#### Table 1.

Expected TMT Optical Element Rigid Body Post-Installation Errorsa

| $d{s}_{z}$ | $d{s}_{x}$, $d{s}_{y}$ | $d{r}_{x}$, $d{r}_{y}$ | $d{r}_{z}$ |

M1 | 0 | 0 | 10 arcsec | 0 |

M2 | 2 mm | 200 μm | 30 arcsec | 0 |

M3 | 0 | 0 | 0 | 0 |

aThe errors are random, uniformly distributed within zero-centered ellipsoids with

$[\mathit{d}{\mathit{s}}_{\mathit{x}},\mathit{d}{\mathit{s}}_{\mathit{y}},\mathit{d}{\mathit{s}}_{\mathit{z}}]=[\mathit{x}\text{-}\mathrm{\text{decenter}},\mathsf{y}\text{-}\mathrm{\text{decenter}},\mathrm{\text{piston}}]$ semi-axes for shift and

$[\mathsf{d}{\mathsf{r}}_{\mathsf{x}},\mathsf{d}{\mathsf{r}}_{\mathsf{y}},\mathsf{d}{\mathsf{r}}_{\mathsf{z}}]=[\mathsf{\text{tip}},\mathsf{\text{tilt}},\mathsf{\text{in}}\text{-}\mathsf{\text{plane rotation}}]$ semi-axes for rotation. Zero values for M2 in-plane rotation and M3

$\mathsf{x}$,

$\mathsf{y}$-decenter correspond to rigid motions with no optical consequences. M3 tip and tilt are used for beam steering and are not considered misalignments. M1 segment

$\mathsf{x}$,

$\mathsf{y}$-decenters, in-plane rotations as well as M3 piston are not controlled and are compensated by other controllable DoFs. M1 segment pistons are not sensed by the WFS, but are controlled by a separate phasing controller and are thus set to zero here

#### Table 2.

Expected TMT Optical Element Surface Warp Post-Installation Errorsa

| M1 | M2 | M3 |

“Finishing error,” Gaussian |

${r}_{0}$ at | | | |

$\lambda =0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ | 1.0 m | 0.325 mb | 0.178 m c |

“Flexure errors”, uniform |

Defocus | 50 nm | 1.0 μm | 1.0 μm |

Astigmatisms | 100 nm | 2.0 μm | 2.0 μm |

Comas | 10 nm | 1.0 μm | 1.0 μm |

Trefoils | 10 nm | 0.2 μm | 0.2 μm |

aThe high spatial frequency “finishing” error is modeled as Gaussian random Kolmogorov turbulence surface; the low order “flexure” errors are modeled as linear combinations of Zernike polynomials with random, uncorrelated, zero-mean, uniformly distributed weights given in the table.

#### Table 3.

Best Achievable System Performance Metrics Computed for the Standard Perturbations Listed in Tables a in Case of Noiseless WFS Measurements and the Turbulence Noise

Metric | I | P | R | TR |

$dw$ rms, nm: M1 | 0 | 76 | 7.4 |

M2 | 0 | 1913 | 23 |

M3 | 0 | 1890 | 32 |

$ds$ rms, μm: M2 | 0 | 90380 | 0.0 |

$dr$ rms, μrad: M1 | 0 | 30 | 0.0 |

M2 | 0 | 94 | 0.0 |

$dW$ rms, nm: M1 | 0 | 10000 | 7.1 |

$dc$ rms, nm: M1 | 0 | 19690 | 4.0 |

OPD rms, nm: | | | | |

axis | 0.2 | 28920 | 83 | 210 |

AO edge | 41 | 28920 | 92 | 217 |

FoV edge | 2297 | 29000 | 2300 | 2300 |

PSSN : axis | 1.00 | 0.00 | 0.985 | 0.981 |

AO edge | 1.00 | 0.00 | 0.986 | 0.982 |

FoV edge | 1.00 | 0.00 | 0.989 | 0.988 |

a100 Monte-Carlo runs averaged. The PSSN values (see Eq.

9) are computed for

$\lambda =1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. The performance metrics notation is described in Section

3.C. The metrics whose values are always zero (see Table

1) are omitted. The OPD rms for ideally aligned telescope are given for comparison. Column title abbreviations: I—ideally aligned telescope, P—perturbed telescope, R—corrected telescope residual errors, TR—corrected telescope residual errors in the presence of turbulence noise.

#### Table 4.

Results of Monte-Carlo Simulation of the Residual Errors due to the Turbulence Noisea

Metric | tRTC | hRTC | RTC |

$dw$ rms, nm: M1 | 9 | 16 | 17 |

M2 | 249 | 0 | 249 |

M3 | 92 | 0 | 130 |

$ds$ rms, μm: M2 | 12.8 | 0 | 12.8 |

$dr$ rms, μrad: M1 | 0.65 | 0 | 0.65 |

M2 | 21 | 0 | 21 |

$dW$ rms, nm: M1 | 33 | 16 | 36 |

$dc$ rms, nm: M1 | 81 | 26 | 85 |

OPD rms, nm: | | | |

axis | 370 | 31 | 371 |

AO edge | 374 | 51 | 375 |

FoV edge | 2345 | 2296 | 2345 |

PSSN: axis | 0.92 | 0.97 | 0.89 |

AO edge | 0.92 | 0.97 | 0.89 |

FoV edge | 0.93 | 0.97 | 0.91 |

a100 random trials averaged. The initial system state is ideal and thus not shown. Column title abbreviations: tRTC—residuals of tomographic part of RTC alone, hRTC—residuals of high order part of RTC alone, RTC—combined RTC residuals

#### Table 5.

Results of Monte-Carlo Simulation of the Aliasing Residual Errorsa

Metric | P | tRTC | hRTC | RTC |

$dw$ rms, nm: M1 | 6 | 42 | 40 | 43 |

M2 | 23 | 130 | 0 | 134 |

M3 | 32 | 140 | 0 | 138 |

$ds$ rms, μm: M2 | 0.007 | 13.2 | 0 | 13.2 |

$dr$ rms, μrad: M1 | 0.004 | 1 | 0 | 1 |

M2 | 0.007 | 18 | 0 | 23 |

$dW$ rms, nm: M1 | 5.5 | 51 | 40 | 45 |

$dc$ rms, nm: M1 | 9 | 140 | 64 | 146 |

OPD rms, nm: | | | | |

axis | 83 | 261 | 77 | 268 |

AO edge | 93 | 262 | 87 | 272 |

FoV edge | 2299 | 2324 | 2298 | 2315 |

PSSN: axis | 0.99 | 0.79 | 0.83 | 0.82 |

AO edge | 0.99 | 0.79 | 0.83 | 0.82 |

FoV edge | 0.99 | 0.81 | 0.85 | 0.83 |

a100 random trials averaged. Column abbreviations are the same as in Tables

3 and

4.

#### Table 6.

Results of Monte-Carlo Simulation for the Full RTC Errors in Case of Noiseless WFS Measurements and the Turbulence Noisea

Metric | P | RTC | nRTC |

$dw$ rms, nm: M1 | 81 | 44 | 48 |

M2 | 8660 | 143 | 282 |

M3 | 2840 | 148 | 192 |

$ds$ rms, μm: M2 | 10400 | 4.1 | 6.5 |

$dr$ rms, μrad: M1 | 31 | 0.12 | 0.15 |

M2 | 128 | 7 | 19 |

$dW$ rms, nm: M1 | 10000 | 46 | 58 |

$dc$ rms, nm: M1 | 19490 | 148 | 170 |

OPD rms, nm: | | | |

axis | 26400 | 295 | 448 |

AO edge | 26400 | 300 | 448 |

FoV edge | 26800 | 2315 | 2392 |

PSSN: axis | 0.0 | 0.80 | 0.75 |

AO edge | 0.0 | 0.80 | 0.75 |

FoV edge | 0.0 | 0.82 | 0.77 |

a100 random trials averaged. Column abbreviations: P—perturbed telescope, RTC—the RTC residuals in case of noiseless WFS measurements, nRTC—the RTC residuals in case of WFS turbulence noise.

#### Table 7.

Results of Monte-Carlo Simulation for the Errors of the Alignment Sequence Involving RTC Followed by M1 Segment Controller and the M1 Phasera

Metric | P | R | nR |

$dw$ rms, nm: M1 | 81 | 13 | 15 |

M2 | 8660 | 137 | 273 |

M3 | 2840 | 153 | 204 |

$ds$ rms, nm: M2 | 10400 | 4.4 | 5.9 |

$dr$ rms, μrad: M1 | 31 | 0.6 | 1.9 |

M2 | 128 | 6.1 | 19.7 |

$dW$ rms, nm: M1 | 10000 | 149 | 265 |

$dc$ rms, nm: M1 | 19490 | 22 | 26 |

OPD rms, nm: | | | |

axis | 26400 | 83 | 307 |

AO edge | 26400 | 120 | 318 |

FoV edge | 26800 | 2326 | 2363 |

PSSN: axis | 0.0 | 0.99 | 0.97 |

AO edge | 0.0 | 0.98 | 0.96 |

FoV edge | 0.0 | 0.97 | 0.96 |

TMT requirement | 0.9 |

a100 random trials averaged. Column abbreviations: P—perturbed telescope, R—the RTC

$+$follow-ups sequence residuals in case of noiseless WFS measurements, nR—the RTC

$+$follow-ups sequence residuals in case of WFS turbulence noise.