An adaptive optics system is being designed for an telescope working at (H band)

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Fourier optics and image formation · Fourier optics and image formation

What an Instrument Can See, Ch Keeping the phase: holography, wavefront modulation and adaptive optics

Problem

An adaptive optics system is being designed for an

8\ \mathrm{m}

telescope working at

\lambda=1.65\ \mu\mathrm{m}

(H band), at a site with Fried parameter

r_0=0.15\ \mathrm{m}

500\ \mathrm{nm}

and wind speed

v=12\ \mathrm{m\,s^{-1}}

. A bright on-axis guide star fixes the subaperture signal-to-noise ratio at

\mathrm{SNR}=10

, and the target sits on the guide star, so anisoplanatism is zero. The design target is a Strehl ratio of at least

S=0.80

at this wavelength. Use the error budget

\sigma_\phi^2=\sigma^2_{\mathrm{fit}}+\sigma^2_{\mathrm{temp}}+\sigma^2_{\mathrm{noise}}+\sigma^2_{\mathrm{aniso}}

with

\sigma^2_{\mathrm{fit}}=0.34(d/r_0)^{5/3}

\sigma^2_{\mathrm{temp}}=(\nu_{\mathrm{G}}/\nu_{3\mathrm{dB}})^{5/3}

\nu_{\mathrm{G}}=0.43v/r_0

and

\sigma^2_{\mathrm{noise}}=1/\mathrm{SNR}^2

, all evaluated at the science wavelength via

r_0(\lambda)=r_0(500\ \mathrm{nm})(\lambda/500\ \mathrm{nm})^{6/5}

. Two design freedoms are still open once the guide star fixes the noise term and the target fixes the total: the actuator spacing

d

and the closed-loop bandwidth

\nu_{3\mathrm{dB}}

. Split whatever variance is left, once the noise term is subtracted from the target total, equally between the fitting and the temporal terms; nothing in the problem favours correcting one of these two faster than the other, so an equal split is the natural design choice. ((a)) Find

r_0

1.65\ \mu\mathrm{m}

. ((b)) Find the total residual variance

\sigma_\phi^2

the target Strehl allows, and the noise term's contribution to it. State how much variance is left for the fitting and temporal terms together, and each one's equal share. ((c)) Using the fitting term's share, find the actuator spacing

d

the deformable mirror needs, and hence the number of actuators across the pupil,

N\approx(D/d)^2\pi/4

. ((d)) Using the temporal term's share, find the Greenwood frequency and hence the minimum closed-loop bandwidth

\nu_{3\mathrm{dB}}

that meets it.

Browse the problem library