , i ) c 1 , i 2 + 4 c 2 , i a i 2 c 2 , i for   a i > 0 c 4 , i + sgn ( c 4 , i ) c 4 , i 2 + 4 c 5 , i a i 2 c 5 , i for   a i 0 } = L ( a i ) ,
M a P N ( Δ b ) = Δ e ,
M a P N ( L ( Δ a ) ) = Δ e ,
M a P N ( L ( v a , i ) ) = Δ e = { v a , i for   1 i n m 0 elsewhere .
a [ k ] = v a , 1 θ in [ k ] ,
θ out [ k ] = v a , 1 T e [ k ] = v a , 1 T M a d [ k ] ,
Θ out ( z ) Θ in ( z ) = G Loop ( z ) = β w z w + β w + 1 z w 1 + + β x 1 z x + 1 + β x z x 1 α 1 z 1 α 2 z 2 α y 1 z y + 1 α y z y .
θ out [ k ] = i = 1 y α i θ out [ k i ] + i = w x β i θ in [ k i ] ,
p ( j ) = i = 1 y α i p ( j i ) + i = w x β i c ( j i ) ,
p ( j ) = E { θ out [ k ] θ in [ k j ] } ,
c ( j ) = E { θ in [ k ] θ in [ k j ] } = E { θ in [ k j ] θ in [ k ] } = c ( j ) .
[ α 1 α y β w β x ] T = [ p ( w 1 ) p ( w y ) c ( 0 ) c ( w x ) p ( x + y 1 ) p ( x ) c ( x + y w ) c ( y ) ] 1 [ p ( w ) p ( x + y ) ] .
w > 0 ,
x w ,
y 0 ,
p ( j ) = 0   for   j < w .
θ in [ k ] = { 1 for   k 0 0 elsewhere ,
i = w x β i = 1 i = 1 y α i .
p ( j ) = { 1 K j k = 0 K j 1 θ out [ k + j ] θ in [ k ] for   j 0 0 elsewhere ;
c ( j ) = c ( j ) = 1 K | j | k = 0 K | j | 1 θ in [ k + | j | ] θ in [ k ] ;
[ p ( 0 ) p ( 1 ) p ( 2 ) p ( 3 ) ] = [ 8.8 × 10 5 0.0212 0.0180 2.9 × 10 4 ] ,
[ c ( 0 ) c ( 1 ) ] = [ 0.0401 5.6 × 10 4 ] .
G Loop ( z ) = 0.5359 z 1 + 0.4559 z 2 .
[ p ( 0 ) p ( 1 ) p ( 2 ) p ( 3 ) p ( 4 ) ] = [ 0.0007 0.0002 0.2243 0.0125 0.0011 ] ,
[ c ( 0 ) c ( 1 ) ] = [ 0.25 0.0018 ] .
G Loop ( z ) = z 2
Φ Err ( z ) Φ Atm ( z ) = 1 1 + G Loop ( z ) G c ( z ) ,

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