## Abstract

We reexamine a simple technique for encoding complex amplitude information onto a phase-only spatial light modulator (SLM). The basis for the approach is to spatially vary the diffraction efficiency of a two-dimensional checkerboard binary phase diffraction grating where the period for the Nyquist grating is two pixels. As the phase depth of this 2D grating changes spatially, the amount of light diffracted into the zero order can be controlled. Unwanted information is encoded onto the first diffraction orders and is directed away from the center. This process uses a very simple coding algorithm to generate a complex beam reconstruction on-axis and allows exploiting the full spatial resolution for encoding amplitude. However, its experimental realization with the current liquid-crystal on silicon (LCOS) technology is strongly affected by the limitations imposed by the fringing effect in these devices. We provide experimental evidence of how this effect impacts the efficiency of diffraction gratings displayed on the SLM. We then show how it affects the encoding technique, both in the near field and in the Fourier transform domain, where the limitations imposed by the fringing effect are clearly visible in the form of a focused peak. These results provide evidence of the usefulness of the technique but also about the limitations imposed by the current LCOS technology, which do not allow fully exploiting their high resolution. Finally, we discuss the performance of these newer LCOS devices compared to other SLMs.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Since their pioneering use in optical processing [1–5], the use of liquid-crystal spatial light modulators (LC-SLM) is nowadays extensive for all kinds of applications requiring displaying diffractive optical elements (DOE) and computer-generated holograms (CGH) [6]. In general, both magnitude and phase information are required for many applications in DOEs. However, most LC-SLMs only encode phase information. One of the first techniques allowing encoding of both kinds of information used a blazed grating approach with spatially variant modulation depth [7]. In this approach, as the phase depth decreases from 2*π*, the amount of light diffracted into the first order decreases while the amount of light in the zero order increases. Thus, magnitude information could be encoded onto phase-only displays by spatially modulating the maximum phase depth of such blazed gratings, and the complex-encoded hologram reconstruction is found in the first diffraction order [7]. Thus, this blazed grating technique is especially appropriate for application with SLMs that require operation off axis because they produce a DC component.

However, there are some problems with this blazed grating approach. The first is that the diffraction efficiency of a blazed grating requires at least 8 pixels/period to reach a diffraction efficiency of 90%. This limits the spatial variation of the amplitude information to be encoded. The second is that the diffraction angle of the first order is small, thus providing a small spatial separation of the desired complex beam from the unmodulated zero order. And finally, the technique requires a look-up table that relates the amplitude information to the diffraction efficiency. Nevertheless, despite these limitations, the blazed grating approach [7], and similar variations [8], have been extremely popular. Since then, a number of other approaches have been developed. We refer the interested reader to Ref. [9] that summarizes much of this work.

Another different approach was proposed by Arrizon [10] for low-resolution devices. This work demonstrated theoretically and by simulation a complex Fourier transform diffractive element with reconstruction on axis. Later, Mendoza-Yero *et al* [11] presented an equivalent technique but applied in a 4f system where non-desired diffraction orders are removed in the Fourier plane and the complex function reconstruction is obtained at the system output after a second Fourier transform. In both cases, the technique uses a checkerboard grating [12].

However, that previous work did not emphasize the important advantages of this technique. In this work, we examine these advantages. First, the grating can operate at the Nyquist limit, where the period is only 2 pixels. In this situation, light removed from the beam is diffracted to the furthest points away from the optical axis. Now as the phase depth of this binary grating decreases, the amount of energy diffracted into the zero order decreases while increases the energy diffracted into the four first diffraction orders. As a result, the amplitude information can be encoded using the maximum available space bandwidth. Second, the encoding algorithm is extremely simple to implement, avoiding the use of a look-up table as required in the blazed grating approach. Third, it provides a complex function reconstruction on-axis.

Nevertheless, this approach has not been as popular as the blazed grating approaches. Probably the fact that newer liquid crystal on silicon (LCOS) SLMs present additional unmodulated light on-axis has contributed to this. In this work we reexamine the approach, give experimental results and discuss these problems with LCOS devices. We show how the fringing effect strongly affects the diffraction efficiency. In this effect, the signal applied to a single pixel affects neighboring pixels. This pixel crosstalk modifies the modulation regime for gratings operating near the Nyquist limit, causing a non-modulated DC component which cannot be ignored. In fact, a recent work [13] highlighted these problems for the complex encoding technique, and the checkerboard grating was combined with a blazed grating to operate the technique in an off-axis first order that is filtered in a 4f system. Here, instead, we illustrate these effects with holograms designed to generate complex beams either in a Fresnel diffraction domain or in the Fourier transform domain. We show how the DC component clearly degrades the hologram response in the Fourier domain, while it can be tolerated in the Fresnel domain where its impact is in the form of a background noise.

The work is organized as follows: after this introduction, Section 2 introduces the complex encoding technique for this approach and shows computer simulations. Section 3 discusses our experimental system and the SLM used in the experiments as well as the problems with the fringing effects. Section 4 shows experimental results where we encode Gaussian beams onto the SLM and verify their generation in the Fresnel diffraction domain. Section 5 shows some additional results in the Fourier domain that clearly illustrate the problems associated with the fringing effect. Section 6 shows results when we encode on the SLM a Fourier transform hologram of a desired output. Section 7 provides a discussion of the results while Section 8 includes the conclusions of the work.

## 2. Theory for the complex encoding technique

Let us now first describe the kind of gratings [12] used in the encoding technique. Figure 1 compares a one-dimensional binary phase grating with a two-dimensional version. To form the Nyquist limit on these gratings, we specify one pixel/phase value thus creating a period of two pixels. Each pixel represented in black is encoded with a phase as $\exp (i\phi /2)$ while each pixel represented in white is encoded with the opposite phase $\exp ( - i\phi /2)$. Consequently, the total phase difference between these areas is defined by $\phi$.

For the one-dimensional grating in Fig. 1(a), all of the energy is in the zero order when the phase difference is zero. As the phase difference increases towards *π* radians, the zero order decreases while the first orders reach maxima of about 40.5% each with the remaining energy going into higher orders. Then as the phase difference increases towards 2*π*, the energy returns to the zero order.

Similar effects occur with the two-dimensional grating of Fig. 1(b) and are shown in Fig. 2. These are experimental results obtained using a Fourier transform optical system as explained in the next section, and the diffraction orders appear as focused spots. Again, when this phase difference is zero, all the energy is concentrated into the zero order as shown in Fig. 2(a). As this phase difference increases, the energy splits between the zero order and the four first diagonal orders as in Fig. 2(b). Finally, when the phase difference reaches a value of about *ϕ* = *π*, all of the energy is sent into the four first orders as in Fig. 2(c). Again, as the phase difference increases towards 2*π*, the energy reverts to the zero order.

In a previous work [12], both theoretical and experimental results showed that the intensity ${I_0}$ in the central zero order varies as

where ${I_{\max }}$ is the maximum intensity, obtained a uniform pattern addressed to the SLM.The goal is to encode in the SLM a desired complex amplitude pattern $F({\bf{x}})$ given by

Here ${\bf{x}} = (x,y)$ denotes the spatial coordinate. The magnitude $M({\bf{x}})$ is normalized in the range from 0 to 1 while the phase $\vartheta ({\bf{x}})$ ranges from –*π*to +

*π*.

Now the efficiency of the binary phase grating is spatially modulated using the arccosine of the magnitude as in Eq. (1). Since the electric field in the zero order depends on the cosine of the phase depth $\phi$ in Eq. (1), the magnitude $M({\bf{x}})$ can be used as the argument to calculate the arccosine function. As a result, the total phase depth $\mathrm{\Phi} ({\bf{x}})$ displayed at any pixel is given by the sum of two terms as

Here $(m,n) \in [ - N/2,\ldots , - 1,0, + 1,\ldots (N/2) - 1]$ are integer indices denoting the pixel coordinates, which relate to the spatial coordinates as $(x,y) = (m\Delta ,n\Delta )$, where Δ is the SLM pixel spacing and $N \times N$ denotes the total number of pixels in the SLM array. Note that Eq. (3) was already proposed in [10] and is equivalent to the function proposed in [11].The term ${( - 1)^{m + n}}$ in Eq. (3) creates the Nyquist checkerboard grating, with a period of two pixels, which is then modulated by the arccosine function. The phase difference between the two levels in the term ${( - 1)^{m + n}}\arccos [M({\bf{x}})]$ is given as $\phi ({\bf{x}} )= 2\arccos [M({\bf{x}})]$. Therefore, when the phase-only function $\exp [i\mathrm{\Phi} ({\bf{x}})]$ in Eq. (3) is displayed in the SLM, the zero-diffraction order reproduces the phase pattern $\vartheta ({\bf{x}})$ with an intensity given by Eq. (1), i.e., with a magnitude proportional to $\cos \{{{\mathop{\rm arcos}\nolimits} [{M({\bf{x}} )} ]} \}= M({\bf{x}} ).$ As a result, the complex information $M({\bf{x}})\exp [{i\vartheta ({\bf{x}})} ]$ is encoded in the zero order. This is an extremely simple approach which, in addition, does not require a look-up table as in the blazed grating approach [7].

We next show in Fig. 3 a computer simulation where we encode the TEM_{00} Gaussian beam. Here we use a Gaussian beam with a very small beam waist size of only 16 pixels. We select this beam to illustrate the checkerboard grating technique since this very narrow pattern would be extremely difficult to encode using the blazed diffraction grating approach [7], that would require periods of 8 pixels. In all simuations, we choose the pixel spacing of Δ=20 microns and the wavelength of 0.6328 microns, equivalent to our experiments. In addition, we choose a Nyquist grating where the period is two pixels.

Figure 3(a) shows the central 128×128 pixels of the binary phase grating made using Eq. (3) where, in this color coding, yellow and magenta represent +*π* and –*π* respectively. The 2D binary phase grating is clearly seen at the corners. In these areas, light is highly diffracted leading to low magnitude values of the on-axis beam. However, at the center of the hologram where the magnitude of the Gaussian beam would be strongest, the black pixels indicates that the phase difference is removed. Consequently, light will not be diffracted in this central area and will remain on axis. The dark area only covers about 30 pixels.

Figures 3(b), 3(c) and 3(d) show the computer simulated intensity of the field obtained at a distance of 1100 mm when the SLM is illuminated with a plane wave. Figure 3(b) shows the Gaussian beam that has expanded to about 80 pixels as expected. Beams with a larger waist size would not expand as much when propagated. Figure 3(c) shows a larger view of the output plane with the small Gaussian beam at the center. Figure 3(c) also shows the unwanted information that has been diffracted into the four diagonal first diffraction orders, as shown in Figs. 2(b) or 2(c) for a regular grating. These first diffracted orders now reproduce a diffracted pattern with inverse magnitude information to that observed on axis. Since we selected the Gaussian beam with such a small waist, the four first diffraction orders carry most of the energy. For shorter propagation distances, these orders overlap the central desired beam. However, as the beam propagates, these four diffraction orders diverge from the central axis leaving a clean optical beam with encoded magnitude. Therefore, in many situations it is not necessary to perform the 4f Fourier filtering as in [11], simply because the diffracted orders would not enter the optical system. Figure 3(d) shows a 3D plot of Fig. 3(c), which more clearly shows the Fresnel diffraction of the square SLM screen aperture.

Next, we discuss the experimental system.

## 3. Experimental system, the checkerboard grating and the fringing effect

The experimental system used in this work is sketched in Fig. 4 and is similar to that reported earlier [14]. Light from a He-Ne laser at the wavelength of *λ* = 0.6328 microns is spatially filtered, collimated, and sent through a non-polarizing beam-splitter (NPBS) to a reflective LCOS-SLM. The input polarization is selected with a linear polarizer (P) to be parallel to the liquid-crystal director of the SLM which is aligned horizontal in the laboratory framework. Light reflected by the SLM passes again through the NPBS to a CCD detector. Although the SLM has an antireflection coating for the visible wavelengths, there is some reflection by the front surface of the SLM. This reflection, together with the DC component generated by the hologram addressed to the SLM, contributes to generate a collimated unmodulated reflected beam. This unmodulated DC component increases strongly for holograms having spatial frequencies close to the Nyquist limit, which are strongly affected by the fringing effect.

We used two variations of the system, as sketched in Fig. 4. In the first one (Fig. 4(a)) we simply let the reflected beam propagate a certain distance to the CCD detector. This distance must be far enough, so the first-order diffracted beams separate from the zero order. Now the unmodulated reflection from the SLM is spread out so that we do not see it if the hologram reconstruction is intense enough.

In the second variation in Fig. 4(b), a lens (L2) is added to the system after the beam splitter to obtain the Fourier transform of the hologram loaded onto LCOS-SLM focused on the CCD detector. Here the unmodulated light reflected from the SLM is also focused and can be easily seen in the form of a bright spot. In this situation, the desired output pattern must be shifted from the focused unmodulated light, as it will be demonstrated later in the experimental results.

The SLM is a Hamamatsu LCOS device (model X10468-01). It has a display of 792×600 pixels with a pixel spacing of Δ=20 microns. The electronics of this device show a very linear relationship between phase shift and gray level. We first calibrated the device by placing it between crossed polarizers oriented at 45° with respect to the liquid-crystal director and measuring the transmission as a function of gray level [15]. For this wavelength, we measured phase modulation depths of *π* and 2*π* using gray scale values of 100 and 200 respectively.

We next compared the phase modulation with the diffraction efficiency of the two-dimensional binary diffractive gratings of Fig. 1(b) displayed onto the SLM and compared results with Eq. (1).

The Hamamatsu device is in principle suitable to operate on-axis since this device is free of flicker, a phase fluctuation that generates a DC component [16]. However, we found that this device is not suited perfectly for this experiment due to the fringing effect. This effect was noticed because the gray levels that provide the *π* phase difference between the two levels vary with the period of the grating.

We first examined the predictions of Eq. (1) by measuring the intensities of the zero order and one of the first orders as a function of the addressed gray level. Figure 5(a) shows the experimental data when we display a two-dimensional grating as in Fig. 1(b) where the grating period consisted of 16 pixels. We observe that the zero order vanishes at a gray level of 100 in agreement with our calibration value for the *π* phase modulation, and increases back to a maximum of only 80% of the original value at a gray level of 200, where the 2*π* phase modulation is expected. We noted additional diffracted orders at this higher gray level and this reduced the intensity compared to the zero gray level case. Nevertheless, in this situation we can conclude that the LCOS-SLM encodes with reasonably good approximation a binary phase grating capable to change the phase difference from zero to 2*π*.

However, we examined the intensities of the zero order and one of the first orders as a function of gray level for the Nyquist grating, i.e., now the period is only of 2 pixels. The corresponding results are shown in Fig. 5(b) and show clearly different behavior. First, the zero order vanishes for a gray level of about 140. In addition, the maximum of the first order does not coincide with the minimum in the zero order and does not reach the expected value at zero gray level. The first diffraction order vanishes at the maximum gray level of 255 where the zero order only returns to only about 35% of the initial value. Now we noted a larger number of other more intense diffracted orders. In this situation, the binary phase grating displayed on the LCOS-SLM cannot reach the required 2*π* phase modulation regime.

These results are very different from those in [12] where a transmission LC-SLM device was used for equivalent experiments. Transmission devices are characterized by a low fill factor, i.e., the transparent region in each pixel is smaller than the spacing between pixels. As a result, the wire grid separating the pixels from the LCD forms a two-dimensional series of diffracted orders. However, the separation between pixels shows the beneficial consequence that there is no crosstalk caused by fringing in these transmission devices.

On the contrary, the results with the LCOS device shown in Fig. 5 are a consequence of the fringing effect [17,18] where the voltage applied to an individual pixel affects the neighboring pixels. This occurs because the liquid crystal layer is not physically separated into different areas. As a result, the voltage applied to one area of the liquid crystal can affect the neighboring areas. According to these previous discussions [17,18], we expect the fringing effect to decrease as the voltage difference between pixels decreases.

Consequently, this affects the complex encoding technique especially in two different aspects. First, when the function to be encoded presents sharp variations of the phase or of the magnitude, the encoding creates a large voltage jump between neighbor pixels. For example, when phase gradients extend over 2*π*, these effects will be very evident. On the contrary for slowly varying magnitude or phase distributions, the fringing should not affect results.

Secondly, the fringing effect creates a problem for the complex encoding because the technique operates at the Nyquist grating limit. A solution to this limitation is to use a larger period in the checkerboard grating, thus diffracting the non-desired light to a smaller angle. This is accomplished by changing the encoding function in Eq. (3) by the following relation:

*p*denotes the grating period in pixels, and the floor(.) function is used to round down the values in the exponent to the closest integer number. Some of the experimental proof here presented is demonstrated with a larger period. SLMs that are free of this fringing effect could implement the technique fully exploiting the highest spatial resolution of the diffraction grating.

Next, we cover experimental results.

## 4. Experimental results encoding Gaussian beams

We first generated various Gaussian beams encoding their corresponding magnitude and phase onto the gratings. Since we generate Gaussian modes, we can use the free propagation scheme in Fig. 4(a) and simply let the beam reflected on the SLM propagate a distance far enough, so the first orders do not overlap the central order [Fig. 4(a)]. In this situation, no Fourier filtering is required since the non-desired light is diffracted off axis and does not enter the detector. In these experiments we used gratings where the period is 4 pixels to reduce the impact of the fringing effect and encoded a Gaussian spot size of 64 pixels. The detector was placed at 1100 mm from the SLM.

Figure 6 illustrates the diffractive masks used to generate the HG_{01}, HG_{02}, and HG_{12} modes. Only the central part of the image is shown. The first and second rows [Figs. 6(a) and 6(b)] show the phase $\vartheta ({\bf{x}})$ and the magnitude $M({\bf{x}})$ for these modes. We do not expect that the fringing effect will be as bad in these cases because the phase discontinuities in $\vartheta ({\bf{x}})$ are only of the extent of *π* radians. In addition, they coincide with the minima for the intensity lobes for these modes as shown in Fig. 6(b). As before, the color code in Fig. 6(a) represents *π* radians in yellow, while black represents a zero phase. Figure 6(c) shows the corresponding phase masks $\mathrm{\Phi} ({\bf{x}})$ given by Eq. (3) with the complex encoded function, when the checkerboard grating with a period of four pixels is employed.

Experimental results are shown in Fig. 7 where we generated a number of Hermite Gaussian modes. The beams agree well with theory. We are using a WinCamD detector where the intensity is color-coded with red being the highest intensity. However, we see some areas where the intensity is not perfectly uniform in the various portions of the beams. We note that in this arrangement, the light reflected from the SLM is not focused, and therefore it only affects as a weak background superimposed to the desired hologram reconstruction.

Figures 8 and 9 show equivalent results for various Laguerre-Gauss modes. Figures 8(a) and 8(b) illustrate respectively the phase function $\vartheta ({\bf{x}})$ and the magnitude function $M({\bf{x}})$ used to generate the LG_{01}, LG_{02}, and LG_{03} modes. They show their characteristic spiral phase and annular magnitude distributions. Figure 8(c) shows the corresponding checkerboard complex encoded grating.

Figure 9 shows the corresponding experimental reconstruction of these and other Laguerre-Gauss modes. These modes are reproduced with a quality equivalent to the Hermite-Gauss modes in Fig. 7. Again, the period of the grating is of 4 pixels to mitigate the fringing effect on the checkerboard pattern. However, in this case these effects are clearly seen along the azimuthal direction. Now the azimuthal phase discontinuities in $\vartheta ({\bf{x}})$ are of the extent of 2*π* radians and they coincide with the circular rings for these modes. Thus, these modes are more affected by the fringing effect.

As examples, note that the rings for the LG_{01}, LG_{02} and LG_{03} modes in Fig. 9 show one, two, and three brighter areas, respectively. These correspond to the 2*π* phase jumps in the spiral phases of $\vartheta ({\bf{x}})$, as shown in Fig. 8. Figure 8(a) shows the spiral phase associated with the angular momentum for these three Laguerre-Gauss modes. Note that we are showing the phase patterns in a color code to better visualize them, but the image addressed to the SLM is a gray level pattern, where the gray level controls the phase modulation. Since the phase is modulo 2*π*, there are one, two and three large phase jumps of 2*π* radians along the azimuthal direction for the LG_{01}, LG_{02} and LG_{03} modes. These phase jumps correspond to the sharp yellow-magenta color transitions in Fig. 8(a). The fringing effects associated with these large phase jumps cause the bright spots in the experimental annular rings in Fig. 9. This was confirmed by rotating the phase pattern and seeing that the bright spots also rotated.

In conclusion, we find good results in encoding these kinds of Gaussian beams. However, the fringing effects are more noticeable when we introduce large phase discontinuities as in the LG modes.

## 5. Effects of the fringing effect in encoding the Fourier transform

In the previous section, we saw some effects of the fringing effects on the complex encoding technique. In order to see them more clearly, we used a second different approach. We changed the optical setup to that in Fig. 4(b). In this other configuration we focus the Fourier transform onto the CCD detector. Therefore, any minor deviation from the ideal complex hologram that creates a DC component that will be noticeable in the form of a bright peak on axis. In addition, as mentioned earlier, some light is reflected by the front surface of the SLM will also be focused by the lens.

Figure 10 shows some experimental results obtained to reproduce a HG_{33} beam but using different periods of the checkerboard grating. We show results with periods of *p*=2, *p*=4 and *p*=10 pixels. For each case we show two images of the Fourier transform plane. Here we used a lens L2 with focal length of 10 cm. In Figs. 10(a), 10(b) and 10(c) we show the wide view of the Fourier transform plane. In each case, we see the zero order information together with four first diffracted orders that carry the light intensity removed from the input beam to generate the complex function. As expected, these first diffraction orders are diffracted with greater angles as the period becomes smaller, reaching the Nyquist limit when *p*=2 [Fig. 10(a)].

Figures 10(d), 10(e) and 10(f) show larger versions of the central reconstruction obtained by adding a microscope objective to the CCD detector and imaging a magnified version of the output plane. These results evidence the generation of the HG_{33} beam, but a bright peak is visible in all cases. This peak is generated by the unmodulated light reflected by the SLM, which stays collimated and is focused by the external lens. As discussed earlier, this includes the light reflected from the front surface of the SLM. But, in addition, any deviation from the ideal hologram modulation results in a DC component that contributes to this peak. Note how the peak is strongest for the Nyquist case, *p*=2 [Fig. 10(d)], when the fringing effect affects most the phase modulation. For *p*=10 the HG beam is better reproduced, but the focused peak is still clearly visible.

Next, we further examine Fourier transform patterns.

## 6. Encoding images with Fourier transform holograms

As a final example, we examine the case where we encode a Fourier transform hologram of a small letter “L” as shown in Fig. 11(a), where we only show the central 64×64 portion of the 1024 × 1024 pixel array. Here we are again using the setup of Fig. 4(b). The letter has an area of 30×46 pixels. Figure 11(b) shows the magnitude of the Fourier transform (here the image is the 128×128 portion of the array). Again, this information is normalized to the largest value so that it satisfies the range of the magnitude from zero to one. We note that the central peak of this Fourier transform is only about 50 pixels wide. Using Eq. (3), Fig. 11(c) shows the 128×128 portion of the phase mask that encodes the complex valued Fourier transform hologram. As in the case of Fig. 3(a), the central region where the magnitude is large results in a zero phase-modulation and removes the grating [visualized as the dark central zone in Fig. 11(c)], while the regions with lower magnitude reproduce the phase grating.

Experimental results are shown in Fig. 12 and the reconstruction is quite good. In this case we formed the Fourier transform using a lens [L2 in Fig. 4(b)] having a focal length of 1000 mm. It is important to note that the phase-only hologram successfully reconstructs the complete “L” pattern, and not only the edges as is characteristic in phase-only holograms. This is an indication of the correct implementation of magnitude information of the hologram. Thus, this eliminates the requirement to add an additional noise pattern to remove the edge enhancement effect that happens in phase-only holography. In this case, we were able to use the Nyquist binary grating with a period of only 2 pixels. We believe this result is due to the fact that the Fourier transform varies slowly and that there are no large phase jumps in the function $\vartheta ({\bf{x}})$, similar to the case of the HG Gaussian modes. Nevertheless, we had to design the hologram with an input image where the letter has been shifted from the center, so the reconstruction did not overlap with the peak. Finally, we again stress that this would be very difficult to implement with the blazed diffraction grating [7] approach because of the small width of the central area of the Fourier transform.

Next, we discuss our conclusions and problems with these new LCOS spatial light modulators.

## 7. Discussion

All these results show the usefulness of the checkerboard grating technique to encode complex holograms onto phase-only SLMs. The motivation for this approach is that we can better encode small objects or objects with rapidly varying intensity profiles because the spatial periods required can be as small as only 2 pixels. However, the performance is reduced because of the fringing effect in the SLM. We found better results when the phase varied more slowly. However, for cases where the phase jump is large, for example when phase gradients extend over 2*π*, these effects are very evident.

The fringing effect also creates a problem for the technique operating close to the Nyquist grating limit. Since the SLM is not capable to operate with the ideal phase-only modulation with 2*π* modulation depth, a non-modulated DC component is reflected. In addition, the antireflection coatings applied to the front surface of the SLM are not perfect. In situations where the reflected beam is not focused, like the Gaussian modes generated in Section 4, this unmodulated component results in some background noise that could be neglected. However, in situations where the reflected beam is focused, the unmodulated component results in a bright focused spot. As shown, the strength of this focused bright spot increases as we get closer to the Nyquist limit in the grating, being an indication that the fringing effect is degrading notably the phase modulation of the SLM.

Since there are these problems with the newer LCOS-SLM, some history might be useful. It appears that the designs for LC-SLMs have come full circle. The optically addressed reflective Hughes liquid-crystal light valve (LCLV) was probably the first device to be utilized in a number of experiments [1]. A CCD based version allowed 2D pixelated electrical addressing [2]. These devices were not pixelated and displayed fringing effects.

These devices were followed by transmissive devices as the Citizen television sold by Radio Shack [3]. Many such devices followed, primarily with twisted-nematic displays that affected both amplitude and phase. Citizen [4] and Epson [5] made parallel-aligned transmissive devices, but these were never commercially available. These devices did not exhibit fringing effects because the pixel areas were physically separated by the addressing electronics.

However, the demand for higher resolution and smaller pixel sizes led to the LCOS devices [19]. Their operation is very similar to the LCLV and present the same fringing problem. In both cases, the electronics affect a specific region of the liquid crystal layer. However, there is no physical isolation between these areas of the liquid crystal. As a result, neighboring areas are affected leading to this fringing effect crosstalk. Ultimately this reduces the effective resolution of the device as seen earlier. By comparison, the transmissive CRL device used in [12] and the Hamamatsu device used here both advertise the same pixel resolution of 20 microns. However due to the fringing effects, the actual resolving power of the Hamamatsu device cannot match the single pixel resolution of the CRL device.

These effects will increase as the pixel size of these devices becomes smaller. Therefore, they will limit any attempts to encode patterns with high spatial frequency and large phase difference, and certainly affects the complex encoding grating approach that cannot operate at the Nyquist limit.

Based on these observations and problems with reflective LCOS devices, we would recommend that the SLM manufacturers return also to the transmissive devices. We consider that such transmissive displays, with an emphasis on the phase-only parallel-aligned varieties, would have great advantages and interest to the DOE community.

## 8. Conclusions

In conclusion, we reexamined a previously published approach [10,11] for encoding complex amplitude onto a single phase-only SLM using a checkerboard Nyquist grating. This technique present several important advantages that is worth highlighting. The most obvious is that the minimum period of the grating is only 2 pixels. This width is important in encoding rapidly varying amplitude information. As an example, the blazed grating technique in Ref. [7] requires grating periods of at least 8 pixels to obtain high enough efficiency. We have shown several examples of the capability of this encoding technique with good results. This represents the highest spatial resolving power for encoding amplitude information onto phase-only patterns.

However, we have shown that the performance is notably reduced because of the fringing effect in the SLM. We have shown an easy way to examine these effects with any SLM by conducting the kinds of experiments as in Fig. 5, where the intensity of the zero and first orders is measured for binary phase diffraction gratings with different periods.

We have then presented experimental proof where the checkerboard grating complex encoding technique is demonstrated with a larger period, thus diffracting the non-desired light to a smaller angle, but then diminishing the impact of the fringing effect. SLMs free of this fringing effect could implement the technique fully exploiting the highest spatial resolution of the diffraction grating.

We look forward to further applications of this very interesting approach for encoding complex amplitude onto a single phase-only device.

## Funding

Ministerio de Ciencia, Innovación y Universidades (RTI2018-097107-B-C33); Conselleria d'Educació, Investigació, Cultura i Esport (PROMETEO-2017-154).

## Disclosures

The authors declare no conflicts of interest.

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