Abstract

We describe a solution to increase the performance of a pyramid wavefront sensor (P-WFS) under bad seeing conditions. We show that most of the issues involve a reduced sensitivity that depends on the magnitude of the high frequency atmospheric distortions. We demonstrate in end-to-end closed loop adaptive optics simulations that with a modal sensitivity compensation method a high-order system with a nonmodulated P-WFS is robust in conditions with the Fried parameter r0 at 0.5μm in the range of 0.050.10  m. We also show that the method makes it possible to use a modal predictive control system to reach a total performance improvement of 0.060.45 in Strehl ratio at 1.6μm. Especially at r0=0.05  m the gain is dramatic.

© 2008 Optical Society of America

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References

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  8. C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
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  10. M. H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, 1996).
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  12. M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, "Modelling astronomical adaptive optics--I. The software package CAOS," Mon. Not. R. Astron. Soc. 356, 1263-1275 (2005).
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2007 (1)

2006 (1)

2005 (1)

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, "Modelling astronomical adaptive optics--I. The software package CAOS," Mon. Not. R. Astron. Soc. 356, 1263-1275 (2005).
[CrossRef]

2004 (2)

2001 (1)

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behavior in partial correction adaptive optic systems," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

1998 (1)

1997 (1)

1994 (1)

E. Gendron and P. Lena, "Astronomical adaptive optics. 1: modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

Appl. Opt. (2)

Astron. Astrophys. (2)

E. Gendron and P. Lena, "Astronomical adaptive optics. 1: modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

S. Esposito and A. Riccardi, "Pyramid wavefront sensor behavior in partial correction adaptive optic systems," Astron. Astrophys. 369, L9-L12 (2001).
[CrossRef]

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, "Modelling astronomical adaptive optics--I. The software package CAOS," Mon. Not. R. Astron. Soc. 356, 1263-1275 (2005).
[CrossRef]

Opt. Commun. (1)

C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

Other (4)

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

O. Wulff and D. Looze, "Nonlinear control for pyramid sensors in adaptive optics," Proc. SPIE 6272, 62721S (2006).

B. Le Roux, "Optimal control law for adaptive optics, application to MCAO and XAO," in EAS Publications Series (EAS Publications Series, 2006), Vol. 22, 139-150.
[CrossRef]

M. H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, 1996).

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Figures (7)

Fig. 1
Fig. 1

Illustration of the sensitivity reduction. Left, radial cuts of incoming phase at a WFS aperture, black plot is the DM shape when a pure mirror mode is applied; gray plot is the mode having an atmospheric residual added onto it. Right, corresponding P-WFS measurement signals of the phases.

Fig. 2
Fig. 2

Illustration of the sensitivity compensation in a closed loop.

Fig. 3
Fig. 3

Illustration of a closed loop AO system with sensitivity compensation.

Fig. 4
Fig. 4

Loop convergence with and without SCCs. The simulations are made with a simple integrator using optimized loop gains (0.6 with SCCs and 3.2 with conventional command matrix) giving corresponding Strehl ratios of 0.32 and 0.06. The vertical lines show the positions at which the SCCs are recomputed. Both cases are made with r 0 = 0.05 , and the P-WFS is simulated with PMA.

Fig. 5
Fig. 5

Radially averaged PSDs of the residuals and their estimates. The thin plots are made at time steps 100 and the thick plots at time steps 700. The simulation was made with r 0 = 0.05   m , and the P-WFS is simulated with PMA.

Fig. 6
Fig. 6

SCCs for the first 100 modes. The coefficients are shown at time steps 100, 200, 700, and 1200. The simulation is made with three seeing values ( r 0 being 0.05, 0.08, and 0.10 m) and using the PMA case with the P-WFS. As a controller here is used the simple integrator with optimized loop gains.

Fig. 7
Fig. 7

Typical Strehl ratios as a function of loop gain at several seeing levels ( r 0 is 0.05, 0.08, and 0.10 m). Solid curves are made without the subbeam interferences (AMA) and dashed curves with the PMA case. Curves with markers are made with the simple integrator (x markers with the SCCs, o markers without the SCCs). The horizontal lines without markers show the Strehl ratio when the modal predictive control is applied with SCCs.

Tables (2)

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Table 1 Simulation Parameters

Tables Icon

Table 2 Optimal Strehl Ratios at 1.6 μm

Equations (21)

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ϕ ˜ resi ( f ) = { ϕ ˜ LF ( f ) , when   f 1 / ( 2 d ) C ( f 2 + L 0 2 ) 11 / 6 , when   f > 1 / ( 2 d ) ,
ϕ DM k ( x ) = i = 1 M y i ( k ) M i ( x ) ,
ϕ LF k ( x ) = i = 1 M c i ( k ) M i ( x ) ,
ϕ ˜ DM ( f ) = 1 N k k = k 1 k 2 | { W ( x ) i = 1 M y i ( k ) M i ( x ) } | 2 ,
ϕ ˜ LF ( f ) = 1 N k k = k 1 k 2 | { W ( x ) i = 1 M c i ( k ) M i ( x ) } | 2 ,
C = 0.1517 r 0 5 / 6 ,
C = b < f < 1 / ( 2 d ) ϕ ˜ DM ( f ) d f b < f < 1 / ( 2 d ) ( f 2 + L ^ 0 2 ) 11 / 3 d f ,
ϕ ^ ( x ) = Re ( 1 { n ( f ) [ ϕ ˜ resi ( f ) ] 1 / 2 } ) ,
m i 0 = [ m 1 ( α M i ( x ) + 0 ) m N ( α M i ( x ) + 0 ) ] [ m 1 ( 0 ) m N ( 0 ) ] ,
m i ϕ = [ m 1 ( α M i ( x ) + ϕ ^ ( x ) ) m N ( α M i ( x ) + ϕ ^ ( x ) ) ] [ m 1 ( ϕ ^ ( x ) ) m N ( ϕ ^ ( x ) ) ] .
ξ i ( ϕ ^ ( x ) ) = [ ( m i 0 ) T m i 0 ( m i ϕ ) T m i ϕ ] 1 / 2 = [ j = 1 N [ m j ( α M i ( x ) ) m j ( 0 ) ] 2 j = 1 N [ m j ( ϕ ^ ( x ) + α M i ( x ) ) m j ( ϕ ^ ( x ) ) ] 2 ] 1 / 2 .
ξ ^ i = ξ i ( ϕ ^ ( x ) ) ,
c ( k ) = B s ( k ) ,
[ c 1 ( k ) c M ( k ) ] = [ ξ ^ i ξ ^ M ] B s ( k ) ,
[ u 1 ( k ) u N a ( k ) ] = Z [ y 1 ( k ) y M ( k ) ] ,
y ( k ) = θ ( k ) T ϕ ( k ) ,
θ ( k ) = [ b 1 ( k ) b p 1 ( k ) a 0 ( k ) a q 1 ( k ) ] T ,
ϕ ( k ) = [ y ( k 1 ) y ( k p + 1 ) ] , [ c ( k 2 ) c ( k q 1 ) ] T ,
ϵ = L j k [ y ( j ) θ T ϕ ( j ) ] 2
Φ = [ ϕ ( k L ) T ϕ ( k ) T ] ,
E = [ y ( k L ) T θ ( k L ) T ϕ ( k L ) y ( k ) T θ ( k ) T ϕ ( k ) ] .

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