Superpixel Technique Enabled Spatial Phase Modulation Equalization for Power-Efficient Vortex Beam Generation

Realizing a spatial phase modulation (SPM) from 0 to 2π is vital for achieving precise light field manipulation, when the phase-only spatial light modulator (SLM) is driven by the voltage. However, insufficient bias voltage degrades the quality of the generated structured light. Here, we propose a superpixel method that consists of 2 × 2 adjacent pixels to equalize the imperfect SPM response, when the bias voltage is insufficient. Meanwhile, a look-up table (LUT) is established between the target light field and the superpixel equalized light field, for reducing the computational complexity of phase hologram generation. When the phase modulation depth is reduced from 2π to 1.67π, we can generate the vortex beams with better quality by the superpixel technique. Characterizing the spatial phase distribution of vortex beams through interferometry verifies the correct function of the superpixel enabled SPM equalization method. In addition, the peak signal-to-noise ratio (PSNR) is used as the metric to quantitatively evaluate the quality of the generated vortex beam. Compared with the phase modulation distortion caused by the insufficient bias voltage, the PSNR after the superpixel enabled SPM equalization is improved by 4.5 dB. Our results would facilitate various applications mediated by power-efficient phase modulation utilizing the SLM.

modulators (SLMs) which can precisely manipulate the spatial phase are vital devices in the generation of vortex beams [4].However, the power consumption of SLM becomes an important consideration for the generation of vortex beams.Manipulating and generating vortex beams by low power-consumption optoelectrical devices is still challenging [5].The spatial phase of light can be electrically modulated by the SLM [6], so the use of low bias voltage is ideal to achieve a power-efficient vortex beam generation.However, the insufficient bias voltage will degrade the phase modulation depth to be less than 2π.The undesired phase response can further affect the performance of SLM [7].To solve the problem of undesired spatial phase modulation (SPM) response of SLM, many research works have been proposed.A method is presented for the phase calibration of SLM based on a hologram of diffraction-grating to measure and compensate for the nonlinear SPM response, so that the device can work linearly within the SPM range of 2π [8].Moreover, in order to calibrate the non-uniform SPM response of liquid crystal SLM (LC-SLM), a Twyman-Green interferometer is proposed to characterize the wave-front distortion [9].However, those phase calibration schemes cannot solve the performance degradation caused by the phase modulation depth of less than 2π.We hope to propose a SPM equalization scheme for the SLM, when the phase modulation depth is less than 2π.
Sawchuk and Hsueh have reported a method of double-phase hologram [10].Thus, any complex amplitude modulation can be decomposed into the sum of two different phase values, which are symmetrically located at the complex plane.The superpixel technique based on the principle of light field superposition generated by adjacent pixels can achieve more complex and accurate modulation [11], [12].In addition, an error diffusion scheme is used to enhance the image quality of superpixel [13].However, when the phase modulation depth is less than 2π, those existing superpixel techniques based on the double-phase hologram method in calculating the phase hologram result in incorrect phase values, leading to a performance penalty of vortex beam generation.Thus, we start to explore the superpixel technique to equalize the SPM response of SLM, when the bias voltage is insufficient.
In the current submission, we propose and experimentally leverage the superpixel scheme for achieving the equalization of SPM response, when the SLM is driven with insufficient bias voltage.Meanwhile, to reduce the computational complexity of phase hologram generation, we establish a look-up table (LUT) between the target light field and all pixel combinations of superpixel.Moreover, we take the generation of vortex beams as an example to verify the scheme of SPM equalization.For the ease of evaluating the quality of vortex beam generation, we calculate the peak signal-to-noise ratio (PSNR) of the generated vortex beam, with and without the proposed equalization scheme, respectively.

II. OPERATION PRINCIPLE
The incident Gaussian beam is spatially phase-modulated by the phase-only SLM.As a result, the light independently controlled by each pixel of the SLM can be described by a two-dimensional vector with a constant amplitude but variable phases at the complex plane.Assuming the unachievable SPM range is θ, as shown by the dashed line in Fig. 1(a1).The imperfect SLM is responsible for the distortion of the targeted light field in Fig. 1(a2).Since the optical phase is periodic where the modulation effects of 0 and 2π are similar.Combining the collective phase modulation effects of adjacent pixels at the SLM plane into one superpixel, the coherent superposition of light reflected from the pixels brings a new variation of both amplitude and phase.Therefore, the phase modulation range is extended to 2π for the imperfect SLM and even the complex amplitude modulation.As schematically shown in Fig. 1, when the phase hologram is loaded into the SLM, the 8-bit grayscale image will be converted into the corresponding bias voltage of the SLM.Then, two lenses constitute a 4f system behind the SLM.The first lens performs the Fourier transformation on the phase-modulated light field and a spatial filter is placed at the back focal plane of the first lens to block high spatial frequencies at the output plane of the 4f system [14].Therefore, the light field generated by n × n pixels is overlapped with each other at the output plane.The electrical field E SP (x, y)of single superpixel by combining n × n pixels is described as where ϕ j represents the phase value at the j th pixel of n × n pixels.A SP is the amplitude of the superpixel that is normalized to the range of [0, 1] by 1/n 2 .ϕ SP is the phase of a superpixel, whose value is within the range [0, 2π].x and y are the horizontal and vertical coordinates of the light field.
Assuming the complex amplitude at a specific position of the equalized target is E t (x, y) = A t exp(iϕ t ).When the phase ϕ SP of the superpixel is equal to the equalized target phase ϕ t , the SPM equalization can be achieved.
When the phase modulation depth is less than 2π, the phase value that combines a superpixel may decompose into the vector within the distortion region, which results in error phase values of the hologram.In order to avoid error phase values during the hologram calculation, we need to choose n × n phases values outside of the unachievable SPM range to realize the vector summing, as shown in Fig. 1(b1).For example, when the SPM response to be equalized is 5π/3 indicated by E t in Fig. 1(b1), we can simply choose E 1 whose phase value is 0π and the vector E 2 whose phase value is 4π/3, in order to realize the SPM equalization.However, when directly calculating the phase hologram from the target light field, decomposing the equalized target Et into n × n unknown phase values will result in a high computational complexity.Therefore, in case the phase modulation depth of SLM is less than 2π, we calculate all possible pixel combinations based on the operation principle of superpixel and establish a LUT to reduce the computational complexity of hologram calculation.In addition, the computational complexity of creating LUT is in relationship with the pixel size.Taking the commonly-used SLM with a phase resolution of 2π/256 as an example, the number of gray levels G is 256.We can define the number of n × n pixel combination results as the computational complexity of LUT, as shown in (2), where the computational complexity of creating LUT increases exponentially with the pixel size, comparing the computational complexity with the pixel size of 2 × 2, 3 × 3, and 4 × 4. In addition, according to the diffraction efficiency of superpixel technique [11], we preserve the pixel combinations with maximum amplitude and various phase values to establish the LUT.Since the superpixel with the same phase but a larger amplitude will bring better modulation effects, the larger amplitude of the superpixel represents a higher diffraction efficiency.

III. SIMULATION RESULTS
In order to evaluate the SPM equalization performance, we numerically compare the PSNR of the interference pattern between the generated vortex beam with and without the use of the proposed superpixel scheme.Meanwhile, the one using a perfect SLM whose SPM range can reach [0, 2π] is chosen as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the benchmark.The equation for calculating the PSNR [15] is shown in (3), (3) where |E t (x, y)| is the interference pattern of the target light field generated by a perfect SLM.|E g (x, y)| is the interference pattern of light field generated by an imperfect SLM with and without the use of the proposed superpixel scheme, respectively.M and N are the length and width of the interference pattern.MAX(|E t (x, y)|) is the maximum value of |E t (x, y)|.We do observe that the reduction of phase modulation depth leads to a degradation of the generated vortex beam in simulation, as shown in Fig. 2. The equalization scheme with a pixel size of 2 × 2 can effectively improve the quality of the light field, indicating a higher PSNR.However, as the phase modulation depth of SLM further decreases, the capability of superpixelenabled equalization becomes weak.The maximum amplitude of superpixel becomes small as the phase modulation depth decreases, leading to a PSNR degradation for the generated light field.In addition, we provide the simulation results based on various pixel sizes for the phase equalization, in case the phase modulation depths are varied, as shown in Fig. 3.The equalization performance is inversely proportional to the pixel size of superpixel.Since the superpixel technique is at the cost of the SLM pixel resolution, large pixel size is unfavorable for equalizing the SPM response.Consequently, we numerically verify the performance of the proposed equalization scheme with a pixel size of 2 × 2, under the condition of phase modulation depth of only 1.67π.

IV. EXPERIMENTAL RESULTS AND DISCUSSIONS
In the experiment, we first characterize the relationship between the phase modulation response of SLM and the bias voltage, with the help of polarization interferometry, as shown in  Fig. 4. The swing of bias voltage to achieve a phase modulation depth of 2π is from 1.14 V to 2.82 V.However, the phase modulation depth of 1.67π corresponds to a swing of bias voltage from 1.14 V to 2.36 V.When the bias voltage is reduced by 0.46 V during the vortex beam generation, it can provide a lower power consumption for all pixels of SLM.
Next, we start to establish the LUT based on all pixel combinations of the size of 2 × 2 and the phase modulation depth of 1.67π.The LUT contains the possible phase equalization targets and the pixel combinations corresponding to the superpixel results.Since the possible phase equalization targets E t (x, y) = A t exp(iϕ t ) holds a random complex amplitude at the unachievable SPM range, but the resolution of phase modulation is finite.The discrete points in the first column represent possible phase equalization targets at the unachievable SPM range, as shown in Table I.When the resolution of superpixel results is approximately 1.67π/256/4 = 5.1 × 10 −3 , we choose the discrete points with a resolution of 5.1 × 10 −3 to represent all possible phase equalization targets.In addition, when the gray level in the pixel combination represents the phase to be modulated by the pixel, we can calculate the complex amplitude of the superpixel results by (1).Meanwhile, in order to ensure that all the superpixels with same maximum amplitude in the LUT, we choose the maximum amplitude is 0.76 when the intensity of incident light is 1.Each superpixel result is calculated with the similarity with all possible phase equalization targets, where the pixel combination corresponding to the superpixel with the highest similarity are selected and stored in the LUT.Finally, we can use the complex amplitude of the phase equalization targets to index the pixel combination with 2 × 2 different gray levels during the phase hologram calculation.
In order to evaluate the performance of superpixel enabled SPM response equalization, the experimental setup of vortex beam generation, together with the interference detection, is shown in Fig. 5. Since combining 2 × 2 pixels into one superpixel requires a higher pixel resolution of SLM, we need to downsample the target light field first during the calculation of phase hologram.The size of the phase hologram of the superpixel loaded in the SLM is the same as the target light field with 1080 × 1080 pixels.In addition, the Gaussian beam emitted from a 633 nm semiconductor laser (MLL-III-633L) is collimated and expanded through two lenses with focal lengths of f = 50 mm and f = 250 mm, respectively.Its state of polarization (SOP)  is linearly polarized by a polarizer (P1).A reflective phaseonly SLM (Holoeye, LETO-3-CFS-127), with a resolution of 1920 × 1080 pixels, is used to generate the vortex beam, after the Gaussian beam is reflected by a mirror (M1) and a beam splitter (BS).To realize the proposed SPM response equalization, we establish a 4f system consisting of two lenses with a focal length of 100 mm, and place an adjustable diaphragm (D1) at the focal plane of the first lens.The CCD (MER2-1220-32U3C) with a resolution of 4024 × 3036 pixels is used to record the intensity profile of the vortex beam.Meanwhile, the Gaussian beam from the same semiconductor laser is propagated through the BS and lens (L5), and then reflected by the mirror (M2) to generate a spherical wave as the reference beam.After the free space transmission, the spherical wave enters the 4f system and interferes with the vortex beam.
As shown in Fig. 6, in comparison with the generated vortex beam under a phase modulation depth of 2π, the intensity profile of the generated vortex beam is deviated from the ideal case, when the phase modulation depth is only 1.67π.After the use of superpixel enabled phase equalization, the intensity profile of the generated vortex beam has a uniform intensity distribution.Fig. 7 shows the interference patterns of the generated vortex beam after the interference with a spherical wave, which denotes the phase distributions of the vortex beam.Since the interference pattern has several spiral stripes extending outward from the vortex singularity, the number of which is related to the topological charge l.Although the noise is found in the interference pattern, it doesn't directly come from the proposed superpixel scheme.Because the spatial phase response is determined by not only the liquid crystal itself but also the inherent wavefront distortion of SLM.The undesired phase response will affect the phase distribution of vortex beam and cause the noise of experimental interference pattern.In addition, when the SPM response is degraded, it is challenging to identify the topological charge of the generated vortex beam.In order to quantitatively evaluate the improvement, we choose the interference pattern of experimental vortex beam in Fig. 7 8, the PSNR comparison between the phase distribution for various vortex beams with and without the proposed equalization scheme is implemented.The quality of interference pattern is substantially improved, by the use of superpixel enabled phase equalization.Since the intensity profile of the vortex beam is more uniform and the phase distribution is more complete, the similarity between the interference pattern and the target light field is higher.The phase distribution of the vortex beam with a topological charge of l = +2 has the maximum PSNR improvement of 4.5 dB.Meanwhile, we experimentally characterize the spatial phase distribution of vortex beams with topological charges of l =+3 and l = +4.The PSNR improvements are 1.36 dB and 2.19 dB, respectively.The PSNR results of interference patterns are almost the same within the monitoring period of one hour.Thus, we believe that, the phase modulation of the proposed superpixel schemes is stable.

V. CONCLUSION
We have proposed a SLM phase modulation response equalization scheme based on the superpixel technique.We combine 2 × 2 SLM pixels into a superpixel, in order to achieve the equalization of the SPM response, when the bias voltage is insufficient.Moreover, the implementation procedure and parameter optimization of the superpixel enabled equalization scheme are discussed, and the feasibility of the power-efficient vortex light generation scheme is experimentally verified.When the phase modulation depth of SLM is reduced from 2π to 1.67π, the bias voltage can be reduced by 0.46 V.In comparison with the vortex beam generated by the SLM with insufficient phase modulation response, the PSNR of the vortex beam can be improved by up to 4.5 dB, after the superpixel enabled SPM equalization.Our scheme is promising for power-efficient vortex beam generation and various spatial phase modulation applications.

Fig. 2 .
Fig. 2. Calculated equalization performance with a pixel size of 2 × 2 with respect to variable phase modulation depths.

Fig. 4 .
Fig. 4. Relationship between the bias voltage and the phase response of SLM at λ = 633 nm.

Fig. 6 .
Fig. 6.Intensity profiles of simulated and experimental vortex beams.(a1)-(c1) are the vortex beams with different topological charges, under the condition of sufficient bias voltage.(a2)-(c2) and (a3)-(c3) are the vortex beams with and without the superpixel equalization, under the condition of insufficient bias voltage, respectively.

Fig. 7 .
Fig. 7. Interference pattern of simulated and experimental vortex beams.(a1)-(c1) are the vortex beams with different topological charges, under the condition of sufficient bias voltage.(a2)-(c2) and (a3)-(c3) are the vortex beams with and without the superpixel enabled phase equalization, under the condition of insufficient bias voltage, respectively.

Fig. 8 .
Fig. 8. PSNR comparison of interference pattern of experimental vortex beams, with and without the superpixel enabled phase equalization.

TABLE I LUT
WITH A SIZE OF 2 × 2 PIXELS UNDER THE CONDITION OF 1.67π MODULATION DEPTH Fig. 5. Experimental setup of vortex beam generation, together with the interference detection.