## Abstract

We demonstrate independent and simultaneous manipulation of light beams of different wavelengths by a single hologram, which is displayed on a phase-only liquid crystal spatial light modulator (SLM). The method uses the high dynamic phase modulation range of modern SLMs, which can shift the phase of each pixel in a range between 0 up to 10*π*, depending on the readout wavelength. The extended phase range offers additional degrees of freedom for hologram encoding. Knowing the phase modulation properties of the SLM (i.e. the so-called lookup table) in the entire exploited wavelength range, an exhaustive search algorithm allows to combine different independently calculated 2*π*-holograms into a multi-level hologram with a phase range extending over several multiples of 2*π*. The combined multi-level hologram then reconstructs the original diffractive patterns with only small phase errors at preselected wavelengths, thus projecting the desired image fields almost without any crosstalk. We demonstrate this feature by displaying a static hologram at an SLM which is read out with an incoherent red-green-blue (RGB) beam, projecting a color image at a camera chip. This is done for both, a Fourier setup which needs a lens for image focusing, and in a ”lensless” Fresnel setup, which also avoids the appearance of a focused zero-order spot in the image center. The experimentally obtained efficiency of a two-colour combination is on the order of 83% for each wavelength, with a crosstalk level between the two colour channels below 2%, whereas a three-colour combination still reaches an efficiency of about 60% and a crosstalk level below 5%.

© 2014 Optical Society of America

## 1. Introduction

High resolution liquid crystal SLMs are currently used in many fields of light manipulation. In most of these applications the SLM displays phase-only diffractive patterns (DPs), which have the advantage that they can, in principle, reach 100% light efficiency. These DPs are typically designed for a certain wavelength, where they shape the diffracted light field with maximal efficiency. However, there are applications where an independent control of different wavelengths would be of great advantage. For consumer products, such elements might be used to display [1], or to project [2] color holograms. In scientific applications such elements could be used in microscopy (or general imaging) applications [3–5], for optical beam steering [6–8], and for aberration correction [9,10] at different wavelengths simultaneously, which is useful in cases where a sample is excited with different laser wavelengths, like in coherent-anti-Stokes-Raman (CARS) microscopy, or in stimulated emission depletion (STED) microscopy, or in certain applications of fluorescence-recovery-after-photobleaching (FRAP) experiments [11]. For two-photon fluorescence experiments it would be possible to steer a laser beam for the excitation of the two-photon fluorescence, and to simultaneously correct the aberrations of the frequency-doubled fluorescence image in the imaging path. Furthermore the elements could be used to control laser tweezers for optical trapping, and at the same time as a Fourier processor in the imaging path, which for example acts as a phase contrast filter for the (different) imaging wavelength [5].

Thus methods have been investigated to control different wavelength with an SLM simultaneously. One approach, suggested for color hologram projection, is to use three separate SLMs, which are read out with separate red-green-blue wavelengths, and to recombine the diffracted intensity images using multichromatic optics [12,13]. Similarly, a single SLM panel can be used to display three adjacent holograms which are reconstruced with their respective light wavelengths, and recombined in the image plane [14]. In a time-multiplexing scheme the holograms calculated for the different colours are displayed sequentially and read out with red-green-blue light sources which are switched on and off in synchronisation with the SLM display [13,15].

Another approach used for optical tweezers control [8], and hologram projection [16, 17], is based on color hologram projection in the Fresnel regime. There, a single DP is calculated, which consists of a superposition of two or more original DPs, which represent the different color channels. The phase function of each DP is superposed by a parabolic phase term, corresponding to the phase of a Fresnel lens. The focal lengths of the individual Fresnel lenses can be programmed such that all color channels are reconstructed sharply at a common selectable distance behind the DP, there producing a color image. However, there is no suppression of crosstalk between the colour channels, i.e. each incident light wavelength reconstructs all colour channels simultaneously, and with the same efficiency, although not in the pre-selected image plane, but at other distances behind the DP. The theoretical maximum efficiency of this approach is thus only 50% for the case of two wavelengths. A similar multiplexing approach consists in the use of gratings rather than parabolic lenses, which, however, also requires wavelength-dependent incidence angles for the SLM illumination [18].

An interesting method, which avoids these disadvantages and yields a high diffraction efficiency with low crosstalk has been investigated and demonstrated for static diffractive optical elements (DOEs). There, the corresponding DOEs are designed with a phase modulation depth which significantly exceeds the normally used 2*π* phase range, thus being deeper structures which require a sophisticated production technique. Noting that the addition of arbitrary integer multiples of 2*π* to the phase values of each pixel does not change the diffracted field at that particular wavelength, but strongly influences the phase when the wavelength is changed, suggests to use this effect for designing diffractive structures, which reconstructs different pre-designed readout fields at different readout wavelengths. The first multi-order DOEs which used this principle were demonstrated in [19–21], and consisted in achromatic diffractive lenses, which had the same focal lengths at different wavelengths. Then it was suggested [22, 23], and later demonstrated [24–27] that two independent 2*π*-DOEs for two-color image reconstruction, or wavelength-dependent fan-out gratings, can be combined using different algorithms to determine the best combinations. In [28] such a method was experimentally used to produce a DOE working at three wavelengths.

To our knowledge no efforts were undertaken to investigate the applicability of such a method for a dynamic device, like an SLM. A typical DP displayed on an SLM consists of a 2*π*-wrapped phase landscape to achieve an optimal diffraction efficiency of up to 100%. However, modern SLMs often offer the possibility to shift the phase in a much wider range, e.g. up to 10*π* at a wavelength of 400 nm. This feature has been already used to demonstrate higher order diffraction at different light wavelengths [29]. Here we show that by using the full dynamic range of such an SLM, the additionally attained degrees of freedom can be used to combine two or more independent DPs (all of them originally optimized in a 2*π* phase range) into a multi-order phase structure, which reconstructs the programmed images at the intended wavelengths, with an efficiency on the order of 84% for two DPs, and of about 60% for three DPs, with only moderate crosstalk.

## 2. Method of multi-color DP calculation

In order to calculate a multi-colour DP, we first calculate independently the individual DPs with a standard optimization algorithm, like a kinoform [30] or a variant of a Gerchberg-Saxton algorithm [31, 32]. As a result we obtain *N* DPs, *D*_{1}(*x*, *y*)..*D _{N}*(

*x*,

*y*), each consisting of a two-dimensional pixel array which is modulated in a phase range between 0 and 2

*π*. The task which has now to be solved is to combine these

*N*DPs into a single, multi-level DP, such that it reconstructs the individual DPs at the preselected wavelengths

*λ*

_{1}..

*λ*. For this purpose it is essential to know the so-called lookup-table of the SLM in the whole exploited wavelength range. Such a lookup-table is either provided by the manufacturer (as for our SLM), or it has to be experimentally measured in advance. The lookup-table relates the voltage values (or alternatively the gray values of an image pixel displayed on the SLM) to the respective phase shifts of the diffracted wave. The corresponding phase shift

_{N}*ϕ*(

*U*,

*λ*) is given by:

There, *λ* is the readout wavelength, *d* is the thickness of the liquid crystal layer, *U* is the voltage applied to the SLM pixel (which is typically encoded as a gray value of the image pixel displayed on the SLM), and *n*(*U*, *λ*) is the refractive index of the birefringent liquid crystal layer for the selected readout polarization, which depends on both the applied voltage, and the readout wavelength.

Obviously, the phase shifts reciprocally with changing wavelength, even if the refractive index were constant. However, this trend is significantly increased for a real liquid crystal layer, since, due to its strong dispersion, the refractive index increases quadratically when approaching the blue spectral range. Thus, the same gray level of an SLM pixel can produce a maximal phase shift of 4*π* at 700 nm, and of almost 10*π* at 400 nm.

An exhaustive search method to calculate a combined multi-level DP is to sequentially apply all available voltage levels, *U _{k}*, (where

*k*= 0..

*k*, and

_{max}*k*corresponds to the number of digitally addressable gray levels, which is typically 256 for an 8-bit SLM) to a certain image pixel, and to determine the corresponding phase values

_{max}*ϕ*

_{1}(

*U*) ..

_{k}*ϕ*(

_{N}*U*) from the lookup table at each of the target wavelengths

_{k}*λ*

_{1}..

*λ*. Then the distances Δ

_{N}_{1}.. Δ

*between the obtained phase values (modulo 2*

_{N}*π*) and the pre-calculated ideal values,

*D*, are determined, i.e. the distance of the phase difference

_{i}*D*−

_{i}*ϕ*(

_{i}*U*) to the next nearest integer multiple of 2

_{k}*π*. This is done by calculating:

There, the ”round[..]” operation means rounding to the nearest integer value. Afterwards an optimization constraint is defined. One option is to minimize the sum (at all target wavelengths) of the squared distances between the ideal and the actually obtained phases, i.e. the variance *V*(*U _{k}*), by calculating:

After this has been performed for all addressable values of *U _{k}* (i.e. all digitally addressable gray levels of a pixel), the minimal value of

*V*(

*U*) is determined, and the corresponding voltage (or gray level) is assigned to the respective pixel. Clearly, other minimization constraints can be designed, e.g. to minimize the sum of the absolute values of the phase distances, or to favour one DP at the cost of the others by including different weights to the minimization constraint. The above procedure is performed for all pixels of the DP individually, which may however be parallelized in a calculation program like MATLAB. In our case the merging of three 600 x 600 pixel DPs to an optimized multi-level DP (assuming 64 practically distinguishable digitized phase levels, i.e.

_{k}*k*=64) requires a computation time of 0.4 s on a standard desktop computer. An advantage of this exhaustive search method is that it can be used to combine any number of DPs for any SLM with a known lookup-table. As a result, each pixel of the original DPs, which are now combined in the multi-level DP, has now a small (and typically random) phase error as compared to the ideal ”master” DP at its target wavelength. This error reduces, if a larger separation between the readout wavelengths is chosen, or if the SLM offers a larger dynamic range and/or a larger dispersion.

_{max}For high speed applications using a fixed set of input wavelengths, it is also possible to pre-calculate a lookup-table in advance, which assigns to each possible phase combination of all used wavelengths (which is a *N*-tupel for *N* colors) a corresponding optimal voltage value (or gray level). For example, for a combination of three DPs at three fixed wavelengths, the lookup table is a three dimensional array *U*(*ϕ*_{1}, *ϕ*_{2}, *ϕ*_{3}), where the coordinates (*ϕ*_{1}, *ϕ*_{2}, *ϕ*_{3}) range from 0 to 2*π* and correspond to the desired phase shifts at the three colours. *U*(*ϕ*_{1}, *ϕ*_{2}, *ϕ*_{3}) is the ideal voltage (or gray level) for displaying the best possible approximation of the corresponding phase triple, which is pre-calculated for all (*ϕ*_{1}, *ϕ*_{2}, *ϕ*_{3})-triples with the method described above. Afterwards, the determination of the optimal gray level for each SLM pixel just reduces to a table look-up in this three-dimensional array. Using this method, a combination of three phase patterns into a multi-level DP with 600×600 pixels is performed in 16 ms with a standard desktop computer, which allows for calculating dynamic DPs at video rate.

## 3. Experiments

Our experimental setup (sketched in Fig. 1) consists of a wavelength tunable, fiber coupled thermal light source (Till Photonics Monochromator Polychrome IV), which delivers monochromatic light with a bandwidth of 8 nm to 15 nm in a wavelength range between 400 nm and 700 nm at the output of a multi-mode fiber. Alternatively, for color hologram projection the monochromator is replaced by a three-color LED light source, which combines the light of a red (633 nm), a green (532 nm) and a blue (460 nm) LED with appropriate dichroic mirrors and couples them into the same multi-mode fiber. Interference filters behind the LEDs reduce the bandwidth of each beam to about 3 nm. The beam emerging from the fiber is expanded and collimated by a set of two lenses (L1 and L2), and illuminates a liquid crystal phase-only reflective SLM (Hamamatsu LCOS SLM X10468-01, 792 × 600 pixels, pixel size: 20 × 20 *μ*m^{2}) under a small tilt angle (15°). A linear polarisation filter after lens 1 optimizes the incident light polarisation for maximal diffraction efficiency. The DPs displayed at the SLM can be modulated in a phase range up to 10*π* at a wavelength of 400 nm, and up to 4.2 *π* at 700 nm. The corresponding lookup-table was provided by the manufacturer in the wavelength range between 400 nm and 700 nm. The light which is reflected by the SLM includes an angle of about 330° with respect to the incident beam. For Fourier DP readout it passes through a second achromatic lens L3 (f=100 mm) and focuses at a CMOS camera (Matrix Vision mvBlueFox 221G, without objective lens). In this case the image recorded by the camera is the Fourier transform of the field diffracted off the SLM. For Fresnel hologram readout, lens L3 is removed and the DPs displayed at the SLM are designed such that their images are focused at a distance of 30 cm in the camera plane. For color hologram recording, the monochrome camera is replaced by a colour camera (Canon 1000D) without objective.

For an experimental demonstration, two ”standard” 600×600 pixel DPs (spanning a phase range of 2*π*) were calculated with a Gerchberg-Saxton algorithm, supposed to reconstruct a test image of the letter ”B”, and of the letter ”R”, respectively. The two DPs were combined into a multi-level DP with the method described above, which was designed to reconstruct the image ”B” at a wavelength of 430 nm, and the image ”R” at 520 nm as Fourier holograms. Thus, for readout, lens L3 was inserted in front of the monochrome camera. Image reconstruction was performed using the monochromator as a light source, by tuning the wavelength between 390 nm and 610 nm with a step size of 10 nm. During readout the exposure time of the reconstructed images was controlled to avoid saturation of any camera pixel. Figure 2 shows the results. The upper row displays some examples of the images recorded by the camera, normalized to the average image intensity in each picture, and then displayed using the same colour scale. As expected, in the vicinity of the ”design” wavelength 430 nm (namely at 420 nm) the image of the letter ”B” is reconstructed with maximal efficiency, whereas near the intended wavelength 520 nm (namely at 510 nm) the letter ”R” is optimally reconstructed. At intermediate wavelengths, both of the images are reconstructed simultaneously, but at the intended wavelengths there is almost no crosstalk. A quantitative evaluation of the readout efficiencies as a function of the wavelength is shown in the left graph below. There the relative efficiencies are plotted, i.e. the integrated intensities of the letters ”B” and ”R”, respectively, as compared to the total integrated intensity in each image. The plot shows that the relative efficiency of the ”B”-image has a peak at a wavelength of 420 nm, where it reaches about 83%, whereas the ”R”-image has its maximal efficiency of 84% at 500 nm. At the same two wavelengths the crosstalk level (i.e. the efficiencies of the ”wrong” images) are below 2%, which yields an image intensity contrast of over 40 at the design wavelengths.

The graph at the right shows a numerical simulation of the expected efficiencies. The calculations were done by starting with the actually displayed multi-level hologram, translating its gray values to the respective phase values at all readout wavelengths by using the lookup table provided by the manufacturer, and by numerical propagation of the respective light fields to their Fourier plane, where the images are reconstructed and the respective intensities of letters ”B” and ”R” were determined. Quantitatively, the simulations predict a readout efficiency at the correct wavelength of 84% for both of the holograms, and a crosstalk level of 0.2% at the intended wavelengths. The experimental curves are shifted to bluer wavelengths as compared to the numerical expectations, which is possibly due to the limited accuracy of the provided lookup tables. But although there are deviations between theoretical and experimental data, the qualitative behaviour, like the shape of the efficiency curves, is reproduced within the expectable accuracy of lookup tables and SLM imperfections.

A similar experiment was performed for a three-color multi-level DP. In this case the DP was designed to reconstruct images of the letters ”B”, ”G”, and ”R” at the wavelengths of 410 nm, 520 nm, and 630 nm, respectively. The images were again reconstructed using the monochromator as a light source in at wavelength range between 390 nm and 660 nm with 10 nm increments. The results are shown in Fig. 3. The upper row shows some exemplary images, with the respective reconstruction wavelengths indicated below. Images reconstructed at the target wavelengths 410 nm, 510 nm and 630 nm are highlighted by a yellow frame. The reconstructed images at these wavelengths still correspond to the intended letters with a good suppression of crosstalk, i.e. at these wavelengths only the targeted images are visible. The quantitative evaluation of the respective efficiencies is plotted below at the left. The maximal efficiencies of the images ”B” (blue curve), ”G” (green curve) and ”R” (red curve) appear at the target wavelengths 410 nm, 510 nm and 630 nm, and have peak values of 63%, 69% and 44%, respectively. The corresponding background from the reconstruction of the ”wrong” images at these wavelengths is 5% (sum of ”wrong” intensities) at 410 nm, 5% at 510 nm, and 6% at 530 nm, which still yields an average contrast ratio of > 11. The graph at the right shows the results of the corresponding numerical simulations. There the expected peak efficiencies at the three target wavelengths 410 nm, 510 nm and 630 nm are 69%, 62%, and 48%, and the corresponding crosstalk levels 1.8%, 3%, and 3%. This yields an average contrast of 25 between target images and background of the wrong reconstructions. Although there are again quantitative differences between the experimental data and the numerical simulations, the overall behaviour of the different efficiency curves agrees sufficiently well within the experimental errors to use the model for the optimized combination of different DPs in the whole visible wavelength range.

To demonstrate this, we modified the setup for the single exposure projection of a full color red-green-blue (RGB) image. For this purpose a readout light source consisting of 3 light emitting diodes (LEDs) at 460 nm, 532 nm, and 633 nm was assembled. The bandwidth of each diode was narrowed by corresponding interference filters to about 3 nm (FWHM). Then their light was combined by dichroic mirrors and coupled into the multi-mode fiber, from then on keeping the previous experimental setup. For image recoding a color CMOS camera (without objective) was used. In order to construct a multi-order RGB DP, we started with a master RGB color image. There the three color channels were separated and used as three new gray-level masters for the calculation of the three DP color channels. The different diffraction angles of the light at the target wavelengths 460 nm, 532 nm and 633 nm were considered by down scaling (i.e. resizing) the green and the red master images as compared to the blue image with a scaling factor corresponding to the ratio of the respective wavelengths, i.e. with 460/532 = 0.86 for the green, and with 460/633 = 0.73 for the red image, respectively. The boundaries of the down-scaled images were then symmetrically padded with zeros, in order to obtain the same image size of 600 x 600 pixels for each of the three images. This procedure assures that at the end the three holograms, reconstructed with different wavelengths, are size-matched. Afterwards the three images are used as masters to compute three corresponding DPs with a GS-algorithm, such that they are either reconstructed in the Fourier plane, or, after removing the imaging lens L3, as Fresnel holograms at a selectable distance *f* behind the SLM. This can be achieved by superposing the phase of the pre-calculated Fourier holograms with a certain lens term, followed by a modulo 2*π* operation. The phase of the lens term is given by:

*r*is the radial polar coordinate measured from the optical axis,

*f*is the intended focal length of the lens, and

*λ*is the intended reconstruction wavelength which differs for the three color channels. This means that for Fresnel hologram reconstruction the three color channels of the DP are superposed with different lens terms, which, however, will focus in the same focal plane at a distance of

_{rgb}*f*, when read out with the respective different wavelengths. Afterwards the three DPs representing the different color channels are combined to a three-color multi-level DP with the above explained method, which is in this case optimized for the target wavelengths 460 nm, 532 nm, and 633 nm.

A first result of such an experiment, aimed for reconstructing a ”white” version of the letter ”R” is shown in Fig. 4 at the left. The DP was designed as a Fresnel DP reconstructing at a distance of *f* =30 cm behind the SLM in the camera plane. The image shows the letter ”R” in white, as a result of the superposition of the three color channels, which are size-matched. Next to the central white ”R”, the blue, red, and green versions of the image become visible. This is due to the pixelation of the SLM, which corresponds to a superposed cross-grating, which diffracts the reconstructed image into its own diffraction orders, and simultaneously splits the three wavelengths. The right side of the figure shows a test image projected by a three-color multi-level DP, which was constructed the same way as the example at the left. However, in this case the size of the image was trimmed to show only the central part (and not the field diffracted off the superposed cross grating).

Another example comparing reconstructed Fresnel and Fourier DPs is shown in Fig. 5. The left column shows the master for the test image in its original RGB colors. With the method described above, the respective three-colour mult-level DPs were generated and read out with full RGB illumination. For reconstruction of the Fourier version of the multi-level DP (middle column), the imaging lens L3 was inserted in the setup, whereas it was removed for the Fresnel DP projection (right column). A comparison between the reconstructed images shows, that the Fourier version comprises a bright zero-order spot in its center. This zero-order spot is avoided in the ”lensless” Fresnel version of the DP. In this case the zero-order intensity is uniformly distributed as a diffuse background. The reconstructed images show the appearance of a slightly increased speckle noise, which is due to the reduced diffraction efficiency as compared to a standard (single-level) DP, i.e. the remaining light is diffusely scattered and interferes with the reconstructed images. This is partly due to the hologram phase errors that are inevitably introduced by combining multiple single colour holograms into a single multi-colour hologram, and partly due to the fact that a phase-only device is used to project an amplitude image, i.e. the used GS algorithm produces only an approximation of the actually needed transmission function. The speckle noise is more pronounced in the Fresnel version of the DPs. This is due to the fact that the resolution required for the Fresnel DPs is at the limit of the SLMs capabilities, because the superposed Fresnel lens term consumes much of the available modulation bandwidth. The resulting phase patterns have finer modulation and show deeper phase gradients, which are more challenging for the SLMs to display. However, it may be assumed that future SLMs with still finer phase resolution and smaller pixel structure may overcome this problem, since they might be operated well below their resolution limits.

## 4. Conclusion

We demonstrated how to control multiple wavelengths simultaneously with a multi-level DP displayed on an SLM, by using the additional degrees of freedom provided by its high phase modulation range. A practical advantage of the method is that SLMs with such a high dynamic range are already available, and used in many labs, for example for adaptive optics, beam control for optical tweezers, fluorescence microscopy, spatial image filtering in Fourier microscopy, and many other applications. By using the demonstrated method, these devices become much more flexible, for example by allowing to perform different tasks (e.g. beam steering and Fourier filtering, as recently demonstrated in [34]) at different wavelengths simultaneously. This may open a new application field in SLM technology. Thus it may be expected that in the future specialised SLMs may be produced, which optimize multi-color performance by providing an even higher dynamic phase modulation range. Another interesting application field may be laser hologram projection by mobile devices with integrated laser diodes, since holographic projection does not ”waste” light by absorption, and therefore takes only a small fraction of the battery power needed by traditional projectors. The possibility to do this for colour images, or even for three-dimensional coloured objects, may also open a new application field in consumer optics.

## Acknowledgments

This work was supported by the ERC Advanced Grant 247024 catchIT, and by the Austrian Science Fund: Project No. P19582-N20.

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