1. Introduction

Phase-only spatial light modulators (SLMs) have been used for beam steering for various optical tasks such as optical switching/routing [1–4], object tracking [5], and in optical tweezers [6, 7]. For beam steering, two characteristics are of general importance; the steering efficiency (fraction of total power in steered beam) and the steering (positioning) accuracy. The steering efficiency has been studied in detail [8, 9], while only little work has been published on the positioning accuracy of actual SLMs yielding a phase-only response [10].

In this paper we look into this issue for a parallel incident beam being steered/deflected by a 1D phase-only SLM. First, in Section 2 we review the conventional method for setting the phase of the SLM pixels and give a simple analytic expression for the approximate positioning accuracy that can be expected with this method. We also point out that this method often yields far from optimal results; for instance, during a continuous sweep there might even be two consecutive settings for which the beam moves in the reverse direction. To avoid the poor accuracy, which is caused by not accounting for the limited number of possible phase values in the SLM modulation, in Section 3 we propose a new scheme to optimize the SLM for a higher precision in the realized steering angle. The simulation results are confirmed with measurements presented in Section 4.

2. Beam steering with conventional setting of the SLM pixels

2.1. SLM pixel setting

Using a blazed grating, the diffractive counterpart of the prism, it is possible to steer a monochromatic single beam into any given direction. However, when using a pixelated SLM, the phase modulation of the blazed grating has to be approximated with a staircase grating with phase resets. This limits the smallest realizable period of the grating to two pixel widths yielding a maximal deflection angle of

according to the grating equation, where λ is the wavelength of the incident light and p is the pixel width. However, as will be shown in this work, while a pixelated SLM with an analog phase response can still steer to any angle within the angular range (limited by ±α _max) a pixelated SLM with a quantized phase response inevitably results in a lowered steering accuracy.

 

Fig. 1. (a) An ideal wavefront with a steering angle α (y=x tanα, x_j=(j-1)p where p is the pixel width) and the wave contributions from the pixels. (b) An ideal phase setting (dashed) and the staircase approximations realizable with a pixelated SLM with an analog (solid) and quantized (dotted), M=4, phase modulation. (c) Two ideal phase settings, corresponding to two wavefronts with a difference in steering angle of δα, and the realizable phase levels in between them.

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When steering a beam to an angle α, the wave leaving the SLM should ideally be a plane wave with a wavefront tilted by the angle α, see Fig. 1(a). The phase modulation over the SLM surface of the ideal tilted plane wave is

where x is the spatial coordinate in the SLM plane and φ ₀ is an arbitrary constant phase offset. It is straightforward to find the optimal phase setting for a pixelated SLM with an analog phase response; the phase values are chosen to coincide with the ideal phase in the pixel centers, see the solid line in Fig. 1(b). The phase is given by

where x _j is the center x-coordinate for pixel j, j=1,2,…,N, where N is the total number of pixels. If the SLM only allows M equidistant phase levels between 0 and 2π, the phase values in the pixel centers are set to the allowed phase level closest to the ideal values at the same spatial position, see the dotted line in Fig. 1(b). This is achieved by the following expression

where “round” simply rounds the value to the closest integer value. Of course, in addressing the SLM all the phase values are taken modulo 2π, yielding wrapped staircase gratings.

2.2. Coarse estimation of steering accuracy

There is a simple way to estimate the typical change of the beam angle when scanning the beam in as small steps as possible. The idea is simply to count the number of changing events S, i.e., the number of times any of the pixels in the SLM has changed its setting - and thereby probably causing some motion of the beam – during during a scan from a nominal steering angle α to α+δα. We can then estimate the typical smallest angular change, and thus typical positioning error, simply as α _error=|δα|/S. Evidently, it is possible to have larger inaccuracies than this; the estimation does not account for the actual movement of the beam and in general the pixel changing events are not evenly distributed over the nominal steering angles. For instance, several pixel changing events might even coincide completely, which in effect reduces the actual number of changing events. To calculate S, we calculate the number of times pixel number j changes its setting when the nominal steering angle changes from α to α+δα, which is given by the total phase change divided by the step size of the phase as

since α and δα are small angles. The total number of changes of any of the N pixels in the SLM is obtained by summing S _j over all the pixels

Thus

Finally, we relate this to the diffraction limited spot size of the beam, which we conveniently define as α _spot=λ/N_p, so the angular inaccuracy is approximately

which is in accordance with the results from the analysis in Ref. [10].

 

Fig. 2. Ratio Q between the actual number of changes S _actual in the SLM setting and the number S given by Eq. (6) when the beam is being steered over an angular interval [α_start,α_start+α_spot] with (a) α _start=0, (b) α _start=0.5 ×α _max, and (c) α _start=0.625×α _max. The number of SLM pixels N=32, 64, 128,…, 2048. The number of phase levels M=2, 4, 8,…, 256.

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As mentioned this value is approximate, partly because we count as multiple events a single event where simultaneously more than one pixel changes. We have simulated a large number of situations in which the beam was steered over the nominal angular interval [α _start,α _start+α _spot] in a continuous scan, i.e., the desired steering angle was changed in very small steps. Thus for most steps there was no change at all in the settings of the realized SLM, but those steps where the SLM setting did change were counted to get the true number of changing events, S _actual. Apart from situations when the SLM setting was nearly periodic, we found as a rule of thumb that S _actual ≈0.8S, as illustrated in Fig. 2, thereby slightly increasing the positioning inaccuracy measure to

 

Fig. 3. (a) Simulations of 500 realized steering angles for an SLM (N=32 and M=4) with φ ₀=0. Pos I and II indicate an angle for which the steering error is positive and negative, respectively. The maximum deviation from the aiming angle, relative to the beam spot size, ${\epsilon }_{\mathrm{norm},max}^{{\phi }_0=0}\approx 0.15$.

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In our analysis we used x ∈ [0, p,…, (N-1)p] for the pixel center positions, and Eq. (4) for finding the realizable phase setting of each pixel. Thus we see that the phase of the leftmost pixel, at x=0, is always kept fixed, while the others are trying to approximate the wavefront at the nominal angle α. Of course, we could choose any other pixel to be the one that is always fixed; however this increases the number of events when two or more pixels are changed simultaneously and thus lowers S _actual leading to higher inaccuracy. This situation is worst if the phase is fixed in the most central pixel in the SLM, which unfortunately is a likely choice for anyone who is not considering inaccuracy effects in their beam steering SLM. Then our simulations show that S _actual is only ~0.2S, and hence our simple estimation would suggest a fourfold increase in positioning inaccuracy. Thus, we have already obtained a very simple design rule: choose the pixel whose phase you keep fixed to be in the periphery, not in the center, of the SLM.

2.3. Actual steering accuracy

To this point we considered the number of different SLM settings during a scan of the beam. This gives a crude measure of the positioning accuracy, but what in fact determines the steering angle is the mean slope of the phase modulation “staircase” generated by the SLM pixels. The mean slope is the slope of the line obtained as the best linear fit to the staircase. In fact there is a virtually exact correspondence, such that the position of the beam from an SLM with a certain value of the mean slope of the phase staircase, measured in radians per unit length along the SLM, is exactly (within a very small fraction of α _spot) the same as that of a beam produced by a perfect wavefront with the same slope. Although this might seem quite reasonable, we also checked this claim thoroughly by comparing the mean-slope method for obtaining the steering angle with rigorous calculations of the spot position using the Fresnel diffraction integral[11] (data not shown), and both yielded in all essence the same results. Thus, all beam positioning simulations that follow are directly based on the calculation of the mean slope of the phase modulation staircase.

 

Fig. 4. Simulated normalized steering error εnorm in the entire possible scanning range for φ ₀=0, (a) N=32 and M=2, (b) N=32 and M=8, (c) N=32 and M=32, (d) N=512 and M=2, (e) N=512 and M=8, and (f) N=512 and M=32.

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As a consequence, for an SLM whose pixels can take on any phase value, i.e., an analog SLM, the phase errors inevitably induced by the staircase approximation due to the pixelation of the SLM causes a decreased steering efficiency while the direction of the steered beam still perfectly coincides with the aiming direction. However, as soon as the phase response is limited to a finite number of phase levels the induced phase errors generally affect the mean slope of the approximate staircase grating, thus steering the beam in a slightly different direction than intended. In Fig. 3, the realized steering angle using Eq. (4) with φ ₀=0, is shown as a function of the aiming angle when the beam is steered over the angular region [0,α _spot]. Here, the normalized steering error ε _norm has been defined as the difference between realized and aiming angle divided by the angular spot size. Also, the maximal normalized steering ε _norm,max is defined as the absolute maximum of ε _norm over the full steering range.

As seen in Fig. 3, the agreement between the aiming and realized steering angles can be surprisingly poor, and in the aiming angle region [0.25,0.40]α _spot the beam even reverses its scanning direction. For the worst aiming angle, indicated as Pos I in the figure, the maximal normalized steering error ${\epsilon }_{\mathrm{norm},max}^{{\phi }_0=0}$ is as large as 15%. However, it is evident that the problem is not primarily that the SLM can only provide certain steering angles but rather that the recipe for realizing the wavefronts steers the beam in a direction sometimes quite far from the aiming direction. For instance, in Pos I we aim for a steering angle of ~0.2α _spot but obtain something quite different, although it is obvious from the figure that the SLM is indeed physically capable of producing a steering angle quite close to 0.2α _spot.

One might argue that the simulation shown here was made for an unrealistically low value of N=32 and that the steering error would decrease for larger values. It is true that the steering error, measured in absolute angle, does decrease as the number of pixels increases, keeping everything else unchanged. However, when the number of SLM pixels increases, the angular spot size α _spot=λ/N_p decreases as well and thus, as it turns out, the maximum angular steering error normalized to the angular spot size is in fact very similar for any choice of N. This can be seen in Fig. 4 showing the results of steering the beam over the entire angular range [0,α _max] achievable with the SLM. The figure shows the simulated normalized steering error ε _norm as a function of the aiming angle for N=32 and 512 with M=2, 8, and 32. Note that while the error peaks are sharper for larger N-values the peak height is independent of N. Also, Fig. 4 clearly shows that the maximal normalized steering error ε _norm,max decreases with an increasing number of phase levelsM, as it should. Further analysis of the simulations showed that ${\epsilon }_{\mathrm{norm},max}^{{\phi }_0=0}\approx 0.625M^{-1}$, which again underscores that the error is independent of the number of pixels. Hence, the steering error and its causes and remedies should be of interest also for the large SLMs (N ≈ 1000) used today. Evidently, one has to be particularly careful when using a binary SLM, for which M=2.

 

Fig. 5. Ideal phase (blue dash-dotted), realized staircase phase modulation (black solid), and the mean phase tilt of the realized modulation (red dashed) for the aiming angles labeled (a) Pos I and (b) Pos II in Fig. 3 for N=32, M=4, and φ ₀=0. The tilt error, corresponding to an error ε in steering angle is indicated. (c) and (d) show the two same cases as in (a) and (b) but for the optimal choice of φ ₀. The black dotted lines indicate the threshold phase levels used for rounding the phase to the closest allowed phase level; they are the same in all figures.

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Simulations also showed that if φ ₀ is set such that φ_N/2=0, pinning the phase modulation of a pixel in the SLM center to a constant value for all aiming angles, a larger normalized steering error of ${\epsilon }_{\mathrm{norm},max}^{\phi \frac{N}{2}=0}\approx M^{-1}$ is obtained. However, the deterioration is thus a factor of 1.6, which is less severe than the factor of 4 based on the simplified analysis of the number of SLM changing events in Section 2.2.

3. Improved steering accuracy by optimizing φ₀ for each aiming angle

From Fig. 3, it is clear that the simple phase mapping with φ ₀=0 used in the previous section is not suitable for accurate beam steering. In Figs. 5(a) and (b) we illustrate why this method fails for the aiming angles labeled Pos I and Pos II in Fig. 3, respectively. These figures show the aimed ideal wavefront for beam steering to these angles, the realized staircase phase modulation, and a wavefront with the same mean tilt as the realized modulation (for display purposes the mean tilt wavefront was displaced vertically to coincide with the ideal wavefront at the left-most pixel but it is only the wavefront slope that is of significance). As seen in Fig. 5(a) too many pixels on the right side of the SLM are set to the higher phase level resulting in a too large mean tilt of the grating, i.e., the steering error ε > 0. Similarly, in Fig. 5(b), too few pixels takes on the higher phase level resulting in a too small mean tilt, i.e., ε < 0.

Note that, according to the quantization rule that the pixel phase takes on the permitted value that is closest to the ideal one, it is the intersections of the ideal phase setting and the imaginary phase threshold levels (positioned centered between any two neighboring phase levels) that determine the shape of the resulting staircase phase modulation. The positions of these intersections can be adjusted by changing φ ₀. Thus, a simple but powerful strategy to increase the steering accuracy is to consider φ ₀ as a free parameter and optimize its value for each steering angle α. Previous work has shown that a similar approach can improve the beam shaping abilities of phase quantized diffractive optical elements (DOEs)[12].

 

Fig. 6. Absolute value of the normalized steering error ε _norm as a function of ϕ ₀ for N=128, M=2, and (a) α=0.47875α _max and (b) α=0.499α _max; the latter case is within the difficult steering angle region around α=0.5α _max.

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Fig. 7. (a) Simulations of 500 realized steering angles for an SLM (N=32 and M=4) when φ ₀ is optimized for each aiming angle. ${\epsilon }_{\mathrm{norm},max}^{\mathrm{opt}{\phi }_0}\approx -0.024$. (b) The optimized value of φ ₀ corresponding to the aiming angle.

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The key to the success of this method is to choose φ ₀ such that the obtained mean slope of the phase staircase grating becomes as close to the aimed slope as possible. In general, ε _norm as a function of φ ₀ shows an almost random behavior, see Fig. 6(a). However, in the vicinity of angles that are particularly difficult to steer to, ε _norm shows a more slowly-varying behavior, see Fig. 6(b). Thus, φ ₀ is efficiently optimized in two steps: first the approximate optimum is found using a discrete set (200 samples were used here) of values φ ₀ ∈ [0,2π/M]; then a smaller region (20 samples were used here) close to the found approximate optimum is used to obtain the final optimum. In Fig. 5(c) and (d), the effects of the described scheme are shown. Evidently, we can obtain a phase staircase with a mean slope that is very close to that of the ideal wavefront, leading to a steering angle very close to the desired one.

 

Fig. 8. Simulated normalized steering error ε _norm using the optimized value of φ ₀ for each aiming angle α, (a) N=32 and M=2, (b) N=32 and M=8, (c) N=32 and M=32, (d) N=512 and M=2, (e) N=512 and M=8, and (f) N=512 and M=32.

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In Fig. 7(a), the simulated beam steering is shown for the same task as in Fig. 3, but now with the optimization of φ ₀ for every aiming angle. The corresponding optimized φ ₀ values are shown in Fig. 7(b). As can be seen, the agreement between aiming and realized angle is far better than for the results shown in Fig. 3. For the worst aiming angle, the difference between aiming and realized angle is only 2.4 % of the spot size (compared to 15% for the case when φ ₀=0). Thus, for this angular interval and choice of M, the maximum steering inaccuracy can be reduced ~6.3 times. The improvement in steering accuracy occurs not only for the specific SLM (N=32 and M=4) and steering interval shown in Figs. 3 and 7, but over the full steering range of any SLM. This is seen in Fig. 8 which is for cases identical to those in Fig. 4, but with φ ₀ optimized for every aiming angle.

Comparing the left and right columns, it is also evident that with the φ ₀ optimization, the accuracy does improve with a larger number of pixels. Further analysis of the simulations showed that ${\epsilon }_{\mathrm{norm},max}^{\mathrm{opt}{\phi }_0}\approx \mathrm{}3N^{-1}{M}^{-1}$. For each aiming angle, it is of course also possible to find a worst value of φ ₀, which maximizes ε _norm, in this case the normalized steering error is given by ${\epsilon }_{\mathrm{norm},max}^{\mathrm{worst}{\phi }_0}\approx \mathrm{}1.5M^{-1}$. Further, the improvement factor for ε _norm,max using the optimized φ ₀ rather than the conventional fixed φ ₀ is given by $\frac{{\epsilon }_{\mathrm{norm},max}^{{\phi }_0=0}}{{\epsilon }_{\mathrm{norm},max}^{\mathrm{opt}{\phi }_0}}\approx \frac{0.625M^{-1}}{3N^{-1}{M}^{-1}}\approx 0.2N$. Thus, the improvement is independent of M.

4. Measurements

The measurements were made with a Boulder Non-linear Systems SLM (BNS 512×512 SLM System) nematic liquid crystal (NLC) SLM, a schematic of the optical setup used is shown in Fig. 9(a). To allow for accurate modulation, the phase and amplitude modulation of the SLM was characterized using a diffraction-based method [13]. During the characterization, the CCD camera shown in the setup was replaced with a photo detector. The measured amplitude and phase modulation of the SLM is shown in Fig. 9(b). Since the amplitude modulation was very small, the SLM was considered to be a purely phase modulating device.

 

Fig. 9. (a) Optical setup; The HeNe-laser beam (λ=543.5 nm) is expanded (lenses L₁ and L₂), attenuated with a neutral density filter (ND) and polarized (P₁) before it falls on the SLM. Lens L₃ forms the Fourier plane (FP) which is then magnified by lens L₄. In the magnified FP (MFP), the steered diffraction spot is captured with a CCD camera. Polarizer P₂ is used to block any non-modulated light. (b) Measured amplitude and phase modulation of the SLM as functions of the pixel setting. (c) Typical SLM-frame in which the central 128×128 pixels are used for the experiment. Outside the central pixels the SLM is set to steer the light into directions not disturbing the measurements. (d) Typical image captured by the CCD camera. The determined beam centroid and the intensity along its x-and y-direction are shown.

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Due to the SLM backplane being slightly curved, a typical feature of the used SLM [14], the full area of the SLM was not used since the shape of the resulting spot became too deteriorated. Instead, only the central N×N pixels of the SLM were used to emulate a 1D grating, while the pixels not involved in the experiment were set to a binary grating deflecting the light to positions outside the measurement area in the beam steering experiment, see Fig. 9(c). Measurements were performed for N=128 and 256.

The far-field pattern of the steered beam was magnified and captured on a CCD camera, see Fig. 9(d). The beam position was determined by calculating the center of mass of the thresholded (> 50% of the peak intensity) image. The region used in the center of mass calculation, i.e., fulfilling the threshold condition, had a diameter of ~150 and ~100 pixels for N=128 and 256, respectively.

To perform the measurements for the full angular rangewould have been too time consuming. Instead, measurements were performed for narrow angular regions centered at α _c=0.5α _max, 0.25α _max, 0.125α _max, and 0.47875α _max. In the listed order, these regions yield a decreasing steering error since beam steering is an increasingly simple task in these regions. Note that since the width of the error peaks decreases as the number of phase levels M increases, see Fig. 4, smaller angular regions were used for larger values of M. For each angular region, 150 equally distant aiming angles (resulting in 150 SLM frames) were used.

In Fig. 10, measured and simulated normalized steering errors ε _norm are shown for the angle α _c=0.5α _max with N=128 and M=2, 8, and 32. As seen, the agreement between measurements and simulations is very good. Naturally, as an increasing M value yields a decreasing ε _norm, the measurements for M=32 are more sensitive to noise, which can also be seen in the figure.

 

Fig. 10. Measured normalized steering error for N=128, α _c=0.5α _max, and (a) M=2, (c) M=8, and (e) M=32. Simulations of the same cases are shown in (b), (d), and (f). The error is shown for φ ₀ taking on values corresponding to φ ₀=0 (red dashed), φ_N/2=0 (green dash-dotted), worst φ ₀ for each aiming angle (black dotted), and optimal φ ₀ for each aiming angle (blue solid). For the used device, an ε _norm value of 1% corresponds to 2.12 µrad and α _max is 13.6 mrad.

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Similar results were obtained for all measurements (N=128 and 256, M=2, 8, and 32, for the regions centered at the α _c values listed above). For larger values of N, the agreement between measurements and simulations was not quite as good, see Fig. 11. This is partly caused by the limited beam stability which affects a smaller spotsize (for N=256) more than a larger one (for N=128). However, measurements showed that the beam stability, i.e., the small fluctuations of the beam position when the SLM was continuously updated with the same frame, was ±1 % and ±3 % of the spot size (still defined as λ/N_p) for N=128 and 256, respectively. Thus, additional effects must contribute to the noise in the measurements. One effect is most probably caused by the backplane of the SLM which is known to be slightly curved for the used model[14]. Also, for the larger N the incident intensity on the peripheral pixels is noticeably lower as a result of the Gaussian laser beam, whereas a constant incident intensity was assumed in the simulations.

5. Conclusions and discussion

We have shown that the commonly used method to set the pixel values in a beam steering phase modulating SLM can lead to unexpectedly large deviations in the actual steering angle from the desired. During a beam scan even reverse tracking can occur over a few consecutive settings of the SLMs.

 

Fig. 11. Measured normalized steering error for N=256, α _c=0.125α _max, and (a) M=2, (c) M=8, and (e) M=32. Simulations of the same cases are shown in (b), (d), and (f). The error is shown for φ ₀ taking on values corresponding to φ ₀=0 (red dashed), φ _N/2=0 (green dash-dotted), and optimal φ ₀ for each aiming angle (blue solid). For the used device, an ε _norm value of 1% corresponds to 1.06 µrad and α _max is 13.6 mrad.

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To overcome this problem and obtain a higher steering accuracy, it is crucial to optimize φ ₀ for each aiming angle such that a good agreement is obtained between the ideal wavefront phase slope and the mean slope of the realized staircase phase modulation. The maximal normalized steering error over the full angular range of the SLM ε _norm,max is ~3N ^-1 M ^-1 or ~0.625M ^-1, if the optimized or commonly used scheme is used to obtain the phase grating, respectively. Thus, an improvement in ε _norm,max of ~0.2N can be obtained.

We also found that even if one keeps to the commonly used method, an extremely simple measure can improve the steering accuracy significantly, namely, setting the peripheral, rather than the central, pixel to a fixed phase modulation value for all steering angles. Anyway, in the light of the φ ₀-optimization method both conventional methods perform poorly; they yield only ~2.4 and ~1.5 times better accuracy, respectively, than if φ ₀ is actively chosen to take on the worst possible value for each steering angle.

The simulations have been verified with experiments in a free-space optics setup. The agreement between experiments and simulations was excellent except for the cases when the theoretical angular error was so small that non-ideal properties of the SLM limited the performance, such as the backplane curvature and errors in the phase modulation characteristics.

It should be noted that the procedure of optimizing φ ₀ does not only affect the peak position of the spot but also have a slight influence on the spot shape and power. Therefore, the exact improvement of the positioning accuracy should depend on the metric used to define the spot position. We investigated the positioning accuracy both with the spot position defined as the position of the peak intensity, and the position defined as the center-of-mass of the power distribution around the spot above a certain cut-off level. Even for the worst case, M=2, there was no significant difference, and so the reported improvement of the accuracy was in all essence identical for either definition. This was also confirmed by the good agreement between simulations, based on peak position, and measurements, based on center-of-mass positions. Only for an application which is particularly sensitive to the detailed features of the side-lobes or a slight asymmetry in the tails of the spot intensity distribution, should there be a reason to do a more detailed analysis, with a spot quality measure including more features, to find the optimal phase offset. Similarly, there might be cases for which some accuracy can be traded for a slight increase in spot power. In such a situation, it should be possible to use a similar approach as presented here to optimize φ ₀ such that the highest spot power close to an aiming position is obtained.

It is important to understand that the problem causing the steering error is a phase mapping problem. Therefore the same type of steering error, with the same typical magnitude, will occur for any design method that yields a continuous phase modulation as its primary output, such as, e.g., the commonly used Gerchberg-Saxton algorithm[15].

Another possible approach to obtain high steering accuracy would be to allow phase slope irregularities, i.e., remove the requirement for a monotonically varying phase modulation, so that we no longer try to approximate an ideal wavefront. With such an approach the theoretical steering accuracy could likely be increased even further, but at the expense of a slower algorithm for finding the optimal setting of the SLM pixels since that would be a multidimensional optimization, whereas the method presented in this paper has only one optimization variable – the phase offset φ ₀.