FIELD OF THE INVENTION This invention relates to spatial light modulator (SLM) devices, in particular for modulating optical phase for the purpose of diffracting light to form images or provide illumination.

BACKGROUND TO THE INVENTION

The formation of images or illumination using a laser-illuminated spatial light modulator to diffract light using phase modulation is an established technique (often termed holographic projection) and has many advantages over conventional image projection techniques, including significantly improved efficiency due to the formation of images using interference (effectively loss-less), compared with conventional techniques (such as LCOS (Liquid Crystal on Silicon) imaging or DLP (Digital Light Processor™) imaging which form images by blocking light. Such a holographic projection approach can be used to form full-colour images in their entirety (e.g. Choi, Jim et al., Practical Holography XVI and Holographic Materials VIII, Proc. SPIE Vol. 4659 [2002]), in which case the SLM resolution may be comparable to, or larger than the image resolution (e.g. 1280 x 1280), or to form an efficient backlight for the illumination of a conventional microdisplay panel (GB Patent 2461894, US Patent Application 12/182,095), in which case the SLM resolution may be significantly smaller than the image resolution (e.g. 128 x 128).

Conventional approaches, in the literature and also on the market, have employed essentially a conventional microdisplay device (based on non-imaging LCOS or MEMS technology) as the spatial light modulator, modified to modulate phase rather than amplitude, and featuring a conventional pixel arrangement, invariably a rectangular array of rectangular pixels.

Adaptive optics for astronomical imaging have employed MEMS (micro- electromechanical systems) SLM (spatial light modulator) technology, and a MEMS device with hexagonal pixels in a hexagonal array is described in US2004/0160 18. Further background prior art is described in US2004/0165243 however these devices are adapted for use in adaptive optics and the requirements of a diffractive imaging system are significantly different and special, as explained further later. In particular the aim of maximising the diffraction efficiency makes certain pixel shapes and array structures especially advantageous. Thus, for example, in a conventional imaging system light loss is proportional to the square of the dead space between pixels whereas in a diffractive imaging system this loss is approximately proportional to the fourth power of the dimension of the dead space between pixels. Further background prior art can be found in JP8122760 A.

SUMMARY OF THE INVENTION

According to a first aspect of the invention there is provided a method of providing a diffractive image display, the method comprising: displaying a diffraction pattern on pixels of a spatial light modulator (SLM), the SLM comprising a plurality of optical phase modulating pixels; and illuminating said SLM with a beam of light to reconstruct a far field image formed by said diffraction pattern; and wherein the method further comprises: providing a hexagonal or diamond shape for said pixels.

In preferred embodiments the pixel shape is selected to substantially optimise a diffraction efficiency of the SLM, in particular by minimising the dead space between pixels (subject to any other design constraints which may be present). This in turn can be achieved by selecting a pixel shape which aims to maximise the ratio of the area of a pixel to the perimeter of the pixel, and this leads to a hexagonal or diamond shape for a pixel (noting that one can broadly speaking squash a hexagon to form a diamond). Furthermore in some diffractive image display applications, in particular in the holographic projector applications we describe later, it is desirable to aim to maximise light diffracted into the first diffraction order, and this is dependent on, in part, the spatial Fourier transform of the pixel shape. Conversely, the pixel shape in the diffracted image is dependent on the Fourier transform of the outline of the pixel array, and in some preferred embodiments this is substantially rectangular. It is preferable that the pixels are rectangularly spaced in the sense that the pixels have a regular spacing in the X- and Y- perpendicular directions in a lateral plane of the device.

In some preferred embodiments the pixels have a shape given by a shape parameter S the calculation of which is described later. In preferred embodiments S is greater than 0.333, 0.4, 0.5, 0.6, 0.7, 0.8 or 0.9. In some preferred embodiments, in particular when incorporated into a holographic projector, a first diffraction order of light diffracted by the SLM as a substantially square shape. Preferably this square shape is achieved when the SLM is illuminated by a beam of light at an angle to a normal to the substrate of the SLM which is greater than 0, for example substantially 45°. This facilitates optical design of the holographic projection system.

In some preferred embodiments a pixel of the SLM includes a phase ramp and/or at least one phase step, for example a binary phase step of approximate^ 30 mn. Because a phase ramp or step in Fourier space corresponds to a position shift in image space, incorporating a phase ramp/step (preferably with a gradient chosen to shift the Y- direction by a quarter of the field height) into the SLM pixel has the effect of shifting the diffraction pattern attenuation envelope (for example a sine envelope) away from a zero order spot and towards a centre of the displayed image. This decreases the attenuation caused by the diffraction attenuation envelope in the Y- direction and thus improves diffraction efficiency.

In some embodiments the pixel pitch of the SLM is less than 30 pm by 20 pm in the horizontal and vertical directions respectively, and, preferably, a (perpendicular or minimum) distance between the pixels is less than 1 pm. For example in a preferred embodiment the pixel pitch in the X- and Y- directions is approximately 15 pm x 10 pm and a space between the pixels is approximately 0.5 pm. Selecting these values of pixel pitch and pixel spacing helps to maximise diffraction efficiency. In some preferred embodiments a pixel comprises a piston-type MEMS optical phase modulation pixel, that is pixel in which a light reflecting surface may be translated in a direction substantially perpendicular to the substrate, for example by application of an electrostatic force. In one preferred holographic projector application the diffraction pattern displayed on the pixels of the SLM comprises a hologram. More particularly and in preferred implementations the hologram is of a first, low spatial frequency component of an input image, and the reconstructed far field image is used to illuminate a second spatial light modulator which amplitude modulates the reconstructed far field image using a second, substantially higher spatial frequency component of the input image. Embodiments of the method/system may thus further comprise (means for) inputting display image data defining an image for display, processing the image data to determine first in which data representing a first spatial frequency portion of the image data and second image data representing a second spatial frequency portion of the image data, where the second spatial frequency is higher than the first spatial frequency. The method may then comprise displaying a hologram of the first image data on an SLM as described above to form a holographically generated intermediate real image, the method then modulating this intermediate real image using the second image data to display the image. In such an approach the computation of the hologram becomes relatively quick and straightforward, the resolution of the projection system may readily be changed by changing the resolution of the second SLM without changing that of the hologram SLM, and other benefits such as miniaturisation of the optical system can also be achieved. Thus in embodiments the active pixel area of the (hologram) SLM may have a lateral dimension of less than 5mm, 4mm, 3mm, 2mm or 1 mm.

In a related aspect the invention provides a diffractive image display system

comprising: a spatial light modulator (SLM) comprising a plurality of optical phase modulating pixels for displaying a diffraction pattern; at least one light source to illuminate said SLM to reconstruct a far field image formed by said diffraction pattern; wherein said SLM pixels have a hexagonal or diamond shape.

In a further related aspect the invention provides a phase-modulating spatial light modulator (SLM) comprising a plurality of optical phase modulating pixels for displaying a diffraction pattern, wherein said pixels have a shape parameter S of greater than 0.333 where S is given by:

4(H - A _y ) ²

Δ., ^■ 4Δ DS

where is a pixel pitch in a horizontal direction in a lateral plane of said SLM, A _y is a pixel pitch in a vertical direction in a lateral plane of said SLM, A _DS is a length of deadspace between pixels, and H is one half of a maximum dimension of a said pixel in a lateral plane of said SLM measured in said vertical direction. BRIEF DESCRIPTION OF THE DRAWINGS These and other aspects of the invention will now be further described, by way of example only, with reference to the accompanying figures in which:

Figure 1 shows a first example of a holographic image projection system for use with an SLM according to an embodiment of the invention;

Figure 2 shows an improved holographic image projection system for use with an SLM according to an embodiment of the invention;

Figures 3a to 3d show an example of a holographic image display system without aberration correction illustrating, respectively, a block diagram of a hologram data calculation system, operations performed within the hardware block of the hologram data calculation system, energy spectra of a sample image before and after multiplication by a random phase matrix, and an example of a hologram data calculation system with parallel quantisers for the simultaneous generation of two sub- frames from real and imaginary components of complex holographic sub-frame data;

Figures 4a and 4b show, respectively, an outline block diagram of an adaptive OSPR- type system, and details of an example implementation of the system; Figure 5 shows a schematic representation of a pixel layout of an SLM according to an embodiment of the invention, comprising a regular array of irregular hexagonal pixels;

Figure 6 shows example techniques for the separation of light incident onto and diffracted from the SLM of Figure 5, by polarization (left) and by angle (right);

Figure 7 shows a schematic view from above of a portion of an SLM according to an embodiment of the invention showing four parameters defining pixel shape; and

Figure 8 shows an example of image placement within the first diffraction order of light diffracted by an SLM according to an embodiment of the invention. DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Broadly speaking we will describe a spatial light modulator (SLM) device comprising an array of non-rectangular, in particular hexagonal, pixels for modulating optical phase, for the purpose of diffracting light to form images or illumination. Such a spatial light modulator is particularly useful for diffractive image formation in a diffractive holographic projector.

To aid understanding of the invention we therefore first describe some preferred implementations of holographic image display systems with which the calibration techniques we describe may be used.

Hologram Generation

We will describe applications of embodiments of the invention to an OSPR-type holographic image display system, and we therefore describe examples of such systems below. However applications of embodiments of the invention are not restricted to this type a hologram generation procedure and may be employed with holographic image display systems employing other types of hologram generation procedure, for example: a Gerchberg-Saxton procedure (R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures" Optik 35, 237-246 (1972)) or other procedure. The techniques may also be employed more generally with systems which display a diffraction pattern on an SLM, and in principle in other fields.

Optical system

Figure 1 shows an example optical layout for a first example of a holographic image projection system 100 to project a 2D image onto a screen 110. In the full colour holographic image projector of Figure 1 there are red R, green G, and blue B lasers. The system also includes the following additional elements:

• SLM is the hologram SLM (spatial light modulator). • L1, L2 and L3 are collimation lenses for the R, G and B lasers respectively (optional, depending upon the laser output).

• M1, M2 and M3 are corresponding dichroic mirrors; they may be implemented as a prism assembly.

· PBS (Polarising Beam Splitter) transmits the incident illumination to the SLM.

Diffracted light produced by the SLM - naturally rotated in polarisation by 90 degrees (with a liquid crystal SLM) - is then reflected by the PBS towards L4.

• Mirror M4 folds the optical path.

• Lenses L4 and L5 form an output telescope (demagnifying optics). The output projection angle is proportional to the ratio of the focal length of L4 to that of L5.

In embodiments L4 may be encoded into the hologram(s) on the SLM, for example using the techniques we have described in WO2007/110668, and/or output lens L5 may be replaced by a group of projection lenses. In embodiments L5 may comprise a wide-angle or fisheye lens, mounted for translation perpendicular to the output optical axis (e.g left-right in Figure 1), to enable configuration of the output optical system as an off-axis system for table- down projection.

• D1 is a diffuser located at intermediate image plane to reduce speckle. It may comprise a plastic plate, and optionally, may be piezoelectrically-actuated so that it can be moved rapidly in two orthogonal directions to reduce streaking

(see our GB2456170). The diffuser increases the etendue of the source, increasing safety and reducing speckle.

A processor 102 acts as a system controller and performs signal processing in either dedicated hardware, or in software, or in a combination of the two, as described further below. Thus processor 102 inputs image data and provides hologram data 104 to the SLM. The different colours are time-multiplexed and the sizes of the replayed images are scaled to match one another, for example by padding a target image for display with zeros (the field size of the displayed image depends upon the pixel size of the SLM not on the number of pixels in the hologram).

In embodiments the SLM may be a liquid crystal device. Alternatively, other SLM technologies to effect phase modulation may be employed, such as a pixellated MEMS-based piston actuator device. Figure 2 shows an optical architecture for a second example of a holographic image projection system 200, in which like elements to Figure 1 are indicated by like reference numerals. This is described in more detail later.

OSPR

We now describe an example OSPR-type hologram generation procedure, with reference to Figures 3a to 3d: Broadly the SLM is modulated with holographic data approximating a hologram of the image to be displayed. However this holographic data is chosen in a special way, the displayed image being made up of a plurality of temporal sub-frames, each generated by modulating the SLM with a respective sub- frame hologram, each of which spatially overlaps in the replay field (in embodiments each has the spatial extent of the displayed image).

Each sub-frame when viewed individually would appear relatively noisy because noise is added, for example by phase quantisation by the holographic transform of the image data. However when viewed in rapid succession the replay field images average together in the eye of a viewer to give the impression of a low noise image. The noise in successive temporal subframes may either be pseudo-random (substantially independent) or the noise in a subframe may be dependent on the noise in one or more earlier subframes, with the aim of at least partially cancelling this out, or a combination may be employed. Such a system can provide a visually high quality display even though each sub-frame, were it to be viewed separately, would appear relatively noisy.

The procedure is a method of generating, for each still or video frame I = l _xy , sets of N binary-phase holograms h ⁽¹⁾ ... h ^(N . In embodiments such sets of holograms may form replay fields that exhibit mutually independent additive noise. An example is shown below:

1. Let = /^ exp^ ^' j^) where is uniformly distributed between 0 and 2π for

\ < n≤NI2 and l < x,^ < w 2. Let g " ^} = F ^'x where F ^~l represents the two-dimensional inverse Fourier transform operator, for 1 < n≤ N/2

3. Let = 5R {g } for 1 < n≤ Nl 2

4. 3{g[?} for l < « < _V/ 2

5. Let where Q ⁽ⁿ⁾ = median ) and 1 < n≤ N .

Step 1 forms N targets G _x ⁽ " ^] equal to the amplitude of the supplied intensity target l _xy , but with independent identically-distributed (i.i.d.), uniformly-random phase. Step 2 computes the N corresponding full complex Fourier transform holograms . Steps 3 and 4 compute the real part and imaginary part of the holograms, respectively. Binarisation of each of the real and imaginary parts of the holograms is then performed in step 5: thresholding around the median of ensures equal numbers of -1 and 1 points are present in the holograms, achieving DC balance (by definition) and also minimal reconstruction error. The median value of may be assumed to be zero with minimal effect on perceived image quality. Figure 3a, from our WO2006/ 134398, shows a block diagram of a hologram data calculation system configured to implement this procedure. The input to the system is preferably image data from a source such as a computer, although other sources are equally applicable. The input data is temporarily stored in one or more input buffer, with control signals for this process being supplied from one or more controller units within the system. The input (and output) buffers preferably comprise dual-port memory such that data may be written into the buffer and read out from the buffer simultaneously. The control signals comprise timing, initialisation and flow-control information and preferably ensure that one or more holographic sub-frames are produced and sent to the SLM per video frame period.

The output from the input comprises an image frame, labelled /, and this becomes the input to a hardware block (although in other embodiments some or all of the processing may be performed in software). The hardware block performs a series of operations on each of the aforementioned image frames, /, and for each one produces one or more holographic sub-frames, h, which are sent to one or more output buffer. The sub- frames are supplied from the output buffer to a display device, such as a SLM, optionally via a driver chip. Use of a ferroelectric liquid crystal SLM can be advantageous because of its fast switching time. The SLM may be binary phase or multi-phase - binary phase devices can be convenient but binary quantization results in a conjugate image whereas the use of a multi-phase SLM suppresses this. Figure 3b shows details of the hardware block of Figure 3a; this comprises a set of elements designed to generate one or more holographic sub-frames for each image frame that is supplied to the block. Preferably one image frame, Ixy, is supplied one or more times per video frame period as an input. Each image frame, Ixy, is then used to produce one or more holographic sub-frames by means of a set of operations comprising one or more of: a phase modulation stage, a space-frequency transformation stage and a quantisation stage. In embodiments, a set of N sub-frames, where N is greater than or equal to one, is generated per frame period by means of using either one sequential set of the aforementioned operations, or a several sets of such operations acting in parallel on different sub-frames, or a mixture of these two approaches.

The purpose of the phase-modulation block is to redistribute the energy of the input frame in the spatial-frequency domain, such that improvements in final image quality are obtained after performing later operations. Figure 3c shows an example of how the energy of a sample image is distributed before and after a phase-modulation stage in which a pseudo-random phase distribution is used. It can be seen that modulating an image by such a phase distribution has the effect of redistributing the energy more evenly throughout the spatial-frequency domain. The skilled person will appreciate that there are many ways in which pseudo-random binary-phase modulation data may be generated (for example, a shift register with feedback).

The quantisation block takes complex hologram data, which is produced as the output of the preceding space-frequency transform block, and maps it to a restricted set of values, which correspond to actual modulation levels that can be achieved on a target SLM (the different quantised phase retardation levels may need not have a regular distribution). The number of quantisation levels may be set at two, for example for an SLM producing phase retardations of 0 or π at each pixel, or more for a multi-phase SLM.

In embodiments the quantiser is configured to separately quantise real and imaginary components of the holographic sub-frame data to generate a pair of holographic sub- frames, each with two (or more) phase-retardation levels, for the output buffer. Figure 3d shows an example of such a system. It can be shown that (depending on the implementation of the procedure) for discretely pixellated fields, the real and imaginary components of the complex holographic sub-frame data are uncorrelated, which is why it is valid to treat the real and imaginary components independently and produce two uncorrelated holographic sub-frames. In other approaches only the real or only the imaginary part may be used.

Adaptive OSPR

In the OSPR approach we have described above subframe holograms are generated independently and thus exhibit independent noise. In control terms, this is an open-loop system. However better results can be obtained if, instead, the generation process for each subframe takes into account the noise generated by the previous subframes - in order to cancel it out, effectively "feeding back" the perceived image formed after, say, n OSPR frames to stage n+1 of the algorithm. In control terms, this is a closed-loop system. One example of this approach comprises an adaptive OSPR algorithm which uses feedback as follows: each stage n of the algorithm calculates the noise resulting from the previously-generated holograms H ₇ to H^, and factors this noise into the generation of the hologram H _n to cancel it out. As a result, it can be shown that noise variance falls as MN ² in comparison to the 1/Λ/ falloff for (non-adaptive) OSPR. An example procedure takes as input a target image T, and a parameter N specifying the desired number of hologram subframes to produce, and outputs a set of N holograms H, to H _N which, when displayed sequentially at an appropriate rate, form as a far-field image a visual representation of T which is perceived as high quality. An optional pre-processing step performs gamma correction to match a CRT display by calculating T(x, y) ^{1 3} . Other pre-processing may include colour space conversion and geometry correction (if projecting at an angle). Then at each stage n (of N stages) an array F (zero at the procedure start) keeps track of a "running total" (desired image, plus noise) of the image energy formed by the previous holograms H ₁ to /-/„., so that the noise may be evaluated and taken into account in the subsequent stage:

F(x,y) A random phase factor φ is added at each stage to each pixel of the target image, and the target image is adjusted to take the noise from the previous stages into account, calculating a scaling factor a to match the intensity of the noisy "running total" energy F with the target image energy (Γ) ² . The total noise energy from the previous n - 1 stages is given by a F - (n - 1 )(Γ) ² , according to the relation a := ·

∑F(x, y)T (x,y) ² and therefore the target energy at this stage is given by the difference between the desired target energy at this iteration and the previous noise present in order to cancel that noise out, i.e. (Tf - [ F - (n - 1 )(Γ) ² ] = n(Tf + a F. This gives a target amplitude |Γ"| equal to the square root of this energy value, i.e.

At each stage n, H represents an intermediate fully-complex hologram formed from the target T" and is calculated using an inverse Fourier transform operation. It is quantized to binary phase to form the output hologram H _n , i.e.

H(x,y) =r- [TXx,y)\

Figure 4a outlines this method and Figure 4b shows details of an example

implementation, as described above.

Thus, broadly speaking, an ADOSPR-type method of generating data for displaying an image (defined by displayed image data, using a plurality of holographically generated temporal subframes displayed sequentially in time such that they are perceived as a single noise-reduced image), comprises generating from the displayed image data holographic data for each subframe such that replay of these gives the appearance of the image, and, when generating holographic data for a subframe, compensating for noise in the displayed image arising from one or more previous subframes of the sequence of holographically generated subframes. In embodiments the compensating comprises determining a noise compensation frame for a subframe; and determining an adjusted version of the displayed image data using the noise compensation frame, prior to generation of holographic data for a subframe. In embodiments the adjusting comprises transforming the previous subframe data from a frequency domain to a spatial domain, and subtracting the transformed data from data derived from the displayed image data. More details, including a hardware implementation, can be found in WO2007/141567 hereby incorporated by reference. One can extend the ADOSPR procedure so that in addition to feeding forward the reproduction error present at each of the M χ M sampling points (x, y), the errors present between the sampling points after stage N - 1, i.e. at (x½, y), (x, y½) and (x½, y½), are also fed forwards and compensated for when calculating the hologram H _w in stage N. In this way super-resolution can be implemented using inter-pixel interference to create structure at increased spatial frequencies. For further details reference may be made to detailed example calculations for Stage 1 , Stage 2 and Stage N in WO 2007/085874.

If we define a parameter p which is a measure of the total background noise energy independent of the image displayed (for OSPR p = 0.6321) and a parameter c which defines the coverage of an image, the energy in the desired image as a proportion of the maximum available energy then the average contrast ratio in a holographically replayed image is 1 +— -— . Different images l _xy have different coverages and c{/ _xy } c(l-p)

is proportional to (l _xy ) ² . Thus to compensate for coverage one can display/illuminate each subframe of image / for a time proportional to c{7} / p and/or modulate the illumination power proportional to c{I} I p (an approximate value for c may be used).

Dual modulation architecture

We now describe the improved holographic image projection system architecture 200 of Figure 2. This employs dual SLM modulation - low resolution phase modulation and higher resolution amplitude (intensity) modulation. This can provide substantial improvements in image quality, power consumption and physical size. The primary gain of holographic projection over imaging is one of energy efficiency. Thus the low spatial frequencies of an image can be rendered holographically to maintain efficiency and the high-frequency components can be rendered with an intensity-modulating imaging panel, placed in a plane conjugate to the hologram SLM. Effectively, diffracted light from the hologram SLM device (SLM1) is used to illuminate the imaging SLM device (SLM2). Because the high-frequency components contain relatively little energy, the light blocked by the imaging SLM does not significantly decrease the efficiency of the system, unlike in a conventional imaging system. The hologram SLM is preferably be a fast multi-phase device, for example a pixellated MEMS-based piston actuator device.

In Figure 2:

• SLM1 is the hologram SLM (spatial light modulator), for example a 160 ^χ 160 MEMS or ferroelectric liquid crystal device with pixels of sizeA ; it may have physically small lateral dimensions, e.g <1mm.

· L1, L2 and L3 are the collimation lenses.

• M1 , M2 and M3 are dichroic mirrors a implemented as prism assembly.

• M4 is a turning beam mirror.

• SLM2 is the imaging SLM and has a resolution at least equal to the target image resolution (e.g. 854 * 480); it may comprise a LCOS (liquid crystal on silicon) panel.

• Diffraction optics 210 comprises lenses LD1 and LD2, forms an intermediate image plane on the surface of SLM2, and has effective focal length f such that /λ / Δ covers the active area of imaging SLM2. Thus optics 210 perform a spatial Fourier transform to form a far field illumination pattern in the Fourier plane, which illuminates SLM2.

• PBS2 (Polarising Beam Splitter 2) transmits incident light to SLM2, and reflects emergent light into the relay optics 212. PBS2 preferably has a clear aperture at least as large as the active area of SLM2.

• Relay optics 212 relay light to the diffuser D1.

· M5 is a beam turning mirror.

• D1 is a diffuser to reduce speckle, as previously described.

• Projection optics 214 project the object formed on D1 by the relay optics 212, and preferably provide a large throw angle, for example >90°, for angled projection down onto a table top (the design is simplified by the relatively low entendue from the diffuser). A system controller and hologram data processor 202 performs signal processing in either dedicated hardware, or in software, or in a combination of the two, as described further below. Thus controller 202 inputs image data and provides low spatial frequency hologram data 204 to SLM1 and higher spatial frequency intensity modulation data 206 to SLM2. The controller also provides laser light intensity control data 208 to each of the three lasers.

In an example procedure:

1. The hologram SLM size is M * M pixels.

2. The input image target amplitude, T, is of size P * P pixels. Amplitude range for the input is between 0 (black) and 1 (white).

3. N ADOSPR subframes are generated.

4. D is a diffraction efficiency boost parameter controlling the trade-off between reconstruction error and diffraction efficiency A value of 1.0 gives theoretically perfect reconstruction; larger values of D increase the optical efficiency at the expense of increasing the noise. (Simulations suggest using a value for D of approximately 1.5).

We assume the illumination incident on the SLM is Gaussian, with the 1/e ² intensity at the edges of the SLM. The steps are:

1. Form a 2M * 2M target image, R, for hologram generation comprising peak values of blocks of the image. Subdivide the input (P * P) image T into 2M * 2M blocks, each of size PI2M ^χ P/2M. Set each pixel of the target R to be the peak amplitude of the image data within the corresponding P/2M * P/2M block of the image.

2. Generate a hologram set H of N holograms of size M * M from R. For example the above ADOSPR algorithm optionally with super-resolution may be employed, optionally iteratively optimising the holograms, for example using a Gerchberg-Saxton procedure.

3. Calculate the reconstruction intensity / of the hologram set, oversampled to P x P pixels. Sum the intensities of the reconstructions of each of the N holograms and divide the final intensity by N. (An example of reconstruction of an image from hologram data is described above, as part of the ADOSPR procedure).

4. Calculate the intensity image F to display on the imaging SLM. Set each pixel of Fto the corresponding pixel of the target image intensity T ² . Divide each pixel in F by the corresponding pixel intensity in /. Let m be the maximum value in the new field F. Then multiply each pixel in F by Dim. Finally, set every pixel greater than 1 in F to 1.

5. The relative laser power K used to display this frame is given by mID.

The image is projected by displaying F on the imaging SL , while sequentially displaying the N hologram subframes on the hologram SLM. For further details reference may be made to WO2010/007404 (hereby incorporated by reference).

Diffractive light formation using an SLM with non-rectangular pixels

Referring now to Figure 5, this shows a schematic representation of an SLM 500 according to an embodiment of the invention comprising a rectangular array of irregular optical phase modulating pixels 502. The usable diffraction efficiency of the device (that is, the light in the first-order diffraction image formed as a proportion of the total light illuminating the SLM) is related to the pixel shape and arrangement in a non-trivial way. It turns out that the optimal pixel shape and arrangement depend on a number of factors, including the angle at which the SLM is mounted in the system, and factors such as pixel dead- space that relate to the resolution of the semiconductor fabrication process employed when the SLM is made, but that in general optimal performance is obtained not from rectangular arrays of rectangular pixels, but from substantially rectangular arrays of pixels which are generally irregular hexagons, as illustrated. Note that, with this arrangement, the image projected by diffraction from such a device is still a rectangular array of rectangularly-spaced pixels - because the diffracted image is given by the Fourier transform of the array, the pixel shape of the diffracted image is not equal or related to the pixel shape of the SLM, but rather given by the Fourier transform of the outline of the entire array, which in this arrangement is still substantially rectangular. This contrasts with an imaging system in which the projected structure is the image of the surface of the display. We have previously described (GB Patent 2454246, US Patent Application 12/740,000 incorporated by reference) how one may provide an out-of-plane relief structure (topography) for each pixel. Thus a pixel can include (for example) a step such that one-half of the pixel is raised above the other by a few tens of nanometres, which can improve diffraction efficiency by up to 25%. We now describe how to optimally choose the in-plane structure (outline shape) of each pixel to further improve diffraction efficiency.

Generally, efficiency depends on pixel structure in two broadly independent ways:

1. Deadspace between pixels reduces efficiency, as light is lost in this space.

Therefore minimising the ratio of pixel perimeter to pixel area leads to optimal efficiency.

2. Pixellated SLMs lose light into higher diffraction orders, essentially due to diffraction that results from the shape of the pixel. Different pixel shapes lead to different relative proportions of light diffracted into the desired first order (where the image is formed) against higher diffraction orders (where any light is not usable and therefore wasted).

For consideration (1) above, the regular hexagon has the lowest perimeter to area ratio of any tesseliating shape, and therefore potentially satisfies this criterion. For consideration (2), the relative distribution of diffraction efficiencies into each order is determined by the Fourier transform of the pixel shape and it also turns out that, for tesseliating shapes, a hexagonal pixel (albeit not necessarily regular) maximises the proportion of light diffracted into the first diffraction order. In practice, the optimal hexagon is indeed irregular, with its shape determined by a number of variables, as described further later. One determining factor is the angle of light illuminating the SLM. Referring to Figure 6, in a typical liquid crystal (LCOS) SLM 602, the device rotates polarisation as well as modulating phase, so incident light can be separated from diffracted light using a polarising beam splitter (PBS) 604, and therefore a normal angle of incidence can be employed. In a typical MEMS-based SLM 606, polarisation is not rotated, and therefore a PBS cannot straightforwardly be used to separate incident from diffracted light; instead the SLM may be mounted at an angle, for example 45 degrees, as shown. Similar considerations apply to other SLM technology used to modulate phase. Referring now to Figure 7, this shows four pixels from an SLM pixel array, illustrating the dimensions used to determine a pixel shape parameter: mathematical modelling can be used to determine the optimal pixel shape for a given illumination condition and SLM pixel array structure. Assuming a hexagonal structure, variables defining a general display pixel layout are the pixel pitch (horizontal Δ and vertical A _y ), pixel deadspace ADS, and the hexagon height H. A normalised shape parameter S can then be defined in ter

Such a definition, while appearing complex, is very useful as it gives a range of meaningful settings of S, corresponding to different situations, which are independent of the exact values of Δ _Χ , A _y and A _D s, as shown in Table 1 , below. Figures given are for a typical display condition: a 16:9 aspect ratio image (e.g. WVGA, 854 x 480) fitting within the top half of the first diffraction order, such that the height of the image occupies 90% of the available height of the top half of the first diffraction order, as shown in Figure 8.

Table 1 - Examples of shape parameters S and their meanings.

The shape parameter S determines the diffraction efficiency of the SLM, and thus for a particular application one can optimise this shape parameter and hence determine an optimum shape for pixels of the SLM, some examples of which are given in Table 1. There is no analytic expression to determine the value of S to optimise diffraction efficiency for a given set of input variables but if desired a value for S can be determined to high accuracy using numerical analysis techniques.

As an alternative to a calculation based upon a particular optical arrangement, for example an arrangement as described previously, an approximate value of the shape parameter may be employed, as indicated in Table 1. The shape parameter effectively defines the pixel shape; for a small pixel dead space the hexagonal shape approaches a diamond shape - that is the vertically orientated sides of the hexagon have a length which approaches zero. In practice the dead space between pixels (which should be small) is limited by the fabrication process and the need to provide clearance between adjacent pixels. Then for a given desired shape parameter knowing the pixel dead space enables values of the horizontal and vertical pixel pitch, and pixel height to be determined. The SLM can then be fabricated with pixels of this shape. In general, it is desirable for the SLM to produce a diffracted image as large as possible, which makes small pixels desirable. This is subject to a potential constraint that the drive electronics fit underneath a pixel - although it is also practical to put the electronics to one side of the SLM. Additionally, it may be desirable for the first diffraction order, 800 in Figure 8 to be square in aspect ratio, because for various reasons (including the desire to avoid the zero-order undiffracted light region 802 in the middle of the first order) it is practical often to use only one half 804 of the first diffraction order. As this half has an aspect ratio of 2:1 this is suitable for containing a wide variety of image formats. For a MEMS-based SLM illuminated at 45 degrees, a square first diffraction order is formed when the pixel pitch (distance between centres of adjacent pixels) is a factor of sqrt(2) longer in the x-direction than in the y-direction. In this configuration, a typical suitable pixel pitch may be around 14 x 9.9 um, and MEMS-type processing considerations may give rise to a deadspace of around 0.5 um between adjacent pixels. For this configuration, if we assume the pixel contains a step (as described in GB Patent 2454246), mathematical analysis can show that the optimal shape parameter to maximise diffraction efficiency is around 0.775, as shown in Table 2 below:

Table 2 - Example of optimal pixel layout configurations for large deadspace

(left) and smaller deadspace (right).

In the configuration described above with 0.5 um deadspace, typical for standard MEMS processes, and assuming a stepped pixel which can modulate eight equally- spaced phase levels, diffraction efficiency is improved from 52.7% (using rectangular pixels) to 56.5% (using optimal hexagonal pixels), an increase of 7.2%. Naturally, those skilled in the art will recognise that this represents just one possible configuration and that many other alternative configurations are possible, with no material change to the invention.

Thus broadly speaking we have described apparatus for forming an image or an illumination structure by diffraction, the apparatus comprising a pixellated spatial light modulator with substantially non-rectangular pixels. The light illuminating the SL is preferably coherent although in principle a light emitting diode with a sufficiently small source size may alternatively be employed. In embodiments the width and height of the pixels may be unequal; more particularly the pixels may be substantially hexagonal in shape. In embodiments the shape of the pixels is chosen to maximise diffraction efficiency of the image/illumination formation, and the pixels act to modulate phase of light incident on the SLM.

The SLM may be fabricated using a range of technologies including, but not limited to: liquid crystal, magneto-optic, acousto-optic, optically-addressed, Kerr or Pockels effect- based, MEMS (micro-electro-mechanical-system), or other technologies. A MEMS SLM may have pixels which translate along an axis substantially perpendicular to the device surface and/or which tilt around one or more axes substantially parallel to the device surface. Movement of the MEMS pixels may be controlled, for example, by electrostatic, magnetic or thermo-mechanical means. No doubt many other effective alternatives will occur to the skilled person. It will be understood that the invention is not limited to the described embodiments and encompasses modifications apparent to those skilled in the art lying within the spirit and scope of the claims appended hereto.