## Abstract

A method for compensating for pixel crosstalk in liquid crystal based spatial light modulators is presented. By modifying a commonly used hologram generating algorithm to account for pixel crosstalk, the intensity errors in obtained diffraction spot intensities are significantly reduced. We also introduce a novel method for characterizing the pixel crosstalk in phase-only spatial light modulators, providing input for the hologram generating algorithm. The methods are experimentally evaluated and an improvement of the spot uniformity by more than 100% is demonstrated for an SLM with large pixel crosstalk.

©2012 Optical Society of America

## 1. Introduction

Phase-only spatial light modulators (SLMs) can be used for holographic control of laser beams and are regularly used in applications such as optical communication [1,2] and holographic optical trapping (HOT) [3,4]. Such applications often require precise control of the intensity in obtained diffraction spots and numerous hologram optimization algorithms have been developed for this purpose [5–13]. The accuracy of the algorithms rely on the ability to numerically predict the intensities in the diffraction spots for a given phase pattern. Generally, the phase retardation obtained from each pixel is assumed to be uniform over its entire surface and independent of the setting of surrounding pixels. However, phase modulating SLMs are usually liquid crystal (LC) based and thus suffer from crosstalk between adjacent pixels, inducing errors in the calculation of diffraction spot intensities.

The phase retardation in an LC-SLM is obtained by applying a variable voltage over each pixel volume in the LC material. Pixel crosstalk is caused by gradual voltage changes (known as ‘fringing fields’) across the border of neighboring pixels [1,14,15] and by elastic forces in the LC material preventing abrupt spatial variations in the phase modulation [1,16]. Both effects cause the realized phase modulation of a pixel to depend also on the voltage applied over adjacent pixels.

Studies on the effect of fringing fields in one dimensional blazed gratings have shown that the result of pixel crosstalk is a spatial low pass filtering of the phase retardation expected for an SLM with no pixel crosstalk. As such, it can be modeled as the convolution of the ideal phase with a point-spread-function (PSF) given by the SLM [14]. If the PSF is known, it is therefore possible to predict and compensate for the effect of pixel crosstalk. This can be done either using a two-step method, in which an ideal hologram is first optimized without regards to the pixel crosstalk and afterwards adjusted according to the PSF, or using an algorithm that includes the pixel crosstalk effect already in the optimization of the hologram.

Starting with a hologram optimized for an SLM with no pixel crosstalk, it is in principle possible to obtain the optimal phase pattern to address on the real SLM by deconvolving the desired phase with the PSF. However, this approach typically gives rise to phase patterns with very large phase range, and is generally not applicable since a typical SLM allows addressing within a range of 0-2π. If a moderately large phase range is available, around 0-4 π, it is possible to adjust the hologram using an iterative approach that keeps the phase within this range while improving the realized phase pattern [14]. In cases where a phase range of less or equal to 0-2π is available, similar improvements can be obtained by letting the conversion between desired phase and applied voltage for each pixel depend on the local spatial frequencies in the hologram [17].

Including a model for the pixel crosstalk in the optimization algorithm generally produces better holograms than two-step methods [18]. One such algorithm, based on simulated annealing, was shown to provide strong suppression of the undesired diffraction orders from a one dimensional blazed grating [19]. Considerable improvements were demonstrated, also for two dimensional spot arrangements, by modeling the pixel crosstalk at a resolution higher than the native resolution of the SLM (oversampling) during optimization with simulated annealing [20]. However, algorithms based on simulated annealing generally require a very large number of iterations and to our knowledge, no such algorithm enabling real-time operation has been demonstrated for spot generating holograms.

For real-time operation, the use of Gerchberg-Saxton (GS) based algorithms is preferable [21]. One approach is to induce an iterative low-pass filtration within the optimization loop of a weighted GS algorithm [18]. The algorithm was shown to yield a much improved uniformity among the spot intensities, but, in common with most other solutions, it requires a higher phase range than is usually available in LC-SLMs.

We have previously described a method for generating holograms compensated for pixel crosstalk, based on a weighted GS algorithm, which limits the required phase range to 0-2π and enables real-time operation [21,22]. The method is here described in greater detail and evaluated experimentally for an SLM suffering from strong pixel crosstalk. We also present a novel method for characterizing the pixel crosstalk in phase-only SLMs to provide input for the optimization method.

## 2. Algorithm for generating compensated holograms

The effect of pixel crosstalk can, as previously mentioned, be described as a spatial low pass filtering of the phase profile displayed on the SLM and modeled as its convolution with a spatially invariant PSF (Fig. 1(a) and 1(b)). It is therefore possible to calculate the intensities in the resulting diffraction spots by numerical propagation of the field after convolution of its phase (Fig. 1(c) and 1(d)). By including this calculation in an iterative algorithm such as GS, we can compensate for the crosstalk and obtain holograms producing optimal spot patterns when displayed on the SLM. To simulate the effect at higher resolution, the hologram can be oversampled using nearest neighbor interpolation before the convolution and numerical forward propagation, while the numerical back propagation is always done at the native resolution of the SLM.

We have modified a weighted GS algorithm to compensate for pixel crosstalk according to this method. The unmodified algorithm has been shown to produce almost perfect phase holograms in terms of obtained diffraction efficiency and uniformity of spot intensities for SLMs with no pixel crosstalk [11]. It is also very fast, the computation time for an optimized hologram can be reduced to a few milliseconds using a parallel computing architecture such as CUDA [21,23]. After modification for crosstalk compensation, the iteration cycle can be described by the flowchart presented in Fig. 2 , and is comprised of the following steps:

- A. An initial phase,
*φ(x,y)*, is selected by reusing a previously used hologram [24] or by generating a random phase pattern. - B. (optional step) The phase pattern is oversampled to a higher resolution by nearest-neighbor interpolation to obtain
*φ(x*, where^{i},y^{i})*x*are the interpolated coordinates of the SLM.^{i},y^{i} - C. The phase is convolved with a PSF representing the pixel crosstalk. The convolution is done by multiplication of the phase pattern and the PSF in Fourier space:
_{${\varphi}^{\prime}\text{(}{x}^{i},{y}^{i}\text{)=}\varphi \text{(}{x}^{i},{y}^{i}\text{)}\otimes a\text{(}{x}^{i},{y}^{i}{\text{)=FFT}}^{\text{-1}}\text{(FFT(}\varphi \text{(}{x}^{i},{y}^{i}\text{))FFT(}a\text{(}{x}^{i},{y}^{i}\text{))),}$}where FFT is a fast Fourier transform and FFT^{−1}its inverse. - D. The convolved phase pattern is combined with the amplitude distribution of the illuminating laser,
_{${E}^{\prime}({x}^{i},{y}^{i}\text{)=}{A}_{laser}({x}^{i},{y}^{i}\text{)}e\text{xp(i}\varphi ({x}^{i},{y}^{i}\text{)),}$}and the resulting field is numerically propagated to the spot positions using an FFT or a finite sum representation of the Fresnel approximation [11,25] to obtain E(*u*), where^{m},v^{m},w^{m}*u*,^{m}*v*^{m},w^{m}are the coordinates of the spot positions. - E. The computation window is reduced to the size of a field propagated from a non-oversampled phase pattern. The amplitudes in the desired spot positions,
_{$A({u}^{m},{v}^{m},{w}^{m}\text{)=abs(}E\text{(}{u}^{m},{v}^{m},{w}^{m}\text{)),}$}are replaced with a weighted combination of the obtained and the desired amplitudes as described in [11,25]. - F. The field is numerically propagated back to the SLM plane to obtain E(
*x*,*y*), the new phase pattern,_{$\varphi \text{(x,y)=arg(E(x,y)),}$}is calculated and the cycle is repeated from step (B).

The amplitude of the field obtained from the numerical forward propagation will now differ from the desired amplitude, both because of the changed amplitude in the SLM plane, as in an unmodified GS-based algorithm, and because of the convolution with the PSF preceding the forward propagation. Replacing the amplitude in the spot positions therefore compensates both for the SLMs inability to provide amplitude modulation and for the pixel crosstalk. The iteration cycle is terminated when the amplitudes obtained in step E are sufficiently close to the desired ones.

The performance of the described algorithm is determined by how accurately we can simulate the pixel crosstalk (step C in the flowchart), which in turn is determined by the choice of PSF used for the convolution. The PSF must therefore be carefully chosen to correspond to the actual behavior of the used SLM. In the following section, we present a method for identifying the PSF best representing the crosstalk in a given SLM.

## 3. Identifying the PSF

#### 3.1 The effect of pixel crosstalk on binary phase gratings

SLMs with no pixel crosstalk give uniform phase retardation of a reflected wavefront over the entire area of each pixel. When a binary phase grating is applied on such SLMs, the intensity in the first order diffraction spot is

where*Δφ*is the phase separation, i.e., the separation between the two phase levels in the grating. When Δ

*φ*takes on values from 0 to 2

*π*, we call this the first diffraction order response curve. The pixel crosstalk distorts the shape of the phase profile and therefore also causes a change in shape of the first diffraction order response curve. Since the crosstalk mainly influences the region close to a phase step, a grating with short period is more affected than a grating with longer period. As shown in Fig. 3 , the realized phase retardation may not reach desired levels anywhere on the pixel if the grating period is of the same magnitude or shorter than the width of the PSF. By studying how the measured first diffraction order response curve changes for different grating periods, we can therefore obtain information about the shape of the PSF representing the crosstalk. We propose a method where the first diffraction order response curves are measured for an SLM and compared to an extensive set of simulations to identify the shape of the PSF best describing the pixel crosstalk for the used SLM.

#### 3.2 Proposed method for identifying the PSF

The intensity in the first diffraction orders are measured for a set of binary phase gratings, _{${\varphi}_{\Lambda ,\Delta \varphi}(x,y),$} with different periods, Λ, and phase separation, Δ*φ*. To reveal differences in pixel crosstalk in the *x* and *y* direction, the measurement is repeated for both directions.

The measured intensities, _{${I}_{\Lambda ,\Delta \varphi}^{mx}$}and _{${I}_{\Lambda ,\Delta \varphi}^{my},$} are compared to simulations where the same set of gratings are convolved with a trial PSF, a(*x,y*), of a specific shape. Guided by previous studies on the effect [14,15], we assume the PSF to be an elliptical generalized Gaussian function,

*r*and

_{x}*r*in the

_{y}*x*and

*y*direction respectively, and shape parameter

*γ*. The simulated intensities,

_{${I}_{\Lambda ,\Delta \varphi ,a}^{sx}$}and

_{${I}_{\Lambda ,\Delta \varphi ,a}^{sy},$}are calculated by convolving

_{${\varphi}_{\Lambda ,\Delta \varphi}({x}^{i},{y}^{i})$}with the trial PSF and taking the square of the Fourier transform,

Since only horizontal and vertical gratings are used for the characterization, the simulations can be done using one dimensional gratings and the integral of the PSF along each direction, significantly reducing the computation time. The goodness of fit for the trial PSF is evaluated by calculating the total root mean square error (RMSE) of measurements and simulations,

*N*is the total number of measured intensities. By varying the parameters,

_{tot}*r*,

_{x}*r*, and

_{y}*γ*, and identifying the set of simulated intensities that minimizes the RMS error we obtain the optimal shape of the PSF for our SLM. As in the hologram generating algorithm, the convolution and numerical propagation may be done at a resolution higher than the native resolution of the SLM.

#### 3.3 Experimental procedure

The intensities of the first diffraction orders were measured by projecting the Fourier plane of our SLM (PPM X8267-15, Hamamatsu Photonics) directly onto the sensor chip of a CCD camera (SCOR-20SO, Point Grey Research, Inc.) using a single lens. The SLM was illuminated by a collimated laser beam with a wavelength of 1070 nm (1070 nm, IPG Photonics). Only the central 10% of the active area of the SLM was illuminated to avoid errors due to spatial variations in the phase response [26]. The SLM used is addressed by sending 8-bit gray scale images over a VGA interface and has been calibrated to give a linear phase response from 0 to 2π for gray level values 0-255. For each grating, Δ*φ* was stepped from 0 to 252 in steps of 4 gray levels. Phase gratings with periods Λ = 2, 4, 6, 8, 10 and 12 pixels were evaluated. The intensities of the first diffraction orders were measured by integrating the pixel values in a region of 40 × 40 pixels centered on the diffraction order after subtraction of a dark image acquired with the laser light blocked. To increase the signal to noise ratio, the pixelwise sum of 40 consecutively acquired images were used for the intensity measurement. The set of measurements for each PSF was normalized to its maximum value. Each measurement was repeated for the *x* and *y* direction of the SLM.

#### 3.4 Simulations and analysis

Simulations of the first diffraction order intensities, for the same set of gratings as used in the measurement, were done by computing the convolution of the gratings with a PSF and propagating the obtained fields numerically to the first diffraction order. The convolution was calculated by multiplying the discrete Fourier transforms of the phase grating and the PSF respectively,

To reduce the number of simulations needed for the optimization, a two dimensional golden section search, minimizing the RMS error, was implemented to find the optimal *r _{x}* and

*r*for different values of

_{y}*γ*. The initial limits for

*r*and

_{x}*r*for were set to [0,4] SLM pixels and

_{y}*γ*was stepped from 0.2 to 4 in steps of 0.05. The optimization was repeated for 2, 4 and 8 times oversampling compared to the resolution of the SLM.The optimal values for of

*r*,

_{x}*r*and

_{y}*γ*for each degree of oversampling are presented in Table 1 . The measured intensities,

_{${\text{I}}_{\Lambda ,\Delta \varphi}^{mx}$}and

_{${\text{I}}_{\Lambda ,\Delta \varphi}^{my},$}are shown in Fig. 4(a) and 4(b) respectively. The RMSE for the optimal

*r*,

_{x}*r*for each simulated

_{y}*γ*is shown in Fig. 4(c) and 4(d) shows the optimal values for

*r*,

_{x}*r*. The simulated intensities for the optimal values of

_{y}*r*,

_{x}*r*and

_{y}*γ*are shown in Fig. 4(e) and 4(f). The simulation and optimization data shown in Fig. 4 was generated with 8 times oversampling.

## 4. Evaluating the performance of the method

To evaluate our methods for characterizing and compensating for the pixel crosstalk, we generate holograms compensated for the optimal PSFs, as found in the previous section, for each degree of oversampling. Optimized, but not compensated, holograms giving the same spot arrangements were also generated, and their resulting spot intensities simulated using the optimal PSF for the case with no oversampling. The spot intensities for the measurements were determined as described in the previous section and the resulting uniformity,

was calculated for each generated hologram.*I*and

_{max}*I*are here the intensity of the strongest and weakest spot respectively. Since the GS algorithm is nondeterministic for random initial phase, 25 holograms were generated with different initial phase patterns for each spot arrangement. The average uniformity and its standard deviation were calculated for the optimal PSF found for each degree of oversampling. Holograms giving 3 × 3 and 4 × 4 spots separated by 36, 48 or 60 times the grid spacing given by the FFT, and centered on the zeroth order were used for the evaluation.

_{min}## 5. Results and discussion

Figure 4 shows a good agreement in shape and period between measured (4(a) and 4(b)) and simulated (4(e) and 4(f)) first diffraction order response curves after minimization of the RMSE function, although the amplitudes vary slightly. We note that the first diffraction order response curve differs significantly in the *x* and *y* direction, indicating a great disparity in crosstalk in the two directions. The cause for this effect was not further investigated but similar behavior have previously been observed for other types of SLMs [27].

Table 2 shows the average uniformity with standard deviation for each PSF evaluated, and Fig. 5 shows examples of the resulting spot patterns for holograms generated without (5(a)-5(f)) and with (5(m)-5(r)) compensation for pixel crosstalk. Figure 5(g)-5(l) show simulations of the resulting spot patterns for the uncompensated holograms after convolution with the optimal PSF for optimization without oversampling. The similarity of the simulations with the measured intensities indicates that the crosstalk can be accurately modeled by convolution with the obtained PSF. We also observe a drastic increase in uniformity of the measured spot intensities for the holograms generated with crosstalk compensation. The combined average uniformity obtained for all spot patterns used in the evaluation increased from ~0.4 to ~0.9. The average obtained uniformity for each spot arrangement is given with standard deviation at the bottom of Fig. 5(m)-5(r). The improvement enables the use of SLMs suffering from large amounts of crosstalk also in applications requiring precise control of the spot intensities. In some of the spot patterns used for the evaluation here, no spot was located in the location of the zeroth order diffraction spot, see Fig. 5(d)-5(f), 5(j)-5(l) and 5(p)-5(r). The intensity in this location is then not regulated by the algorithm and may in some cases be very strong.

If suppression of the zeroth order is desired, we suggest the addition of an active zeroth order reduction method to the iteration cycle, as presented in [20]. Physical blocking of the zeroth order is also common practice in some applications [13].

Interestingly, for the used SLM, the performance of our method is virtually independent of the degree of oversampling of the holograms used for simulating the crosstalk, presumably an effect of the relatively long range of the crosstalk in our SLM. The native resolution of the SLM could therefore be used with maintained control of the spot intensities; yielding a significant reduction in computation time for the algorithm compared the use of oversampled holograms. The inclusion of the convolution step in the iteration cycle naturally increases the computation time of the algorithm, however. Compared to a regular weighted GS algorithm using FFTs in the propagation step, the computation time per iteration cycle is approximately doubled, from 0.3 ms to 0.6 ms when executed on an Nvidia GTX 580 GPU. Our modification also affects the convergence of the GS algorithm. Weighted GS algorithms normally converge to uniformities of 0.99 or higher within 25-50 iterations using random initial phase, and around 10 iterations reusing previous holograms as initial phase [24]. The introduced convolution step impairs the convergence and using the PSF found for our SLM, the number of required iterations increased to more than 100 in many cases when using random initial phase. The number of required iterations is in general lower when spots are located near the zeroth diffraction order, due to lower amount of high spatial frequencies in the hologram. By limiting the maximum spot displacement, real time operation at rates of 20-50 Hz may be obtained. SLMs with smaller amounts of crosstalk, e.g. with narrower PSFs also requires fewer iterations for hologram generation with crosstalk compensation. Such SLMs may, however, require oversampling during the hologram generation in order to accurately simulate, and compensate for, the crosstalk.

In addition to the significantly improved uniformity, we note a slight increase in diffraction efficiency using our algorithm compared to using holograms optimized with a regular weighted GS algorithm. The increase was in the order of 10% for the spot arrangements used in our evaluation.

Although the PSF found in this work yielded a significantly increased control over spot intensities, we note that a perfect fit has not been found when comparing the measured, see Fig. 4(a) and 4(b), and simulated first diffraction order response curve for the optimal PSF, see Fig. 4(e) and 4(f). This may indicate that the PSF for our SLM does not have the shape of a generalized Gaussian function as assumed here. To further improve the fit, and hence, the control of the diffraction spot intensities, trial PSFs with other shapes may be used. For this purpose, the method may be extended to using a set of more complex test patterns, including two dimensional spot arrangements, for determination of the PSF.

## 6. Conclusions

We have demonstrated how the intensities in the diffraction spots for phase only holograms can be more precisely controlled by including a simulation of the crosstalk between adjacent pixels in the hologram optimization algorithm. We have also introduced a novel characterization method for phase-only SLMs providing input for the simulation.

For our SLM, the modified algorithm was found to increase the uniformity of the diffraction spot intensities from ~0.4 to ~0.9 compared to a commonly used hologram optimization algorithm. We also found an increase in diffraction efficiency by ~10%.

The method enables the use of SLMs suffering from strong pixel crosstalk in applications requiring precise control of diffraction spot intensities.

## Acknowledgments

This work was supported by the Swedish Research Council (M.G.).

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