Shack-Hartmann wavefront sensor

A Shack-Hartmann wavefront sensor measures the shape of a wavefront by dividing it into small patches and finding the tilt of each one. A two-dimensional array of microlenses (lenslets), typically 100–500 μm across, sits in front of a camera at one lenslet focal length. A flat wavefront produces a regular grid of focal spots; a distorted wavefront shifts each spot sideways by an amount proportional to the average slope of the wavefront over that lenslet. Measuring all the spot displacements and integrating the slopes gives the wavefront error across the beam, without a reference beam and with ordinary or broadband light. It is the most common sensor in adaptive optics.

Spot displacement and slope

For a lenslet of focal length $f_L$, a wavefront $W(x, y)$ with local slope $\partial W/\partial x$ moves the spot centroid by

$$\Delta x = f_L \,\frac{\partial W}{\partial x}, \qquad \Delta y = f_L \,\frac{\partial W}{\partial y}.$$

As a worked example, take a lenslet pitch $p = 150$ μm, $f_L = 5$ mm and camera pixels of 5 μm. A spot shift of one pixel corresponds to a slope of 5 μm / 5 mm = 1 mrad, and over one 150 μm lenslet this slope is a wavefront tilt of 150 nm, or 0.24 wave at 633 nm. Centroiding algorithms locate a well-sampled spot to a tenth of a pixel or better, so the slope resolution is about 0.1 mrad, a tilt of 15 nm per lenslet.

Sensitivity and dynamic range

The same lenslet geometry sets both limits. The diffraction-limited spot of a square 150 μm lenslet at 633 nm is $2\lambda f_L/p$ = 42 μm wide between first zeros, and each spot must stay within its own subaperture so that it can be assigned to the right lenslet. The largest measurable slope is therefore about $(p/2)/f_L$ = 15 mrad, or 10.8 mrad once half the spot width is subtracted. For a 10 mm beam, a slope of 10.8 mrad at the edge corresponds to a spherical wavefront with a radius of curvature of about 0.46 m, so a strongly converging beam has to be relayed or partly collimated before it reaches the sensor.

Doubling the focal length to 10 mm improves the sensitivity to 0.5 mrad per pixel, but the spot also doubles to 84 μm, so the dynamic range falls by more than half, to (75 − 42) μm / 10 mm, about 3.3 mrad. The pitch sets the spatial resolution: aberrations that vary on a scale smaller than one lenslet are averaged out, so a 10 mm beam sampled at 150 μm has about 67 slope measurements across its diameter.

Reconstruction

Slopes are converted to a wavefront in one of two ways. Zonal reconstruction solves a least-squares problem for the wavefront values at grid points whose differences match the measured slopes. Modal reconstruction fits the slopes with the derivatives of a set of basis functions, usually Zernike polynomials, and reports their coefficients, which separate tilt, defocus, astigmatism and the other aberrations directly. Piston, a constant phase over the whole pupil, produces no slope and cannot be measured. The rms wavefront error from either method gives an estimate of the Strehl ratio.

Where it is used

In adaptive optics a Shack-Hartmann sensor views a guide star or a reference point and drives a deformable mirror in closed loop at hundreds to thousands of frames per second. In laser work, the sensor measures the wavefront and intensity of a beam together, and from the reconstructed field the software can estimate beam quality from a single frame, where a scanning measurement with a beam profiler at several planes would otherwise be needed. In ophthalmology it measures the aberrations of the eye, and in optical testing it checks lenses and assemblies where an interferometer would be disturbed by vibration or needs coherent light.

Pitfalls

• Pupil conjugation. The lenslet array must be at an image of the plane of interest; otherwise the measured wavefront includes propagation from that plane.
• Reference calibration. Lenslet manufacturing errors and camera alignment are removed by recording a reference spot pattern with a known flat or spherical wavefront.
• Spot crossover. Large aberrations push spots into neighboring subapertures, and the reconstruction then fails abruptly.
• Background and saturation. Stray light, camera offset and saturated pixels bias the centroids; background subtraction and a threshold are needed, but too high a threshold also biases them.

Common questions

How does a Shack-Hartmann sensor compare with an interferometer?

An interferometer measures the wavefront height directly with nanometer resolution and dense spatial sampling, but it needs a coherent reference and a stable setup. A Shack-Hartmann sensor measures slopes at lower spatial resolution; it is far less sensitive to vibration, works with incoherent and broadband light, and can run at camera frame rates.

What is the difference between a Hartmann test and a Shack-Hartmann sensor?

J. Hartmann introduced in 1900 a screen with an array of holes placed over a telescope objective, and the positions of the ray bundles behind it revealed the objective's errors. R. V. Shack and B. C. Platt replaced the holes with an array of lenslets around 1971, which collects nearly all the light and focuses each patch into a compact spot, making the method usable with faint sources and electronic cameras.

References: B. C. Platt and R. Shack, "History and principles of Shack-Hartmann wavefront sensing," J. Refract. Surg. 17, S573 (2001); D. Malacara (ed.), Optical Shop Testing, 3rd ed. (Wiley, 2007); J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).