TMT-AGE: numerical simulation of a new tomographic
reconstruction method for wide FoR MOAO
Yoshito H. Onoa,b , Akiyama Masayukia , Shin Oyaba Astronomical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi

980-8578, Japan;
ohoku Place, Hilo, Hawaii 96720, USA

b Subaru Telescope, NAOJ, 650 North A

ABSTRACT
We are conducting a concept study on a wide ﬁeld of regard (FoR) Multi-Object Adaptive Optics (MOAO) system
for Thirty Meter Telescope (TMT-AGE: TMT-Analyzer for Galaxies in the Early universe). The main sciencetarget of TMT-AGE is high-redshift galaxies. Considering the small number density of high-redshift galaxies,
enlarging the FoR of an MOAO system up to around 100 is critical. In order to increase the FoR of an MOAO
system, we propose a new tomographic reconstruction method. In the new method, we use atmospheric windproﬁles and WFS measurements at previous time steps to increase the number of virtual measurement points ofatmospheric turbulence layers for tomographic reconstruction. We present the results of numerical simulations
with the new tomography method. The simulations show the new method can reduce the tomographic error ina wide FoR.
Keywords: TMT, Multi-Object Adaptive Optics, tomography

1. INTRODUCTION
Multi-Object Adaptive Optics (MOAO) is an approach of wide-ﬁeld Adaptive Optics (AO). MOAO applies AO
corrections for multiple objects in a ﬁeld of regard (FoR) and realize high spatial resolution observations of
the multiple objects. For Thirty Meter Telescope (TMT), an instrument incorporating MOAO and multiple
integral ﬁeld units is proposed to reveal properties of high-redshift galaxies, called near-InfraRed Multi-Object
Spectrograph (IRMOS) [1]. One of key parameters of an MOAO system is the size of a FoR. Since very highredshift galaxies at redshift larger than 5 have small number density, wide FoR of around 100 diameter is necessary
to utilize the high multiplicity of an MOAO system for such targets. However, the MOAO system of IRMOS isconsidered capability over a 50 FoR. Therefore, we are conducting a concept study on a wide-ﬁeld MOAO system
for TMT, which has a wide FoR of around 100 diameter (TMT-AGE: TMT-Analyzer for Galaxies in the Early
universe, [2]).
The key technique for enlarging the FoR of an MOAO system is a tomographic reconstruction which estimatesthe 3-dimensional atmospheric wavefront disturbance using information from multiple guide stars (GS). The
size of a FoR and the accuracy of a tomographic reconstruction of an MOAO system depend strongly on a
conﬁguration of multiple GSs. Wider GS separation can realize wider FoR, but cause large tomographic error
because the area of the gap regions of the high-altitude atmospheric turbulence that is covered only by one or
no GS meta-pupil increases with the separation (Figure 1). In the case of laser guide stars (LGS), the inﬂuenceis worse due to the cone eﬀect. Therefore increasing the FoR of an MOAO system with keeping the tomographic
reconstruction accuracy is diﬃcult. In this paper, we present a new tomographic reconstruction method toincrease the FoR of an MOAO systems.
The new method uses atmospheric wind proﬁles and WFS measurements at previous time steps to increase
the number of measurement points of atmospheric turbulence layers for tomographic reconstruction. Thanks
to the displacement of the previous measurements due to winds, we can ﬁll in the gap regions with previous
measurements and expanding the FoR of an MOAO system without increasing the tomographic error. We
Further author information: (Send correspondence to Yshito H. Ono)
Yoshito H. Ono: E-mail: ono@naoj.org, Telephone:+1-808-934-5910

also present a method to estimate wind speed and direction at each altitude, which is necessary for the new
tomographic reconstruction method. Through numerical simulations of an MOAO system for 30m aperture
telescope at Mauna Kea, we demonstrate that the new method reduces a tomographic error compared to aclassical tomographic method at the same conﬁguration of LGS.

F o R 	
LGS	

LGS	

LGS	

Un-sensed Region	

Atmospheric	
Turbulence	Less-sensed Region	
Telescope	
Figure 1. Schematic geometry of atmospheric turbulence layers and LGS meta-pupils. There are the gap regions in thehigh-altitude atmospheric turbulence that is covered only by one or no GS meta-pupil.

2. METHOD
In this section, we explain a classical tomographic method at ﬁrst. Then, we present the new tomographic
method and the method to estimate wind speed and direction at each altitude.

2.1 Measurement Model
Using Shack-Hartmann Wave-Front Sensor (WFS), the linear equation between the incoming wavefront wj fromjth LGS of Nlgs and the gradients measured by jth WFS observing the jth LGS at time t is given bysj (t) = Gj wj (t) + ηj (t),(1)

where Gj is a gradient operator of Fried geometry for jth WFS, and ηj (t) is a measurement noise vector.
Assuming that the turbulent atmosphere consists of thin layers in total Nlayer located in diﬀerent altitudes hl ,the phase wj of light from jth LGS at time t is distorted as∑

Nlayerwj (t) =Pl,j φl (t) = Pj φ(t).

(2)

l=1

Here, Pl,j is a projection matrix which extracts the wavefront within the jth LGS meta-pupil from the lth
atmospheric turbulence φl at altitude hl with a bilinear interpolation; φ and Pj are a vector and matrix
concatenating the atmospheric turbulences of Nlayer and projection matrices, respectively. Combining Eq.(1)and Eq.(2) and considering the all LGSs, the equation connecting the measurements of the all WFSs at time t
through the all atmospheric turbulences at time t can be represented as follows:s(t) = GP φ(t) + η(t).

(3)

where s and η are expansion vector of sj and ηj respectively, and G is a block diagonal matrix, which is
G = diag(Gj , · · · , GNlgs ).

2.2 Tomographic Reconstruction
We use Minimum Variance Reconstructor[3] (MVR) to estimate the atmospheric turburences. The MVR provides
an optimal solution for Eq.(3) using the statistics of the atmospheric turbulence and the WFS measurement noise.
We can estimate φ by minimizing
E(||φ − φ̂||2 ).
(4)
The hat symbol indicates estimated quantities. The solution to this problem is given by(P T GT GP + σ 2 LT L)φ̂(t) = P T GT s(t),(5)

2where L is proportional to a discretization of the Laplacian operator and σ is the variance of WFSs measurements. Eq.(5) is of the form Ax̂ = b and the all matrices included in Eq.(5) are sparse. Such problems canbe solved eﬃciently by using iterative methods[4][5]. In this paper, we use a Conjugate Gradient (CG). In thefollowing, we call Eq.(5) a classical method.
After the tomographic reconstruction, the command (or the shape) of each Deformable Mirror (DM) in
direction of a science target is calculated with the estimated atmospheric turbulences. In this simulation, theDM wavefronts ŵdm are estimated with a bilinear interpolation.
ŵdm,k = Pk φ̂(t)(k = 1, · · · , Ntarget )(6)

2.3 New Tomographic Reconstruction
Under the assumption of Taylor’s frozen-ﬂow hypothesis, evolution of atmospheric turbulences can be describedas the shift of each layer due to the wind v = (vx , vy ) at each altitude. Introducing a projection matrix Pl,j (∆t)
which takes account of the displacements of jth LGS meta-pupil at altitude hl due to a wind vl within a timeinterval ∆t, the relation between measured gradients of the all WFSs at time (t − ∆t) and the all atmosphericturbulences at time t is given bys(t − ∆t) = GP (∆t)φ(t) + η(t − ∆t).

(7)

Concatenating Eq.(7) with Eq.(3) and using MVR, we can estimate the atmospheric turbulence with measurements taken both current and previous time step as follows.
(P 0T G0T G0 P 0 + σ 2 LT L)φ̂(t) = P 0T G0T s0 ,
0

0

0

(8)

where s = [s (t) s (t − ∆t)] , G = diag(G, G), and P = [P P (∆t)]. The DM wavefronts is estimatedwith Eq.(6). Wind speeds and directions at each altitude are necessary in order to make P 0 (∆t). Next, we
present how to estimate wind information.
T

T

T

T

T

2.4 Estimation of Wind Speeds and Directions
To date, two methods for estimate wind speeds and directions of multiple atmospheric layers. One is using
spatio-temporal cross correlation of measured WFS data[6], which was developed from the SLODAR method[7].
In the SLODAR method, the peaks of spatial cross correlation for two WFSs data with no-time delay givethe information of turbulence altitudes. Performing spatio-temporal cross correlation with time-delayed WFSs
data, each correlation peak on the correlation map moves according to the wind speed and direction. We canestimate the wind speeds and directions by tracking this movement of each peak. However, it is diﬃcult to detectand track multiple peaks on a correlation map. The other way is using fourier technique, called Fourier Wind
Identiﬁcation (WFI)[8], but this method doesn’t evaluate turbulence altitudes.
In this paper, we propose a method to estimate wind speed and wind direction at each altitude using the
turbulent atmosphere estimated by a tomographic reconstruction. Tomographic reconstruction provides thephase map at each altitude. Using a projection matrix Pl,j and a gradient operator Gj , the estimated slope
map at each altitude in each LGS direction can be calculated.
ŝl,j (t) = GPl,j φ̂(t),(9)

ŝl,j (t) is the estimated slope at altitude of lth atmospheric screen in direction of jth LGS. Performing temporalcorrelation to the estimated slopes ŝl,j and tracking the peaks give the wind direction and speed at altitude ofhl . Since ŝl,j includes only one atmospheric layer component, there is only one peak expected on the correlation
map ideally and tracking the peak is easier than the above like method.

3. SIMULATIONThe setup of our numerical simulations is as follows. The diameter of the circular aperture is set to 30m. An
atmospheric model of seven layers with r0 = 0.156m and L0 = 30m is adopted, which is typical condition at the2summit of MaunaKea and used in [9]. The altitudes, CN
fraction, and wind speeds are listed in Table 1. Eachatmospheric layer is assumed 100% frozen ﬂow layer. The wind directions are random and the wind speeds do
not change with time. Shack-Hartmann WFSs are used with 0.5m subaperture and RMS read error of σe = 5
electrons. The brightness of LGSs corresponds to 700 electrons per subaperture. Tomographic reconstruction
is performed with two conﬁgurations of eight LGSs shown in Figure 2: one conﬁguration is the nominal TMTIRMOS design(Open green star), the other is wider conﬁguration(Filled red star).
2
Although the altitudes and CN
values of atmospheric layers must be measured on-sky during observation,for example with SLODAR, we assume the information is known perfectly in this simulation. The time-delay
due to calculation time and WFS exposure are not considered.

Altitude [km]

0

0.5

1

2

4

8

16

2
fractionCN

0.59600.09630.03250.03720.08690.06840.0826Wind speed [m/s]

7.817

8.690

9.738

12.35718.95324.215

5.783

2
Table 1. Altitude, CN

fraction, and wind speeds of atmospheric layers.

300

★

200

✩
Y (arcsec)

100

★

✩★ ✩

0

★

★
✩ ✩

✩
✩ ★ ✩

-100
-200

★

★

-300
-300

-200

-100
0
100
X (arcsec)

200

300

Figure 2. LGS configuration. Open green star: nominal TMT-IRMO configuration which is optimize for a 50 FoR. Filled
red star: wider configuration.

4. RESULTS
4.1 Estimation of Wind Speeds and Directions
Wind estimation is performed using 10s simulated WFSs data. The estimation error of wind speeds are less than
1 [m/s] for all the layers, except 2.3 [m/s] at 8km. It is due to the highest wind speeds of 24[m/s] at the layer
in this simulation. The errors of wind directions are very small for all altitude, which are less than 1 degree.
Note that this is a result with 100% frozen ﬂow assumption and perfect information of atmospheric layer
altitudes. Indeed, the time scale of frozen ﬂow is limited and the correlation peak becomes fainter with time[6]. This aﬀects the accuracy of the detection of the correlation peaks. Also, if a tomographic reconstruction
for a layer is performed at middle of two real atmospheric layers, the estimate can mix the two atmosphericlayers. In this case, two peaks corresponding to the atmospheric layers may appear on a correlation map andidentifying multiple peak can be diﬃcult. We will evaluate the accuracy of the wind estimation method with theuncertainties of the altitudes of atmospheric layers in a future work.

4.2 Tomographic Reconstruction
The simulation results are presented in Figure 3. We perform the classical tomography method for the two
conﬁgurations of LGSs and the new tomography method for only the wider conﬁgurations. For the new method,
the measurements at 0.1s previous time step are used, ∆t = 0.1s, and the estimated wind speeds and directions
are also used. The accuracy of tomographic reconstructions is evaluated with residual wavefront error(WFE).
Using the classical method for the nominal TMT-IRMOS conﬁguration(open green circle in Figure 3), the WFE
within the central 50 is less than 200nm but the WFE of the outside of the central 50 FoR increases steeply withradius much higher than 300nm, because there is no LGS at the outer region of the central 50 . In the case of thewide conﬁguration(ﬁlled red circle in Figure 3), the WFE of the outer region of FoR is smaller than that of the
TMT-IRMOS conﬁguration but the central WFE increases up to 250nm due to more gap regions which are lesscovered with LGSs meta-pupils.
With the new tomographic reconstruction method for the wider conﬁguration (ﬁlled red square in Figure 3),
the WFE within the central 50 FoR reduces to less than 200nm and the WFE in 80 FoR is less than 250nm. It
is because the wider conﬁguration has the gap regions which are un-sensed or less-sensed with LGS meta-pupils
and the new method can ﬁll the gap regions with measurements at previous time step. In this simulation, theestimation errors of both speeds and directions do not aﬀect on the results of tomography. The displacement
error within 0.1s due to wind estimation error is less than 25cm. Since this is half of DM segment size, atomographic error due to the error of wind estimation is smaller than the ﬁtting error. These results show the
new tomographic reconstruction can increase the accuracy of reconstruction within the FoR and is beneﬁcial toexpand the FoR of MOAO.
400

Residual WFE [nm]TMT-IRMOS configuration, ClassicalWider Configuration, Classical350
Wider Configuration, New
300
250
200
150
100

0

1

2

3

4

Distance from the center of FoR [arcmin]

5

Figure 3. The residual WFE as a function of field radius. Open green and filled red symbol indicate results with theTMT-IRMOS configuration and the wider configuration, respectively. The circle and square show the results with classicaland new tomographic reconstruction method, respectively.

5. CONCLUSIONS AND FUTURE WORK
We propose a new tomographic reconstruction method using measurements at previous time steps and windinformation to increase the FoR of MOAO systems and evaluate the method with numerical simulations. We
also introduce a method to estimate wind speeds and directions at each altitude. The wind proﬁles can be
estimated with suﬃcient accuracy under assumption of 100% frozen ﬂow and perfect knowledge of altitudes ofatmospheric turbulences. With estimate of wind speeds and directions, the new tomographic reconstruction
method reduces WFE to less than 250nm for the wider LGS conﬁguration in 80 FoR.

A future work is to perform a SLODAR simulation to evaluate the estimation accuracy of atmosphericturbulence altitudes. Then, we will be able to evaluate the wind estimation method in a more realistic situation.
Further, we will try these new methods with on-sky WFS data which were taken by Subaru Telescope withRAVEN[10], which is an MOAO technology and science demonstrator developed by University of Victoria andHIA.

ACKNOWLEDGMENTS
This work is supported by NAOJ Japanese TMT oﬃce strategic research and development funding of FY2012,
FY2013, and FY2014. M.A. is supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 23740140
and Grant-in-Aid for Scientiﬁc Research (B) 26287027.

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