WO2008120122A2

WO2008120122A2 - Wavefront constructor and method of constructing wavefronts

Info

Publication number: WO2008120122A2
Application number: PCT/IB2008/051017
Authority: WO
Inventors: Peter Christiaan Schmale
Original assignee: Koninklijke Philips Electronics, N.V.; U.S. Philips Corporation
Priority date: 2007-03-30
Filing date: 2008-03-18
Publication date: 2008-10-09
Also published as: WO2008120122A3; TW200907614A

Abstract

A wavefront constructor (10) comprises a double balanced mixer (12) having an input (22) and an output (24), the input for receiving a wavefront matrix of complex number pixel values, wherein the double balanced mixer is configured for converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self- interference of the input and (ii) a conversion carrier frequency fc, and outputting the matrix of real number pixel values on the output (24), wherein the output matrix of real number pixel values is suitable for use by a spatial light modulator (SLM) (14) for providing a corresponding reproduced wavefront (30).

Description

WAVEFRONT CONSTRUCTOR AND METHOD OF CONSTRUCTING WAVEFRONTS

The present embodiments relate generally to image reproduction systems and more particularly, to a wavefront constructor and method of constructing wavefronts.

Photographic plate based holography, and computer generated holography, suffer from common artifacts. The artifacts are additional images that an observer sees when looking through the hologram. Apart from distorting the picture, the artifacts also lead to a low efficiency in (re)generating the desired picture.

Accordingly, an improved method and system for overcoming the problems in the art is desired.

Figure 1 is a block diagram view of a wavefront constructor according to one embodiment of the present disclosure;

Figure 2 is a cross-sectional diagram view of a portion of a spatial light modulator of the wavefront constructor according to an embodiment of the present disclosure;

Figure 3 is a diagram view illustrating an amount of transmitted light through a spatial light modulator with a given polarizer direction, a maximal liquid crystal (LC) helix rotation, and analyzer direction of a first example;

Figure 4 is a diagram view illustrating an amount of transmitted light through a spatial light modulator with a given polarizer direction, a partially unfolded liquid crystal (LC) helix rotation, and analyzer direction of a second example;

Figure 5 is a diagram view illustrating an amount of transmitted light through a spatial light modulator according to an embodiment of the present disclosure with a given polarizer direction, a completely unfolded liquid crystal (LC) helix rotation, and analyzer direction of a third example;

Figure 6 is a diagram view illustrating an amount of transmitted light through a spatial light modulator according to an embodiment of the present disclosure with a given polarizer direction, a maximal liquid crystal (LC) helix rotation, and analyzer direction of a fourth example;

Figure 7 is a diagram view illustrating a wavefront capturing arrangement;

Figure 8 is a diagram view illustrating a wavefront reproduction arrangement;

Figure 9 is a illustrative view of an original, 2-dimensional, object used as an example herein; Figure 10 is an illustrative view of a holographic reproduction of the object of Figure 9, reproduced using known computer generated holography techniques;

Figure 11 is an illustrative view of a reconstruction result obtained by eliminating the reference beam during wavefront capturing;

Figure 12 is an illustrative view of an un- sharp diffraction of a conjugate image;

Figure 13 is an illustrative view of a conjugate image corresponding to a dual version of an object that is regenerated at an opposite side of a hologram;

Figure 14 is a spatial frequency distribution view of an object that is modulated by an interference pattern that functions as a carrier with frequency £ = 1 / λ_interference;

Figure 15 is a spatial frequency distribution view of an interference pattern obtained with a reference beam under an angle θ_c > 0;

Figure 16 is a spatial frequency distribution view of an interference pattern illuminated with a reference beam at an angle to the hologram plane θ_c and where the object image is reproduced in a beam at an angle θ_r = θ_c to the interference pattern;

Figure 17 is an illustrative view of an example of an object reproduced in a beam at an angle θ_r = θ_c to the interference pattern;

Figure 18 is the same illustrative view of Figure 17 showing highlighted portions of the example of the object reproduced in a beam perpendicular to the interference pattern;

Figure 19 is a spatial frequency distribution view of an interference pattern calculated from a desired wavefront distribution using double-balance mixing with a reference beam according to one embodiment of the present disclosure, to be compared with the spatial frequency distribution of Figure 15;

Figure 20 is an illustrative view of an example of a simulated reproduction of an object reproduced or calculated from a desired wavefront distribution using double -balance mixing with a reference beam according to one embodiment of the present disclosure;

Figure 21 is an illustrative view of an example of a simulated reproduction of an object reproduced or calculated from a desired wavefront distribution using double -balance mixing with a reference beam and parameters for an optimum separation of the conjugate image from a desired image in a sampled system according to another embodiment of the present disclosure;

Figure 22 is a spatial frequency distribution view of an effect of positioning the conjugate image on the Nyquist frequency (for simplicity sake only shown in one direction), which folds its spatial frequency band around the Nyquist frequency, to achieve a splitting effect according to another aspect of the embodiments of the present disclosure; and

Figure 23 is a simulated pictoral view of an image of the object of Figure 9 wherein the conjugate images of Figure 21 have been suppressed via an optical low-pass filter and a contrast of the desired image is improved according to another aspect of the embodiments of the present disclosure.

In the figures, like reference numerals refer to like elements. In addition, it is to be noted that the figures may not be drawn to scale.

A method is disclosed herein for generating a grid of light wavefronts, the method comprising (1) calculation of real SLM (spatial light modulator) coefficients, (2) coherent light beam modulation with an SLM, and (3) optical low pass filtering. The method avoids the artefacts common to computer generated holograms. Applications include 3-D imaging, mobile displays and lighting. Advantageous elements of the method and apparatus include (1) use of double balanced mixing to eliminate carrier pattern and self- interference artefacts of conventional hologram, (2) use of an SLM with light -polarizer, rotator and analyzer, permitting positive and negative real light output, (3) use of a spatial frequency shift of a conjugate image artefact towards an edge of the Nyquist band, and (4) an elimination of a conjugate image artefact by means of an optical low pass filter.

A wavefront constructor generates a grid of light wavefronts. The wavefront constructor is provided with: (1) double balanced mixing means to eliminate the carrier pattern and self-interference artifacts of conventional holograms, (2) spatial light modulator ("SLM") with light-polarizer, rotator and analyzer, permitting positive and negative real light output, (3) means for spatial frequency shifting of conjugate image artifacts towards an edge of a Nyquist band and (4) an optical low pass filter to eliminate conjugate image artifacts. Any light source may be implemented as a form of the wavefront constructor. The invention avoids artifacts common to computer generated holograms. Applications include 3-D imaging, mobile displays, lighting, and the like.

One object of the embodiments of the present disclosure is to be able to (re)produce wavefronts. This ability permits the reproduction of scenes including all the depth information, in contrast to normal photography and television, which only produce a flat image at the plane of the display itself. The ability to reproduce wavefronts is currently associated with the use of holography.

Turning now to the drawings, Figure 1 is a block diagram view of a wavefront constructor 10 according to one embodiment of the present disclosure. Wavefront constructor 10 comprises a double balanced mixer 12, a spatial light modulator (SLM) 14, a collimated light beam source 16, a spatial light modulator controller 18, and an optical low-pass filter 20. Double balanced mixer 12 has an input 22 and an output 24, the input 22 for receiving a wavefront matrix of complex number pixel values. In one embodiment, the double balanced mixer 12 is configured for converting the wavefront matrix of complex number pixel values on input 22 into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self- interference of the input and (ii) a conversion carrier frequency fc, as will be discussed further herein below. The double balanced mixer 12 outputs the matrix of real number pixel values on the output 24.

The output matrix of real number pixel values is suitable for use by the spatial light modulator (SLM) 14 for providing a corresponding reproduced wavefront 30. SLM 14 comprises an optical input 28 and an optical output 30. SLM 14 is coupled to the double balanced mixer 12, to be discussed further herein below, and is configured for providing the reproduced wavefront on the optical output 30 as a function of the matrix of real number pixel values 24. SLM 14 further includes means responsive to a reference beam incident on the optical input 28 at a reference beam angle and a matrix of real number pixel values 24 for providing a reproduced wavefront on the optical output 30 as a function of the matrix of real number pixel values. In one embodiment, the matrix of real number pixel values comprises positive and negative values.

Light source 16 is optically coupled to the optical input 28 of the SLM 14 for providing the incident reference beam at a reference angle θ with respect to the optical input 28 of the SLM 14. SLM controller 18 is configured for controlling the SLM 14 according to the matrix of real number pixel values 24 via suitable drive signals on signal line 26. Signal line 26 is illustrated by a single line; however, it could also include multiple lines. Responsive to the reference beam 28 and to the controller control signals on signal line 26, the SLM 14 provides the reproduced wavefront on the optical output 30 of the SLM. In one embodiment, the reference angle θ comprises an optimum angle θ_opt_imum in both an xz plane and a yz plane of the optical input 28. In another embodiment, the light source 16 comprises a laser and the reference beam comprises a collimated laser beam.

Optical low pass filter 20 is adapted to filter out conjugate images from the reproduced wavefront on the optical output 30 of the SLM 14, as will be discussed further herein below. In one embodiment, the optical low pass filter 20 includes a cut-off frequency on the order of one-half of a Nyquist frequency (f>j/2) of the reproduced wavefront on the optical output 30 of the SLM 14. In another embodiment, the optical low pass filter 20 comprises a portion of the SLM 20.

The method and apparatus of the present embodiments advantageously use low pass filtering. The apparatus of the present disclosure further includes a low pass filter that operates in the optical domain on the image resulting from illuminating the holographic pattern on the display. The use of low pass filtering suppresses the unwanted conjugate image which is associated with holographic image reconstruction from a real-valued holographic pattern. Furthermore, modulation in the embodiments of the present disclosure is used for advantageously separating the spectra of the desired image and the conjugate image.

Figure 2 is a cross-sectional diagram view of a portion 32 of the spatial light modulator 14 of the wavefront constructor 10 according to an embodiment of the present disclosure. Portion 32 is representative of a single display cell of the spatial light modulator 14. For example, in one embodiment, the matrix of real number pixel values comprises positive and negative values, wherein the SLM 14 comprises a signed range LCD configured to produce positive and negative light output, as will be discussed further herein below. The positive light output can comprise, for example, a first phase of light and the negative light output can comprise, for example, a second phase of light, wherein the first phase is different from the second phase by a phase difference of 180 degrees.

The signed range LCD comprises a polarizer 34, a liquid crystal (LC) material 36, and an analyzer 38, wherein the liquid crystal material 36 is characterized by a liquid crystal helix rotation having an operating range between a first rotation and a second rotation, as will be discussed further herein with reference to Figures 5 and 6. In addition, the polarizer 34 and analyzer 38 are configured with respect to the liquid crystal material 36 for positioning the middle of the liquid crystal helix rotation operating range to yield a minimum light transmission of the SLM 14. Portion 32 further includes electrodes 40 and 42. Incoming light is represented by arrows 44 and modulated light is represented by arrows 46.

As noted herein, the output 24 of the double balanced mixer 12 comprises a real number that can have positive and negative values. The positive and negative values output by the double balanced mixer of the method provides an advantage to suppress unwanted carrier and self-interference components, to be discussed further herein below. While negative outputs can be propagated fine in the simulation model, the embodiments of the present disclosure advantageously provide a method and apparatus for enabling negative outputs to work with a real Spatial Light Modulator comprising an improved LCD; however, a normal LCD only accepts a positive input signal.

To understand the concept of "negative light", consider the following discussion. "Negative light", as used herein, is a phase reversal compared to the positive counterpart. As with any definition of a negative, when you add light of a certain intensity and its negative (with opposite phase), the result is zero. In one embodiment herein, the positive light output comprises a first phase of light and the negative light output comprises a second phase of light, wherein the first phase is different from the second phase by a phase difference of 180 degrees. As a result, the LC based SLM device 14, as discussed herein, is capable of rendering positive and negative light values. Such an LCD is called a signed range LCD.

Referring again to Figure 2, incoming light 44 is passed through a polarizer 34 which filters out only light that is polarized in one direction. The light then passes through the LC material 36 which has a helix structure when no voltage is applied on the electrodes 40,42. The helix rotates the polarization of the light. This helix can be dimensioned such that the polarization is rotated by 90 degrees. The analyzer 38 has a filtering direction which is perpendicular to that of the polarizer 34.

Figure 3 is a diagram view 48 illustrating an amount of transmitted light 50 through a spatial light modulator with a given polarizer direction 52, a maximal liquid crystal (LC) helix rotation 54, and analyzer direction 56 of a first example. Because of the polarization rotation caused by the helix, incoming light can pass through the analyzer; the transmission in this situation is maximal (Figure 3). In other words, the transmitted light 50 is maximal. Figure 4 is a diagram view 58 illustrating an amount of transmitted light 60 through a spatial light modulator with a given polarizer direction 52, a partially unfolded liquid crystal (LC) helix rotation 64, and analyzer direction 56 of a second example. When an electrical voltage is applied at the electrodes (40,42), the helix unfolds and the rotation becomes less. The result is that less light passes (See Figure 4). In other words, the amount of transmitted light 60 is reduced. When sufficient voltage is applied that the helix unfolds completely, then the transmission is minimal.

According to one embodiment of the present disclosure, a signed range LCD can be constructed by positioning the polarizer and analyzer such that the middle of the operating range of the helix yields minimum transmission. An example of this is presented with reference to Figures 5 and 6. Figure 5 is a diagram view 68 illustrating an amount of transmitted light 70 through a spatial light modulator according to an embodiment of the present disclosure with a given polarizer direction 72, a completely unfolded liquid crystal (LC) helix rotation 74, and analyzer direction 76 of a third example. In other words, the helix is totally unfolded and the transmitted light 70 has the polarization vector to the left.

Figure 6 is a diagram view 78 illustrating an amount of transmitted light 80 through a spatial light modulator according to an embodiment of the present disclosure with a given polarizer direction 72, a maximal liquid crystal (LC) helix rotation 84, and analyzer direction 76 of a fourth example. In other words, the helix causes rotation of the polarization and the transmitted light 80 has the polarization vector to the right. The phase of the transmitted light in Figure 6 is the opposite of that of the transmitted light in Figure 5. It can be considered "negative". With a helix rotation in between these two positions (74 and 84), the light transmission will become minimal, reaching the "zero" point.

Turning now to Figure 7, the figure is a diagram view illustrating a wavefront capturing arrangement 90. With respect to wavefront capturing and reproduction technology, holography was invented in 1947 by Dennis Gabor, for which he received the Nobel Prize in physics in 1971. Holography permits capturing and reproducing the wavefronts arising from an object or scene illuminated with coherent light and thereby reproducing the complete 3 dimensional scene for an observer. Figure 7 shows a typical capturing arrangement 90. The illumination of the object 92 and the reference beam 94 are typically derived from a common laser source (not shown). The object beam 96 and reference beam 94 create an interference pattern on the hologram plane 98. This interference pattern can be captured, for example, on a photographic plate.

Figure 8 is a diagram view illustrating a wavefront reproduction arrangement 100. In Figure 8, the hologram is the developed photographic plate 98 used in the capturing process, with an interference fringe pattern on it. This hologram is illuminated with the same reference beam 94 as was used during capturing. The light beam shining through the hologram contains the diffracted object beam 104, just like it would be when the object was really in its original place. When looking through the hologram, an observer will see the reproduced object image 102 (dotted in the drawing).

More recently, Computer Generated Holography was introduced. This technique simulates the holographic capturing process using mathematical models of objects and light propagation in a computer. In the model, the light intensity distribution on the plane of the hologram is calculated. For reproduction, an SLM (Spatial Light Modulator), such as an LCD, is used instead of the photographic plate. The SLM is controlled to display the calculated light intensity distribution. This gives the possibility to dynamically generate wavefronts representing 3-dimensional objects.

Both the original, photographic plate based holography, and the computer generated holography, suffer from some common artefacts. The artefacts are additional images that the observer sees when looking through the hologram. Apart from distorting the picture, the artefacts also lead to a low efficiency in (re)generating the desired picture. Figures 9 and 10 illustrate this. Figure 9 shows an original (2-dimensional) object 106 and Figure 10 shows the object reproduced with conventional computer generated holography (where the result was simulated with a computer model of light propagation, with θ = 0).

In particular, Figure 10 is an illustrative view of a holographic reproduction 108 of the object 106 of Figure 9, reproduced using known computer generated holography techniques. Figure 10 shows a reduced contrast (and therefore a low efficiency of the generated image) because of the reference beam adding a background illumination to the original picture. Elimination of the reference beam during capturing gives a reconstruction result 110 as shown in Figure 11. Due to the self- interference of the object beam, the holographic principle works also without a reference beam during capturing. The artefacts that are now visible are due to the so-called conjugate image. This is a dual version of the object that is regenerated at the other side of the hologram (see Figure 13). The artefact seen is the un-sharp diffraction of this image towards the plane where an observer observes the reproduced picture. Figure 12 shows this un-sharp diffraction 112 of a conjugate image. This is clearly recognizable as the artefacts in Figure 11.

Figure 13 is an illustrative view of a conjugate image 116 corresponding to a dual version of an object 118 that is regenerated at an opposite side of a hologram 120. This hologram is illuminated with reference beam 122. The light beam shining through the hologram contains the diffracted object beam 124, just like it would be when the object was really in its original place. When looking through the hologram, an observer will see the reproduced object image 118 (dotted in the drawing) behind the hologram 120, as well as the conjugate image 116 in front of the hologram 120. Note that these results were obtained with θ=0, which is not usual. More commonly, use is made of the fact that the conjugate image beam comes out of the hologram at an angle of 2Θ with respect to the wanted image. With θ sufficiently large (several tens of degrees), the conjugate image can be rotated out of the viewing area. For SLM based holography; however, the maximum value of θ is currently limited to only a few degrees because of the relatively large pixel size of currently available SLM 's, and the conjugate image can not be rotated out of sight.

Holography was discovered more or less by coincidence and not by design. The exact explanation why it works and why this is the "best" way for reproducing wavefronts is hard to find in literature. The usual explanation is that the holographic photograph captures not only the amplitude but also the phase of the incident light, which enables reproduction of the wavefront. The following describes a somewhat different view which is believed to involve a new understanding or insight.

In fact, a photographic plate is not able to capture any phase information of light. It only captures amplitude (or rather, intensity) information. The capturing process resets the phase of all captured wavefronts to the same value, as it were. This loss of phase information is the fundamental cause for the generation of the conjugate image, which can be understood as follows: resetting the phase is like adding another wavefront with opposite phase (and this is a wavefront of the conjugate image). So we add the conjugate image while we throw away the phase information. This conjugate image gives considerable distortion in the reproduction as can be seen for example in Figure 11.

The need for a reference beam during holographic capture, as is common practice, is not immediately obvious. Indeed, the example of Figure 11 shows that reproduction, albeit with artefacts, is entirely possible without using a reference beam during capturing, and it has the advantage of not reducing the contrast of the reproduced picture. Making a holographic recording without reference beam depends on the so called self-interference component, which is sometimes identified in literature as one of the unwanted outputs of the holographic reproduction process.

It turns out that the only use for a reference beam during capturing, is to separate the original and the conjugate image. This process can be analysed in the spatial frequency domain. For simplicity sake, the following discussion considers only one dimension; this can easily be extended to the two dimensions of the plane on which a real holographic interference pattern is captured. When a reference beam with angle θ_c > 0 is used, the addition of the reference beam and the object beam causes an interference pattern to appear on the plane of the hologram, with wavelength interference = λi_lgrit / sin θ_c. This interference pattern subsequently functions as a carrier with frequency f_c = 1 / λ_interference on which the spatial frequency distribution of the object is modulated. Figure 14 is a spatial frequency distribution view 126 of an object 128 that is modulated by an interference pattern that functions as a carrier 130 with frequency f_c = 1/ λ_interference. The modulation occurs because of the non-linear process of taking the intensity (a quadratic function) when exposing the photographic plate (or the computer simulation of this process). The modulation process causes the generation of two sidebands, where one sideband represents the captured original object, and the other sideband represents the conjugate image. Figure 15 is a spatial frequency distribution view 132 of an interference pattern obtained with a reference beam under an angle θ_c > 0. In Figure 15, the spatial frequency distribution of the conjugate image is identified by reference numeral 134, the spatial frequency distribution of the self- interference of the object is identified by reference numeral 136, and the spatial frequency distribution of the object is identified by reference numeral 138.

Note that the spatial frequency scale corresponds to the beam angle θ. So if we were to illuminate the interference pattern (hologram), having the spatial frequency distribution of Figure 15, with a beam perpendicular to the hologram plane (θ_r = 0), the object image beam would appear under an angle θ_c. To obtain the image beam under an angle θ=0, it is common practice to place the illumination beam during reproduction under the angle θ_r = θ_c. In the spatial frequency domain, this corresponds to a shift like shown in Figure 16. Figure 16 is a spatial frequency distribution view 140 of an interference pattern illuminated with a reference beam at an angle to the hologram plane θ _c and where the object image is reproduced in a beam at an angle θ_r = θ_c to the interference pattern, corresponding to a shift in the spatial frequency distribution. In Figure 16, the spatial frequency distribution of the conjugate image is identified by reference numeral 134, the spatial frequency distribution of the self- interference of the object is identified by reference numeral 136, the spatial frequency distribution of the object is identified by reference numeral 138, and the dc component is identified by reference numeral 142.

Now the object is reproduced in a beam perpendicular to the hologram. An example of such image reproduction is shown in Figure 17 (θ = 0.73 degrees). In other words, Figure 17 is an illustrative view 144 of an example of an object reproduced in a beam at an angle θ_r = θ_c to the interference pattern. Figure 18 is the same illustrative view of Figure 17 showing highlighted portions of the example of the object reproduced in a beam perpendicular to the interference pattern.

Referring now to Figures 17 and 18, we can note the following: Gray background which reduces contrast, caused by the reference beam. The dominating reproduced image 146 (not exactly in the middle) is the unwanted self- interference. The vague square 148 to the upper-left from the self-interference is the wanted, reproduced object image. The vague square 150 on the other side of the self- interference is the unwanted conjugate image. Both reproduced object image and conjugate image show heavy interference lines caused by the energy from the self-interference image

The poor reproduction performance (even foregoing the conjugate image for the moment) is caused by the carrier /1, which transforms into a dc component, and the self- interference. These two components are typical of the non-ideal mixing of object beam and reference beam which takes place in the photographic plate and is simulated in prior-art Computer Generated Holography.

According to the embodiments of the present disclosure, Computer Generated Holograms need not follow the limitations of conventional photography. According to one embodiment of the present disclosure, a Computer Generated Hologram is calculated from the desired wavefront distribution by applying double-balance mixing with a reference beam. Figure 19 is a spatial frequency distribution view of an interference pattern calculated from a desired wavefront distribution using double-balance mixing with a reference beam according to one embodiment of the present disclosure, to be compared with the spatial frequency distribution of Figure 15. In Figure 19, the spatial frequency distribution of the conjugate image is identified by reference numeral 154, the spatial frequency distribution of the self- interference of the object has been suppressed (not shown), the spatial frequency distribution of the object is identified by reference numeral 156, and the suppressed carrier f_c is identified by reference numeral 158.

A simulation of reproduction obtained by applying double-balance mixing with a reference beam is shown in Figure 20 (θ = 0.73 degrees). That is, Figure 20 is an illustrative view 160 of an example of a simulated reproduction of an object reproduced or calculated from a desired wavefront distribution using double-balance mixing with a reference beam according to one embodiment of the present disclosure. The only artefact remaining in Figure 20 is the conjugate image, as indicated by reference numeral 162. Note the greatly improved contrast of the wanted image, compared to that of Figure 17.

Now we want to determine the optimum angle θ, which will give us as much separation of the conjugate image as possible. Note that in practice, all Computer Generated Hologram systems will employ a matrix structure for reproducing the hologram. This turns the display system into a sampled system, to which Nyquist's formula for the maximum sampled bandwidth applies. When the dimension of the matrix display is "/" and the number of pixels is "«", then the Nyquist spatial frequency of the system is: f_N = n / (2 * l) .

Since all spatial frequencies must be contained in the interval -fy ... + /_N, and according to Figure 19, both the wanted object image and the conjugate image must fit in this interval, the optimum and biggest distance between the two spectra is equal to /_N. Note that the separation between the two spectra is equal to 2 * f_c, and according to earlier formulae,/^ = sin θ_c / λ_llgllt.

According to the invention, optimum separation of the conjugate image from the wanted image in a sampled system is achieved for θ_opt_imum = arcsin {{n * λhght) / (4 * I)). Due to the sampled nature of the system, there exist more θ for which optimum separation is achieved. Each such θ must satisfy the following condition: sin θ = (2*k + 1) * (n * λ_llgllt) / (4 * 1), where k is any integer.

In our examples we have been using n = 1024, λ_llgllt = 10^"6 and / = 10^"2. Then θ_optimum = 1.47 degrees. The distance between object plane and hologram plane d = 4 * 10^"2. The result of generating an image with these parameters is shown in Figure 21. In particular, Figure 21 is an illustrative view 164 of an example of a simulated reproduction of an object reproduced or calculated from a desired wavefront distribution using double-balance mixing with a reference beam and parameters for an optimum separation of the conjugate image from a desired image in a sampled system according to another embodiment of the present disclosure.

In Figure 21, the conjugate image is split into four parts identified by reference numerals 1621, 1622, 1623, and 1624. We see that the conjugate image is now split into four parts, each with a quarter of the original intensity. This peculiar effect is caused by positioning the conjugate image exactly on the Nyquist frequency both in x and y direction, which folds its spatial frequency band around the Nyquist frequency, see Figure 22. In other words, Figure 22 is a spatial frequency distribution view of an effect of positioning the conjugate image on the Nyquist frequency (for simplicity sake only shown in one direction), which folds its spatial frequency band around the Nyquist frequency, to achieve a splitting effect according to another aspect of the embodiments of the present disclosure. In Figure 22, one half spatial frequency distribution of the conjugate image at -^_N is identified by reference numeral 168, the spatial frequency distribution of the object is identified by reference numeral 170, and one half spatial frequency distribution of the conjugate image at _/_N is identified by reference numeral 172. This splitting effect is achieved whenever the sum of the angles of the reference beam and the reconstruction beam equals twice the optimum angle: tfc ' t/r ^ t/optimum ■

Figure 22 also points the way on how to take the conjugate image out of the reproduced image.

According to another embodiment of the present disclosure, the conjugate image is eliminated from the reproduced image as follows: (i) use double balanced mixing, as discussed herein, (ii) use θ_opt_imum for the reference beam angle during calculation of the Computer Generated Hologram, (iii) also use θ_opt_imum as the angle of the illumination beam during reproduction of the Computer Generated Hologram, and (iv) apply an optical low- pass filter with an approximate cut-off frequency of (fN / 2) to the hologram displayed by the SLM.

The result of generating the image is shown in Figure 23. In other words, Figure 23 is a simulated pictoral view 174 of an image of the object of Figure 9 wherein the conjugate images of Figure 21 have been suppressed via an optical low-pass filter and a contrast of the desired image is improved according to another aspect of the embodiments of the present disclosure. The conjugate image is completely eliminated and the contrast of the picture is further improved. The low-pass filter can be dimensioned to obtain the best trade-off between suppressing the conjugate image and the sharpness of the wanted image.

In operation, a system for using the embodiments of the present disclosure can include a wavefront constructor subsystem and a wavefront calculator subsystem. The wavefront calculator subsystem produces a matrix of desired wavefront values, representing a desired scene, which serves as input to the wavefront constructor. The wavefront constructor produces light, approximately in accordance with the matrix of desired wavefront values. An observer looks at this light and obtains the visual impression of the desired scene.

Since the system can, within certain design limits, display arbitrary wavefronts, the observer can see a scene that can be 3 -dimensional or a scene consisting of a 2-dimensional plane that is positioned before or after the plane of the display of the waveform constructor, etc.

The system can be constructed to show static pictures or moving pictures. In case of moving pictures, the wavefront calculator is to generate a sequence of wavefront matrices, with sufficient refresh rate, that are to be displayed in succession by the wavefront constructor, like in conventional television.

The way in which the desired scene and its calculated wavefront representation are obtained is beyond the scope of the embodiments of the present disclosure. However, for the examples discussed herein and shown, a 2-dimensional object of lcm* lcm is taken which is at 4 cm from the (holographic) display surface. The wavefront at the object plane is trivially obtained from the bitmap representation. This wavefront is then numerically propagated over 4 cm using the frequency transfer characteristic of free space according to the Fresnel approximation. This renders the matrix of desired wavefronts that is the input to the wavefront constructor. The wavefront constructor reproduces these wavefronts and the observer sees the object 4 cm behind the plane of the display. Alternative ways to obtain the matrix of desired wavefronts include: ray-tracing from a computer-graphics rendered 3-d scene, derive it from a 3 -dimensional camera system which captures wavefronts, or other suitable method. In the example, the wavefront constructor has an input of 1024 * 1024 wavefront values. Each wavefront value is expressed as a complex number, allowing representation of both amplitude and phase of the wavefront. Note that a scalar representation of the light wave is used, in other words the vector direction is assumed to be irrelevant. In addition, the described system is monochrome (using one wavelength of light). The embodiments of the present disclosure can also be applied to colour systems, which would employ 3 parallel versions of the monochrome system, one for each primary colour, and have a facility for merging the 3 pictures into one.

According to another embodiment, the wavefront constructor comprises optical components and electrical components. The optical part consists of a collimated laser beam that illuminates a signed range LCD. The angle under which the laser beam reaches the LCD is equal to θ_opt_imum, both in the xz plane and in the yz plane (See Figure 8). The output of the LCD passes through an optical low -pass filter to form the final wavefront. The electronic part comprises a computational unit "Double Balanced Mixer" and the LCD control unit. The double balanced mixer receives the desired wavefront matrix values as input. The output of this computational unit is the matrix of LCD pixel values. The LCD control unit drives the LCD in accordance with these values, in a conventional manner. The detailed operation of the computational unit and the explanation of the signed range LCD are discussed elsewhere herein.

In operation, the double balanced mixer operates as follows. The input wavefront matrix of complex number pixel values comprises for each pixel (x,y), a complex number represented by the expression:

W (X, y) = W_re (X, v) + Z W_lm (x, y) , where w_re and w_im are real numbers, where x and y are non-negative integers in a range of (0,0) to (m-l,n-l), and where m and n are dimensions of the matrix of complex number pixel values. In one embodiment, m=1024 and n=1024. Further wherein, for each pixel of the input, the output of the double balanced mixer comprises: o (x, y) = w_re (x, y) * cos (φ_x,_y) + w_im (x, y) * sin (φ_x,_y) , where output o (x, y) is a real number, where φ_x,_y = α * (x + y) , and where α is a phase difference of the reference beam from one pixel to the next due to its inclination angle θoptimum. The phase difference can be determined as the proportion of the path difference between beams reaching adjacent pixel to the wavelength: α = 2*π*/*sin(θ_Optimum)/(«*λi_lght) , and substituting the earlier found value for θ_opt_imum, we obtain α = π/2. In other words, a phase of the reference beam under the optimum angle θ_opt_imum changes 90 degrees from pixel to pixel.

In addition, converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values by the double balanced mixer comprises an operation of multiplying the input wavefront matrix of complex number pixel values with the normalized complex amplitude value at each pixel of a simulated reference beam, incident at a reference beam angle θ_opt_imum with respect to a corresponding hologram in both an xz plane and a yz plane. In one embodiment, the double balanced mixer comprises a sampled system and wherein the reference beam angle θ_opt_imum comprises an angle equal to arcsin((/?*λi_lght)/(4*/)), where n is the number of pixels along one side of the SLM with dimension /, and λ_llgllt is the wavelength of light. Still further, the double balanced mixer further comprises a spatially sampled system that is frequency band limited by Nyquist criterion.

According to a further embodiment, in a simplified embodiment of the double balanced mixer, wherein the sin and cos of multiples of 90 degrees is easily determined, the double balanced mixer operates according to the pattern: o (0, 0) = w_re (0, 0) o (0, 1) = W_1n (0, 1) o (0, 2) = -w_re (0, 2) ... o (0, n) = -W_1n (0, n) o (1, 0) = W_1n (1, 0) o (1, 1) = -w_re (1, 1) o (1, 2) = -w_in (1, 2) o (2, 0) = -w_re (2, 0) o (2, 1) = -w_m (2, 1) o (2, 2) = w_re (2, 2) o (3, 0) = -w_ιn (3, 0) o (3, 1) = w_re (3, 1) o (3, 2) = w_ιn (3, 2) o (4, 0) = w_re (4, 0) o (4, 1) = W_1n (4, 1) o (4, 2) = - w_re (4, 2) o (m, 0) = -w_ιn (m, 0) ... ... o (m, n) = w_re (m, n).

Note that the above embodiment could also have taken α= -π/2, which would lead to a corresponding sign reversal in the pattern. Furthermore, shifts left or right of the pattern by one or more positions are also possible. According to one embodiment, a wavefront constructor apparatus comprises a double balanced mixer having an input and an output, the input for receiving a wavefront matrix of complex number pixel values, wherein the double balanced mixer is configured for converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self- interference of the input and (ii) a conversion carrier frequency fc, and outputting the matrix of real number pixel values on the output; a spatial light modulator (SLM) coupled to the double balanced mixer and having an optical output, the SLM configured for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values; a light source optically coupled to an optical input of the SLM for providing a reference beam at a reference angle θ with respect to the optical input of the SLM; a controller for controlling the SLM according to the matrix of real number pixel values, wherein responsive to the reference beam and to the controller control, the SLM provides the reproduced wavefront on the optical output of the SLM; and an optical low pass filter adapted to filter out conjugate images from the reproduced wavefront on the optical output of the SLM, wherein the optical low pass filter includes a cut-off frequency on the order of one-half of a Nyquist frequency (f>j/2) of the reproduced wavefront on the optical output of the SLM.

According to a further embodiment, a method of constructing a wavefront comprises: receiving a wavefront matrix of complex number pixel values; double balanced mixing the received wavefront matrix of complex number pixel values by converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self- interference of the input and (ii) a conversion carrier frequency fc; and outputting the matrix of real number pixel values, wherein the output matrix of real number pixel values is suitable for use by a spatial light modulator (SLM) for providing a corresponding reproduced wavefront. In one embodiment, wherein the wavefront matrix of complex number pixel values comprises for each pixel (x,y), a complex number represented by the expression:

W (X, y) = W_re (X, v) + Z W_lm (x, y) , where w_re and w_im are real numbers, where x and y are non-negative integers in a range of (0,0) to (m-l,n-l), and where m and n are dimensions of the matrix of complex number pixel values; and wherein for each pixel of the received wavefront matrix of complex number pixel values, the output matrix comprises:

O (X, y) = W_re (X, v) * COS ((p_x,_y) + W_lm (x, y) * SUl ((p_x,_y) , where output o (x, y) is a real number, where φ_x,_y = α * (x + y) , and where α is a phase difference of the reference beam from one pixel to the next due to its inclination angle

"optimum-

In a further embodiment, converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values by double balanced mixing comprises an operation of multiplying the received wavefront matrix of complex number pixel values with the normalized complex amplitude value at each pixel of a simulated reference beam, incident at a reference beam angle θ_optlmum with respect to a corresponding hologram in both an xz plane and a yz plane. Furthermore, the double balanced mixing comprises using a sampled system and wherein the reference beam angle θ_optlmum comprises an angle equal to arcsin((/?*λi_lght)/(4*/)), where n is the number of pixels along one side of the SLM with dimension /, and λi_lght is the wavelength of light. The method still further comprises: using a spatial light modulator (SLM) having an optical output and being responsive to the matrix of real number pixel values for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values, wherein the matrix of real number pixel values comprise positive and negative values, and wherein the SLM comprises a signed range LCD configured to produce positive and negative light output.

Although only a few exemplary embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings and advantages of the embodiments of the present disclosure. The embodiment can be advantageously applied in a wide range of holographic display applications. Some examples, include, but are not limited to: medical 3-D imaging, 3-D entertainment (television, theatre, and the like), head mounded displays (HMD) that are configured to display images close to the eye(s) of an observer, mobile displays (e.g., portable music / video player and mobile telephone), and lighting (e.g., controllable light beam for projecting ambient experience lighting, adjustable car headlights, etc.). Accordingly, all such modifications are intended to be included within the scope of the embodiments of the present disclosure as defined in the following claims. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents, but also equivalent structures.

In addition, any reference signs placed in parentheses in one or more claims shall not be construed as limiting the claims. The word "comprising" and "comprises," and the like, does not exclude the presence of elements or steps other than those listed in any claim or the specification as a whole. The singular reference of an element does not exclude the plural references of such elements and vice-versa. One or more of the embodiments may be implemented by means of hardware comprising several distinct elements, and/or by means of a suitably programmed computer. In a device claim enumerating several means, several of these means may be embodied by one and the same item of hardware. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to an advantage.

Claims

CLAIMS:

1. A wavefront constructor apparatus comprising: a double balanced mixer having an input and an output, the input for receiving a wavefront matrix of complex number pixel values, wherein the double balanced mixer is configured for converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self-interference of the input and (ii) a conversion carrier frequency fc, and outputting the matrix of real number pixel values on the output, wherein the output matrix of real number pixel values is suitable for use by a spatial light modulator (SLM) for providing a corresponding reproduced wavefront.

2. The apparatus of claim 1, wherein the wavefront matrix of complex number pixel values comprises for each pixel (x,y), a complex number represented by the expression:

W (X, y) = W_re (X, v) + Z W_lm (x, y) , where w_re and w_im are real numbers, where x and y are non-negative integers in a range of (0,0) to (m-l,n-l), and where m and n are dimensions of the matrix of complex number pixel values.

3. The apparatus of claim 2, further wherein for each pixel of the input, the output of the double balanced mixer comprises:

O (X, y) = Wre (X, y) * COS (φ_x,_y) + W_lm (x, y) * SUl ((p_x,_y) , where output o (x, y) is a real number, where φ_x,_y = α * (x + y) , and where α is a phase difference of the reference beam from one pixel to the next due to its inclination angle θ_opt_imum.

4. The apparatus of claim 1, wherein converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values by the double balanced mixer comprises an operation of multiplying the input wavefront matrix of complex number pixel values with a normalized complex amplitude value at each pixel of a simulated reference beam, incident at a reference beam angle θ_opt_imum with respect to a corresponding hologram in both an xz plane and a yz plane.

5. The apparatus of claim 4, wherein the double balanced mixer comprises a sampled system and wherein the reference beam angle θ_opt_imum comprises an angle equal to arcsin((/?*λi_lght)/(4*/)), where n is the number of pixels along one side of the SLM with dimension /, and λ_llgllt is the wavelength of light.

6. The apparatus of claim 4, wherein a phase of the reference beam under the optimum angle θ_opt_imum changes 90 degrees from pixel to pixel.

7. The apparatus of claim 4, wherein the double balanced mixer operates according to the pattern: o (0, 0) = w_re (0, 0) o (0, 1) = w_m (0, 1) o (0, 2) = -w_re (0, 2) ... o (0, n) = -w_m (0, n) o (1, 0) = W_1n (1, 0) o (1, 1) = -w_re (1, 1) o (1, 2) = -w_im (1, 2) o (2, 0) = -w_re (2, 0) o (2, 1) = -w_m (2, 1) o (2, 2) = w_re (2, 2) o (3, 0) = -w_m (3, 0) o (3, 1) = w_re (3, 1) o (3, 2) = w_in (3, 2) o (4, 0) = w_re (4, 0) o (4, 1) = W_1n (4, 1) o (4, 2) = -w_re (4, 2) o (m, 0) = -w_ιn (m, 0) ... ... o (m, n) = w_re (m, n).

8. The apparatus of claim 1, wherein the double balanced mixer further comprises a spatially sampled system that is frequency band limited by Nyquist criterion.

9. The apparatus of claim 1, further comprising: a spatial light modulator (SLM) coupled to the double balanced mixer and having an optical output, the SLM configured for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values.

10. The apparatus of claim 9, wherein the matrix of real number pixel values comprises positive and negative values, and wherein the SLM comprises a signed range LCD configured to produce positive and negative light output.

11. The apparatus of claim 10, wherein the positive light output comprises a first phase of light and the negative light output comprises a second phase of light, wherein the first phase is different from the second phase by a phase difference of 180 degrees.

12. The apparatus of claim 10, wherein the signed range LCD comprises a polarizer, a liquid crystal material, and an analyzer, wherein the liquid crystal material is characterized by a liquid crystal helix rotation having an operating range between a first rotation and a second rotation.

13. The apparatus of claim 12, further wherein the polarizer and analyzer are configured with respect to the liquid crystal material for positioning the middle of the liquid crystal helix rotation operating range to yield a minimum light transmission of the SLM.

14. The apparatus of claim 9, further comprising: a light source optically coupled to an optical input of the SLM for providing a reference beam at a reference angle θ with respect to the optical input of the SLM; and a controller for controlling the SLM according to the matrix of real number pixel values, wherein responsive to the reference beam and to the controller control, the SLM provides the reproduced wavefront on the optical output of the SLM.

15. The apparatus of claim 14, wherein the reference angle θ comprises an optimum angle θ_opt_imum in both an xz plane and a yz plane of the optical input.

16. The apparatus of claim 14, wherein the light source comprises a laser and wherein the reference beam comprises a collimated laser beam.

17. The apparatus of claim 9, further comprising: an optical low pass filter adapted to filter out a conjugate image from the reproduced wavefront on the optical output of the SLM.

18. The apparatus of claim 17, further wherein the optical low pass filter filters out conjugate images from the reproduced wavefront on the optical output of the SLM.

19. The apparatus of claim 17, wherein the optical low pass filter comprises a portion of the SLM.

20. The apparatus of claim 17, wherein the optical low pass filter includes a cut-off frequency on the order of one-half of a Nyquist frequency (f>j/2) of the reproduced wavefront on the optical output of the SLM.

21. A wavefront constructor apparatus comprising: a double balanced mixer having an input and an output, the input for receiving a wavefront matrix of complex number pixel values, wherein the double balanced mixer is configured for converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self-interference of the input and (ii) a conversion carrier frequency fc, and outputting the matrix of real number pixel values on the output; a spatial light modulator (SLM) coupled to the double balanced mixer and having an optical output, the SLM configured for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values; a light source optically coupled to an optical input of the SLM for providing a reference beam at a reference angle θ with respect to the optical input of the SLM; a controller for controlling the SLM according to the matrix of real number pixel values, wherein responsive to the reference beam and to the controller control, the SLM provides the reproduced wavefront on the optical output of the SLM; and an optical low pass filter adapted to filter out conjugate images from the reproduced wavefront on the optical output of the SLM, wherein the optical low pass filter includes a cut-off frequency on the order of one-half of a Nyquist frequency (f>j/2) of the reproduced wavefront on the optical output of the SLM.

22. The apparatus of claim 21, wherein the wavefront matrix of complex number pixel values comprises for each pixel (x,y), a complex number represented by the expression: w (x, y) = w_re (x, y) + i w_im (x, y) , where w_re and w_im are real numbers, where x and y are non-negative integers in a range of (0,0) to (m-l,n-l), and where m and n are dimensions of the matrix of complex number pixel values; and wherein for each pixel of the input, the output of the double balanced mixer comprises:

O (X, y) = Wre (X, y) * COS ((p_x,_y) + W_lm (x, y) * SUl ((p_x,_y) , where output o (x, y) is a real number, where φ_x,_y = α * (x + y) , and where α is a phase difference of the reference beam from one pixel to the next due to its inclination angle θ_opt_imum.

23. The apparatus of claim 22, wherein converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values by the double balanced mixer comprises an operation of multiplying the input wavefront matrix of complex number pixel values with a normalized complex amplitude value at each pixel of a simulated reference beam, incident at a reference beam angle θ_opt_imum with respect to a corresponding hologram in both an xz plane and a yz plane, wherein the double balanced mixer comprises a sampled system and wherein the reference beam angle θ_opt_imum comprises an angle equal to arcsin((/?*λi_lgrit)/(4*/)), where n is the number of pixels along one side of the SLM with dimension /, and λi_lgrit is the wavelength of light, and wherein a phase of the reference beam under the optimum angle θ_opt_imum changes 90 degrees from pixel to pixel.

24. The apparatus of claim 21, wherein the matrix of real number pixel values comprise positive and negative values, and wherein the SLM comprises a signed range LCD configured to produce positive and negative light output, wherein the positive light output comprises a first phase of light and the negative light output comprises a second phase of light, wherein the first phase is different from the second phase by a phase difference of 180 degrees.

25. The apparatus of claim 24, wherein the signed range LCD comprises a polarizer, a liquid crystal material, and an analyzer, wherein the liquid crystal material is characterized by a liquid crystal helix rotation having an operating range between a first rotation and a second rotation, further wherein the polarizer and analyzer are configured with respect to the liquid crystal material for positioning the middle of the liquid crystal helix rotation operating range to yield a minimum light transmission of the SLM.

26. A method of constructing a wavefront comprising: receiving a wavefront matrix of complex number pixel values; double balanced mixing the received wavefront matrix of complex number pixel values by converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values while suppressing from the matrix of real number pixel values (i) a self- interference of the input and (ii) a conversion carrier frequency fc; and outputting the matrix of real number pixel values, wherein the output matrix of real number pixel values is suitable for use by a spatial light modulator (SLM) for providing a corresponding reproduced wavefront.

27. The method of claim 26, wherein the wavefront matrix of complex number pixel values comprises for each pixel (x,y), a complex number represented by the expression:

28. The method of claim 26, wherein converting the wavefront matrix of complex number pixel values into a matrix of real number pixel values by double balanced mixing comprises an operation of multiplying the received wavefront matrix of complex number pixel values with a normalized complex amplitude value at each pixel of a simulated reference beam, incident at a reference beam angle θ_opt_imum with respect to a corresponding hologram in both an xz plane and a yz plane.

29. The method of claim 28, wherein the double balanced mixing comprises using a sampled system and wherein the reference beam angle θ_opt_imum comprises an angle equal to arcsin((/?*λi_lght)/(4*/)), where n is the number of pixels along one side of the SLM with dimension /, and λ_llgllt is the wavelength of light.

30. The method of claim 26, further comprising: using a spatial light modulator (SLM) having an optical output and being responsive to the matrix of real number pixel values for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values, wherein the matrix of real number pixel values comprise positive and negative values, and wherein the SLM comprises a signed range LCD configured to produce positive and negative light output.

31. A spatial light modulator (SLM) apparatus comprising: an optical input; an optical output; and means responsive to a reference beam incident on the optical input at a reference beam angle and a matrix of real number pixel values for providing a reproduced wavefront on the optical output as a function of the matrix of real number pixel values, wherein the matrix of real number pixel values comprises positive and negative values.

32. The apparatus of claim 31, wherein the SLM further comprises a signed range LCD configured to produce positive and negative light output.

33. The apparatus of claim 32, wherein the positive light output comprises a first phase of light and the negative light output comprises a second phase of light, wherein the first phase is different from the second phase by a phase difference of 180 degrees.

34. The apparatus of claim 32, wherein the signed range LCD comprises a polarizer, a liquid crystal material, and an analyzer, wherein the liquid crystal material is characterized by a liquid crystal helix rotation having an operating range between a first rotation and a second rotation.

35. The apparatus of claim 34, further wherein the polarizer and analyzer are configured with respect to the liquid crystal material for positioning the middle of the liquid crystal helix rotation operating range to yield a minimum light transmission of the SLM.

36. The apparatus of claim 31 , further comprising: a light source optically coupled to the optical input of the SLM for providing the incident reference beam at a reference angle θ with respect to the optical input of the SLM; and a controller for controlling the SLM according to the matrix of real number pixel values, wherein responsive to the reference beam and to the controller control, the SLM provides the reproduced wavefront on the optical output of the SLM.

37. The apparatus of claim 36, wherein the reference angle θ comprises an optimum angle θ_opt_imum in both an xz plane and a yz plane of the optical input.

38. The apparatus of claim 36, wherein the light source comprises a laser and wherein the reference beam comprises a collimated laser beam.

39. The apparatus of claim 31 , further comprising: an optical low pass filter adapted to filter out conjugate images from the reproduced wavefront on the optical output of the SLM.

40. The apparatus of claim 39, wherein the optical low pass filter includes a cut-off frequency on the order of one-half of a Nyquist frequency (f>j/2) of the reproduced wavefront on the optical output of the SLM.