This invention relates to the field of adaptive optics, in particular adaptive-optics apparatus for controlling the wavefront of a beam of light in response to changes in the wavefront of light incident on a sensor.

In typical adaptive-optics apparatus, the wavefront properties of light incident on a sensor are measured and processed and then used to drive a deformable mirror such that the shape of the reflective surface compensates for distortions in (or otherwise takes into account) the incident wavefront shape. Such apparatus may be used, for example, for correction of distortions of received wavefronts resulting from passage of a beam through the atmosphere or through an optical system. It can also be used to correct an outgoing laser beam for its own wavefront distortions as well as pre-distorting the laser wavefront for predicted optical and atmospheric distortions as measured by analysis of received wavefronts. Francois Roddier proposed at pp 1223-1225 of Applied Optics Vol. 27 No. 7, 1988 ("Curvature sensing and compensation: a new concept in adaptive optics") that for membrane or bimorph mirrors the signal from a curvature sensor can be amplified and directly applied to the mirror. A wavefront curvature sensor measures the 2-dimentional Laplacian ("curvature") V ² Z(r, 0) of a received wavefront over the aperture of the sensor, together with the wavefront

normal slope —z(r,θ) at the aperture periphery. That dn information allows the shape of the wavefront to be calculated to within a constant using, for example, a

Green's function solution with Neumann boundary conditions

^f"Wavefront sensing by use of a Green's function solution to the intensity transport equation", S.C Woods and A.H Greenaway, J. Opt. Soc.Am.A/Vol .20, No.3/March 2003) . The constant is a trivial piston error in the wavefront shape. However, as proposed by Roddier, for bimorph mirrors and electrostatic membrane mirrors, much of the intermediate computation necessary in calculating the entire sensed wavefront shape may be eliminated by making use of the mirror's applied voltage/curvature deformation characteristics. Reducing the computation involved in determining the required shape for the deformed mirror is of particular importance in the case of real-time operation of compensating systems.

Schwartz et al, at pp 895-902 of J.Opt. Soc.Am. A, Vol.11, No.2, Feb.1994, showed that for a bimorph mirror under static conditions (where V ⁴ Z(r _/ θ)~ V ^z V{r,θ), with V = applied voltage) the requirement for the local Laplacian of the deformed mirror surface to be proportional to the local applied voltage was satisfied "under conditions of a simple support, i.e. Z = 0 on the edge with no external moments".

US 2001/0040743 Al (Graves et al . ) describes a deformable curvature mirror with a set of outer electrode segments for controlling the overall slope of a wavefront and a set of inner electrode segments for controlling the curvature of the wavefront .

EP 0 779530 Al (Yalestown Corp.) describes an adaptive optical module based in part on Roddier' s recognition that the output signal from a curvature sensor will in general be proportional to the local curvature of an incident wavefront and proportional to the local slope of the wavefront around its periphery. However, constraints arising from the

^^reatment of the mirror edge condition preclude precise reproduction of the incident wavefront by the module.

An object of the invention is to provide an improved adaptive-optics apparatus and method. The basis of the technique described herein may be described as a "closed-form" approach in which the boundary conditions of the wavefront to be replicated are calculated and imposed in a particular way. It therefore differs from an influence function approach which uses a "best-fit" technique to calculate the combinations of voltages that need to be applied to the various mirror segments to yield a "best-fit" to the wavefront surface to be replicated.

The technique is particularly helpful in enabling derivation of an optimal specification for a deformable mirror to meet particular requirements; for example, quantification of the number, size and shape of mirror- surface segments, the resonant frequency, and the maximum travel of the piezoelectric actuators. To that end, example apparatus according to the invention can be operated "open loop" in order to assess the performance of a particular mirror specification; once a desired specification is identified, however, it is expected that the apparatus will in practice be operated with feedback (i.e. closed loop) . According to a first aspect of the invention there is provided adaptive-optics apparatus for controlling the wavefront of a beam of light, which is incident on a deformable mirror, in response to properties of a wavefront of light that is incident on a sensor, the apparatus comprising: a) a sensor for sensing a signal dependent upon the curvature of the wavefront of the incident light at regions across an area of the wavefront and a sensor for sensing a signal dependent on the slope of the wavefront

iround the periphery of said area in a direction normal to the periphery of said area; b) a processor for calculating the relative amplitude and normal slope of the wavefront around the periphery of said area from the sensed wave-front signals; and c) the deformable mirror for reflecting the beam of light, the mirror comprising an inner zone and an outer zone, which meet at an inner-zone outer perimeter, the mirror being locally deformable in a plurality of regions in response to applied control signals, there being deformable regions in the inner zone arranged to receive control signals provided by the processor such that the mirror is deformed in the inner zone according to the sensed wavefront curvature; wherein deformable regions in the outer-zone are arranged to receive control signals calculated by the processor such that the mirror is deformed at the inner-zone outer perimeter according to the calculated peripheral wavefront relative amplitude and normal slope of said area, thereby providing correct peripheral boundary conditions for the inner-zone. In general, the wavefront surface to be imposed on the mirror over the inner zone will be such that the amplitude of deformation Z does not equal zero on the inner-zone outer perimeter. However, the surface of the outer zone, which may be annular, may be used to transition the true boundary conditions at the inner-zone outer perimeter, corresponding to the measured wavefront edge, down to other boundary conditions at the outer-zone outer perimeter. The deformable regions in the outer zone may be arranged to receive, from the processor, control signals that have been calculated to provide a smooth change (i.e. without discontinuities) in the relative amplitude and slope of deformation from the deformed inner-zone outer perimeter to

%^h.e outer perimeter of the outer zone. The amplitude of deformation at the outer-zone outer perimeter may be planar, (that is zero, constant or tip-tilted) . Under those conditions and with no external moments then the boundary requirements determined by Schwartz (see page 2) are met and the distortion of the entire mirror surface under applied voltage is such that the local Laplacian of the mirror surface is proportional to the applied voltage.

Thus the surface shape of the outer zone may be defined by requiring that the relative amplitude and normal slope of the outer zone at its inner boundary (i.e., the inner-zone outer boundary) match those of the measured wavefront and that the relative amplitude of the outer zone at its outer boundary is planar. The local Laplacian distribution of the defined outer zone surface shape may then be calculated.

Thus the Laplacian distribution for the entire extended surface (inner zone, corresponding to the wavefront to be replicated, plus the outer zone) is then known, derived from the sensed wavefront parameters for the inner zone and by calculation from the boundary conditions for the outer zone. Applying a voltage distribution to an idealised curvature control mirror which is proportional to this overall Laplacian distribution will result in replicating both the desired wavefront surface over the inner zone of the mirror and the calculated transition shape over the outer zone. If the measurement sensor is a curvature sensor the only necessary wavefront reconstruction calculation is to determine the relative amplitude and slope at the edge of the measured wavefront. The wavefront curvature and slope information may conveniently be derived from a curvature sensor but any sensory means from which the information may be generated is

^rpplicable. The processor may also calculate radial curvature of the wavefront around the periphery of the area of the sensed wavefront, and supply control signals to the outer-zone deformable regions such that the radial curvature at the outer-zone inner perimeter matches the calculated radial curvature at the inner-zone outer perimeter.

The mirror may be constrained at the outer-zone outer perimeter to enforce the planar perimeter condition, providing that no external moments are applied. The mirror may be allowed to have zero or non-zero slope at the outer-zone outer perimeter.

The area to be replicated will be the area defined by the smallest stop in the optics or an area within that. However, the signals applied to the mirror need not simply reproduce the measured wavefront. They may for example be used to introduce a desired curvature to produce a desired focusing or defocusing effect on light reflected from the mirror. Thus, the control signals received by the mirror may be arranged to eliminate or introduce curvature of the light reflected by the mirror.

The control signals received by the mirror may be arranged to cancel distortions caused in the wavefront of the beam of light by passage of the light through the atmosphere. The apparatus may further comprise a laser, which outputs a beam of light. The control signals received by the mirror may be arranged to pre-distort the wavefront of the laser light to cancel optical and atmospheric distortions. The laser may output light that is incident on the sensor. The control signals received by the mirror may then

^fe arranged to correct distortions in the wavefront of the laser light.

The apparatus may be arranged such that the light incident on the sensor has been reflected by the mirror. The apparatus may be arranged such that the light incident on the sensor has not been reflected by the mirror when it is so incident.

Ideally, the deformable mirror needs to be mounted in the optical system in such a way that the mounting does not affect the replication of the desired surface shape. Typical mountings of curvature-effect mirrors include clamping or "hinging" around the periphery or supporting the mirror at its centre. In the scheme described above any of these mounting methods could be used provided that no external movements resulted. Such arrangement, however, may result in practical difficulty in achieving adequate mirror deformation together with an adequately high response bandwidth/resonant frequency.

Consider the concept of point support of the mirror rather than perimeter support. If three non-co-linear fixed points are used to support the mirror then the overall shape of the mirror (assuming local deformation due to mounting points is minimised by using an appropriate mounting technology such that local flattening does not occur and local slopes are not constrained) is not affected by constraints arising from the mount itself even when the mounting points are placed under the region of the mirror where the desired surface shape is being replicated. Thus, preferably, the mirror further comprises three or more mirror supports arranged to support the mirror, three of the mirror supports being non-co-linear and defining a mirror- orientation plane. All the mirror supports may be within

inner and/or outer zones. More preferably, the mirror comprises three such supports.

Such an arrangement allows additional design flexibility, beyond that of a mirror supported only around its periphery, when trading off mirror surface displacement, response bandwidth and natural frequency.

If the three supports are fixed, then the planar orientation of the mirror is constrained by such a mounting arrangement but the error in the planar orientation can be compensated by calculating the wavefront value to within a trivial constant at the positions corresponding to the three mounting points. Those planar data can then be used to correct the wavefront planar orientation by for example tip/tilting the bimorph itself by means of piezoelectric stacks at the three mounting points or, if the optical context permits or dictates, using a separate tip/tilt mirror.

Thus, the apparatus may comprise tip/tilt actuators for changing the mirror-orientation plane, the tip/tilt actuators being arranged to maintain the mirror-orientation plane by compensating for changes in the mirror-orientation plane which would otherwise result from the deformation of the mirror. The mirror supports may comprise the tip-tilt actuators. The actuators may be piezoelectric stacks. The apparatus may further comprise a tip/tilt mirror arranged to compensate for changes in the mirror-orientation plane resulting from the deformation of the mirror.

Also, if the ordinates of the mounting points are made continuously variable (for example, by incorporating piezo- electric stacks) then more than three mounting points can be used. Such additional supports beyond the three defining the mirror orientation plane may comprise extensible

^supports that support the mirror but do not affect its shape.

If necessary to achieve the required system performance characteristics, peripheral and multi-point mounting may be used simultaneously but with fixed points limited to three in number.

According to a second aspect of the invention there is provided a method of controlling the wavefront of a beam of light, which is incident on a deformable mirror, in response to properties of a wavefront of light that is incident on a sensor, the method comprising: a) receiving the incident light at the sensor; detecting or calculating a signal dependent upon the curvature of the wavefront of the incident light at regions across an area of the wavefront and detecting or calculating a signal dependent on the slope of the wavefront around the periphery of said area in a direction normal to the periphery of said area; b) calculating the relative amplitude and normal slopes of the wavefront around the periphery of said area from the detected wave-front signals; c) deforming the deformable mirror, the mirror comprising an inner zone and an outer zone, which meet at an outer-zone inner perimeter (which may alternatively be referred to as an inner-zone outer perimeter) , the mirror being locally deformable in a plurality of regions in response to applied control signals, the method including applying to deformable regions in the inner zone control signals dependent on measurements provided by the sensor such that the mirror is deformed in the inner zone according to the detected wavefront curvature; and d) reflecting the beam of light from the mirror; wherein the method further comprises applying to deformable regions of the outer-zone control signals such

the mirror is deformed at the outer-zone inner perimeter according to the calculated inner zone peripheral wavefront amplitude and normal slope, (and, if desired, radial curvature) thereby providing correct peripheral boundary conditions for the inner zone. Preferably, the mirror is maintained planar at the outer-zone outer perimeter.

The method may further comprise applying to deformable regions of the outer-zone control signals such that the mirror is deformed at the outer-zone inner perimeter according to the calculated inner zone peripheral wavefront normal curvature.

The method may further comprise supporting the mirror at three or more points within the inner and/or outer zone, three of the mirror supports defining a mirror-orientation plane.

By way of example only, an example of embodiments of the invention will now be described with reference to the accompanying drawings of which: Fig. 1 is a schematic of an adaptive-optics wavefront controller according to the invention;

Fig. 2 is (a) a front view of a deformable mirror; (b) a side view of the mirror; and (c) a schematic front view of the mirror. Frangois Roddier at pp 1223-1225 of Applied Optics vol.

27, no. 7 (1988) describes a curvature sensor based on the measurement of the intensity of a light beam at two planes that are placed on the axis of the beam, equally spaced, at opposite sides of a focal point. The normalised difference between the intensities

Ii (r) , I ₂ (r) at the two planes is given by:

/ ₂ (r*) -/, (r*)

S (r) = — z(y) δ _c + Ψz{r) J ₂ (^) ₊ Z ₁ Cr*) on where r* and r are co-ordinates in the measurement planes and the focal plane of the sensor, respectively, δ _c is a linear Dirac distribution around the aperture

perimeter, —z(r) is the slope of the wavefront in a dn direction n normal to the aperture perimeter and V ² z(r) is the curvature of the wavefront .

Curvature-effect deformable mirrors (whose static characteristics are defined by V ⁴ Z(X)-V ² F(X)) have a local curvature that is proportional to a locally applied voltage V providing that appropriate boundary conditions are applied, that is

V ² Z(x)~F(x) where Z is the amplitude of the mirror surface deflection at x =(x,y) .

So applying a voltage signal that is proportional to S (r) to such a deformable mirror will mean that :

V ² Z(X)-F(x)-S{x)~Ψz{r) away from the perimeter of the aperture subject to appropriate boundary conditions.

In the arrangement of Fig. 1, incoming light, carrying information of interest, is received by input optics 10 (light beams are shown in the drawings by dashed lines) . The light is focused by input optics 10 towards steering mirror 20, which directs the light to bimorph mirror 30, which reflects the light to beamsplitter 40.

A fraction of the light incident on beamsplitter 40 is directed to wavefront curvature sensor 60. The remainder of the light incident on beamsplitter 40 passes through

40 to detector 50, where the light provides an image, which is recorded for further processing for whatever purpose the light is being monitored.

Wavefront curvature sensor 60 measures light intensity across the aperture. Wavefront sensor 60 passes data describing the sensed wavefront to signal processor 70.

Signal processor 70 then calculates and outputs control signals to bimorph mirror 30. The control signals are calculated to provide a desired effect at detector 50, for example, elimination of curvature from the light beam incident on detector 50.

Bimorph mirror 30 (Fig. 2) comprises two wafers 130, 135 of a piezoelectric material. Wafer 130 is provided with a mirrored outer surface. The wafers 130, 135 are bonded together (Fig. 2b) such that they are polarised in opposite directions, parallel to their axes. Between the wafers 130, 135 are deposited an array of electrodes 110 (Fig. 2a, 2b), to which voltage signals are applied in use via leads 125. The outer surfaces of wafers 130, 135 are connected to ground via lead 120.

The electrodes 110 divide the mirror 40 into a plurality of regions in the form of radial-circular segments (Fig. 2a) . By applying a voltage to an electrode 110, local distortions are produced in mirror 40; the applied voltage causes one wafer to contract and the other to expand in the region of the electrode to which the voltage is applied.

Signal Processor 70 treats electrodes 110 as divided into two zones (Fig. 2c) : a circular inner zone 250 and an annular outer zone 260 that surrounds the inner zone 250. The inner zone 250 corresponds to the aperture stop of the system (or an area just inside it) ; specifically, signal processor 70 sends to electrodes 110 in the inner zone

%>:ltages that are proportional to the curvature distribution detected in the wavefront incident on the aperture of sensor 250; the local curvature of the mirror produced by the voltage is proportional to the applied voltage (provided that appropriate boundary conditions are applied) and so the local curvature of the mirror in the inner zone 250 is then proportional to the curvature of the wavefront incident on the sensor 60.

The appropriate inner-zone outer perimeter boundary conditions are imposed as follows: signal processor 70 calculates and sends to electrodes at outer-zone 260 voltage signals that cause wafers 130, 135 to deform locally to a relative amplitude and slope at the outer perimeter 280 of the inner zone 250 that equate to the relative amplitude of the incident wavefront and the wavefront's slope normal to the perimeter of the system aperture stop (or just inside it) .

The amplitude w(r) of the wavefront around the inner zone outer perimeter is calculated to within a constant from w{r) ~lS(r')G(r,r')d ² r' , where G(r,r') is a Green's function satisfying V ² G(r,r') = <5(r-r') with Neumann boundary conditions, where δ is the Dirac distribution (S.C Woods and

A.H Greenaway, J. Opt. Soc. Am/A/VoI 20 No. 3/March 2003

"Wavefront sensing by use of a Green's function solution to the intensity transport equation") . The normal slope around the perimeter is calculated from two closely-spaced amplitudes .

For a number of points around the inner zone outer perimeter, a radial polynomial curve is calculated that satisfies the calculated (from sensor data) relative amplitude and normal slope at the inner zone outer perimeter

the desired planar perimeter amplitude at the outer zone outer perimeter. A sufficient number of radial polynomial curves are generated to adequately define the surface shape of the outer zone. The distribution of the Laplacian of the outer zone is then calculated and used as the basis of the signals to be applied to the outer zone electrodes. If required, the polynomial curve can also be made to match the radial curvature at the inner zone outer perimeter.

The plane of the mirror 40 is defined by the three points of attachment of three supports 150. The supports 150 are attached to the rear of the mirror 40, and are arranged with their points of attachment forming a triangle within the inner zone 250 (Figs 2a, 2b) . The supports 150 act to hold the mirror 40 stationary relative to a fixed mount 170.

In general, deformations caused by electrodes 110 alter the plane of mirror 40 relative to fixed mount 170

To counteract that tip/tilt effect, which would otherwise potentially cause misalignment of the light beam from the remainder of the optics of the wavefront controller, a tip/tilt mechanism is provided to mirror 40. In the present embodiment, the tip/tilt mechanism is provided by means of piezo-electric stacks 160 that are provided in supports 150 at their points of attachment to mirror 40. A tip/tilt effect caused by deformation of the mirror 40 is thus removed by an equal and opposite tip/tilt effected by stacks 160. The required compensatory tip/tilt is calculated by signal processor 70 from the wavefront data provided by sensor 60.