Abstract

We show that a liquid crystal spatial light modulator (LCOS-SLM) can be used to display amplitude images, or phase holograms, which change in a pre-determined way when the display is tilted, i.e. observed under different angles. This is similar to the tilt-effect (also called ”latent image effect”) known from various security elements (”kinegrams”) on credit cards or bank notes. The effect is achieved without any specialized optical components, simply by using the large phase shifting capability of a ”thick” SLM, which extends over several multiples of 2π, in combination with the angular dependence of the phase shift. For hologram projection one can use the fact that the phase of a monochromatic wave is only defined modulo 2π. Thus one can design a phase pattern extending over several multiples of 2π, which transforms at different readout angles into different 2π-wrapped phase structures, due to the angular dependence of the modulo 2π operation. These different beams then project different holograms at the respective readout angles. In amplitude modulation mode (with inserted polarizer) the intensity of each SLM pixel oscillates over several periods when tuning its control voltage. Since the oscillation period depends on the readout angle, it is possible to find a certain control voltage which produces two (or more) selectable gray levels at a corresponding number of pre-determined readout angles. This is done with all SLM pixels individually, thus constructing different images for the selected angles. We experimentally demonstrate the reconstruction of multiple (Fourier- and Fresnel-) holograms, and of different amplitude images, by readout of static diffractive patterns in a variable angular range between 0° and 60°.

© 2015 Optical Society of America

1. Introduction

Liquid crystal on silicon spatial light modulators (LCOS-SLMs) have found a broad range of applications in imaging and image projection modalities. LCOS-SLMs are reflective devices with the advantage over transmissive SLMs of a higher spatial resolution and a larger fill factor, as the electronic circuitry is placed behind the LC layer, resulting in a diffraction efficiency of up to 90%. Diffractive patterns displayed on an SLM are calculated as two-dimensional arrays to pixel-wise shift the phase of a monochromatic incident light field, typically in a range between 0 and 2π.

Certain SLMs with thicker liquid crystal layers, however, can exert much larger phase shifts. Recently it was demonstrated that an extended phase modulation range of a phase-only (i.e. longitudinally aligned) LCOS-SLM can be exploited for display of efficient second order diffractive patterns [1], to increase the update rate of holograms [2], to project color holograms [3], for broadband suppression of the zero diffraction order [4], for simultaneous holographic optical trapping and image processing [5], and as a color display [6].

In [3] and [5] the phase modulation range of multiples of 2π was used to display independent diffractive patterns for several wavelengths simultaneously. The method is based on the fact that the transverse phase profile of a monochromatic light beam, which is relevant for diffraction, is only defined modulo 2π. Thus only the ”2π-wrapped phase” has to be considered. An SLM actually displays a ”landscape” of optical path length modulations, determined by the (voltage controlled) orientation of the liquid crystal molecules. An optical path length shift of one wavelength λ corresponds to a phase shift of 2π. If the wavelength is changed, the same optical path length corresponds to a different phase. Thus the 2π-wrapped phase pattern depends on the wavelength. If the optical path length modulation of the SLM covers several wavelengths it is possible to calculate a ”thick” optical path length profile, which reconstructs independent 2π-wrapped phase profiles at selected wavelengths. This was used in [3] to reconstruct independent holograms with nearly full efficiency and negligible crosstalk at different wavelengths.

A similar effect was demonstrated in [6], using a LCOS-SLM panel as a non-holographic image display. In this case a polarization filter is attached to the front panel of the SLM such that both incident beam and modulated beam are being filtered at the same diagonal polarization direction. In this special operation mode amplitude modulation is achieved, as only the component of the diagonal field vector parallel to the long axis of the LC molecules is shifted in phase, by an amount depending on the applied voltage level, whereas the perpendicular component ideally remains unaffected. A shift in phase of one of the orthogonal components can be regarded as a rotation of the field vector’s polarization direction. Passing the polarization filter twice leads to a voltage-dependent attenuation of the amplitude.

Thus a 2π-phase shift of the active optical axis produces one intensity oscillation period of the readout light. Phase shifts over multiples of 2π produce multiple intensity oscillation periods. The wavelength selectivity of the oscillation period is the same as that for the 2π-wrapped phase shift of holograms. Therefore one can use an analogous method to display wavelength-dependent images. This was used in [6] to display color images with white light illumination, which did not need the usually employed color filter masks.

Our present investigation is closely related to these effects. Here we show that independent holograms can be reconstructed from the same diffractive pattern, if the SLM is tilted (i.e. rotated). This effect is similar to a wavelength dependent readout, since a tilt of the SLM at a constant wavelength produces the same phase shift as a change of the wavelength at constant incidence angle. By changing the angle from perpendicular incidence to flatter incidence angles, the phase retardation is decreased by a factor corresponding to the cosine of the angle, independent of the control voltage. Thus it is possible to construct ”thick” diffractive patterns which project different holograms if they are illuminated under different incidence angles, with a similar method than that described in [3] for wavelength-dependent hologram construction. Furthermore, in amplitude modulation mode an analogous method can be used to display (non-holographic) images which change when tilting the SLM, similar (but based on another principle) to the tilt-effect known from certain security elements (kinegrams) [7].

In the experimental demonstration, we project three individual Fourier- and Fresnel-holograms by a single diffractive SLM pattern at illumination angles of 0°, 30°, and 60°, respectively. We also show that two different intensity images can be displayed at readout angles of 0° and 30°.

2. Angle-dependent phase response of a SLM

Diffractive patterns are typically calculated as 8-bit gray scale images, where gray shades from black to white correspond to phase shifts ranging between 0 and 2π, respectively. Depending on the displayed gray levels, voltages U are applied to the respective LC cells in the SLM panel, which leads to reorientation of the LC molecules, resulting in a change in refractive index n(U, λ) for light which is linearly polarized in the direction of the ”director” (i.e. the orientation of the long axis) of the LC molecules. Incident light with the correct polarization experiences a pixel-wise retardation, observable as a shift in phase

ϕ(U,λ)=2πdλn(U,λ),
where d is the thickness of the liquid crystal layer. The change in refractive index with increasing U is non-linear in general. For the work presented here a longitudinally aligned LCOS-SLM (Hamamatsu X10468-07) designed for the infra-red wavelength regime was used, providing a largely extended phase modulation range up to 13π at a wavelength of 445 nm. Its voltage levels (gray levels) can be addressed either with 8-bit or 12-bit incrementation, i.e. with 256 or 4096 levels, respectively.

In Figs. 1(a) and 1(b) the orientations of liquid crystal molecules in the active layer of a reflective LCOS-SLM are illustrated for U = 0 and U > 0, respectively, for normal incidence α = 0 (green beam), and for α > 0 (blue beam). The polarization direction of the field vector E⃗ is in the same plane as the extraordinary axis of the LC molecules (long axis, active axis). In both cases, α = 0 and α > 0, the phase response can be determined by taking the difference of the shift in phase between LC cells at U = 0 and U > 0. With the relation for the optical path length nd = ϕλ/2π of a light wave passing an optical material of thickness d and refractive index n, the phase response at α = 0 can be written as

ϕ0(β¯)=22πdnλ22πdnλcosβ¯=4πdnλ(1cosβ¯),
where d is the thickness of LC layer, and n is the refractive index along the extraordinary axis of the LC molecules (long axis). The factor 2 in both terms in the upper line of Eq. (2) accounts for light travelling the LC layer twice. An increased mean rotational angle β̄ of the LC molecules (by increasing the control voltage) leads to a smaller phase delay, since it reduces the projection of the electric field vector onto the director of the LC molecules.

 

Fig. 1 Illustration of the orientation of liquid crystal molecules in the active layer of an LCOS-SLM at applied voltages (a) U = 0 and (b) U > 0. The polarization direction of the electric field vector E⃗ is in the same plane as the long axis of the LC molecules. The total phase shift results from the passage of the incoming, and the reflected beam, whose wavevectors k⃗ (indicated in (c) and (d)) have an angle of ±α with respect to the SLM surface. For the case of an applied voltage (b) the director of the LC molecules changes by an angle of β, which results in a total angle between LC director and light polarization of α + β for the incoming beam, and βα for the reflected beam (see (d)).

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In Figs. 1(c) and 1(d) the scenario of a field vector E⃗ in plane with the director of a single LC molecule, incident at an angle α > 0 is illustrated for U = 0 and U > 0, respectively. The phase response is proportional to the projection of the electric field vector E⃗ at the director (long axis) of the LC molecules, and thus takes on the form

ϕ(α,β¯)=22πdncosαλ2πdnλ[cos(β¯+α)+cos(β¯α)]=4πdλ(ncosαncosαcosβ¯)=4πdnλcosα(1cosβ¯).
The pre-factor 2 in the first term in the top row of Eq. (3) accounts again for light travelling the LC cell twice at U = 0, while the second and third term correspond to the shift in phase of light travelling towards the reflective back surface of the LC cell, and light being reflected from it, respectively. In the middle row of Eq. (3) the identity cos(β̄ ± α) = cosβ̄ cosα ∓ sinβ̄ sinα was used. The meaning of Eq. (3) is that the phase shift depends on three adjustable parameters, namely the wavelength λ, the mean tilt angle of the molecules β̄ (which is controlled by the applied voltage), and the incidence angle α. However, most importantly, the dependence is factorized. Therefore, two phase look-up tables measured for different wavelengths are identical up to a constant scale factor. Similarly, the phase lookup-tables for different incidence angles also differ by just a constant factor. The factorization also is the reason why angular multiplexing of holograms is analogous to wavelength multiplexing, since both, an angular tilt and a wavelength change, influence the phase by just multiplying it with a constant factor.

In a first series of measurements we investigated the phase response of the LC-panel for angles of incidence α ranging from 0° to 60°. An illustration of the experimental setup is shown in Fig. 2. The polarization direction of a diode laser at a wavelength of 445 nm is adjusted with a halfwave plate (λ/2) to be parallel with the director of the liquid crystals in the SLM panel (horizontal direction, corresponding to rotation direction). With a first telescope (not shown) consisting of two plano-convex lenses of focal lengths f1 = 10 cm and f2 = 25 cm the laser beam is expanded by a factor of 2.5. Furthermore, a pinhole with a diameter of 30 μm is placed in the focal plane between the two lenses as a spatial filter, to increase the uniformity of the beam profile. In order to optimally illuminate the SLM’s active area, the beam is further expanded with a telescope by a factor of 5. A polarization filter ensures that only light with horizontal polarization is incident on the SLM.

 

Fig. 2 Topview illustration of the experimental setup used to measure the phase response of an LCOS-SLM at multiple angles of illumination α, and for multidirectional Fresnel hologram projection. For the projection of Fourier holograms a lens (not shown) with focal length of f = 40 cm is placed between beam splitter cube (BS) and polarization filter.

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The phase response as a function of the angle of incidence α for λ = 445 nm was measured with a double-slit method similar to that described in [8]: A uniform gray level is displayed on one half of the SLM’s active area and a diffraction grating with a period of 32 pixels is displayed on the other half. The grating deviates the corresponding fraction of the laser beam to interfere with light incident on the area where the uniform gray level is displayed. At a distance of approximately one meter the two plane waves sufficiently overlap and give rise to an interference fringe pattern (see inset in Fig. 3(b)). Increasing the uniform gray level from 0 (black) to 4096 (white) leads to a shift of the interference fringes, which is proportional to the phase shift ϕ introduced by the SLM. For each gray level the fringe pattern was captured with a camera (Point Grey Grasshopper GS3-U3-23S6C-C) in an automatized loop in synchronization with the SLM. Sequences of 4096 fringe patterns where acquired for 15 angles of illumination ranging from α = 0° to α = 60°. For a comparison of the phase response at a different wavelength, the entire measurement was repeated with light from a diode laser at λ = 638 nm.

 

Fig. 3 Interferometric measurements (solid lines) and simulations (dashed lines) of the phase responses for (a) λ = 445 nm and (b) λ = 638 nm as a function of the a applied voltage level U for three different angles of illumination α. For an overview, in (c) and (d) the simulated phase responses at the same set of incidence angles is shown for the blue (c) and the red (d) wavelength, respectively.

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In Figs. 3(a) and 3(b) the phase response as a function of the applied voltage level U is shown for three different angles of incidence at λ1 = 445 nm and λ2 = 638 nm, respectively. The solid lines are generated by integration over a vertical section in the interference pattern, which is indicated as a vertical red line in the inset of Fig. 3(b). Increasing the voltage level leads to a non-linear shift of the phase. For each angle of incidence the data is normalized to unity with respect to the maximum intensity at voltage level zero. The local maxima in the normalized intensity oscillations correspond to phase shifts of multiples of 2π. At zero angle of incidence (bottom rows in Figs. 3(a) and 3(b)) the phase modulation range spans from 0 to 13π for λ1 = 445 nm, and from 0 to 7π for λ2 = 638 nm. A significant reduction of the phase range is evident in the middle and upper row of the graphs for angles of incidence α = 40° and α = 60°, respectively.

The dashed lines in the plots are numerical simulations of the measured phase response, based on the phase look-up tables of the SLM for the blue and the red wavelengths, ϕ0(U, λ1) and ϕ0(U, λ2), respectively. For the red wavelength the look-up table (i.e. the phase response as a function of the voltage level ϕ0(U, λ2)) is provided by the manufacturer for zero angle of incidence (upper row in Fig. 3). For the blue wavelength this look-up table is proportionally up-scaled, as look-up tables for different wavelengths to a good approximation only differ by a constant factor. The theoretically expected functions for the intensity oscillations (dashed lines in Figs. 3(a) and 3(b)) are

I(α,λ1)=12(cos[ϕ0(U,λ1)b(α)]+1)
and
I(α,λ2)=12(cos[ϕ0(U,λ2)r(α)]+1),
where the scale factors b(α) and r(α) are the actual quantities to be measured, i.e. they are determined by fitting the data. In Figs. 3(c) and 3(d) the simulated oscillations, Eq. (4) and Eq. (5), respectively, for α = 0°, 40°, and 60° are superimposed to summarize the different phase responses as a function of α.

In Figs. 4(a) and 4(b) the scale factors b(α) and r(α), determined from the data by fitting the phase response with Eqs. (4) and (5), are plotted versus the angle of illumination α. The error bars account for uncertainties in the fit of the specific scale factors. The data show that b(α) and r(α) are equal within the experimental accuracy, and scale with the cosine of the angle of incidence (solid lines), i.e.

ϕ(U,λ,α)=ϕ0(U,λ)cosα.
As expected by Eq. (3) this scaling factor is independent of wavelength and control voltage. The dependence of the phase retardation from the increasing incidence angle was also investigated in [9–11].

 

Fig. 4 Scale factors of the phase response bi and ri as a function of the tilt angle for (a) λ = 445 nm and (b) λ = 638 nm. The solid lines correspond to the cosine of the tilt angle.

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By attaching a linear polarization filter at a 45 °-angle with respect to the LC orientation in front of the SLM, it is converted into an amplitude display. In this case, an incoming beam is considered to be composed of two linear polarization components, one parallel to the active optical axis, and one perpendicular to it. The parallel component is phase modulated according to Eq. (3), whereas the perpendicular component is unmodulated. Behind the SLM, the polarization filter projects the two components at a common (45°-angle) polarization direction where they interfere. Thus the brightness oscillates by one period if the phase along the active optical axis is shifted by 2π. Therefore, the amplitude of each pixel in amplitude modulation mode has the same dependence on wavelength, incidence angle, and control voltage, as the 2π-wrapped phase in phase-modulation mode (without polarizer), which is described by Eq. (3).

3. Angle-dependent holograms

The following description of the generation of multiplexed diffractive patterns is restricted to three interleaved holograms with read-out angles α = 0°, 30°, and 60°, but can straightforwardly be extended to more projections at different viewing angles. The basic procedure is to first calculate the three holograms independently with standard methods. Thus each hologram corresponds to a pattern of phase pixels in a range between 0 and 2π. The next task is to find for each SLM pixel a certain control voltage (i.e. a gray level), producing an optical path length modulation which generates (after the angle-dependent modulo 2π operations) the desired phase values of the three holograms at their respective readout angles. For that purpose the phase retardation of the SLM has to be known as a function of the control voltage, and of the three readout angles. The corresponding data measured for our SLM is shown in Fig. 5(a). Relevant for the diffraction of an incident beam is only the 2π-wrapped phase, plotted in Fig. 5(b). For the illustrated scenario of three angles of incidence each voltage level corresponds to a triplet of 2π-wrapped phases, depending on the three directions of illumination. The task of finding the best voltage to create a certain triplet of phases at the three incidence angles is solved by a ”brute force” method, comparing the desired phase triplet of the three holograms with each of the 4096 phase triplets in Fig. 5(b) within the ”phase space” of the SLM. The best match can be defined by a certain metric, in our case by minimizing the sum of the quadratic phase deviations of the three phase values. This procedure is performed for each SLM pixel individually. Due to the large phase modulation range of the SLM it is possible to find for each possible phase triplet a control voltage which matches the three phase values with a mean phase error on the order of only 0.3π (standard deviation), which was verified by numerical simulations. The numerically simulated diffraction efficiency of the three holograms is on the order of 80%.

 

Fig. 5 (a) Phase response in radians versus the applied voltage level at different angles of incidence. (b) Phase response taken modulo 2π. With each voltage level a triplet of phase shifts can be addressed simultaneously for different angles of incidence.

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In order to calculate the holograms we chose three target images of the letters ”R”, ”G” and ”B” and use a variant of the Gerchberg-Saxton algorithm [12, 13] to independently calculate the respective diffractive patterns. This results in three two-dimensional arrays Hi(nx, ny) containing the required ideal relative phase shifts in the interval (0, 2π), where in our case i = 1, 2, and 3, corresponds to the three readout angles, and nx and ny denote the pixels in horizontal and vertical direction, respectively.

The procedure described in the flow diagram of Fig. 6 assumes that the three phase patterns Hi(nx, ny) (which will reconstruct three calculated holograms at their respective readout angles αi) were already calculated, i.e. they are phase patterns defined in an interval between 0 and 2π. For reasons of simplicity we describe the procedure for the optimization of a single SLM pixel, which has then to be repeated for all of the pixels. The loop starts with k = 0, corresponding to the first voltage level Uk. For each of the three phases (of a single pixel) of H1,2,3(nx, ny) the difference Δi(αi, Uk) between the actually diffracted phase ϕ0(Uk) cosαi, and the desired phase Hi is computed according to:

Δαi,Uk=Hiϕ0(Uk)cosαi2πround[Hiϕ0(Uk)cosαi2π].
Here, the round operation rounds the argument to the nearest integer value. This procedure of computing phase differences accounts for the modulo-2π ambiguity of the phase. In the next step, the quadratic sum of these phase errors is computed, corresponding to the variance V(Uk), according to:
VUk=Δα1,Uk2+Δα2,Uk2+Δα3,Uk2.
In the next step it is checked, whether VUk is smaller that the minimal variance Vmin of all previous loop cycles (note that for the first run Vmin is initialized with an ”infinite” number, such that for the first run this condition is always fulfilled). In this case, the minimal error Vmin is replaced by the new variance value VUk, and the voltage level Uk is assigned to the corresponding SLM pixel H*. In the other case, nothing is done. The loop then proceeds by trying the next voltage level Uk+1. After trying all 4096 voltage values the optimal value (producing a minimal route-mean-square phase error when approaching the three desired phase values Hi) is found, and assigned to the corresponding pixel of H*(nx, ny). The loop then has to start again in order to optimize the next pixel, until the whole array of H*(nx, ny) is obtained. Note that the same calculation can also be performed in parallel to obtain the whole array H*(nx, ny) simultaneously. In this case, Δi, Vk, Vmin, and H* have to be also arrays with sizes (nx, ny).

 

Fig. 6 Algorithm for the optimization of a single pixel in the multiplexed hologram H*. For each voltage level k the deviations Δαi,Uk between the respective SLM induced phase-shift, and the ideal phase values in the individual computer-generated holograms Hi (which have been previously calculated with a GS-algorithm), are calculated according to Eq. (7). In the multiplexed hologram H* the pixels are assigned with the voltage levels Uk for which the variance Vk (calculated according to Eq. (8)) is minimal. The procedure can be also performed for all pixels of H*(nx, ny) in parallel by assigning two dimensional arrays to the respective variables Δi, Vk, Vmin, and H*.

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A further acceleration of the calculation procedure is possible in analogy to the method described in [3] for color-multiplexed holograms. There, the phase values φi of each hologram are quantized (e.g. to 16 levels in the interval between 0 and 2π), and a complete table of optimal voltage values for all possible combinations of the three 16-level phase values (reconstructed at the three angles αi), namely U(φ1, φ2, φ3), is calculated in advance. The task of combining three arbitrary holograms Hi then just corresponds to a table-lookup in this precomputed list, i.e. H* = U(H1, H2, H3), which is done in a few milliseconds on a standard personal computer.

For image projection the multiplexed diffractive pattern is either designed as a Fourier hologram or as a Fresnel hologram of a certain focal length f, which determines at what distance from the SLM the respective image is sharply projected.

3.1. Multi-angle Fourier hologram projection

In Fig. 7 three projections of a multiplexed Fourier hologram at angles of 0°, 30° and 60° are shown. They were captured in the blue channel of a digital color sensor (Canon EOS 5d, 5616 × 3744 pixels). The presented images are sections of 1000 × 1000 pixels of the captured frames. The multiplexed pattern is designed for λ = 445 nm at read-out angles α = 0°, 30°, and 60°. Here, a Fourier transforming lens with a focal length of f = 40 cm is placed between the polarizer and the beam splitter cube (not illustrated in Fig. 2), and the camera is placed in its focal plane. The original images are white capital letters on a black background, arranged around the center, to provide a comparison of diffraction efficiencies and to investigate the cross-talk among the projections.

 

Fig. 7 Optical reconstructions from a multiplexed Fourier hologram at different angles of illumination. The colored rectangles indicate the areas that have been integrated to estimate relative diffraction efficiencies.

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At zero angle of incidence the zero diffraction order, visible in the center of Fig. 7(a), is superimposed by a significant fraction of light being reflected off the protective front glass of the LC-panel, since the anti-reflection coatings of the LCOS-SLM in use are designed for IR-wavelengths. In Figs. 7(b) and 7(c) the reflection is separated from the zero diffraction order, due to the tilt of the SLM. The exposure time was equal in all frames and was chosen such that the zero diffraction order is not in saturation. Note that for a better visualization of the crosstalk the intensity distributions shown in Fig. 7 are normalized to half of the intensity distribution captured at zero angle of incidence.

For each of the three individual projections the relative diffraction efficiencies were determined by division of the integrated brightness levels in the respectively indicated colored rectangles in Fig. 7 by the integrated brightness levels in the entire image. It turns out that they are on the order of 10%, 8.5% and 5% for the three incidence angles 0, 30°, and 60°, respectively. This efficiency is significantly lower than that expected from numerical simulations, which should be on the order of 80% for all three holograms. We assume that this low efficiency is due to the fact that the SLM is actually designed for the infra-red wavelength range between 620 nm and 1500 nm, and thus the optical coatings of the front glass, the reflective mirror at the back side, and also the type of LC molecules are not well suited for the currently used blue wavelength (445 nm). This also agrees with the low efficiency of a single control hologram, which we optimized for zero incidence angle. There it reached an efficiency of just 23%, which is much lower than the 90% efficiency, which can be reached with modern SLMs if they are used within their specified wavelength range. Nevertheless, even with this low efficiency the cross-talk between the three readouts is rather low, i.e. each hologram is displayed only at its programmed readout angle, and the respective other holograms are strongly suppressed. As a quantitative measure of the cross-talk, the integrated intensity within each of the colored rectangles in Fig. 7 was divided by the sum of all three rectangles. The result shows that about 85% of the diffracted intensity (without including the zero-order light) appears in the hologram designed for the respective diffraction angle.

3.2. Multi-angle Fresnel hologram projection

Fresnel holograms displayed at an SLM have several advantages over Fourier holograms. No additional focusing lens is required, the projection distance can be tuned under computer control, and the zero diffraction order is dispersed on a dilute background. For the generation of a multiplexed Fresnel hologram the individual holograms are superimposed with the transmission function of a Fresnel lens,

Hi,Fresnel(nx,ny)={Hi(nx,ny)+2πλf[(xcosα)2+y2]},mod2π,
where x and y are the spatial coordinates in horizontal and vertical directions, respectively, and f is the focal length, which corresponds to the projection distance. The spatial coordinate perpendicular to the rotation axis of the SLM scales with cosα, satisfying the lens to appear as a symmetric function under a certain angle of incidence. Along the same spatial coordinate the original images are squeezed by the factor cosα before generation of the individual holograms to compensate for stretching of the projections. Note that these operations to produce an ”inclined” Fresnel hologram by starting with an optimized Fourier hologram, adding a lens term, and taking the inclined readout into account by stretching the hologram in the inclination direction, are only approximations to a more advanced calculation method, where inclinations of input and output planes are already considered during the GS-algorithm [10]. However, for our situation of long projection distances (i.e. a small numerical aperture of the projection system) the corresponding approximations are well justified, producing no apparent artefacts.

In Fig. 8 projections of capital letters at a distance f = 25 cm are shown. They were captured with the same digital color camera as the Fourier hologram projections. In the insets of Fig. 8 the modified lens terms, according to Eq. (9), are indicated. The relative diffraction efficiencies in Figs. 8(a)–8(c) reach 13%, 8%, and 11%. They were estimated by division of the brightness levels in the letters by the brightness levels in the full frames. For comparison we measured the relative diffraction efficiency of a conventional (non-multiplexed) hologram under the ideal perpendicular incidence direction, which reaches 23%. Thus the efficiency of the multiplexed holograms is on the order of 50% with respect to a standard hologram displayed at the same SLM.

 

Fig. 8 Projection of Fresnel holograms from a multiplexed hologram at read-out angles (a) 0°, (b) 30°, and (c) 60°.

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One issue of angle-multiplexed hologram reconstruction is that a light ray, which is incident at a higher inclination angle, may pass more than one SLM pixel, thus averaging over the corresponding phases of adjacent pixels. In our case the quadratic SLM pixels have an edge length of 20 μm, which is on the same order as the thickness of the liquid crystal layer (about 18 μm, estimated from its optical parameters). For the case of random, erratic phase variations, where a ray should pass through only one pixel, the inclination effectively produces a reduction of the SLM ”fill-factor” (corresponding to the ”active area of a pixel”), since only light hitting the pixel in a certain area will stay in it. This would correspond to a strong reduction of the diffraction efficiency. In our experiments this effect turns out to be much weaker than expected. The reason is that the multiplexed hologram consists of individual ”sub-holograms”, which were calculated by a GS-algorithm, which produces quite smooth phase variations. Thus, even the multiplexed hologram shows large areas (with a mean size of about 30 × 30 pixels from altogether 600 × 600 pixels) where the phase changes only smoothly, followed by an abrupt phase jump to another smooth area. In the case of small phase gradients, averaging through adjacent pixels does not significantly influence the reconstructed phase, and the corresponding diffraction efficiency. We note however, that sharp edges within an image, which require a large spatial bandwidth for reconstruction (i.e. fast phase changes) are expected to produce aberrations at higher readout angles, since in this case phase averaging over adjacent pixels becomes more problematic.

In order to further evaluate the quality of the projected holograms, we calculated their root-mean-square (rms) errors with respect to the master images. For that purpose, all images (experimental, and masters) were normalized to a maximum intensity level of 1, and the master images were overlaid by an image registration procedure with their corresponding experimental reconstructions before calculating the respective intensity differences. It turns out that the three holograms at readout angles of 0, 30°, and 60° have relative rms-errors of 6.4%, 11.5%, and 7.6%, respectively. These results are slightly worse than the rms-error of 6.2% of a ”conventional” (non-multiplexed) hologram of the letter ”B”, which was reconstructed for comparison in the same setup under normal incidence. Due to the only moderate increase of the rms-error of angular multiplexed holograms, the reconstructed images show only negligible crosstalk.

4. Spatial light modulator as a ”latent image” display

In this section we present the use of an LCOS-SLM as an angle sensitive reflective LC display with incoherent illumination. Amplitude modulation mode is achieved by attaching a polarization filter to the front panel of the SLM such that both incident beam and modulated beam are being filtered at the same polarization direction at an angle of 45° with respect to the horizontal plane (which corresponds to the plane in which the director of the LC molecules rotates upon application of a control voltage, and in which the SLM is rotated). For display of independent images at different viewing angles by a single pattern, we employ the significantly reduced phase response at large angles of illumination as described by Eq. (6). For a proof-of-principle we demonstrate switching between two images by tilting the angle by 30°.

Figure 9(a) shows the amplitude modulation for light of a red LED (LED ENGIN LZ4-00MD00) at a dominant wavelength of 623 nm and a FWHM of 20 nm as a function of the applied voltage level ranging from 0 to 255 at the two incidence angles α = 0° and α = 30°. Data was acquired by sequentially increasing a uniform gray value and sharply imaging the LC panel with an objective lens (Tamron 1:2.8 50 mm,ø 25.5 mm) at a digital sensor (Point Grey Grasshopper GS3-U3-23S6C-C) at a distance of 25 cm. In this setup the 792×600 pixels of the LC panel are imaged onto approximately 445×355 pixels in 960×600 pixel sized frames. Data points in Fig. 9 are brightness values averaged over a central section of the LC panel comprising 50×50 pixels in the acquired frames. Note that maxima and minima are scaled to 1 and 0, respectively.

 

Fig. 9 (a): Amplitude modulation as a function of the applied voltage level U with a diagonally oriented polarization filter attached to the LCOS-SLM for two angles of incidence α. Lower part: Experimental performance of an LCOS-SLM as an angle sensitive display. Upper row: Original images. Lower row: Experimental reproductions. (f) and (h) are captured at zero angle of incidence and (g) and (i) are captured at α = 30°.

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Using the measured amplitude modulation characteristics, a search algorithm can be used to attribute a control voltage U (i.e. gray-level) to each SLM pixel, which best approaches the intensity levels of the two different images at their respective readout angles. This can be done for combining two arbitrary gray scale images, but for a first demonstration we chose two binary (black and white) images.

Figure 9 (lower images) provides the performance of the presented dual viewing angle display at incoherent illumination (aforementioned red LED). Original images and respective experimental reproductions are shown in the upper row, and lower row, respectively. The experimental images of the the capital letters and the key pad constitute 164×164 and 261×261 pixels, respectively, of the acquired full frames (960×600 pixels). Figures 9(f) and 9 (g) show acquired images of capital letters at zero angle of incidence, and at a 30° degree tilt angle, respectively. As an example for gray-level images (dithered) we chose two 8-bit photographs of a key pad viewed from different perspectives (d) and (e). These images were converted to 1-bit images with a Floyd-Steinberg dithering algorithm [14]. The experimentally recorded images at viewing angles of 0 (h), and 30° (i) show that also these more complex patterns are reconstructed with acceptable quality.

5. Discussion and conclusions

Our results show that diffractive patterns displayed on an SLM, which provides an optical path length modulation of multiples of the optical wavelength, are angle-selective and can be multiplexed due to this effect. In earlier experiments it has been shown that these SLM patterns are also wavelength-selective, allowing for wavelength multiplexed holograms. Thus, similar to volume or Bragg holograms (although based on a different mechanism), both wavelength and angular selectivity are features of a ”thick” SLM display, and are interchangeable to a certain extent, i.e. a wavelength multiplexed hologram can be read sequentially at a single wavelength by tilting the display, whereas an angular multiplexed hologram can be reconstructed at a single angle by changing the wavelength. The current experimental results for holographic readout demonstrate only a low diffraction efficiency on the order of 10% for the multiplexed holograms, which is far below the theoretically expected relative efficiency on the order of 80%, as has already been obtained with two wavelength multiplexed holograms using another SLM [3]. Our current low efficiency is due to the fact that the SLM is not designed for the visible wavelength range, but for the infra-red regime up to 1500 nm. Thus the anti-reflection coatings are inappropriate for the visible range, and presumably the implemented LC layer may have a significantly increased scattering in the visible range. Furthermore, for higher angles of incidence the efficiency is assumed to decrease, since in this case the obliquely incident beam travels through adjacent SLM pixels, and averages their respective phase retardations. Another serious reduction of the efficiency (not included in our simulations) is due to crosstalk of adjacent LC pixels, also known as fringing-effect [15], which particularly degrades the performance if high phase jumps (like in our case) appear. Anyhow it may be assumed that significantly higher efficiencies are possible for better adapted SLMs, which may approach the efficiencies obtained in wavelength multiplexing experiments ([3]), as e.g. about 60% for the case of three wavelength-multiplexed holograms.

The tilt effect is not limited to hologram projection, but it can also be used as an angle selective (non-holographic) display by attaching a polarization filter to the SLM. Such a filter, at an appropriately chosen polarization angle, transforms the phase-only SLM into an amplitude modulator, based on the adjustable optical birefringence of each pixel. This birefringence is based on the phase delay between horizontal and vertical polarizations, and thus depends in the same way on the control voltage of a pixel as phase-only modulation. Analogously, the feature of a ”thick” SLM to produce phase shifts of multiples of 2π means that the SLM pixels can be controlled to act as multi-order waveplates, depending on the incidence (or observation) angle. This can be used to display multiplexed amplitude images with an optical tilt effect.

Applications of the method might arise for the design of angle-selective ”standard” diffractive optical elements (DOEs), i.e. diffractive structures which are etched in glass, or imprinted in polymer substrates. If they are used in transmission, or in reflection mode (e.g. by attaching a reflective coating), the phase shift depends in a similar way on the readout angle as that of an SLM. Thus for reflective DOEs an optical tilt effect can be achieved by using an analogous method for hologram combination into a phase pattern with increased optical path length modulation, which extends over multiples of the employed light wavelength.

Acknowledgments

This work was supported by the ERC Advanced Grant 247024 catchIT, and by the Christian Doppler Laboratory CDL-MS-MACH.

References and links

1. J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013). [CrossRef]  

2. G. Thalhammer, R.W. Bowman, G.D. Love, M.J. Padgett, and M. Ritsch-Marte, “Speeding up liquid crystal SLMs using overdrive with phase change reduction,” Opt. Express 21, 1779–1797 (2013). [CrossRef]   [PubMed]  

3. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Colour hologram projection with an SLM by exploiting its full phase modulation range,” Opt. Express 22, 20530–20541 (2014). [CrossRef]   [PubMed]  

4. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Broadband suppression of the zero diffraction order of an SLM using its extended phase modulation range,” Opt. Express 22, 17590–17599 (2014). [CrossRef]   [PubMed]  

5. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Combined holographic optical trapping and optical image processing using a single diffractive pattern displayed on a spatial light modulator,” Opt. Lett. 39, 5337–5340 (2014). [CrossRef]  

6. W. Harm, A. Jesacher, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “How to use a phase-only spatial light modulator as a color display,” Opt. Lett. 40, 581–584 (2014). [CrossRef]  

7. S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002). [CrossRef]  

8. C. Kohler, X. Schwab, and W. Osten, “Optimally tuned spatial light modulators for digital holography,” Appl. Opt. 45, 960–967 (2006). [CrossRef]   [PubMed]  

9. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17, 8491–8505 (2009). [CrossRef]   [PubMed]  

10. Tomasz Kozacki, “Holographic display with tilted spatial light modulator,” Appl. Opt. 50, 3579–3588 (2011). [CrossRef]   [PubMed]  

11. J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014). [CrossRef]  

12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

13. B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009). [CrossRef]  

14. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

15. C. Lingel, T. Haist, and W. Osten, “Optimizing the diffraction efficiency of SLM-based holography with respect to the fringing field effect,” Appl. Opt. 52, 68776883 (2013). [CrossRef]   [PubMed]  

References

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  1. J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
    [Crossref]
  2. G. Thalhammer, R.W. Bowman, G.D. Love, M.J. Padgett, and M. Ritsch-Marte, “Speeding up liquid crystal SLMs using overdrive with phase change reduction,” Opt. Express 21, 1779–1797 (2013).
    [Crossref] [PubMed]
  3. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Colour hologram projection with an SLM by exploiting its full phase modulation range,” Opt. Express 22, 20530–20541 (2014).
    [Crossref] [PubMed]
  4. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Broadband suppression of the zero diffraction order of an SLM using its extended phase modulation range,” Opt. Express 22, 17590–17599 (2014).
    [Crossref] [PubMed]
  5. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Combined holographic optical trapping and optical image processing using a single diffractive pattern displayed on a spatial light modulator,” Opt. Lett. 39, 5337–5340 (2014).
    [Crossref]
  6. W. Harm, A. Jesacher, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “How to use a phase-only spatial light modulator as a color display,” Opt. Lett. 40, 581–584 (2014).
    [Crossref]
  7. S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
    [Crossref]
  8. C. Kohler, X. Schwab, and W. Osten, “Optimally tuned spatial light modulators for digital holography,” Appl. Opt. 45, 960–967 (2006).
    [Crossref] [PubMed]
  9. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17, 8491–8505 (2009).
    [Crossref] [PubMed]
  10. Tomasz Kozacki, “Holographic display with tilted spatial light modulator,” Appl. Opt. 50, 3579–3588 (2011).
    [Crossref] [PubMed]
  11. J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
    [Crossref]
  12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  13. B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
    [Crossref]
  14. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).
  15. C. Lingel, T. Haist, and W. Osten, “Optimizing the diffraction efficiency of SLM-based holography with respect to the fringing field effect,” Appl. Opt. 52, 68776883 (2013).
    [Crossref] [PubMed]

2014 (5)

2013 (3)

2011 (1)

2009 (1)

2006 (1)

2002 (1)

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

1976 (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Albero, J.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Bernet, S.

Bowman, R.W.

Campos, J.

del Mar Sánchez-López, María

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

Estapé, M.

Fernández, E.

Floyd, R. W.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

García-Martínez, P.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Haist, T.

Harm, W.

Iemmi, C.

Jesacher, A.

Kohler, C.

Kozacki, Tomasz

Kress, B. C.

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

Lingel, C.

Lizana, A.

Love, G.D.

Márquez, A.

Martín, N.

Martínez, J. L.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Meyrueis, P.

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

Moreno, I.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17, 8491–8505 (2009).
[Crossref] [PubMed]

Osten, W.

Padgett, M.J.

Ritsch-Marte, M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schwab, X.

Sinzinger, S.

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

Steinberg, L.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

Thalhammer, G.

Vargas, A.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

Yzuel, M. J.

Appl. Opt. (3)

Opt. Commun. (1)

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SID (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

Other (1)

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Illustration of the orientation of liquid crystal molecules in the active layer of an LCOS-SLM at applied voltages (a) U = 0 and (b) U > 0. The polarization direction of the electric field vector E⃗ is in the same plane as the long axis of the LC molecules. The total phase shift results from the passage of the incoming, and the reflected beam, whose wavevectors k⃗ (indicated in (c) and (d)) have an angle of ±α with respect to the SLM surface. For the case of an applied voltage (b) the director of the LC molecules changes by an angle of β, which results in a total angle between LC director and light polarization of α + β for the incoming beam, and βα for the reflected beam (see (d)).
Fig. 2
Fig. 2 Topview illustration of the experimental setup used to measure the phase response of an LCOS-SLM at multiple angles of illumination α, and for multidirectional Fresnel hologram projection. For the projection of Fourier holograms a lens (not shown) with focal length of f = 40 cm is placed between beam splitter cube (BS) and polarization filter.
Fig. 3
Fig. 3 Interferometric measurements (solid lines) and simulations (dashed lines) of the phase responses for (a) λ = 445 nm and (b) λ = 638 nm as a function of the a applied voltage level U for three different angles of illumination α. For an overview, in (c) and (d) the simulated phase responses at the same set of incidence angles is shown for the blue (c) and the red (d) wavelength, respectively.
Fig. 4
Fig. 4 Scale factors of the phase response bi and ri as a function of the tilt angle for (a) λ = 445 nm and (b) λ = 638 nm. The solid lines correspond to the cosine of the tilt angle.
Fig. 5
Fig. 5 (a) Phase response in radians versus the applied voltage level at different angles of incidence. (b) Phase response taken modulo 2π. With each voltage level a triplet of phase shifts can be addressed simultaneously for different angles of incidence.
Fig. 6
Fig. 6 Algorithm for the optimization of a single pixel in the multiplexed hologram H*. For each voltage level k the deviations Δαi,Uk between the respective SLM induced phase-shift, and the ideal phase values in the individual computer-generated holograms Hi (which have been previously calculated with a GS-algorithm), are calculated according to Eq. (7). In the multiplexed hologram H* the pixels are assigned with the voltage levels Uk for which the variance Vk (calculated according to Eq. (8)) is minimal. The procedure can be also performed for all pixels of H*(nx, ny) in parallel by assigning two dimensional arrays to the respective variables Δi, Vk, Vmin, and H*.
Fig. 7
Fig. 7 Optical reconstructions from a multiplexed Fourier hologram at different angles of illumination. The colored rectangles indicate the areas that have been integrated to estimate relative diffraction efficiencies.
Fig. 8
Fig. 8 Projection of Fresnel holograms from a multiplexed hologram at read-out angles (a) 0°, (b) 30°, and (c) 60°.
Fig. 9
Fig. 9 (a): Amplitude modulation as a function of the applied voltage level U with a diagonally oriented polarization filter attached to the LCOS-SLM for two angles of incidence α. Lower part: Experimental performance of an LCOS-SLM as an angle sensitive display. Upper row: Original images. Lower row: Experimental reproductions. (f) and (h) are captured at zero angle of incidence and (g) and (i) are captured at α = 30°.

Equations (9)

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ϕ ( U , λ ) = 2 π d λ n ( U , λ ) ,
ϕ 0 ( β ¯ ) = 2 2 π d n λ 2 2 π d n λ cos β ¯ = 4 π d n λ ( 1 cos β ¯ ) ,
ϕ ( α , β ¯ ) = 2 2 π d n cos α λ 2 π d n λ [ cos ( β ¯ + α ) + cos ( β ¯ α ) ] = 4 π d λ ( n cos α n cos α cos β ¯ ) = 4 π d n λ cos α ( 1 cos β ¯ ) .
I ( α , λ 1 ) = 1 2 ( cos [ ϕ 0 ( U , λ 1 ) b ( α ) ] + 1 )
I ( α , λ 2 ) = 1 2 ( cos [ ϕ 0 ( U , λ 2 ) r ( α ) ] + 1 ) ,
ϕ ( U , λ , α ) = ϕ 0 ( U , λ ) cos α .
Δ α i , U k = H i ϕ 0 ( U k ) cos α i 2 π round [ H i ϕ 0 ( U k ) cos α i 2 π ] .
V U k = Δ α 1 , U k 2 + Δ α 2 , U k 2 + Δ α 3 , U k 2 .
H i , Fresnel ( n x , n y ) = { H i ( n x , n y ) + 2 π λ f [ ( x cos α ) 2 + y 2 ] } , mod 2 π ,

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