Abstract

Real time transverse wind estimation contributes to predictive correction which is used to compensate for the time delay error in the control systems of adaptive optics (AO) system. Many methods that apply Shack-Hartmann wave-front sensor to wind profile measurement have been proposed. One of the obvious problems is the lack of a fundamental benchmark to compare the various methods. In this work, we present the fundamental performance limits for transverse wind estimator from Shack-Hartmann wave-front sensor measurements using Cramér–Rao lower bound (CRLB). The bound provides insight into the nature of the transverse wind estimation, thereby suggesting how to design and improve the estimator in the different application scenario. We analyze the theoretical bound and find that factors such as slope measurement noise, wind velocity and atmospheric coherence length r0 have important influence on the performance. Then, we introduced the non-iterative gradient-based transverse wind estimator. The source of the deterministic bias of the gradient-based transverse wind estimators is analyzed for the first time. Finally, we derived biased CRLB for the gradient-based transverse wind estimators from Shack-Hartmann wave-front sensor measurements and the bound can predict the performance of estimator more accurately.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998)
  2. W. Jiang and H. Li, “Hartmann-Shack wave-front sensing and control algorithm,” Proc. SPIE 1271, 82–93 (1990).
    [Crossref]
  3. C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
    [Crossref]
  4. L. Poyneer and J.-P. Véran, “Predictive wavefront control for adaptive optics with arbitrary control loop delays,” J. Opt. Soc. Am. A 25(7), 1486–1496 (2008).
    [Crossref] [PubMed]
  5. R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27(11), A235–A245 (2010).
    [Crossref] [PubMed]
  6. L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Bulk wind estimation and prediction for adaptive optics control systems,” J. Opt. Soc. Am. A 28(8), 1566–1577 (2011).
    [Crossref] [PubMed]
  7. R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
    [Crossref]
  8. M. Schöck and E. J. Spillar, “Measuring wind speeds and turbulence with a wave-front sensor,” Opt. Lett. 23(3), 150–152 (1998).
    [Crossref] [PubMed]
  9. K. Yuan, “Measurement of Path-Averaged Transverse Wind Speed with a Shack-Hartmann Wave-Front Sensor,” Acta Opt. Sin. 29(2), 303–307 (1998).
    [Crossref]
  10. M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
    [Crossref]
  11. S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
    [Crossref]
  12. L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Bulk Wind Estimator Performance for AO Systems,” in Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB5.
  13. C. Plantet, S. Meimon, J.-M. Conan, and T. Fusco, “Revisiting the comparison between the Shack-Hartmann and the pyramid wavefront sensors via the Fisher information matrix,” Opt. Express 23(22), 28619–28633 (2015).
    [Crossref] [PubMed]
  14. F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).
  15. G. Rousset, “Wave-front Sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University., 1999), pp. 91–130.
  16. M. Schöck and E. J. Spillar, “Method for a quantitative investigation of the frozen flow hypothesis,” J. Opt. Soc. Am. A 17(9), 1650–1658 (2000).
    [Crossref] [PubMed]
  17. L. Poyneer, M. van Dam, and J.-P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. A 26(4), 833–846 (2009).
    [Crossref] [PubMed]
  18. E. Gendron and P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
    [Crossref]
  19. B. Kern, T. A. Laurence, C. Martin, and P. E. Dimotakis, “Temporal coherence of individual turbulent patterns in atmospheric seeing,” Appl. Opt. 39(27), 4879–4885 (2000).
    [Crossref] [PubMed]
  20. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  21. H. L. V. Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968).

2016 (1)

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

2015 (1)

2012 (2)

S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
[Crossref]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

2011 (1)

2010 (2)

M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
[Crossref]

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27(11), A235–A245 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

2000 (2)

1998 (2)

M. Schöck and E. J. Spillar, “Measuring wind speeds and turbulence with a wave-front sensor,” Opt. Lett. 23(3), 150–152 (1998).
[Crossref] [PubMed]

K. Yuan, “Measurement of Path-Averaged Transverse Wind Speed with a Shack-Hartmann Wave-Front Sensor,” Acta Opt. Sin. 29(2), 303–307 (1998).
[Crossref]

1996 (1)

E. Gendron and P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[Crossref]

1992 (1)

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

1990 (1)

W. Jiang and H. Li, “Hartmann-Shack wave-front sensing and control algorithm,” Proc. SPIE 1271, 82–93 (1990).
[Crossref]

Abado, S.

S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
[Crossref]

Akondi, V.

M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
[Crossref]

Conan, J.-M.

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

C. Plantet, S. Meimon, J.-M. Conan, and T. Fusco, “Revisiting the comparison between the Shack-Hartmann and the pyramid wavefront sensors via the Fisher information matrix,” Opt. Express 23(22), 28619–28633 (2015).
[Crossref] [PubMed]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

Dimotakis, P. E.

Doelman, N.

Fraanje, R.

Fusco, T.

Gavel, D. T.

Gendron, E.

E. Gendron and P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[Crossref]

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

Gordeyev, S.

S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
[Crossref]

Jiang, W.

W. Jiang and H. Li, “Hartmann-Shack wave-front sensing and control algorithm,” Proc. SPIE 1271, 82–93 (1990).
[Crossref]

Johnson, L. C.

Jumper, E. J.

S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
[Crossref]

Juvénal, R.

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

Kern, B.

Kulcsár, C.

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

Laurence, T. A.

Lena, P.

E. Gendron and P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[Crossref]

Li, H.

W. Jiang and H. Li, “Hartmann-Shack wave-front sensing and control algorithm,” Proc. SPIE 1271, 82–93 (1990).
[Crossref]

Martin, C.

Meimon, S.

Petit, C.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

Plantet, C.

Poyneer, L.

Raghavendra, P. B.

M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
[Crossref]

Raynaud, H.-F.

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

Rice, J.

Rigaut, F.

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

Roopashree, M. B.

M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
[Crossref]

Schöck, M.

Spillar, E. J.

van Dam, M.

Véran, J.-P.

Verhaegen, M.

Wiberg, D. M.

Yuan, K.

K. Yuan, “Measurement of Path-Averaged Transverse Wind Speed with a Shack-Hartmann Wave-Front Sensor,” Acta Opt. Sin. 29(2), 303–307 (1998).
[Crossref]

Acta Opt. Sin. (1)

K. Yuan, “Measurement of Path-Averaged Transverse Wind Speed with a Shack-Hartmann Wave-Front Sensor,” Acta Opt. Sin. 29(2), 303–307 (1998).
[Crossref]

Appl. Opt. (1)

Astron. Astrophys. (1)

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).

Astrophys. Space Sci. (1)

E. Gendron and P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[Crossref]

Automatica (1)

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48(9), 1939–1954 (2012).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

S. Abado, S. Gordeyev, and E. J. Jumper, “Approach for two-dimensional velocity mapping,” Opt. Eng. 52(7), 071402 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (3)

R. Juvénal, C. Kulcsár, H.-F. Raynaud, and J.-M. Conan, “LQG adaptive optics control with wind-dependent turbulent models,” Proc. SPIE 9909, 99090M (2016).
[Crossref]

M. B. Roopashree, V. Akondi, and P. B. Raghavendra, “Real-time wind speed measurement using wave-front sensor data,” Proc. SPIE 7588, 75880A (2010).
[Crossref]

W. Jiang and H. Li, “Hartmann-Shack wave-front sensing and control algorithm,” Proc. SPIE 1271, 82–93 (1990).
[Crossref]

Other (5)

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998)

L. C. Johnson, D. T. Gavel, and D. M. Wiberg, “Bulk Wind Estimator Performance for AO Systems,” in Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB5.

G. Rousset, “Wave-front Sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University., 1999), pp. 91–130.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

H. L. V. Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968).

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Figures (13)

Fig. 1
Fig. 1 Principle of AO predictive correction.
Fig. 2
Fig. 2 SHWS subapertures layout ( 14×14).
Fig. 3
Fig. 3 Principle of SHWS.
Fig. 4
Fig. 4 RMSE as it relates to r 0 .
Fig. 5
Fig. 5 RMSE as it relates to slope noise.
Fig. 6
Fig. 6 RMSE as it relates to wind direction: (a) rectangular coordinate; (b) polar coordinate.
Fig. 7
Fig. 7 RMSE versus slope noise.
Fig. 8
Fig. 8 Bias versus wind velocity in different r 0 .
Fig. 9
Fig. 9 The gradient kernel: (a)-(b) central difference operator; (c)-(d) Prewitt operator; (e)-(f) Sobel operator.
Fig. 10
Fig. 10 Bias versus wind velocity using different gradient operator.
Fig. 11
Fig. 11 Absolute bias versus wind velocity using different gradient operator.
Fig. 12
Fig. 12 RMSE versus slope noise as it relates to wind velocity.
Fig. 13
Fig. 13 RMSE versus slope noise as it relates to r 0 .

Tables (1)

Tables Icon

Table 1 Summary of parameters of simulation

Equations (40)

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s ^ ( z,t )=Hϕ( z,t )+ s n ( z,t )
 ϕ( z,t )=ϕ( zw,t1 )
s ( z , t ) = s ( z w , t 1 )
s ^ ( z , t ) = s ( z , t ) + s n ( z , t )
s ^ ( z , t + 1 ) = s ( z + w , t ) + s n ( z , t + 1 )
s t (z)=s( z+w,t )s( z,t )+ε
s( z+w,t )s( z,t )= s z w+R( z,w )
s t =   s z w + ε
w ^ = x,y s z T s t s z s z T
MSE( w ) E[ w ^ ] w J 1 ( w ) E [ w ^ ] T w +( E[ w ^ ]w ) ( E[ w ^ ]w ) T
b ( w ) = E [ w ^ ] w
MSE ( w ) J 1 ( w )
ln P ( s ^ ; w ) = 1 2 σ 2 x , y [ s ^ ( z , t ) s ( z , t ) ] 2 + [ s ^ ( z , t ) s ( z w , t 1 ) ] 2 + const
J ( w ) = E [ 2 ln P ( s ^ ; w ) w w T ]
J ( w ) = [ a 1 a 2 a 2 a 3 ]   
a 1 = x , y E [ 2 log P ( s ^ ; w ) u 2 ] = x , y 1 σ 2 ( s ( z w , t 1 ) u ) 2
a 2 = x , y E [ 2 log P ( s ^ ; w ) u v ] = x , y 1 σ 2 ( s ( z w , t 1 ) u ) ( s ( z w , t 1 ) v )
a 3 = x , y E [ 2 log P ( s ^ ; w ) v 2 ] = x , y 1 σ 2 ( s ( z w , t 1 ) v ) 2  
a 1 = x,y 1 σ 2 ( s( z,t ) x ) 2 =  x,y 1 σ 2 s x 2 ( z,t )
a 2 = x,y 1 σ 2 ( s( z,t ) x )( s( z,t ) y )= x,y 1 σ 2 s x ( z,t ) s y ( z,t )
a 3 = x,y 1 σ 2 ( s( z,t ) y ) 2 =  x,y 1 σ 2 s y 2 ( z,t )
s( z )= f x f y S( f )exp(j f T )Δ f x Δ f y
J ( w ) =   1 σ 2 f x f y | S ( f ) | 2 f f T Δ f x Δ f y
S( f )=h(f) 0.023 r 0 5/3 f 11/3
J ( w ) =   0.023 σ 2 r 0 5 / 3 f x f y | h ( f ) | 2 f 11 / 3 f f T Δ f x Δ f y
w φ =  d T w=ucos( φ )+vsin( φ )
MSE ( w φ ) σ 2 a 1 a 3 a 2 2 [ a 3 cos 2 ( φ ) + a 1 sin 2 ( φ ) 2 a 2 sin ( φ ) cos ( φ ) ]
S t (f)=S( f )[ exp( j f T w )1 ]+Z(f)
w ^ =  Q 1 f x f y | S( f ) | 2 jG(f) S t * (f)Δ f x Δ f y
E [ w ^ ] =   Q 1 f x f y | S ( f ) | 2 G ( f ) sin ( f T w ) Δ f x Δ f y
b ( w ) =   Q 1 f x f y | S ( f ) | 2 [ G ( f ) sin ( f T w ) G ( f ) G ( f ) T w ] Δ f x Δ f y
b ( w ) =   f x f y | h ( f ) | 2 f 11 / 3 [ G ( f ) sin ( f T w ) G ( f ) G ( f ) T w ] Δ f x Δ f y f x f y | h ( f ) | 2 f 11 / 3 G ( f ) G ( f ) T Δ f x Δ f y
b ( w ) =   x , y   s z T s t   s z s z T w
b ( w ) =   Q 1 f x f y | S ( f ) | 2 G ( f ) [ f T w 1 6 ( f T w ) 3 ] Δ f x Δ f y w
b ( w ) = w Q 1 f x f y | S ( f ) | 2 G ( f ) [ f T G ( f ) ] n Δ f x Δ f y 1 6 w 3 Q 1 f x f y | S ( f ) | 2 G ( f ) ( f T n ) 3 Δ f x Δ f y
b ( w ) = w f x f y | h ( f ) | 2 f 11 / 3 G ( f ) [ f T G ( f ) ] n Δ f x Δ f y f x f y | h ( f ) | 2 f 11 / 3 G ( f ) G ( f ) T Δ f x Δ f y 1 6 w 3 f x f y | h ( f ) | 2 f 11 3 G ( f ) ( f T n ) 3 Δ f x Δ f y f x f y | h ( f ) | 2 f 11 3 G ( f ) G ( f ) T Δ f x Δ f y
w * =± ( 6 f x f y | h(f) | 2 f 11/3 G(f)[ f T G(f) ]nΔ f x Δ f y f x f y | h(f) | 2 f 11/3 G( f ) ( f T n) 3 Δ f x Δ f y ) 1 2
η = b ( w ) w = f x f y | h ( f ) | 2 f 11 / 3 G ( f ) [ f T G ( f ) ] n Δ f x Δ f y f x f y | h ( f ) | 2 f 11 / 3 G ( f ) G ( f ) T Δ f x Δ f y 1 6 w 2 f x f y | h ( f ) | 2 f 11 / 3 G ( f ) ( f T n ) 3 Δ f x Δ f y f x f y | h ( f ) | 2 f 11 / 3 G ( f ) G ( f ) T Δ f x Δ f y
E [ w ^ ] w T = x , y   s z s z ( z + w ) T   s z s z T
MSE ( w ) ( x , y   s z s z ( z + w ) T   s z s z T ) J 1 ( w ) ( x , y   s z s z ( z + w ) T   s z s z T ) T + ( x , y   s z T s t   s z s z T w ) ( x , y   s z T s t   s z s z T w ) T

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