A high-contrast imaging bench on a 4 m telescope corrects wavefront errors in two…

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

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

Problem

A high-contrast imaging bench on a 4 m telescope corrects wavefront errors in two stages: a deformable mirror removes low-order atmospheric turbulence, and a phase-only liquid-crystal spatial light modulator (SLM), pixel pitch

p = 8\ \mu\mathrm{m}

, applies a computer-generated correction for a static aberration left in the relay optics. The static aberration was mapped before installation with a Michelson interferometer illuminated by a filtered lamp of central wavelength

\lambda_0 = 550\ \mathrm{nm}

and spectral width

\Delta\lambda = 2\ \mathrm{nm}

(FWHM); the coherence length of a source of this width is

\ell_c \approx \lambda_0^2/\Delta\lambda

. The mapped aberration is a pure defocus,

W(r) = W_{\max}(r/a)^2

, with zero phase at the centre and a total phase excursion at

\lambda_0

14\pi

rad at the edge of the SLM aperture. The SLM is illuminated over a clear aperture of radius

a = 100

pixels. ((a)) Find the coherence length of the lamp, and state the greatest arm-length difference the interferometer could tolerate and still show fringes. ((b)) Find the local spatial frequency (in cycles per millimetre) that the SLM must reproduce at the edge of the defocus pattern (

r=a

), and check it against the modulator's addressable limit

1/(2p)

. ((c)) With the static term corrected to a residual variance of

0.01\ \mathrm{rad}^2

(left over from the SLM's finite phase quantisation) and the deformable mirror leaving an atmospheric residual of

\sigma_{\mathrm{atm}}^2 = 0.30\ \mathrm{rad}^2

, both quoted at the

2.2\ \mu\mathrm{m}

science wavelength, predict the Strehl ratio of the combined system.

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