A hologram of an on-axis point object is to be recorded with a diode-pumped Nd:YAG…

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 hologram of an on-axis point object is to be recorded with a diode-pumped Nd:YAG laser at λ1=1064 nm\lambda_1 = 1064\ \mathrm{nm}, using a plane reference wave tilted at θR\theta_R to the axis (the off-axis geometry of this chapter). The object's spatial-frequency content extends up to νmax⁡=60\nu_{\max} = 60 cycles per millimetre. Once developed, the plate will be viewed by illuminating it, at the same angle of incidence θR\theta_R, with the same laser's frequency-doubled output at λ2=532 nm\lambda_2 = 532\ \mathrm{nm}; the laboratory owns only this one laser system, so the illuminating angle cannot be re-aimed for viewing. ((a)) Recording at λ1\lambda_1 with a reference at angle θR\theta_R and replaying at the same angle but at λ2\lambda_2, show that the reconstructed object beam now leaves the plate at an angle θ′\theta' to the axis, with sin⁡θ′=sin⁡θR(1−λ2λ1),\sin\theta' = \sin\theta_R\left(1-\frac{\lambda_2}{\lambda_1}\right), by tracking the linear phase the replay beam adds to the term that, at matched wavelengths, reconstructs OO exactly. Confirm that θ′=0\theta'=0 when λ2=λ1\lambda_2=\lambda_1, recovering the on-axis reconstruction already met in this chapter. ((b)) Find the smallest reference angle θR,sep\theta_{R,\mathrm{sep}} that satisfies the usual separation condition νcar≥3νmax⁡\nu_{\mathrm{car}}\ge3\nu_{\max} at the recording wavelength λ1\lambda_1. ((c)) Using part 1, find the angular separation Δ=θR−θ′\Delta=\theta_R-\theta' between the reconstructed object beam and the zero-order beam (which continues along the replay beam's own direction, at θR\theta_R) when the hologram is recorded at θR,sep\theta_{R,\mathrm{sep}} from part 2 and later replayed at λ2\lambda_2. A beam block placed after the plate needs at least 6∘6^\circ of angular clearance to intercept the zero order without clipping the wanted beam. Does θR,sep\theta_{R,\mathrm{sep}} provide it? ((d)) The optical mount is calibrated only in whole degrees. Check whether θR=15∘\theta_R=15^\circ satisfies both the separation condition at λ1\lambda_1 and the 6∘6^\circ replay clearance, and state whether it is an acceptable design choice.

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