3 Wavefront residual error for adaptive optics system

Most adaptive optics compensation systems apply the conjugate surface of the phase measured by WFS with a pair of active elements consisting of a high-speed TM and a DM. TM is usually used to correct the tilt component of the wavefront distortions induced by atmospheric turbulence. The figure (tilt-removed) component of the wavefront perturbation is compensated by DM. In the practical AOS, the compensation is not perfect because of the individual errors such as the uncompensated turbulence-induced error due to the finite bandwidth of the servo system and the measurement noise.

3.1 Figure residual error

The main figure residual errors include: (1) uncompensated turbulence-induced figure error due to the bandwidth of the figure compensation servo system; (2) closed-loop figure noise error due to the measurement error of the wavefront sensor; (3) fitting error due to the finite corrector elements; (4) anisoplanatic error due to the angular anisoplanatism and the focal anisoplanatism.

3.1.1 Uncompensated turbulence-induced figure error

The uncompensated turbulence-induced figure error depends on the temporal power spectrum of the figure (tilt-removed) component of the turbulence $F_{\rm fig}(f)$ and the servo control transfer function $H_{\rm c}(jf)$:

$$\sigma_{\rm fig\_tur}^2=\int_0^{\infty}F_{\rm fig}(f)\vert 1-H_{\rm c}(jf)\vert^2{\rm d}f.$$ (5)

For Kolmogorov turbulence, the power spectrum of the figure (tilt-removed) component of the turbulence with the unit of rad²/Hz, developed by Greenwood and Fried (Greenwood & Fried 1976), has the following form:

$$F_{\rm fig}(f)=$$

$$\left\{ \begin{array}{c} 0.132sec(\xi)k_0^2D^4\mu_0^{12/5}\nu_{5/... ...{5/3}f^{-8/3}, f\geq0.705 D^{-1}\mu_0^{-3/5}\nu_{5/3}^{3/5}. \end{array}\right.$$ (6)

In practical AOS, there exists a time delay between the wavefront sensing and wavefront correcting. Considering the effect of the time delay on servo control loop, the closed-loop transfer function of AOS can be expressed as (Rao et al. 2000)

$$H_{\rm c}(jf)=\frac{e^{-j2\pi f\tau}}{1+jf\big /f_{\rm 3dB}}$$ (7)

where $f_{\rm 3dB}$ is the closed-loop -3dB bandwidth and $\tau$ is the time delay.

For high-bandwidth figure compensation, the uncompensated turbulence-induced figure error at the imaging wavelength $\lambda$ can be obtained with the high-frequency form of the figure spectrum over the entire frequency range, so that (Li et al. 2000):

$$\sigma_{\rm fig\_tur}^2 \approx [f_{\rm G}(\lambda)/f_{\rm 3dB}]^{5/3} [1+2.2156(2\pi\tau f_{\rm 3dB})^{2/3}$$

$$+1.3292(2\pi\tau f_{\rm 3dB})^{5/3}]$$

$$(\lambda_0/\lambda)^2[f_{\rm G}(\lambda_0)/f_{\rm 3dB}]^{5/3}[1+2.2156(2\pi\tau f_{\rm 3dB})^{2/3}$$ =

$$+1.3292(2\pi\tau f_{\rm 3dB})^{5/3}].$$ (8)

3.1.2 Closed-loop figure noise error

In AOS, the noise resulting from WFS mainly consists of the photon shot noise due to the finite light intensity and the read-out noise of the detector such as a CCD. Assuming the noise is white, the closed-loop figure noise error can be obtained as (Rao et al. 2000):

$$\sigma_{\rm fig\_noise}^2 \int_0^{f_{\rm s}/2}F_{\rm fn}\vert H_{\rm c}(jf)\vert^2{\rm d}f$$ =

$$F_{\rm fn}f_{\rm 3dB}~{\rm arctg}[f_{\rm s}/(2f_{\rm 3dB})]$$ = (9)

where $F_{\rm fn}$ is the noise power spectrum and $f_{\rm s}$ is the sampling frequency of the system.

Note that the wavefront measurement noise $\sigma_{\rm fn}^2$ relates the noise power spectrum to:

$$\sigma_{\rm fn}^2=F_{\rm fn} \cdot f_{\rm s}/2.$$ (10)

Substituting Eq. (10) into Eq. (9), the closed-loop figure noise error at the imaging observation wavelength $\lambda$ can be expressed as

$$\sigma_{\rm fig\_noise}^2=(\lambda_{\rm W}/\lambda)^2\pi f_{\rm 3dB}\sigma_{\rm fn}^2/f_{\rm s}$$ (11)

where $\lambda_{\rm W}$ is the detecting wavelength of WFS and ${\rm arctg}[f_{\rm s}/(2f_{\rm 3dB})]\approx\pi/2$ is used.

3.1.3 Fitting error

The uncorrected phase error due to the figure (tilt-removed) component of turbulence at the imaging wavelength $\lambda$, namely figure fitting error, is conveniently expressed in terms of the ratio of the subaperture spacing and the turbulence coherence length (Hardy 1998):

$$\sigma_{\rm fig\_fit}^2=0.34[d_{\rm s}/r_0(\lambda)]^{5/3}=0.34(\lambda_0/\lambda)^2[d_{\rm s}/r_0(\lambda_0)]^{5/3}$$ (12)

where $d_{\rm s}$ is the subaperture spacing.

3.1.4 Anisoplanatic error

When the beacon of AOS and the target object do not lie in the same direction, the error due to angular anisoplanatism at the imaging wavelength $\lambda$ can be expressed as (Fried 1982)

$$\sigma_{\rm fig\_angle}^2=(\theta/\theta_0(\lambda))^{5/3}=(\lambda_0/\lambda)^2(\theta/\theta_0(\lambda_0))^{5/3}$$ (13)

where $\theta$ is the angle between the beacon and the target object.

Furthermore, because the atmospheric turbulence is only partially detected by the beacon light, the use of laser beacons in AOS produces the error due to focal anisoplanatism, which may be expressed as (Tyler 1994)

$$\sigma_{\rm fig\_focus}^2=(D/d_0(\lambda))^{5/3}=(\lambda_0/\lambda)^2(D/d_0(\lambda_0))^{5/3},$$ (14)

where D is the diameter of the telescope. For the beacon with height H, the value of d₀ implied by Eq. (14) is

$$d_0(\lambda_0) \left\{k_0^2[0.057\mu_0^+(H)+0.500\mu_{5/3}^-(H)/H^{5/3}\right.$$ =

$$\left.-0.452\mu_2^-(H)/H^2]\right\}^{-3/5},$$ (15)

where $\mu_m^+(H)=\int_H^{\infty}{\rm d}h~sec^{m+1}(\xi)C_n^2(h)h^m$ and $\mu_m^-(H)=\int_0^{H}{\rm d}h~sec^{m+1}(\xi)C_n^2(h)h^m$ and are the upper and lower turbulence moments at height H respectively.

3.1.5 Total figure residual error

The total figure residual error of AOS with the unit of phase rad² at the imaging wavelength $\lambda$ may be expressed as the sum of individual errors in the form

$$\sigma_{\rm fig}^2 \sigma_{\rm fig\_tur}^2+\sigma_{\rm fig\_noise}^2+ \sigma_{\rm fig\_fit}^2+\sigma_{\rm fig\_anglet}^2+\sigma_{\rm fig\_focus}^2$$ =

$$(\lambda_0/\lambda)^2\left\{[f_{\rm G}(\lambda_0)/ f_{\rm 3dB}]^{5/3}[ 1+2.2156(2\pi\tau f_{\rm 3dB})^{2/3}\right.$$ =

$$+1.3292(2\pi\tau f_{\rm 3dB})^{5/3}]$$

$$+(\lambda_{\rm W}/\lambda_0)^2\pi f_{\rm 3dB} \sigma_{\rm fn}^2/f_{\rm s}+0.34[d_{\rm s}/r_0(\lambda_0)]^{5/3}$$

$$\left. +(\theta/\theta_0(\lambda_0))^{5/3}+(D/d_0(\lambda_0))^{5/3}\right\}\cdot$$ (16)

3.2 Tilt residual error

In AOS, the tilt component of turbulence is often measure by quadrant detectors and centroid trackers. This tilt belongs to G-tilt, namely average gradient over the pupil. The main errors involved in compensating the tilt component of turbulence are: (1) uncompensated turbulence-induced tilt error due to the bandwidth of the tilt compensation servo system; (2) closed-loop tilt noise error due to the measurement error of the tracking sensor.

According to the tracking frequency $f_{\rm T}$ defined by Tyler (Tyler 1994), see Eq. (4), the uncompensated turbulence-induced tilt error at the imaging wavelength $\lambda$ can be expressed as

$$\sigma_{\rm tilt\_tur}^2=\left[f_{\rm T}(\lambda)/f_{\rm c}\right]^2(\lambda/D)^2=\left[f_{\rm T}(\lambda_0)/f_{\rm c}\right]^2(\lambda_0/D)^2$$ (17)

where $f_{\rm c}$ is the closed-loop -3dB bandwidth of the tracking servo loop.

Similar to the closed-loop figure noise error, the closed-loop tilt noise error at the imaging wavelength $\lambda$ relates the tilt measurement error $\sigma_{\rm tn}^2$, the sampling frequency of the tracking system $f_{\rm st}$ and the closed-loop -3dB bandwidth of the tracking servo loop to

$$\sigma_{\rm tilt\_noise}^2=(\lambda_{\rm T}/D)^2\pi f_{\rm c}\sigma_{\rm tn}^2/f_{\rm st}$$ (18)

where $\lambda_{\rm T}$ is the detecting wavelength of the tracking sensor. The tilt measurement error $\sigma_{\rm tn}^2$ is scaled as the unit of $(\lambda_{\rm T}/D)^2$.

The total tilt residual error with the unit of tilt rad² at the imaging wavelength $\lambda$ is given by the sum of the uncompensated turbulence-induced tilt error and the closed-loop tilt noise error,

$$\sigma_{\rm tilt}^2 \sigma_{\rm tilt\_tur}^2+\sigma_{\rm tilt\_noise}^2$$ =

$$(\lambda_0/D)^2\left\{[f_{\rm T}(\lambda_0)/f_{\rm c}]^2+(\lambda_{\rm T}/\lambda_0)^2\pi f_{\rm c}\sigma_{\rm tn}^2/f_{\rm st}\right\}\cdot$$ = (19)