Abstract

We describe modeling and simulation results for the Thirty Meter Telescope on the degradation of sodium laser guide star Shack–Hartmann wavefront sensor measurement accuracy that will occur due to the spatial structure and temporal variations of the mesospheric sodium layer. By using a contiguous set of lidar measurements of the sodium profile, the performance of a standard centroid and of a more refined noise-optimal matched filter spot position estimation algorithm is analyzed and compared for a nominal mean signal level equal to 1000 photodetected electrons per subaperture per integration time, as a function of subaperture to laser launch telescope distance and CCD pixel readout noise. Both algorithms are compared in terms of their rms spot position estimation error due to noise, their associated wavefront error when implemented on the Thirty Meter Telescope facility adaptive optics system, their linear dynamic range, and their bias when detuned from the current sodium profile.

© 2006 Optical Society of America

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  1. B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).
  2. P. S. Argall, R. J. Sica, O. Vassiliev, and M. M. Mwangi, "Lidar measurements taken with a large-aperture liquid mirror: sodium resonance-fluorescence system," Appl. Opt. 39, 2393-2399 (2000).
    [CrossRef]
  3. C. d'Orgeville, F. Rigaut, and B. L. Ellerbroek, "LGS AO photon return simulations and laser requirements for the Gemini LGS AO program," Gemini Observatory preprint #55, available online at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.
  4. B. L. Ellerbroek and G. M. Cochran, "Wave optics propagation code for multiconjugate adaptive optics," in Adaptive Optics Systems and Technology II, R. K. Tyson, D. Donaccini, and M. C. Roggemann, eds., Proc. SPIE 4494, 104-120 (2002).
    [CrossRef]
  5. B. L. Ellerbroek, "Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes," in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg and T. Andersen, eds., Proc. SPIE 5382, 478-489 (2004).
    [CrossRef]
  6. G. A. Tyler and D. L. Fried, "Image-position error associated with a quadrant detector," J. Opt. Soc. Am. 72, 804-808 (1982).
    [CrossRef]
  7. L. Gilles, "Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multiconjugate adaptive optics," Appl. Opt. 44, 993-1002 (2005).
    [CrossRef] [PubMed]
  8. J. W. Beletic, "Follow the yellow-orange rabbit: a CCD optimized for wavefront sensing a pulsed sodium laser guide star," in Optical and Infrared Detectors for Astronomy, J. D. Garnett and J. W. Beletic, eds., Proc. SPIE 5499, 302-309 (2004).
    [CrossRef]

2005 (2)

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

L. Gilles, "Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multiconjugate adaptive optics," Appl. Opt. 44, 993-1002 (2005).
[CrossRef] [PubMed]

2004 (2)

B. L. Ellerbroek, "Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes," in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg and T. Andersen, eds., Proc. SPIE 5382, 478-489 (2004).
[CrossRef]

J. W. Beletic, "Follow the yellow-orange rabbit: a CCD optimized for wavefront sensing a pulsed sodium laser guide star," in Optical and Infrared Detectors for Astronomy, J. D. Garnett and J. W. Beletic, eds., Proc. SPIE 5499, 302-309 (2004).
[CrossRef]

2002 (1)

B. L. Ellerbroek and G. M. Cochran, "Wave optics propagation code for multiconjugate adaptive optics," in Adaptive Optics Systems and Technology II, R. K. Tyson, D. Donaccini, and M. C. Roggemann, eds., Proc. SPIE 4494, 104-120 (2002).
[CrossRef]

2000 (1)

1982 (1)

Argall, P. S.

Beletic, J. W.

J. W. Beletic, "Follow the yellow-orange rabbit: a CCD optimized for wavefront sensing a pulsed sodium laser guide star," in Optical and Infrared Detectors for Astronomy, J. D. Garnett and J. W. Beletic, eds., Proc. SPIE 5499, 302-309 (2004).
[CrossRef]

Britton, M.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Cochran, G. M.

B. L. Ellerbroek and G. M. Cochran, "Wave optics propagation code for multiconjugate adaptive optics," in Adaptive Optics Systems and Technology II, R. K. Tyson, D. Donaccini, and M. C. Roggemann, eds., Proc. SPIE 4494, 104-120 (2002).
[CrossRef]

Dekany, R.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

d'Orgeville, C.

C. d'Orgeville, F. Rigaut, and B. L. Ellerbroek, "LGS AO photon return simulations and laser requirements for the Gemini LGS AO program," Gemini Observatory preprint #55, available online at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.

Ellerbroek, B. L.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

B. L. Ellerbroek, "Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes," in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg and T. Andersen, eds., Proc. SPIE 5382, 478-489 (2004).
[CrossRef]

B. L. Ellerbroek and G. M. Cochran, "Wave optics propagation code for multiconjugate adaptive optics," in Adaptive Optics Systems and Technology II, R. K. Tyson, D. Donaccini, and M. C. Roggemann, eds., Proc. SPIE 4494, 104-120 (2002).
[CrossRef]

C. d'Orgeville, F. Rigaut, and B. L. Ellerbroek, "LGS AO photon return simulations and laser requirements for the Gemini LGS AO program," Gemini Observatory preprint #55, available online at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.

Fried, D. L.

Gavel, D.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Gilles, L.

Herriot, G.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Macintosh, B.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Mwangi, M. M.

Rigaut, F.

C. d'Orgeville, F. Rigaut, and B. L. Ellerbroek, "LGS AO photon return simulations and laser requirements for the Gemini LGS AO program," Gemini Observatory preprint #55, available online at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.

Sica, R. J.

Stoesz, J.

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Tyler, G. A.

Vassiliev, O.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Proc. SPIE (4)

B. L. Ellerbroek and G. M. Cochran, "Wave optics propagation code for multiconjugate adaptive optics," in Adaptive Optics Systems and Technology II, R. K. Tyson, D. Donaccini, and M. C. Roggemann, eds., Proc. SPIE 4494, 104-120 (2002).
[CrossRef]

B. L. Ellerbroek, "Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes," in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg and T. Andersen, eds., Proc. SPIE 5382, 478-489 (2004).
[CrossRef]

B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive Optics for the Thirty Meter Telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

J. W. Beletic, "Follow the yellow-orange rabbit: a CCD optimized for wavefront sensing a pulsed sodium laser guide star," in Optical and Infrared Detectors for Astronomy, J. D. Garnett and J. W. Beletic, eds., Proc. SPIE 5499, 302-309 (2004).
[CrossRef]

Other (1)

C. d'Orgeville, F. Rigaut, and B. L. Ellerbroek, "LGS AO photon return simulations and laser requirements for the Gemini LGS AO program," Gemini Observatory preprint #55, available online at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.

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Figures (8)

Fig. 1
Fig. 1

Illustration of subaperture focal-plane radial geometry CCD arrays.

Fig. 2
Fig. 2

Left panel, mean sodium profile obtained by averaging and centering 88 contiguous frames of lidar measurements of the sodium layer with spatial resolution equal to 24   m (Ref. 2). Right panel, sample sodium profile frame.

Fig. 3
Fig. 3

Left panel, Nyquist sampled subaperture short-exposure PSF. Right panel, Nyquist sampled LLT aperture short-exposure PSF. The subaperture size was taken equal to d SA = 0.5   m (order 60 × 60 wavefront sensor), the LLT diameter equal to d LLT = d SA = 0.5   m , and the 1 / e 2 Gaussian laser beam diameter equal to 0.6 d LLT = 0.3   m . These quantities were computed in the Fourier domain by using a 32 × 32 subaperture grid embedded into a 64 × 64 FFT grid. The Fried parameter is r 0 ( λ 0 = 500   nm ) = 0.15   m and the turbulence outer scale is infinite.

Fig. 4
Fig. 4

Left panels, Nyquist sampled normalized average beacon radial and azimuthal cross sections as seen from a subaperture 1 and 14 .5   m away from the LLT. Right panels, total subaperture spot obtained by convolving the beacon with the short-exposure subaperture PSF at a distance of 1 and 14.5 m.

Fig. 5
Fig. 5

Radial and azimuthal photon and readout noise propagation levels associated with the matched filter (left panels) and centroid spot (right panels) position estimators, as a function of the subaperture-to-LLT separation. These curves are for the median sodium profile displayed in Fig. 2. The beacon brightness has been scaled to provide a mean signal level equal to N = 10 3 photodetected electrons per subaperture per integration time, and the cases of σ e = 0 (top) and σ e = 5 (bottom) electrons rms readout noise are compared for a 16 × 4 subaperture focal-plane CCD pixel array with θ pix = 0.5   arc   sec pixel subtense and θ blur = θ pix / 4 pixel blurring due to charge diffusion. The corresponding SNRs are of the order of 31 and 19, respectively. Blue and red curves refer to the null point set, respectively, at the origin (center) of the subaperture focal plane and at a null position shifted by half a pixel in both the radial and azimuthal directions (as might be the case with sample noncommon path wavefront errors). Such a null point offset has no impact on the noise properties of the algorithms.

Fig. 6
Fig. 6

Average spot position estimation error curves, θ ^ ( ) θ ( ) in , for a central and an edge subaperture as a function of input tilt level when the null point of the subaperture focal plane is at the origin. The curves for the matched filter algorithm (left panels) are for a mean signal level of 1000 photodetected electrons per subaperture (0.5 and 14.5 m, top and bottom, respectively, for left and right panels) and per integration time and a read noise of either 0 or 5 e rms. For the centroid algorithm (right panels), the curves are independent of signal and read noise levels since the algorithm does not incorporate statistical prior information.

Fig. 7
Fig. 7

Same as Fig. 6 but when the null point of the subaperture focal plane is at half a pixel in both radial and azimuthal directions.

Fig. 8
Fig. 8

Average spot position estimation error curves, θ ^ ( ) θ ( ) in , for a central and an edge subaperture (0.5 and 14.5 m, left and right panels, respectively, for 1-frame latency) for the centroid (top panels) and matched filter (bottom) algorithms as a function of input tilt level when the null point of the subaperture focal plane is at the origin and the algorithms have 72   s (i.e., 1 frame) update latency. Different curves correspond to the 87 different pairs of contiguous sodium profile frames. Azimuthal curves are identical for all pairs of profiles for the centroid algorithm as a consequence of the symmetry properties of the algorithm.

Tables (2)

Tables Icon

Table 1 Radial, Azimuthal, and rss Photon and Readout Noise Propagation Levels a

Tables Icon

Table 2 Wavefront Error for the TMT Facility AO System a

Equations (65)

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i ( θ x , θ y ) = PSF SA ( θ x , θ y ) i beacon ( θ x , θ y ) .
i beacon ( θ x , θ y ) = PSF LLT ( - θ x , - θ y ) i Na ( θ x , θ y ) .
d θ x d θ y i ( θ x , θ y ) = N .
i ^ ( u x , u y ) = OTF SA ( u x , u y ) i ^ beacon ( u x , u y ) ,
i ^ beacon ( u x , u y ) = OTF* LLT ( u x , u y ) i ^ Na ( u x , u y ) .
PSF SA ( LLT ) ( θ x , θ y ; λ ) = 1 { OTF SA ( LLT ) ( u x , u y ; λ ) } d u x d u y OTF SA ( LLT ) DL ( u x , u y ; λ ) ,
  OTF SA ( LLT ) ( u x , u y ; λ ) = d x d y f SA ( LLT ) d x d y | U SA ( LLT ) ( x , y ) | 2 ,
f SA ( LLT ) = U SA ( LLT ) ( x , y ; λ ) U * SA ( LLT ) ( x + λ u x , y + λ u y ; λ ) exp [ 1 2 ( 2 π λ ) 2 D OPD ¯ SA ( LLT ) ( x , y ; x + λ u x , y + λ u y ) ] ,
U SA ( LLT ) ( x , y ; λ ) = W SA ( LLT ) ( x , y ) × exp [ j 2 π λ OPD SA ( LLT ) ( x , y ) ] ,
W SA ( x , y ) = rect ( x d SA ) rect ( y d SA ) ,
OPD SA ( x , y ) = θ x ref x + θ y ref y + θ x in x + θ y in y ,
W LLT ( x , y ) = circ ( 2 x 2 + y 2 d LLT ) × exp [ ( x 2 + y 2 ) / ( 2 σ laser 2 ) ] 2 π σ laser 2 ,
OPD LLT ( x , y ) = 0 ,
[ θ θ ] = Rot SA ( φ SA ) [ θ x θ y ] ,
Rot SA ( φ SA ) = [ cos ( φ SA ) sin ( φ SA ) sin ( φ SA ) cos ( φ SA ) ] .
i Na ( θ x , θ y ) = 1 r SA δ ( θ ) P Na ( h ( θ ) + h Na ) ,
h ( θ ) δh + h LGS 2 θ r SA ,
δh = h LGS - h Na ,
i ^ Na ( u x , u y ) = 1 h LGS 2 P ^ Na ( f = x SA u x + y SA u y h LGS 2 ) exp ( j 2 πf h LGS ) .
I avg = d θ x d θ y i ( θ x , θ y ) B ( θ x , θ y )
= d u x d u y i ^ ( u x , u y ) B ^ * ( u x , u y ) ,
B ( θ x , θ y ) = rect ( θ θ θ pix ) rect ( θ θ θ pix ) exp [ ( θ 2 + θ 2 ) / ( 2 θ blur 2 ) ] 2 π θ blur 2 ,
I = I avg + η ,
η = Poisson ( I avg ) I avg + σ e   Normal ( 0 , ) ,
C η = η η T η η T = diag ( I avg + σ e 2 ) .
1 T I avg = d θ x d θ y i ( θ x , θ y ) 1 T B ( θ x , θ y ) = γ N ,
SNR = 1 T I avg Tr ( C η ) = γ N γ N + N pix N pix σ e 2 .
θ ̂ ( ) in = ω ( ) T ( α I - I 0 avg ) ,
I 0 avg = I avg ( θ in = 0 ) ,
ω ( ) = θ ( ) B 1 T I 0 avg θ ( ) ,
α = 1 T I 0 avg 1 T I ,
      θ ( ) B 1 T I 0 avg = [ d θ ( ) T ( α avg I avg I 0 avg ) d θ ( ) in | θ ( ) in = 0 ] 1 1 g ( ) T θ ( ) ,
        g ( ) = I avg θ ( ) in | θ ( ) in = 0 .
θ ^ bias ( t ; t + δ ) = ω ( ) T ( t ) [ α ( t ; t + δ ) I 0     avg ( t + δ ) I 0     avg ( t ) ] ,
α ( t ; t + δ ) = 1 T I 0     avg ( t ) 1 T I 0     avg ( t + δ ) .
ϕ ^ bias ( r ) = 0 r d r θ ^ bias ( r ) .
ϕ ^ bias ( 1 ) ( r ) = ϕ ^ bias ( r ) - c 1 Z 1 ( 2 r D ) ,
ϕ ^ bias ( 4 ) ( r ) = ϕ ^ bias ( 1 ) ( r ) c 4 Z 4 ( 2 r D ) ,
c 1 ( 4 ) = 0 2 π d φ 0 D / 2 r d r ϕ ^ bias ( r ) Z 1 ( 4 ) ( 2 r / D ) π D 2 / 4 ,
σ 1 ( 4 ) 2 = 0 2 π d φ 0 D / 2 r d r { ϕ ^ bias ( 4 ) ( r ) } 2 π D 2 / 4 .
σ θ ( ) 2 = ( θ ( ) B 1 T I 0 avg ) 2 var [ θ ( ) T ( α I 0 avg + α η I 0 avg ) ] ,
α = 1 1 + ϵ 1 ϵ ,
ϵ = 1 T η 1 T I 0 avg .
σ θ ( ) 2 = ( θ ( ) B ) 2 [ ξ ( ) SNR 2 ( θ in = 0 ) + q ( ) 2 SNR 2 ( θ in = 0 ) 2 q ( ) 2 1 T I 0 avg ] ,
q ( ) = θ ( ) T I 0 avg 1 T I 0 avg ,
ξ ( ) = Tr [ θ ( ) θ ( ) T C mod ] Tr ( C mod ) = k [ θ ( ) ( k ) ] 2 [ I 0 avg ( k ) + σ e 2 ] k [ I 0 avg ( k ) + σ e 2 ] ,
C mod = C η ( θ in = 0 ) .
( θ in ̂ , θ in ̂ , δ N ̂ ) = arg min ( θ in , θ in , δ N ) J ( θ in , θ in , δ N ) ,
J ( θ in , θ in , δ N ) = y T C mod - 1 y ,
y = I ( I 0 avg + g θ in + g θ in + I 0 avg N δ N ) ,
C mod = C η ( θ in = 0 , θ in = 0 , δ N = 0 ) ,
θ ̄ ( ) in = ω ( ) T ( I I 0 avg ) ,
ω = σ θ 2 C mod - 1 ( g μ I 0 avg ) ,
μ = g T C mod - 1 I 0 avg I 0 avg T C mod - 1 I 0 avg ,
ω = σ θ 2 C mod - 1 g .
σ θ 2 = 1 g T C mod - 1 ( g μ I 0 avg ) ,
σ θ 2 = 1 g T C mod - 1 g .
P SA ( LLT ) = I M SA ( LLT ) M SA ( LLT ) ,
M SA ( LLT ) = ( M SA ( LLT ) T U SA ( LLT ) M SA ( LLT ) ) 1 × M SA ( LLT ) T U SA ( LLT ) .
OPD ¯ SA ( LLT ) = P SA ( LLT ) OPD SA ( LLT ) .
[ D OPD ¯ SA ( LLT ) ] k l = [ OPD ¯ SA ( LLT ) ( k ) OPD ¯ SA ( LLT ) ( l ) ] 2
= [ C OPD ¯ SA ( LLT ) ] k k + [ C OPD ¯ SA ( LLT ) ] l l 2 [ C OPD ¯ SA ( LLT ) ] k l ,
[ C OPD ¯ SA ( LLT ) ] k l = 1 2 [ P SA ( LLT ) D OPD SA ( LLT ) P SA ( LLT ) T ] k l ,
[ D OPD SA ( LLT ) ] k l = [ OPD SA ( LLT ) ( k ) OPD SA ( LLT ) ( l ) ] 2
= 6.88 ( x ( k ) x ( l ) r 0 ( λ 0 ) ) 5 / 3 ( λ 0 2 π ) 2 .

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