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Curvature sensing with a Shack-Hartmann

sensor

Oleg Soloviev*1,2, Michel Verhaegen1 &Gleb Vdovin1,2,3

1. DCSC, TU Delft, 2628 CD, Delft, The Netherlands

2. Flexible Optical BV, Polakweg 10-11, 2288 GG Rijswijk, the Netherlands

3. ITMO University, Kronverksky 49, 197101 St Petersburg, Russia

*o.a.soloviev@tudelft.nl

Abstract: Shack-Hartmann (SH) sensor, based on sampling

of wavefront tilts in subapertures, is a simple, reliable, and

widely used in adaptive optics wavefront sensor. A

wavefront curvature sensor has the advantage of providing

the results suitable for direct control of membrane and

bimorph deformable mirrors [1], but requires linear

registration of intensity in two planes. SH sensor

modifications using astigmatic microlens array [2] and three

SH sensors [3] provide measurement both in the form of

wavefront gradients and Laplacian curvatures. In this work,

we consider a simple arrangement that turns a standard SH

sensor into a curvature sensor by moving the camera chip

of the SH sensor into the optical plane conjugated to a

deformable mirror. This establishes a direct geometric

correspondence between the coordinates on the DM surface

and the sensor chip. Then, change in the local centroid

density corresponds to the Laplacian curvature of the

mirror, and the phase at the boundary can be found from

the centroid displacements along the edge of the pupil. We

investigate the feasibility of this approach for direct control

of membrane deformable mirror by measuring the

dependence of the calculated centroid density on the control

signal applied to the mirror actuators. The experimental

results demonstrate a good linear dependence.

Keywords: Adaptive optics; Shack-Hartmann sensor; curvature sensor;

membrane deformable mirror

Laplacian curvature from a Shack-Hartmann sensor

Shack-Hartmann (SH) sensor is a popular wavefront sensor (WFS) used in

adaptive optics (AO), which measures the averaged over subapertures

wavefront gradients

∇ϕ

, related to the spot shifts in the Hartmann pattern.

The methods based on finding centroids of the spots are low sensitive to the

irradiance variation over the pupil and to the linearity of the camera.

A wavefront curvature sensor [1] measures the Laplacian curvature of

the wavefront

Δϕ=∇2ϕ.

This provides a good match with the physics of

curvature aberration correctors such as membrane and bimorph deformable

mirrors (DM) and can be used directly for control. To deal with intensity

variations, the standard curvature sensor registers the intensity in two

planes, before and after the focus.

The hybrid curvature and gradient SH sensor modification [2] uses

single detector plane; the sensors provides measurement both in the form of

wavefront gradients and Laplacian curvatures, but requires special

astigmatic microlens array. The differential SH curvature sensor [3] uses

three usual SH sensors for measuring twist and Laplacian curvatures. In all

SH-based method, the range of the wavefront gradient is limited by the ratio

of the microlens array (MLA) pitch to its focal distance p/f, or some special

algorithms are required to index the spots [e.g., 4]. In this work, we propose

a simple arrangement that turns a standard SH sensor into a curvature sensor

and also benefits from low p/f values.

In a traditional aligned AO system, both the DM and the MLA of the

WFS are conjugated to the system pupil. In such configuration, every

subaperture corresponds to a particular pupil subaperture and to the

corresponding patch of the DM. The beam crosses the MLA in a fixed area

(pupil image); the spot pattern can move over the camera chip, but (under

certain obvious conditions) contains the same number of spots.

An interesting configuration appears if the focal plane of the MLA, i.e.

the surface of the camera chip is located in the plane conjugated to the

pupil. In this case a direct geometric correspondence exists between the

coordinates on the DM surface and the sensor chip. The MLA is now

located in front of the aperture, and the beam can cross it in different places,

depending on the WF shape. The number of spots in the Hartmann pattern is

not fixed anymore, but the region they occupy is fixed. The change in local

density of the spots is directly proportional to the local Laplacian curvature,

as in the intensity transport sensors [5, 6], but is almost independent of the

intensity variations. As a consequence, the control signal (CS) applied to an

actuator is proportional to the integral of the spots density over its area. For

the Zernike modes having zero Laplacian curvature [7], the boundary

conditions are given by the centroids displacements along the pupil edge.

Description of the method and experiment

We have used a 15-mm membrane deformable mirror with 39 actuators as

show on Fig.1 and a SH sensor with a MLA featuring 12 mm focal length

and 150 um pitch over full aperture of 4.5 mm, and UI1540 camera with

5.2um pixels. A Galilean telescope with 200 mm and 75 mm lenses was

used to reimage the mirror to the WFS. The measurements have been done

for 10 mm working aperture of the mirror, and full 15-mm aperture. In the

last case some parts of the mirror image were clipped by the MLA aperture.

The frames corresponding to three reference global curvatures of the mirror

(CS = -1 or 0 or 1 sent to all the actuators simultaneously) have been

processed by a usual centroid-finding procedure of a SH-sensor. For the 10-

mm working aperture, the total number of found centroids was 500, 532,

and 568 centroids respectively. This provides an estimate to the density

change range ± 6%. For 19 actuator of approximately equal area this results

in 27±1.66 centroid in the actuator.

After that, the frames corresponding to CS = +1 or -1 set to the i-th

actuator and 0 to the rest actuator have been processed to find the centroid

array. The next step was to pixilate the centroid arrays by representing each

centroid by a group of 4 adjacent pixels having the same center of gravity.

This procedure simplifies the further calculations of the spot density.

The spot density was calculated by convolving of the pixelated

centroids with a truncated Gaussian kernel of size 4p, and with σ=1.1p,

where p is the pitch of the microlens array expressed in pixel width, and

compared with the bias centroid density (technically, it's more efficient to

Figure 1. Actuator structure of 39-ch 15-mm MMDM and the response

functions (calculated density difference) for actuators 1,2, 13, and 21. The

response for the actuator 13 is shown for negative control signal. The edge

artifacts do not influence further results.

Figure 2. Normalised centroid density difference for a linear control signal

applied to actuator 9 and the fitting error from the best-fit linear model.

calculate the difference of the pixelated centroids first and then perform the

convolution). The results are present in the Fig.1. One can see the increased

centroid density in the regions corresponding to the mirror actuators.

To test the linearity of the calculated density difference with respect to

the control voltage, we grabbed and processed 21 frame corresponding to

the CS signals -1,-0.9, ..., 1 to one of the actuators. The pixel values of the

obtained density were then integrated over the area of the actuator by

multiplying the density image by the mask corresponding to the actuator,

summing all values together, and dividing by the actuator area. The obtained

results shown in Fig.2 demonstrate linear dependence with 2% accuracy.

Conclusion

In this work we present a simple arrangement that turns a Shack-Hartmann

sensor into a curvature sensor. The Laplacian curvature of the wavefront can

be found directly by placing the SH sensor camera chip in the plane

conjugated to the optical pupil and calculating the local spot density of the

spot pattern.

A preliminary tests show good linear dependence between the measured

with this method local curvature and the control signal applied to an actuator

of a membrane deformable mirror, which makes this method a good

candidate for fast adaptive optical systems.

This work is sponsored by the European Research Council, Advanced

Grant Agreement No. 339681.

References

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