A near-infrared adaptive optics system is being designed for a 4.2 m telescope at a…

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

A near-infrared adaptive optics system is being designed for a 4.2 m telescope at a site with

r_0 = 0.10

m at 500 nm and a wind speed of

6\ \mathrm{m\,s^{-1}}

; the science camera works at 1.2

\mu

m. Separately, the instrument team wants to reuse a spare liquid-crystal spatial light modulator, built to give a full

2\pi

phase stroke at 633 nm, with 1600 by 1200 pixels of pitch 12.5

\mu

m, as a static beam-shaping element at 1550 nm to flood-illuminate the wavefront sensor's lenslet array with a top-hat beam of diameter 8 mm. ((a)) Using the

\lambda^{6/5}

scaling of the Fried parameter, find

r_0

at 1.2

\mu

m. ((b)) Estimate the number of deformable-mirror actuators needed across the pupil, using

N \approx (D/r_0)^2\times\pi/4

. ((c)) Find the Greenwood frequency for this wind speed and

r_0

, then find the minimum closed-loop bandwidth that keeps the temporal error variance below

0.05\ \mathrm{rad^2}

, taking

\sigma_t^2 = (\nu_G/f)^{5/3}

. ((d)) For the calibration modulator, find the number of pixels spanning the 8 mm beam, then find the actual peak phase shift (in radians) the device provides at 1550 nm, given that it was built to give exactly

2\pi

at 633 nm and its retardance for a fixed layer scales as

1/\lambda

. State whether the device can still realise an arbitrary phase pattern (a full

2\pi

wrap) at 1550 nm.

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