1. Introduction

Exploration on systems and methods for displaying digital complex-valued holograms with existing devices, which are only capable of displaying either the amplitude or the phase component, has been a topic of immense interest for several decades. Although a complex hologram can be displayed with a pair of spatial light modulators (SLM) [1–3], each handling one of the orthogonal components of the hologram, the optical setup is often too complicated and difficult to realize in practice. Similar problems are also associated with the split SLM approach [4,5], whereby the display area of a single SLM is equally partitioned into 2 sections each presenting one of the orthogonal components of the complex-valued hologram. A more simple approach, is to generate a phase-only hologram, so that it can be displayed on a single phase-only SLM. However, merely discarding the magnitude of a hologram will lead to severe distortion on the reconstructed image. To address this problem an iterative methods [6–8], based on the classical Gerchberg Saxton algorithm (GSA) [9] is employed to derive a phase-only hologram by repetitively adjusting the hologram pixels, until the reconstructed image is similar to a given target. Despite the success of these methods, the iterative process is computationally intensive and less suitable for operations at video rate. In [10–13], error diffusion is applied for converting complex-valued hologram into POH. Due to the recursive nature of error diffusion, it is difficult to realize the process at video rate with commodity computers. A non-iterative and non-recursive method, known as complex modulation, for generating monochrome and color POH have been proposed in [14] and [15], respectively. The reconstructed image of a hologram obtained with complex modulation is favorable, but optical filtering is required to extract the correct holographic signal from the optical wavefront. A lens-free approach for generating POH is developed in [16] through adding random phase noise into the source image. However, the reconstructed image is heavily contaminated with noise, and multiple POHs (generally over 10 POHs) of the same source image, each added with a different random phase noise are generated, and displayed at high frame rate. Another similar method has been proposed in [17], in which the sequence of holograms is generated from a set of images. Each image is derived from the same source image, but which has been down-sampled by a different uniform point-sampling lattice. The GSA is then applied to convert each hologram in the sequence into a POH. The above 2 methods both provide reconstructed image of high visual quality, but the complexity of the hologram generation process is increased, and a high speed display system is also needed to present the sequence of POHs. Recently, a solution for simplifying both the POH generation process and the display system has been proposed in [18,19]. Briefly, the intensity of a source image is down-sampled with a uniform grid-cross lattice, and converted into a complex Fresnel hologram. The phase component of the hologram is extracted as a sampled phase-only hologram (SPOH). By illuminating the SPOH with a coherent beam, a reconstruct image of acceptable visual quality will be reconstructed. Due to the down-sampling process, the reconstructed image is sparse (with lots of empty voids) and masked with a foreign texture. In this paper, we shall propose a method to generate a hologram known as the complementary sampled phase-only hologram that will overcome the shortcomings of the existing SPOH. We shall present our method in the following sections of the paper. First, a brief outline on the SPOH method is presented in section 2. Next in section 3, we shall describe our propose method. Experimental evaluation and a conclusion on the paper will be given in sections 4 and 5, respectively.

2. Sampled phase-only hologram

For the sake of clarity, we shall provide a concise outline on the principles of sampled phase-only hologram in [18]. Consider a three-dimensional (3-D) scene with the intensity and depth of each object point represented by the image $I\left(x,y\right)$ and the depth map $z\left(x,y\right)$, respectively, where $\left(x,y\right)$ is the position in a rectangular coordinate space. Generation of the SPOH is comprised of 2 major steps as shown in Fig. 1.

 

Fig. 1 Generation of sampled phase-only hologram (SPOH).

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Step 1: Down-sampling of the intensity image

The intensity image $I\left(x,y\right)$ is down-sampled with respect to a lattice $D\left(x,y\right)$ given by

In C1 and C2, $M$ is the down-sampling factor, and ‘mod’ is the modulus operator that returns the remainder of the division of 2 number. A small section of $D\left(x,y\right)$ covering points is shown in Fig. 2. From the dotted lines that linked up the sample points, we can envisage that the sampling lattice is in the form of repetitively grid and cross patterns.

 

Fig. 2 A small section covering points of the down-sampling lattice.

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Step 2: Generating the phase-only hologram

In this step, we first apply the Fresnel diffraction equation to generate a complex hologram from the down-sampled image. We assume that the hologram and the image planes are identical in size, comprising of $X$ columns and $Y$ rows of pixels. We have

3. Proposed complementary sampled phase-only holograms (CSPOHs)

Our proposed method is shown in Fig. 3. Briefly, we generate a pair of sampled phase-only holograms, each derived from a different sampling lattice. The pair of holograms is referred to as the complementary sampled phase-only holograms (CSPOHs). Subsequently, the 2 CSPOHs are displayed alternately on a SLM that is illuminated with a coherent beam. When the switching speed is fast enough, the reconstructed images of the holograms will appear to be merged to the observer due to ‘persistence of vision’ (POV). This effectuates the illusion that the image $I\left(x,y\right)$ is down-sampled with a dense sampling lattice that are contributed by the union of $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$, equivalent to a decrease of the sampling factor $M$. Next we shall describe the steps in generating the pair of sampling lattice $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$ for the CSPOHs.

 

Fig. 3 Proposed complementary sampled phase-only hologram for display.

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The first sampling lattice $D_1\left(x,y\right)$ is a grid-cross lattice that is generated from the method described in section 2, based on the criteria C1 and C2. The second sampling lattice $D_2\left(x,y\right)$, is generated through shifting $D_1\left(x,y\right)$ by $\Delta =M/2$ units along both the horizontal and the vertical directions, i.e.,

 

Fig. 4 (a) A small section of the sampling lattice D₂(x,y), (b) Union of the sampling lattices in Figs. 2 and 4(a).

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4. Experimental evaluation

A $512\times 320$ test image “Lena” shown in Fig. 5(a), which is parallel to and located at an axial distance of 0.3m from the hologram plane, is employed to illustrate our proposed method. The optical setup, comprising of the Holoeye HEO1080 SLM and a beamsplitter, is shown in Fig. 5(b). A pair of sampling lattices, $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$, are generated with sampling factor $M=10$. The test image is down-sampled by the 2 down-sampling lattice, resulting in a pair of down-sampled images. Equations (2) and 3 are then applied to generate the pair of CSPOHs with $\lambda =633$nm, $\delta =8.1$um, $X=1920$, and $Y=1080$. The optical reconstructed image of each of the CSPOHs is displayed with the optical setup in Fig. 5(b) and shown in Figs. 6(a) and 6(b).

 

Fig. 5 (a) Test image “Lena”, (b) Optical setup for displaying the hologram.

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Fig. 6 (a) (b) Optical reconstructed images of the CSPOH derived from the test image down-sampled with $D_1\left(x,y\right)$, and $D_2\left(x,y\right)$, respectively, (c) Optical reconstructed image of the CSPOHs derived from the test image down-sampled with $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$, merged with POV, (d) Numerical reconstructed image of a complex-value hologram, generated from the test image that is down-sampled with the union of the down-sampling lattices $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$.

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We observe that the visual quality of the reconstructed images is quite acceptable in both cases, but the pattern casted by the down-sampling lattice is prominent. Subsequently, we display the CSPOHs alternately on the SLM, and prolong the exposure time of the recording camera to simulate the POV effect. The merged reconstructed image is shown in Fig. 6(c). We observe that the coarse textural patterns that appear in Figs. 6(a) and 6(b) have been reduced substantially, resulting in a reconstructed image that is clearer and sharper than the ones obtained from the individual SPOHs. Next, we generate a hologram (with both magnitude and phase components) from the test image that has been down-sampled with the union of the lattices $D_1\left(x,y\right)$ and $D_2\left(x,y\right)$, and the numerical reconstructed image, (which we have taken as the reference image) is shown in Fig. 6(d). We observed that the reference image is very similar to the reconstructed image of the CSPOH in Fig. 6(c). The correlation score between the reference image and the numerical reconstructed image of the CSPOH is around 0.95, reflecting that the CSPOH generated by our proposed method is capable of preserving the test image favorably.

5. Conclusion

In this paper we report a low complexity, but effective method for generating phase-only hologram. Our proposed method is based on an existing technique known as the sampled phase-only hologram (SPOH), and the persistence of vision (POV). We note that although a SPOH is capable of reconstructing a reasonable image of the object scene, the reconstructed image is sparse and masked with the pattern of the down-sampling lattice. However, this problem cannot be rectified by simply decreasing the down-sampling interval, as this will jeopardize the shaded area of the image. In our proposed method, we generate a pair of SPOHs based on 2 down-sampling lattices. When the pair of CSPOHs is displayed alternately in rapid succession, their reconstructed images will appear to be merged to an observer because of the POV effect. As such, the visual quality of the combined reconstructed images of the CSPOHs is superior to either one of its constituting SPOHs. The performance of our proposed method is evaluated with optical reconstruction, demonstrating high quality reconstruction of the source image from the CSPOHs. The complexity of generating the CSPOHs is doubled that of a single SPOH. However, this does not impose much problem in practice as a typical SLM is capable of refreshing is content at twice the video rate, and the generation of a SPOH in itself is a low complexity process that can be realized with commodity computing devices.