A phase-only liquid-crystal spatial light modulator (operating wavelength

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

\lambda = 633\ \mathrm{nm}

, pixel pitch

p = 8\ \mu\mathrm{m}

) sits after an adaptive-optics loop that leaves a residual fitting-error wavefront of rms

\sigma_1 = 0.06

waves. Downstream, a Pockels cell of half-wave voltage

V_\pi = 1.8\ \mathrm{kV}

, with electrodes shaped so that its phase shift grows as the square of the distance from the axis, is driven at

V = 300\ \mathrm{V}

to add a controlled defocus term. The electro-optic phase shift at the edge of the beam is

\Delta\phi = \pi V / V_\pi = k_0 W_0

, with

k_0 = 2\pi/\lambda

and

W_0

the peak wavefront amplitude of the defocus term; a pure Zernike defocus term of peak amplitude

W_0

has rms wavefront error

\sigma_{\text{defocus}} = W_0 / (2\sqrt3)

. Separately, the same modulator is tested by writing a blazed phase ramp of spatial frequency

\nu = 40

cycles per millimetre and focusing the beam (diameter

D = 4\ \mathrm{mm}

) with a lens of focal length

f = 250\ \mathrm{mm}

. ((a)) Using the thin-lens Fourier-transform relation, find the displacement

x

of the diffracted spot in the focal plane, and compare it with the diffraction-limited (Rayleigh) spot radius

1.22\lambda f / D

. Is the steered spot resolved from the undiffracted order? ((b)) Find

W_0

(in waves) from the Pockels-cell data, and the resulting

\sigma_{\text{defocus}}

in waves. ((c)) Treating the fitting-error term and the defocus term as independent, use them with the small-error Strehl formula

S = e^{-\sigma^2}

(with

\sigma

in radians) to predict the Strehl ratio at 633 nm.