A 10 m telescope observes at through air with m at 500 nm and a wind speed of 15 m s…

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 10 m telescope observes at

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

through air with

r_0 = 0.20

m at 500 nm and a wind speed of 15 m s

^{-1}

across the pupil. Its adaptive optics system has a deformable mirror with actuator spacing

d = 0.50

m projected on the pupil, a closed-loop bandwidth of 40 Hz, and an on-axis guide star giving a subaperture signal-to-noise ratio of 6. Use the fitting-error term

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

, the temporal term

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

with

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

, and the noise term

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

, with anisoplanatism zero on axis;

r_0

scales as

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

. ((a)) Predict the Strehl ratio at 1.65

\mu

m from this error budget. ((b)) Using the diffraction-limited angular resolution of a lens acting as a Fourier transformer,

\theta_{\mathrm{diff}} = 1.22\lambda/D

, compute the Rayleigh-limited angular resolution of the unaberrated 10 m aperture at 1.65

\mu

m, in milliarcseconds. ((c)) State, with a one-sentence reason drawn from the pupil-sum picture of a low-order wavefront error, whether the corrected image's core width should be close to this diffraction-limited value even though the Strehl ratio found in part (a) is below 1.

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