1. Introduction

The capability of liquid crystal spatial light modulators (LC-SLM) for modulating either the phase or the amplitude of an incoming wave has led to a wide variety of systems and applications, in areas such as digital holography [1], microscopy [2], optical visual simulators [3], programmable spectrum synthesizers or pulse shaping [4], optical superresolution [5] or optical tweezers [6], among others. These devices offer a unique and powerful way of controlling the electromagnetic radiation in the optical range.

Currently, liquid-crystal on silicon (LCOS) displays are one of the dominant SLM technologies. Parallel-aligned liquid crystal on silicon (PAL-LCOS) SLMs offer an exceptionally simple phase-only modulation control on light beams, with very high spatial resolution and reduced pixel sizes. The phase-only modulation regime is obtained for the linear polarization component parallel to the liquid crystal director axis. Nevertheless, LCOS-SLMs suffer from limitations to generate arbitrary complex fields, since amplitude and phase cannot be independently controlled at each pixel. Similar limitations can be considered in general diffractive optical elements (DOE) fabricated onto phase-only substrates.

There are numerous previous approaches in order to surpass this limitation. At the beginning of digital holographic era, techniques were developed to produce printed amplitude-only holograms that could also be applied in the context of new phase-modulation SLM devices or even to other technologies such as digital light processor displays (DLP) to successfully codify complex values into modulation restricted devices. In the case of parallel alignment (PAL) SLM, such as the one used in this work, these elements act naturally as phase-only devices and they can easily be set to produce amplitude modulation by means of simply two 45° oriented linear polarizers, before and after the SLM. In this sense, Burckhardt [7], Lee [8] or Lohmann [9] hologram encoding techniques could also be used to produce complex-encoding holograms. There were also similar techniques devised for phase only modulation substrates such as the double phase hologram (DPH) [10] which could also be applied to current SLM technology. The drawbacks in these cases are less expectable efficiency than phase modulation approaches, multiple order replicas, significant loss of spatial bandwidth (since information is encoded by using ‘macro-pixels’, i.e., several pixels per complex data point to be encoded) and the need, in general, of significant computation charge per pixel.

Another first obvious approach consists on using two SLMs, one to generate the phase pattern and a second one to generate the amplitude pattern [11–13]. This approach requires imaging one SLM onto the second one with very high precision, so that corresponding pixels of the phase and the amplitude patterns exactly overlap. Simpler schemes are those based on a single SLM, which incorporate amplitude modulation encoded onto phase-only masks. A first such approach was developed in the frame of correlation filters for pattern recognition in the mid-nineties, by Cohn and associates [14–16]. In these works, the technique used a random spatial phase pattern whose statistical properties at each pixel were governed by the desired amplitude. It was shown that averaging many of these patterns produced an amplitude modulation, which corresponds to the characteristic function [17] of random distributions. This random method, however, was not further applied due to the success of other alternative methods. Note that intensity averaging can be performed with SLMs by sequentially displaying different hologram realizations and time integrating in the detector, this being useful for reducing the speckle noise in the reconstruction [18,19].

In the late 90’s, Davis and associates developed a different approach [20], similar to that proposed years before in the frame of DOE [21]. There, complex patterns were generated by combining a linear phase grating with the desired phase pattern, and the amplitude information was encoded through the contrast (maximum phase modulation depth) of the carrier phase. The complex pattern is reproduced off-axis, on the first diffraction order. Later, in the early 2000’s, a different approach based on a double-phase computer generated holograms (DP-CGH) was proposed [22–24]. This approach produces on-axis reconstructions, while the generated noise is redistributed to lateral diffraction orders. This approach gets good results as well, but at the cost of a loss in the spatial bandwidth, since two pixels are required for coding of each complex value. A related technique was presented by Arrizón in [25], where the phase of the intended hologram was combined with a correction term containing a modulated 2D binary-phase grating. This way, amplitude modulation was introduced by redirecting undesired light to out of diffraction orders, and the complex function is recovered on axis with maximized signal-to-noise ratio without bandwidth losses, as discussed by the author. A comparison of some of these previous techniques is presented in [26]. Finally, another scheme based on a binary checkerboard grating and Fourier low-band pass filtering in a 4f system was proposed and demonstrated in [27].

Other significant advance towards real-time holographic generation systems was carried out in [28]. In that work, a transmissive SLM panel was combined with external polarizing elements so that each two consecutive pixels of the panel modulated the phase of two orthogonal polarization components of light. The light of alternate pixels was rotated 90 degrees by means of a structured half-wave plate and beams of two consecutive pixels were overlapped in the same direction thanks to a polarization-sensitive component (PSC). The complex modulation was finally achieved by projecting the two orthogonal components onto a 45° oriented analyzer polarizer. The set-up provided high quality results, and permitted direct encoding of complex-values by means of a single SLM panel allowing the system for real-time holographic generating systems. As some disadvantages, the system requires precisely designed external polarization elements, it implies a bandwidth loss because of using two pixels per encoded value, and some amount of computation charge is required because it is implicitly based on the use of the double phase hologram (DPH) representation [10], and, in order to introduce amplitude information, arccosine function computations are required per each encoded point.

In this work, we seek for a fast to calculate, simple and non-iterative encoding technique, valid for on-axis hologram reconstruction with reduced background noise, with minimum external required optical elements. The technique also provides a high degree of control to both the reconstruction and the shape and position of the undesired light out of axis, permitting this way the user to avoid further filtering operations, and being suitable for time integration (and thus noise reduction). Since very simple operations are involved at each calculated pixel, the technique is also suitable for fast computation systems. Advances in LCOS technology has renovated the interest of hologram designs working on axis, since flicker-free LCOS-SLMs that allow operating on axis [29] are available. Here, we propose a new encoding approach, where we recover the idea of random encoding of complex valued DOEs. However, we improve the previous random scheme in [14–16], by combining it with an additional element redirecting non-desired light out of axis to avoid overlapping noise, in a similar way as in [27]. In our approach we generate a spatially multiplexed phase hologram incorporating two different optical elements: 1) the complex-valued hologram, whose reconstruction is on axis, and 2) a diverging optical element, in this case a diverging axicon lens (although other light redirecting phase DOEs are perfectly suitable for this task), whose mission is to redirect undesired light out of axis, to avoid overlapping with the reconstruction. In our approach, the amplitude to be encoded serves as the threshold level to choose at each pixel between the two optical functions. As we show, the resulting hologram shows on-axis reconstruction, with low level of overlapping noise that can be additionally reduced by time integrating different realizations.

The paper is organized as follows: In section 2, the proposed technique is described. Section 3 provides some simulation results and studies for characterizing the performance of the technique. Section 4 presents experimental results regarding Fourier transform hologram reconstruction. Next, in Section 5, we further experimentally verify the method generating Hermite-Gauss and Laguerre-Gauss modes, in order to show the effective generation of intensity but also phase fields. This is demonstrated with a phase-shifting polarization interferometer technique directly implemented in the same setup. Finally, section 6 presents the conclusions of the work. An appendix has been included to provide theoretical support to the validity of the proposed method.

2. Description of the encoding technique

In this section, we introduce the principle of the encoding technique. The first concept is the random multiplexing of two different phase-masks [30]. The second idea implies the introduction of a variable threshold for the multiplexing approach in order to encode the amplitude information into the hologram. This is the key concept in this work and provides an improvement with respect to the previous random approaches reported in [14–16].

First, let us consider a DOE with complex transmittance given by

Next, let us recall the principle of random multiplexing of two phase masks introduced in [30]. In that work, two different phase-only DOEs, F(u) and D(u), were directly encoded into the same plane by generating a random multiplexing binary mask R(u) and its complementary version $\overline{R}\left(u\right)=1-R\left(u\right)$. In these masks, pixels take values either 1 or 0, and both masks are orthogonal to each other, i.e., their product yields zero $\overline{R}\left(u\right)R\left(u\right)=0$. Mathematically, this scheme can be expressed as

The method to introduce the desired spatially varying amplitude can be understood easily and intuitively as follows. We assume a normalized amplitude information A(u), defined in the range [0,1]. If A(u) is close to 1 at a given location u, it is better represented by the phase-only term of the intended hologram F(u), and thus R(u) = 1 should be selected. On the contrary, at locations where A(u) is close to 0, light should be removed and the diverging element D(u) performs this effect. For intermediate values of A(u), we randomly choose between the phase values of F(u) and D(u), depending on the value of A(u). Mathematically, this is defined by the binary random mask R(u) as follows

Next, in order to analyze one single realization of the phase-only DOE, which is the practical situation in most cases, let us decompose the random binary mask as the addition of two terms

In the case of a function rnd(u) with uniform distribution, approximated analytical expressions can be derived. In order to analyze the effect of the last (on-axis) noise term, we can evaluate its Fourier transform, which is a background noise given by$BG\left(\nu \right)={\tilde{N}}_A\left(\nu \right)\ast \tilde{F}\left(\nu \right)$, where * denotes here convolution operation. N_A(u) can be modeled approximately as random signal with zero mean, non-uniform triangularly shaped distribution in the amplitude interval [-1,1], and variance equal to 1/6, as it is discussed in the Appendix. Its Fourier transform can be approximately modeled as (see Appendix A2):

This implies the following background diffracted field, given by the convolution

3. Numerical simulations

In this section, we perform some numerical simulations to check if the designed DOE performs as expected. For simplicity, this first example is limited to evaluate only the intensity reconstruction (in a next section it is shown that the technique is valid also to generate complex-valued reconstructions). We calculated a Fourier transform hologram that reconstructs a letter ‘Ñ’ whose amplitude is shown in Fig. 1(a). It is very well-known that a simple phase-only hologram generates an edge-enhanced version of the object. A usual technique to avoid this effect is to add a random phase (Fig. 1(b)) before performing the Fourier transform [30]. The corresponding amplitude and phase of the inverse Fourier transform represent the functions A(u) and ϕ(u) = arg{F(u)} respectively. These functions are shown in Figs. 1(c) and 1(d) respectively. Note how the synthetic hologram amplitude A(u), see Fig. 1(c), is spread in the Fourier plane thanks to the random phase, Fig. 1(b), added in the input plane. This intentionally added random phase noise ensures that the amplitude distribution Fig. 1(c), does not show very large variations. Also note how the phase distribution in the Fourier plane, shown in Fig. 1(d), is a slowly varying function, since the object ‘Ñ’ to reconstruct is located on axis.

 

Fig. 1 (a) Amplitude distribution of the desired reconstruction. (b) Random phase distribution added to spread its Fourier transform. Corresponding (c) amplitude A(u) and (d) phase ϕ(u) = arg{F(u)}, of its Fourier transform. (e) 160 cycle negative axicon used as the diverging element D(u). (f) A realization of the corresponding random multiplexed phase mask M(u).

Download Full Size | PDF

Figures 1(c) and 1(d) represent respectively the amplitude distribution A(u) and the phase distribution F(u) of the complex function to be encoded in the hologram, which is designed to reconstruct the intensity pattern in Fig. 1(a). These functions are combined with the diverging element D(u), which was selected as a 160 cycle (phase jumps of 2π) negative axicon, as shown in Fig. 1(e). Note that this axicon shows much greater spatial frequency than the phase distribution in Fig. 1(d). These three functions are combined through Eqs. (2) and (3) to provide a multiplexed phase mask M(u) (Fig. 1(f)). For this example, the function rnd(u) is selected in the simplest case uniformly distributed in the interval [0,1]. All images are designed with 512 × 512 pixels, but only the central portion of 190 × 190 pixels is shown to better appreciate the details.

Figure 2 shows some computer simulations of the expected reconstruction of the phase-only hologram, obtained by inverse Fourier transforming M(u). In order to clearly appreciate the different features, these intensity images have been represented oversaturated exactly in the same conditions. In Fig. 2(a) we represent the reconstruction of the simple Fourier transform phase-only hologram in Fig. 1(d). Note how this reconstruction effectively reconstructs the complete letter ‘Ñ’, but shows the characteristic speckle noise overlapping the desired hologram reconstruction. This can be eliminated by using well-known iterative methods [33,34], or other advanced approaches that require operation out of axis, and further spatial filtering [35,36], or they complement other existent iterative techniques [37]. However, as said before, here we want to avoid using iterative algorithms or operate off-axis.

 

Fig. 2 Hologram reconstructions (the intensity has been equally oversaturated in them all to clearly appreciate the noise features) for (a) the phase-only simple hologram shown in Fig. 1(d), (b) a single realization of the random multiplexed phase hologram given at Fig. 1(f), and (c) the average of 10 different realizations of the random multiplexed phase hologram.

Download Full Size | PDF

Figure 2(b) shows the case when a single realization of the random multiplexed hologram, Fig. 1(f), is used. Note how the speckle noise level is reduced significantly because a considerable amount of non-desired light is concentrated on the outer ring characteristic of the Fourier transform of an axicon [31,34]. This ring of light could be easily filtered. Finally, Fig. 2(c) shows the simulated result when the intensity reconstruction of 10 different realizations of the random multiplexed hologram in Fig. 1(f) is averaged. In order to obtain this averaged reconstruction, a different realization of the added random phase pattern, Fig. 1(b), is generated in each realization, and a corresponding multiplexed hologram, Fig. 1(f), is calculated for the same axicon function. As expected, the speckle noise level is reduced. This intensity averaging can be done with SLMs, by sequencing different realizations and integrating in time [18,19].

Next we provide a signal-to-noise ratio (SNR) and diffraction efficiency numerical analysis. This has been performed by controlling the deviation (σ) and mean value (μ) of the multiplexing signal rnd(u), whose distribution is considered now Gaussian (available C + + libraries and routines for this distribution can be found on the internet). This way we are able to control and optimize the design. Such signal is compared against amplitude A(u). This numerical study has been performed by sweeping both μ and σ in the range [0,1].

The SNR parameter has been defined as follows [38]:

For the reconstruction efficiency we considered the following expression

Results of the study for SNR are depicted in Fig. 3(a), while diffraction efficiency is shown in Fig. 3(b). SNR data have been obtained by clipping a 50 × 60 pixel box which contains the whole ‘Ñ’ letter. The maximum SNR value is around 41dB (1/10000 signal strength over noise power) when σ = 0.15 and μ = 0.25. In this case, diffraction efficiency is around 35%. The case when both σ and μ are 0.5 produces SNR of 29dB and efficiency becomes approximately 23%. In the limit case when σ and μ are both zero, i.e., the case in Fig. 2(a), the maximum efficiency of 80% is achieved, but SNR is only of 23dB. When the uniform distribution in range [0,1] is used for the signal rnd(u) (situation assumed in Fig. 1(f) and 2(b)), SNR becomes 33dB, but the efficiency reduces to 9.6%. Finally, we have compared these values to the proposal by Arrizón [25], which is the closest to our method since it is also non-iterative, and provides an axial reconstruction. This method yields a much better SNR close to 80dB, but an efficiency of 9.2%.

 

Fig. 3 (a) SNR and (b) diffraction efficiency for the proposed method in the case of the “Ñ” image reconstruction. Data points are obtained by sweeping both mean value μ and deviation σ of signal rnd(u) in the interval [0,1].

Download Full Size | PDF

Finally, let us point out some computational aspects of this procedure, since it allows fast generation of complex holograms. The arguments are the following. The data values of the random values rnd(u) and of the diverging element D(u) can be calculated only once and stored in memory. Input data each time is given by desired complex information A(u) and F(u), and updating the output M(u) is rapid since it only involves one loop to sweep pixels and one choosing operation per pixel. This performance is clearly in contrast with other non-random techniques, where many floating point and trigonometric operations for each desired data pixel are required and are, indeed, much more computationally costly. In particular, to illustrate it, we compared our ptoposal with the simplest version of Arrizon’s method [25] Using 512 × 512 pixel holograms, the mean execution time for Arrizon’s method after averaging the time of 100 consecutive runs has been estimated as 58.75ms, whilst the presented method takes only 2.35ms. These estimations have been performed using a laptop computer based on a i5-6200U processor at 2.4GHz, 4Gb RAM memory, without any kind of parallelization in the code.

Table 1 summarizes SNR, efficiency results and computational time for the phase-only simple hologram (Fig. 2(a)), the current proposal with rnd(u) uniformly distributed in range [0,1], the optimized current proposal (tuned mean and deviation of rnd(u) with Gaussian distribution) and the method of Arrizon [25].

Table 1. Comparison of SNR, efficiency and computation time with different approaches.

View Table

4. Experimental verification of Fourier transform hologram reconstruction

To verify these results, a simple experimental system depicted in Fig. 4 has been utilized. A 633nm He-Ne laser beam is spatially filtered and collimated, and directed to the SLM. The spatial filter is composed of a 10X microscope objective and a 25 μm pinhole aperture. A 40 cm focal length lens is placed after the pinhole to collimate the beam. This device is a PAL-LCOS display from Hamamatsu, model X10468-01, with 800 × 600 pixels, a pixel pitch of 20 μm, an effective area of 15.8 × 12 mm², and a fill factor of 98%, 5 ms rise time and 25 ms fall time, 8 bit depth and linear phase response with gray level, reaching 2π phase at gray level g = 200 approximately. Linear polarizer P₁ is oriented horizontally to illuminate the SLM with the polarization parallel to the liquid-crystal director, in order to produce phase-only modulation. Aperture D selects the desired beam diameter. Polarizer P₂ is not strictly necessary for the system to produce desired modulation, but it has been included for phase measuring purposes, as explained in Section 5. In normal operation, it can be either removed or positioned to select the horizontal polarization. Finally, light is collected by a Basler CCD camera, model scA1390-17fc, with 1390 × 1040 pixels and 4.65 μm pixel size. Additionally, in some cases, we place a second lens after the SLM to produce the Fourier transform of the SLM phase patterns over the camera. The SLM is positioned with a tilt angle with the impinging beam of less than 10°.

 

Fig. 4 Schematic representation of the optical system used to test the designed masks.

Download Full Size | PDF

Figure 5 provides an example of the experimental reconstruction of this phase-only DOE. Let us mention that in this first set of experiments an additional lens (2f system) has been added after the SLM so that the CCD camera captures the Fourier transform of the phase pattern displayed in the SLM. This way, all the diffraction contributions of the generated phase-only hologram can be observed at once in the same plane. The feasibility to split these contributions from each other in different spatial positions will be further proven in next section. Figure 5(a) shows the diffracted field when a single realization of the multiplexed phase mask M(u) reconstructing the letter ‘Ñ’ and using the 160 cycle diverging axicon is addressed to the SLM. The experiment shows how the desired distribution is generated on axis, while the annular light distribution characteristic of the axicon in the Fourier plane appears separated from the axis reconstruction (with its radius proportional to the number of cycles of the axicon). The lateral ring portions observable in Fig. 5(a) are replicas of the reconstruction occasioned by pixelated structure of the SLM. The ‘Ñ’ reconstruction appears with some speckle-like background noise originated from the random encoding technique. The central reconstruction is shown clipped in Fig. 5(b) for better comparison with other cases. We note that, in comparison to the simulation in Fig. 2(b), there is a small DC zero order component, originated from imperfections in the phase-only modulation provided by the display. Although the Hamamatsu SLM is flicker free, any small fringing effect or deviation from the linear phase modulation regime generates this zero order contribution in the form of a bright peak. Nevertheless, these results show a very reduced DC level, with intensity value similar to the hologram reconstruction. This is very weak compared to cases in which SLMs exhibit flickering effect [39]. We also want to note that ‘Ñ’ letter reconstruction in Fig. 5(b) exhibits some granularity. This is an undesired experimental effect originated because of the random phase mask, Fig. 1(b), used to generate the hologram. Please, note this effect is nothing related to the complex encoding method itself. This phenomenon could be avoided by selecting a different hologram generation approaches which guarantee constant phase [35–37] or smooth phase at the reconstruction plane, or even by changing the propagation plane where the hologram is encoded so that amplitude to be encoded gets wide in the plane being registered. This way, introduction of any random phase pattern to spread propagated amplitude would not be necessary, and phase noise experimental effects could be diminished. Nevertheless, we only want to check the ability of the method for reconstructing hologram data. And provided results serve this purpose.

 

Fig. 5 (a) Experimental verification of hologram reconstruction plane. (b) Central detail of the diffracted field. (c) Distorted reconstruction happening when amplitude and phase corresponding to different hologram realizations are bundled together into a single mask. (d) Average integration of 10 realizations. (e) Average integration of 10 different realizations of a simple phase-only hologram.

Download Full Size | PDF

Let us show that amplitude information is successfully encoded. Figure 5(c) shows what happens if the multiplexed hologram is generated using a wrong amplitude distribution. In order to check it, two ‘Ñ’ images have been generated with different input phase noise distributions, like that in Fig. 1(b). Next, their Fourier transforms have been calculated, and a multiplexed mask is generated bundling together the amplitude, remember Fig. 1(c), of the first one with the phase, see Fig. 1(d), of the second one. If the correct amplitude and phase are used, one gets a result as that in Fig. 5(b). In contrast, if the incorrect amplitude is used, one gets results like that in Fig. 5(c), which is clearly worse and distorted compared to the previous one.

Finally, let us inspect what happens when averaging the reconstruction of different realizations of the hologram. Figures 5(d) and 5(e) show the experimental result after averaging 10 realizations for the current proposal, and for a simple phase-only hologram respectively. This is done by consecutively displaying different realizations of the hologram in the SLM and integrating their reconstruction. As expected, the speckle noise level is reduced significantly with respect to Fig. 5(b). However, although the light amount diverted to the ‘Ñ’ letter is higher in Fig. 5(e), the surrounding noise is also much higher than the current proposal in Fig. 5(d), leading to a worse signal to noise ratio. To provide an estimation of noise reduction, we have computed the experimental noise variance within a region of 100x100 pixels of noise out of axis. For a single image reconstruction, such as in Fig. 5(b), noise variance equals 31.24. For the 10 times averaged image in the same region, i.e. Figure 5(d), noise variance equals 3.69. Thus noise power reduction is expected to be close to 8.5 times or equivalently 9.3dB.

5. Experimental generation of axial Hermite-Gauss and Laguerre Gauss modes

Once we verified the usefulness of the proposed technique to effectively reconstruct Fourier transform intensity patterns, we further test its capability to generate complex valued distributions. We want to verify that the proposed technique successfully generates a desired phase in the reconstructed field. As an example, in [27] a Hartmann-Shack was used to analyze the phase of the generated field. Here, instead, we use a phase-shift polarization interferometer implemented in the same setup in Fig. 3, simply by rotating the polarizers.

We do this by generating the complex distributions of different laser modes. The use of SLMs to generate laser modes has been receiving great interest [40]. This can be done by using a reflective SLM as one of the mirrors of the laser cavity [41], but it is more common to use external SLMs [42–44]. Next, we show the effective realization of Hermite-Gauss (HG) modes and Laguerre-Gauss (LG) modes. These field distributions are solutions of the paraxial scalar wave equation [40] and they are both eigenfunctions of Fourier transform. Therefore, they propagate without changing their intrinsic amplitude and phase shape, just their scale. This property allows us to examine if the phase distribution is also well preserved with the proposed encoding technique. HG modes are described as

In both cases, we normalized the amplitude distributions from 0 to 1. In these experiments we also added a 2m focal length lens phase distribution to the phase of the hologram generating the mode. This way, the planes where the axicon ring is reconstructed (infinity) and the plane of the mode minimum waist (at 2 m) are axially separated, thus minimizing the effect of the axicon reconstruction on the reconstructed modes. This procedure allows for splitting even more the different diffraction contributions of the phase element. Note also that this way, the Fourier transform of the desired pattern T(u) appears at a precise location of 2m from the SLM, without requiring an external lens. A mode waist of 64 pixels (equivalent to 1,280 μm in the used SLM) was used for generating these patterns.

Figure 6 shows the experimental results obtained for the HG modes. The first and second columns show the theoretical intensity and phase distributions. The third column shows the corresponding experimental results, showing a very good agreement with the expected results. The fourth column represents the phase patterns for each case. These experimental phases have been calculated with a phase shifting technique, retrieved from four experimental interferograms shown in the four remaining columns. These interferograms are obtained by rotating first and second polarizers in Fig. 4 to angles different from SLM director orientation.

 

Fig. 6 Experimental reconstruction of different HG modes with indices pq indicated on the left. From left to right, columns show the expected intensity and phase distributions, and the corresponding measured intensity and phase distributions. The four remaining columns to the right represent phase-shifted interferograms used to retrieve phase patterns of each mode.

Download Full Size | PDF

This way a portion of modulated light interferes with non-modulated light (background) after passing through the second polarizer. Angles are experimentally selected so that all interferograms do not saturate the camera. Then, we add uniform phases to the patterns displayed at the SLM, with steps of π/2 radians, in order to implement the phase-shifting procedure [45]. The retrieved experimental phase distributions, although distorted in the areas with low intensity, also agree very well with the expected programmed phase distributions, thus validating the encoding technique also for generating complex fields.

Figure 7 shows the same kind of results but obtained for LG modes. Again, we can see how both the experimentally captured intensity and the retrieved phase patterns are in very good agreement with theoretical patterns. The encoding technique proves again its validity with complex and continuous phase patterns. Note that the combination of the proposed encoding technique, together with the use of the Hamamatsu LCOS-SLM, allowed us to generate complex-field patterns on axis, in contrast with off-axis methods [20,42,43].

 

Fig. 7 Experimental reconstruction of different LG modes with indices pl indicated on the left. From left to right, columns show the expected intensity and phase distributions, and the corresponding measured intensity and phase distributions. The four remaining columns to the right represent phase-shifted interferograms used to retrieve phase patterns of each mode.

Download Full Size | PDF

Finally, readers can check Visualization 1 for a video file with the simulation of the field propagation corresponding to the hologram generating the HG₁₁ mode. It runs from 0m to 4m away from the SLM surface. It shows how the uniform field amplitude initially becomes a random-like pattern. As the field propagates, this mixed pattern is spread over and the HG₁₁ field amplitude distribution starts to appear, reaching its minimum waist at 2m away from the SLM. An initial light ring spreading out away from axis is also appreciable. From this point to 4m the waist of the field increases again and the residual noise keeps becoming spread.

6. Conclusions

We have presented a novel procedure for randomly encoding complex-valued fields into a single diffractive optical element, displayed onto a phase-only spatial light modulator. We have provided a theoretical discussion and also experimental results evidencing that both amplitude and phase of the encoded patterns are reproduced with good enough fidelity on the designed diffraction planes. This new encoding technique presents interesting characteristics compared to other existing methods in literature. The technique does not require iterative algorithms, and also avoids using computational costly calculus. Complex modulation masks are simply calculated by choosing pixels among stored or input data, thus potentially being useful in very fast systems. We point out also the capability of working on axis, since undesired light is redirected to angles far away from the center. This is done by including a diverging element D(u) in the hologram design that can be programmed to spread undesired light far away from the optical axis, where the desired distribution is generated. Different diffraction masks could be investigated for this element, but we selected a negative conical lens or axicon. The random nature of the technique also allows generating multiple realizations to further reduce noise levels by using time integration.

We have experimentally verified the technique using a flicker-free LCOS-SLM which, therefore, can be used to work on-axis. A very low level of DC component has been verified in Fourier transform holograms. And an efficient on-axis generation of HG and LG modes has been demonstrated.

In summary, the method can be very useful for the fast and simple encoding of complex holograms onto phase-only SLMs. Nevertheless, further and deeper understanding of the limitations and disadvantages of the method would be required in future, in order to be applicable to other signals, including functions with narrow amplitude distributions, holograms that reconstruct gray level images, or their application in holographic displays or increasing the SNR to reach higher values comparable to Arrizón’s method.

Appendix - Statistical properties of the encoding technique

In this appendix, we analyze the properties of a single realization of the random multiplexing mask in order to provide theoretical support to the method. We consider a procedure where signals are treated as discrete functions and the results derived in the discrete domain are then further extended to the continuous case.

A.1 Expected value of random masks

First, we analyze the ensemble average of different random masks R(u) and we show that that it matches the amplitude A(u). Let us consider Eq. (3), and also regard A(u) and rnd(u) as uniformly distributed in the interval [0,1). Next, let us consider that both are discrete variables whose values differ in steps of 1/N, being N the total number of realizations, which is considered large enough, i.e., A(u), rnd(u)∈{0,1/N, 2/N, 3/N, …, (N−1)/N}, with N→∞. Now, since the number of attempts is large enough, we afford neglecting the random character of the numbers and we regard that each possible number will always appear once in a test of N trials, according to the assumed uniform distribution property of the random variables.

Consider Fig. 8(a). In this figure all possible values of rnd(u) are depicted. Then, a given value of amplitude, A(u)=n/N, with n∈[0,N), is also depicted as a horizontal line (n takes value equal to 4 in the particular representation of this figure). From this figure, one observes that there will be n cases in a set of N trials where the function rnd(u) takes values below the amplitude value, A(u)=n/N. In these mentioned attempts, R(u)=1, since A(u)>rnd(u). These cases, where the mask produces non-zero values, are represented by the shadowed area in Fig. 8(a). R(u)=0 in all the remaining cases, i.e., in N−n cases over N trials; these cases are represented by the non-shadowed area in Fig. 8(a).

 

Fig. 8 (a) Graphic of possible values of the random number rnd(u). The probability of having A(u)>rnd(u), i.e., the probability of R(u) = 1 is represented by the number of steps over the shadowed area (including the ground or ‘0’ step). (b) Traces of a given realization of 1024 samples of binary multiplexing mask R(u) (first, blue trace), amplitude pattern A(u) (second, red trace), and N_A(u) (third, green trace), and (c) averaged and normalized (30-level) histogram for 10 realizations of N_A(u).

Download Full Size | PDF

According to this discussion, the probability of having R(u)=1 is given by

(14)$$P\left[R\left(u\right)=1\right]\equiv P\left[rnd\left(u\right)<A\left(u\right)\right]=\frac{n}{N}\equiv A\left(u\right),$$

which exactly corresponds the function A(u). Consequently, the probability of R(u) = 0 is

(15)$$P\left[R\left(u\right)=0\right]=\frac{N-n}{N}\equiv 1-A\left(u\right).$$

With this information we obtain the aimed result: the expected value of the random mask, which is the addition of each possible value of the mask multiplied by the probability of having it, provides the desired amplitude, i.e.,

(16)$$E\left\{R\left(u\right)\right\}=1·P\left[R\left(u\right)=1\right]+0·P\left[R\left(u\right)=0\right]\equiv A\left(u\right),$$

which verifies the result appealed in Eq. (4). Following the same kind of derivation, it can be shown that the expected value of orthogonal mask is ${\overline{R}}_{}(u)=1-A(u)$.

A.2 Statistical properties of a single realization

Let us now consider a single realization of the binary random mask R(u). Considering Eq. (5), R(u)=A(u)+N_A(u), we are interested in analyzing its statistical properties to evaluate the quality of the reconstruction of the desired amplitude A(u). R(u) is determined by Eq. (3) through the random function rnd(u), which is a random variable, dependent on the location u, and uniformly distributed in the range [0,1). We also assume that the desired amplitude A(u) takes values also uniformly distributed in the range [0,1). This function A(u) corresponds to the amplitude in the Fourier plane of a given object. Previous uniform distribution assumption is reasonable whenever this amplitude is spread or wide enough. If it is not, a random phase can be added to the object plane to spread its transform amplitude A(u), and thus approach to this uniform distribution condition, as shown in the example of Fig. 1.

It is desirable to derive the probability density of N_A(u) to understand its impact in the diffraction pattern. We use again the information depicted in Fig. 8(a), and we consider again that A(u) and rnd(u) take discrete values in steps of 1/N. We next perform N² trials, so that each possible value of A(u)=n/N appears N times, provided that a high enough number or trials is performed. Then, the number of times that R(u) takes values ‘1’ or ‘0’ for each possible amplitude is directly given by the product of its corresponding probability by N.

According to Eq. (5), N_A(u)=R(u)−A(u). Since R(u) is binary valued, for each given amplitude only two kind of values are possible, either N_A(u)=−A(u) or N_A(u)=1−A(u). The first case is strictly negative, while the second case is strictly positive. If we consider a particular amplitude value A(u)=n/N, see Fig. 8(a), a negative value N_A(u)=−n/N is obtained whenever we obtain R(u)=0, provided that amplitude took that value. This coincides with all the cases above the depicted line in figure. Thus, dividing this number of cases by the number N² of trials, one gets the probability of getting a certain negative value N_A(u)=−n/N as

(17)$$P\left[N_A\left(u\right)=-\frac{n}{N}\right]=P\left[R\left(u\right)=0|A\left(u\right)=n/N\right]=\frac{N\left(1-A\left(u\right)\right)}{N^2}=\frac{N\left(1-\frac{n}{N}\right)}{N^2}.$$

Following the same procedure, a positive value N_A(u) = n/N will occur whenever R(u) = 1 Provided that A(u) = 1−n/N. Thus, the probability in these cases is

(18)$$P\left[N_A\left(u\right)=\frac{n}{N}\right]=P\left[R\left(u\right)=1|A\left(u\right)=1-\frac{n}{N}\right]=\frac{NA\left(u\right)}{N^2}=\frac{N\left(1-\frac{n}{N}\right)}{N^2},$$

which is the same expression as Eq. (14), regardless of the sign of the inspected values. Both expressions can be written as a single one as

(19)$$P\left[N_A\left(u\right)=\frac{n}{N}\right]=\frac{1-\left|\frac{n}{N}\right|}{N},n\in \left[-N,N\right].$$

In the limit when N→∞, we may regard the value 1/N as the infinitesimal of a continuous variable x, i.e., 1/N→dx, and also we may regard n/N as a given value x inside the set [−1,1], since n ranges from −N to N in the case of signal N_A(u). In that case, previous probability becomes the product of the desired probability density function f_NA(x) and an infinitesimal variable, i.e.,

(20)$$P\left[N_A\left(u\right)=x\right]=\underset{N\to \infty }{\lim }\left(\frac{1-\left|\frac{n}{N}\right|}{N}\right)=\left(1-\left|x\right|\right)dx\equiv f_{NA}\left(x\right)dx.$$

We can check that function f_NA(x)=1−|x| also comprises unity area when integrated in the range [−1,1], as required by definition for probability density functions.

Using this result, the statistical properties of the unwanted noise can be evaluated. First, the mean value of this signal is

(21)$${\mu }_{NA}={\displaystyle {\int }_{-1}^{1}x\left(1-\left|x\right|\right)dx}\equiv 0.$$

Thus, the signal has zero expected mean.

Next, the variance or mean power of the signal, which is an important parameter to specify the energy of the further diffracted signal, is expected to be

(22)$${\sigma }_{NA}^2={\displaystyle {\int }_{-1}^1{\left(x-{\mu }_{NA}\right)}^2\left(1-\left|x\right|\right)dx}\equiv \frac{1}{6}.$$

If we assume that the signal is ergodic, then these statistical estimators match estimators calculated using positions u or samples of the signal. This assumption will be used in next section to evaluate spatial autocorrelation of signal, and indirectly from it, its main diffraction spectrum properties.

Let us now provide some numerical simulations to support previous statements. To verify them, Fig. 8(b) shows a numerical simulation of 1024 samples (1D discrete signals) for a given realization of the random binary mask R(u) (blue trace) and the corresponding signals A(u) (red trace) and N_A(u) (green trace). Figure 8(c) shows the 30 level averaged and normalized histogram after 10 different realizations of signal N_A(u). Its overall shape is always repeated after every trial with more or less fidelity. This normalized histogram is directly related to the previously mentioned density distribution function and it can be interpreted as a sampled version of it. Furthermore, after these 10 trials, the numerical values of the mean and the variance of N_A(u) have been averaged to obtain 0.00034 and 0.1715, correspondingly, which are close to expected theoretical ones of 0 and 1/6 respectively.

A.3 Power spectrum properties of the undesired term

Finally, here we analyze the power spectrum associated to signal N_A(u). Let us calculate its autocorrelation function. To do so, we consider that the different locations of this noise-like signal are also uncorrelated between each other. We will also regard, as it is the actual situation when working with SLMs, that signals are sampled at regular spatial intervals given by P, being this the pixel or sampling period. For simplicity we also consider that pixels fully occupy this size P. Consider as well that signal consists of a finite number of samples N_s. So, its total physical width is given by W=N_sP. Then, the auto-correlation of the signal will be given approximately by

(23)$$R_{NN}\left(\xi \right)={\displaystyle {\int }_{-\infty }^{\infty }{N}_A^{\ast }\left(u\right)N_A\left(u+\xi \right)du}\approx W{\sigma }_{NA}^2{\Lambda }_P\left(\xi /2P\right),$$

where Λ_P(ξ/2P) = 1−|ξ/P| for |ξ|≤P, and 0 elsewhere is the triangle function normalized by P.

The latter autocorrelation magnitude has been numerically simulated. Figure 9(a) shows a signal image N_A(u), using P=8 (arbitrary space units) and a square grid of only N_s=32×32 pixels. The obtained autocorrelation samples are shown in Fig. 9(b), and its corresponding central cross section is depicted in Fig. 9(c). Here we can observe a weak background noise level and clearly a peak at the center of the signal. The magnitude of this peak reaches 11099.10, which divided by the area of the image (W=256²) yields 0.16935. This value is in close agreement with the normalized expected peak value of 1/6=0.16667. Aside from that, we can also check in Fig. 9(e) how the autocorrelation function peak is clearly triangularly shaped. The origin in this triangular shape can be understood as follows. Since correlation operation involves the multiplication of the displaced signal N_A(u+ξ) by its centered version N_A(u) and integration in the range u∈[−∞,∞] for each possible value of ξ and, since the pixels have a finite size, there will be an interval ξ∈[−P,P] where the pixels of the same index in N_A(u+ξ) and N_A(u) will coincide. Only in these regions the autocorrelation will yield additive results. In the complementary regions where one pixel partially overlaps with the neighbor, the autocorrelation provides null result. This degenerates the peak into the triangle function Λ_P(ξ/2P), which is precisely the normalized area convolution of two pulse functions Π(ξ/P), modeling the square size of each pixel.

 

Fig. 9 In left column, images corresponding to: (a) pixelated structure of N_A(u) (pixel size P = 8 and N_s = 32x32 samples), (b) autocorrelation R_NN(ξ), and (c) spectral power density |Ñ_A(ν)|², (d) modulus of diffraction spectrum Ñ_A(ν). In the right column: (e) cross section of the central part of autocorrelation function, exhibiting a triangular peak shape with maximum value of 11099.10, and (f) histogram representing the phase value occurrence of Ñ_A(ν).

Download Full Size | PDF

The Fourier transform of the autocorrelation function R_NN(ξ) explains how is the average diffracted intensity spectrum or power spectral density. This is the function |Ñ_A(ν)|² we want to characterize. This property is followed from the well-known in signal processing field Wiener-Khinchin theorem [17], and it is given by

(24)$${\left|{\tilde{N}}_A\left(\nu \right)\right|}^2=\text{FT}\left\{R_{NN}\left(\xi \right)\right\}\approx \frac{WP}{6}\sin {\mathrm{c}}^2\left(\nu P\right)\approx \frac{WP}{6},P\to 0.$$

If finite pixel sizes are regarded instead, most of the diffracted energy is concentrated in frequencies ν∈[−1/P, + 1/P], which is the main central lobe of squared the sinc function. This signal |Ñ_A(ν)|² is represented in Fig. 9(c). We appreciate that spectrum is effectively concentrated in a defined region with the appearance of a lobe. If pixel size is decreased, the width of the lobe increases more and more. These two facts are accordance with previous statements.

With this discussion, we cannot deduce the concrete expression for the complex-valued spectrum Ñ_A(ν) either, but at first glance, we can approach it by the square root of the latter. The modulus of the numerically calculated Ñ_A(ν) is represented in Fig. 9(d). It exhibits a sinc-shaped envelope closely related to the pixel size. This envelope is consistent with the square root of theoretical |Ñ_A(ν)|². Furthermore, from such simulation, phase values seem to behave as a random variable uniformly distributed in the range [0,2π], indicating that diffraction spectrum might also include an exponential phase term of random values, i.e.,

(25)$${\tilde{N}}_A\left(\nu \right)\approx \sqrt{\frac{WP}{6}}\text{sin}c\left(\nu P\right)\exp \left(i2\pi ·rnd\left(\nu \right)\right).$$

This fact is verified by the histogram shown at Fig. 9(f) showing phase occurrence of the numerical version of Ñ_A(ν). The speckled amplitude distribution seems to behave as a random variable with a Gaussian distribution, maybe another random correction to amplitude would be necessary, but we regard so far this particular expression may summarize enough the main characteristics of this unwanted diffraction term.

Funding

Ministerio de Economía y Competitividad of Spain and FEDER funds (grant ref.: FIS2015-66328-C3-3-R); Generalitat Valenciana, Conselleria d’Educació, Investigació, Cultura i Esport (PROMETEO-2017-154).

Acknowledgments

We thank Prof. Joaquín Sánchez Soriano for helpful discussion checking the validity of statistical results.

References and links

1. T. Haist and W. Osten, “Holography using pixelated spatial light modulators—part 1: theory and basic considerations,” J. Micro/Nanolith. MEMS MOEMS 14(4), 041310 (2015). [CrossRef]  

2. C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011). [CrossRef]  

3. E. J. Fernández, P. M. Prieto, and P. Artal, “Adaptive optics binocular visual simulator to study stereopsis in the presence of aberrations,” J. Opt. Soc. Am. A 27(11), A48–A55 (2010). [CrossRef]   [PubMed]  

4. A. M. Weiner, “Femtosecond pulse shaping using spatial Light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]  

5. A. Hussain, J. L. Martínez, A. Lizana, and J. Campos, “Super resolution imaging achieved by using on-axis interferometry based on a spatial light modulator,” Opt. Express 21(8), 9615–9623 (2013). [CrossRef]   [PubMed]  

6. A. Hermeschmidt, S. Krüger, T. Haist, S. Zwick, M. Warber, and W. Osten, “Holographic optical tweezers with real-time hologram calculation using a phase-only modulating LCOS-based SLM at 1064 nm,” Proc. SPIE 6905, 690508 (2008). [CrossRef]  

7. C. B. Burckhardt, “A simplification of Lee’s method of generating holograms by computer,” Appl. Opt. 9(8), 1949 (1970). [CrossRef]   [PubMed]  

8. W. H. Lee, “Sampled Fourier transform hologram generated by Computer,” Appl. Opt. 9(3), 639–643 (1970). [CrossRef]   [PubMed]  

9. B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Develop. 13(2), 160–168 (1969). [CrossRef]  

10. C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. 17(24), 3874–3883 (1978). [CrossRef]   [PubMed]  

11. D. Roberge, L. G. Neto, and Y. Sheng, “Full-complex modulation spatial light modulator using two coupled-mode modulation liquid crystal televisions,” Proc. SPIE 2490, 407–415 (1995). [CrossRef]  

12. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Programmable two-dimensional optical fractional Fourier processor,” Opt. Express 17(7), 4976–4983 (2009). [CrossRef]   [PubMed]  

13. L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4(1), 7441 (2015). [CrossRef]   [PubMed]  

14. R. W. Cohn and M. Liang, “Approximating fully complex spatial modulation with pseudorandom phase-only modulation,” Appl. Opt. 33(20), 4406–4415 (1994). [CrossRef]   [PubMed]  

15. R. W. Cohn and M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35(14), 2488–2498 (1996). [CrossRef]   [PubMed]  

16. L. G. Hassebrook, M. E. Lhamon, R. C. Daley, R. W. Cohn, and M. Liang, “Random phase encoding of composite fully complex filters,” Opt. Lett. 21(4), 272–274 (1996). [CrossRef]   [PubMed]  

17. D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing. Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing (Artech House, MA, 2005).

18. J. Amako, H. Miura, and T. Sonehara, “Speckle-noise reduction on kinoform reconstruction using a phase-only spatial light modulator,” Appl. Opt. 34(17), 3165–3171 (1995). [CrossRef]   [PubMed]  

19. A. Martínez, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatially multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jpn. J. Appl. Phys. 47(3), 1589–1594 (2008). [CrossRef]  

20. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38(23), 5004–5013 (1999). [CrossRef]   [PubMed]  

21. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am 61(8), 1023–1028 (1971). [CrossRef]  

22. P. M. Birch, R. Young, D. Budgett, and C. Chatwin, “Two-pixel computer-generated hologram with a zero-twist nematic liquid-crystal spatial light modulator,” Opt. Lett. 25(14), 1013–1015 (2000). [CrossRef]   [PubMed]  

23. V. Arrizón, “Improved double-phase computer-generated holograms implemented with phase-modulation devices,” Opt. Lett. 27(8), 595–597 (2002). [CrossRef]   [PubMed]  

24. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28(15), 1359–1361 (2003). [CrossRef]   [PubMed]  

25. V. Arrizón, “Optimum on-axis computer-generated hologram encoded into low-resolution phase-modulation devices,” Opt. Lett. 28(24), 2521–2523 (2003). [CrossRef]   [PubMed]  

26. T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016). [CrossRef]   [PubMed]  

27. O. Mendoza-Yero, G. Mínguez-Vega, and J. Lancis, “Encoding complex fields by using a phase-only optical element,” Opt. Lett. 39(7), 1740–1743 (2014). [CrossRef]   [PubMed]  

28. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. 37(11), 1955–1957 (2012). [CrossRef]   [PubMed]  

29. J. L. Martínez, P. García-Martínez, and I. Moreno, “Microscope system with on axis programmable Fourier transform filtering,” Opt. Lasers Eng. 89, 116–122 (2017). [CrossRef]  

30. J. A. Davis and D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19(7), 496–498 (1994). [CrossRef]   [PubMed]  

31. J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. 44(8), 592–597 (1954). [CrossRef]  

32. J. A. Davis, J. Guertin, and D. M. Cottrell, “Diffraction-free beams generated with programmable spatial light modulators,” Appl. Opt. 32(31), 6368–6370 (1993). [CrossRef]   [PubMed]  

33. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7(6), 961–969 (1990). [CrossRef]  

34. I. Moreno, A. Martínez-García, L. Nieradko, J. Albero, and C. Gorecki, “Low cost production of computer-generated holograms: from design to optical evaluation,” J. Eur. Opt. Soc. Rapid. Pub. 5, 10011 (2010). [CrossRef]  

35. X. Li, J. Liu, J. Jia, Y. Pan, and Y. Wang, “3D dynamic holographic display by modulating complex amplitude experimentally,” Opt. Express 21(18), 20577–20587 (2013). [CrossRef]   [PubMed]  

36. Y. Qi, C. Chang, and J. Xia, “Speckleless holographic display by complex modulation based on double-phase method,” Opt. Express 24(26), 30368–30378 (2016). [CrossRef]   [PubMed]  

37. C. Chang, J. Xia, L. Yang, W. Lei, Z. Yang, and J. Chen, “Speckle-suppressed phase-only holographic three-dimensional display based on double-constraint Gerchberg-Saxton algorithm,” Appl. Opt. 54(23), 6994–7001 (2015). [CrossRef]   [PubMed]  

38. F. Fetthauer, C. Stroot, and O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136(1-2), 7–10 (1997). [CrossRef]  

39. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008). [CrossRef]   [PubMed]  

40. A. Forbes, Laser Beam Propagation. Generation and Propagation of Customized Light, (CRC, Pretoria, 2014).

41. S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013). [CrossRef]   [PubMed]  

42. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006). [CrossRef]   [PubMed]  

43. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre-Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34(1), 34–36 (2009). [CrossRef]   [PubMed]  

44. D. Aguirre-Olivas, G. Mellado-Villaseñor, D. Sánchez-de-la-Llave, and V. Arrizón, “Efficient generation of Hermite-Gauss and Ince-Gauss beams through kinoform phase elements,” Appl. Opt. 54(28), 8444–8452 (2015). [CrossRef]   [PubMed]  

45. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988). [CrossRef]