1. Introduction

Holographic projection using pure phase spatial light modulator (SLM) has been under intense investigation for its high diffraction efficiency, low power consumption, high-contrast and wide color gamut images during the past several years [1–4]. Normally, the number of the sampling points on the Fourier plane is equal to that on the hologram plane by iteration algorithms with fast Fourier transform (FFT) [5, 6]. The sampling size on the Fourier plane is L_f = λf/Δx₀ (in one dimension for convenience), and the sampling pitch on the Fourier plane is Δ = λf/D [7]. Here λ is the incident laser wavelength, f is the focal length of the Fourier transform lens, Δx₀ and D are the pixel size and the whole size of the hologram, respectively, Δx₀ also denotes the pixel size of the SLM. Hence the spatial resolution and the projection area are limited by the whole size and the pixel size of the SLM, respectively, and the spatial resolution can hardly be increased once parameters of the SLM are given. In addition, speckle noise always appears when laser is used as the illumination source [8–10], which induces the degradation of image quality.

The time-integrating speckle technique was used in holographic projection to minimize speckles by sequentially loading sub-holograms calculated with different initial phase distributions on the SLM [11, 12], while the spatial resolution as well as the projection area is still unchanged. In addition, this method requires very high frame rate devices to display these sub-holograms for the effect to be seen by a human eye.

Some algorithms were proposed to realize zoomable holographic projection. Zoomable pixel size of the reconstructed image was obtained in lensless zoomable holographic projection by using ARSS-Fresnel diffraction algorithm [13, 14]. Double-step Fresnel diffraction algorithm for reconstruction of digital holograms with adjustable magnification was proposed [15], and the aliasing noise was mitigated by band-limited method [16]. By using these algorithms, the spatial resolution can be increased when the sampling pitch on the Fourier plane is decreased, while the reconstructed image is demagnified accordingly, and the speckle noise still exists.

In this paper, we present a holographic projection with higher image quality, and the projection area keeps unchanged. By further developing the modified Gerchberg-Saxton (GS) algorithm mentioned in [17], the spatial resolution of the reconstructed image is 2 times of that of the existing holographic projection, and speckles are suppressed well at the same time. Finally, we verify the effectiveness in experiments.

2. The schematic of the holographic projection

The schematic of the holographic projection is similar to the schematic of the beam shaping in [17], and the projected images are observed by displaying holograms with the use of the SLM. The modified GS algorithm in [17] proposed to design diffractive optical element (DOE) for beam shaping indeed reduces speckles on the Fourier plane by controlling 3N² extra points with zero amplitude besides the original N × N sampling points on the DOE plane, and finally the sampling pitch on the Fourier plane is λf/(2D). The speckles of the 3N² extra points on the Fourier plane will also be controlled, and true beam shaping will be realized by reducing speckles.

However with the constraint of zero amplitude of the padding points, the iteration is easy to fall into a local minimum and leads to the stagnation of the convergence with iteration algorithms. Hence good optimization result is hard to be obtained. Therefore the constraint is relaxed to set the amplitude of the padding points to be a given and suitable value at the beginning. The whole optimization process includes J loops, j = 1, 2 … J are the loop times. And there are K_j iteration times of the GS algorithm in loop j, k = 1, 2…K_j are the iteration times. The constraint will be stricter with j increasing. α_j, denoting the value of the amplitude of the padding points in loop j, are used for the constraint. α_j are given constants, 0 < α_j < 1, α_j decreases with j increasing. The optimal phase distribution in a loop is used as the initial value in the next loop with a stricter constraint. The gradually strengthened constraint would lead to good optimization results.

In this paper, in order to increase the spatial resolution, the sampling pitch on the Fourier plane Δ is also changed to λf/L₀, where L₀ = 2D is the new size of the hologram after padding with 3N² zeros around the hologram. Whether padding with points or not, the sampling range L_f on the Fourier plane keeps unchanged according to the sampling rule Δx₀L_f = λf [7]. The speckles of the 3N² extra points will be controlled to be replaced by useful signals, and the pixels of the original images are 4N², so true higher resolution can be realized as well as speckles be reduced.

The modified GS algorithm in [17] was proposed for beam shaping, and the desired patterns are often binary. The equation of the modified amplitude distribution in the GS iteration was | U_{f, k+1}| = ε^k+1B, ε^k+1 = ε(B/| U_{f, k} |)^γ, where U_f and B are the complex amplitude distribution and the desired amplitude distribution on the Fourier plane, respectively, ε is the adaptive weight, and γ is a given constant. Good performance of the beam shaping is obtained with the equation above. While the original images used in the holographic projection are always grayscale, and the value of some points is close to zero. After several hundred times of iteration, ε^k+1 of these points may be a very large value. Then the optimization result becomes worse, or even the iteration cannot be convergent. Hence the equation is developed as follows,

3. Two-dimensional simulation results

The size of the hologram is D × D = 4.32 mm × 4.32 mm, the sampling points are N × N = 540 × 540, the pixel size of the hologram is 8 μm × 8 μm. λ = 660 nm, f = 200 mm. The size of the amplitude distribution B of the required image on the Fourier plane is L_f × L_f = 16.5 mm × 16.5 mm, and the pixel number of B is 2N × 2N = 1080 × 1080. So the sampling interval on the Fourier plane is Δ = λf/D/2 = 15.3 μm × 15.3 μm. In order to avoid zero-order background noise [4], an original image with the sampling points of only 512 × 512 shown in Fig. 1(a) is used as a part of B, and the amplitudes of the other sampling points of B are zeros. In order to investigate the optimization performance on increasing the spatial resolution of the diffracted image as well as reducing speckles, the original image is consisted of a grayscale image and a resolution test target. The resolution test target is composed of 4 pair of lines with different width. The width and the interval of the narrowest lines are both Δ. After the optimization with the developed algorithm, the intensity distribution on the Fourier plane with smaller sampling pitch is re-calculated with the optimized hologram to expose speckles on the non-sampling points. Here the sampling pitch on the Fourier plane is 3μm × 3 μm ($\ll$Δ). The re-calculated result is shown in Fig. 1(b).

 

Fig. 1 (a) The resolution test target with 15.3μm × 15.3μm interval, (b) the re-calculated result of the developed algorithm with 3μm × 3μm interval.

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Here we define ERR as the evaluation criteria as follows,

As a comparison, the results of the GS algorithm [5] are also given here. The number of the sampling points are both 540 × 540 on the hologram plane and the Fourier plane. The size of the sampling area on the Fourier plane is still L_f × L_f = 16.5 mm × 16.5 mm. The sampling pitch on the Fourier plane is Δ = λf/D = 30.6 μm × 30.6 μm, it is 2 times the size of that with the developed algorithm. The original image with the sampling points of only 256 × 256 shown in Fig. 2(a) is used to make sure that the size of the resolution test target used in the GS algorithm is the same with that in the developed algorithm. The resolution test target in the original image is composed of only 3 pairs of lines with different width. The simulation result of the GS algorithm is shown in Fig. 2(b). In order to expose speckles on the non-sampling points, the intensity distribution on the Fourier plane is also re-calculated with the sampling pitch of 3μm × 3μm. The re-calculated result is shown in Fig. 2(c), and the ERR result of the resolution test target is 77.2%. The image in Fig. 2(c) is so dark because there are so strong speckles that signals are suppressed into a lower gray level when the intensity distribution data is transformed into image. The comparison between Fig. 1(b) and Fig. 2(c) shows that the spatial resolution of the diffracted image with the developed algorithm can be double of that with the GS algorithm. At the same time, the diffracted image is clear and speckles are well suppressed with the developed algorithm.

 

Fig. 2 (a) The resolution test target with 30.6μm × 30.6μm interval, (b) The simulation result with the GS algorithm with 30.6μm × 30.6μm interval, (c) The re-calculated result with 3μm × 3μm interval.

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4. Experimental results

A Holoeye PLUTO SLM is used here. The pixel number of the SLM is 1920 × 1080 and the pixel size is 8 μm × 8 μm. The used pixel number of the SLM is 540 × 540. A COHERENT LaserCam-HR CCD is employed to record the reconstructed images for further analysis. The pixel number of the CCD is 1280 × 1024 and the pixel size is 6.7 μm × 6.7 μm. Other parameters are the same with those used in the simulation.

Experimental result of Fig. 1(b) optimized by the developed algorithm is shown in Fig. 3(a). The result was captured by CCD with an exposure time of 20 ms. In contrast, the experimental result of Fig. 2(c) optimized by the GS algorithm is shown in Fig. 3(b). The width of the narrowest two lines of the target image in Fig. 3(a) is about 20 μm. It is larger than the sampling pitch with the developed algorithm (Δ = 15.3 μm). The difference is because the width of the narrowest line is larger than the width of two pixels (13.4 μm) of the CCD while smaller than that of three pixels (20.1 μm). The width of the narrowest two lines of the target image in Fig. 3(b) is about 34 μm. It is a little larger than the sampling pitch with the GS algorithm (Δ = 30.6 μm), and is also because of the pixel size of the CCD. The ERR results of the resolution test targets in Figs. 3(a) and 3(b) are 26.3% and 49.8%. Because of the spatial integral effect of CCD pixel, the experimental ERRs are better than the simulated ERRs. Speckles in Fig. 2(c) are more and sharper than those in Fig. 1(b), so the spatial integral effect of CCD has a considerably larger impact on the experiment for Fig. 3(b). The comparison shows that spatial resolution with the developed algorithm is double of that with the GS algorithm. And the diffracted image is clear as well as speckles are obviously suppressed with the developed algorithm.

 

Fig. 3 Experimental results obtained (a) by the developed algorithm and (b) by the GS algorithm.

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In addition, if the time-integrating speckle averaging technique [11] to further suppress speckle noise is used here, the integrating time should be shorter by using less sub-holograms for the effect to be seen by a human eye with the developed algorithm compared to that of the GS algorithm. When time-integrating speckle averaging technique is used, n sub-holograms optimized with different initial phase distributions are sequentially loaded on the SLM with the speed of 20 ms per frame, and the exposure time of CCD is 20n ms. The incident beam intensity is adjusted by an attenuator to ensure that CCD is not saturated. The experimental average result of 4 sub-holograms by the developed algorithm and the one of 10 sub-holograms by the GS algorithm are shown in Figs. 4(a) and 4(b), respectively. The experimental results show that the reconstructed image in Fig. 4(a) is a little bit clearer than that in Fig. 4(b). Hence the integrating time can be shorter by using less sub-holograms for the effect to be seen by a human eye with the developed algorithm, and the holographic projection for motion pictures may be realizable by the spatial light modulators with higher frame rate.

 

Fig. 4 Experimental results obtained from (a) 4 holograms by the developed algorithm and (b) 10 holograms by the GS algorithm.

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5. Conclusion

In conclusion, we demonstrate a holographic projection with higher image quality. The limitation of the resolution by the size of the SLM is broken, and the spatial resolution of the reconstructed image is 2 times of that of the existing holographic projection. Simulation results and experimental results show that the spatial resolution is increased and the details of the reconstructed grayscale image are clearer. So less sub-holograms will be needed and the requirement of high frame rate devices can be loosen, which may be useful to the holographic projection for motion pictures.