## Abstract

A single SLM (spatial light modulator) full-color holographic 3-D display based on sampling and selective frequency-filtering methods is proposed. Spatially-sampled *R*(red), *G*(green) and *B*(blue)-holograms can provide periodic 3 × 3 arrays of their frequency spectrums. Thus, by allocating three groups of three spectrums to each color hologram, and selectively filtering out those spectrums with their own spectrum filtering masks (SFMs), frequency-filtered *R*, *G* and *B*-holograms can be obtained. These holograms are synthesized into a single color-multiplexed hologram, and optically reconstructed into a color distortion-free full-color 3-D object on the 4-*f* lens system, where color-dispersion due to the pixelated structure of the SLM can be removed with the optical versions of SFMs. Fourier-optical analysis and experiments with 3-D color objects in motion confirm the feasibility of the proposed system in the practical application.

© 2017 Optical Society of America

## 1. Introduction

Thus far, the holographic technique has been regarded as one of the most attractive approaches to implement a realistic three-dimensional (3-D) display because it can create the most authentic illusion of observing 3-D scenes without any special glasses [1]. Furthermore, the holographic 3-D display has been considered as an alternative to the current stereoscopic display with drawbacks of eye fatigue and dizziness due to the human factor problem [1,2]. This holographic method, however, has several practical problems. One of them is the unavailability of a full-color holographic camera for capturing the daylight-illuminated outdoor scenes, even though the incoherent types of holographic cameras were also suggested recently [3–5]. Another problem is the unavailability of a large-scale and high-resolution spatial-light-modulator (SLM) to reconstruct the holographic data into a 3-D image since the hologram resolution is in the order of the light wavelength. The diagonal size and pixel-resolution of the current SLM are in the ranges of about 0.55”-1.8” and 1,024 × 768 - 4,096 × 2,464, respectively, which are too small in size and too low in resolution to display the full-color 3-D hologram [6]. In addition, the optical configuration of a full-color holographic display appears to be very complex due to the separated processing of those three color holograms of an input 3-D scene [7,8]. Meanwhile, in the electro-holographic 3-D display, the computational complexity in real-time generation of holographic 3-D videos, as well as the unavailability of practical SLMs, has been considered as critical issues. Thus, many research works in electro-holography have been focused on the development of fast CGH (computer-generated hologram) algorithms [9–12].

In fact, several problems mentioned above prevent the current holographic 3-D display from being widely accepted in many practical application fields [13]. Until now, to realize the full-fledged full-color holographic 3-D display, several kinds of approaches have been proposed [14]. Since the holographic display operates based on interference and diffraction optics, three color holograms of an input 3-D scene need to be separately generated and reconstructed. Thus, for the full-color display, three SLMs for each color hologram are required [15,16], which means that this three SLMs-based holographic 3-D display system becomes too bulky in size and too high in cost [16].

As an alternative, time and spatial-multiplexed single SLM full-color holographic 3-D displays were proposed [17–20]. In the time-multiplexed system, red(*R*), green(*G*) and blue(*B*)-color reference beams are time-sequentially illuminated into the single SLM, where corresponding color holograms are also loaded in sequence [17–19]. This system, however, requires the SLM with a very high frame-rate to overcome the flickering problem, and also undergoes the operational difficulty in synchronization [17–19]. On the other hand, in the spatial-multiplexed system, *R*, *G* and *B*-holograms are placed side-by-side on a single SLM, and then corresponding reference beams are illuminated on each of them [20]. It may alleviate a couple of issues of the time-multiplexed system, but it also suffers from other problems of color dispersion and resolution loss [20].

For solving those drawbacks, another type of the color-multiplexed hologram (CMH)-based single SLM color-holographic 3-D display was proposed [21–24]. Here, the CMH represents a multiplexed three-color hologram using the so-called color-encoding method, where the most challenging issue is color dispersion. There are two kinds of color dispersions such as CMH and SLM-related color dispersions, which are called CMH-CD and SLM-CD, respectively, here. These color dispersions may occur due to optical diffraction from the fringe pattern of the CMH and pixelated structure of the SLM, respectively [25], which causes the reconstructed 3-D image to be severely distorted in color. For removing those two kinds of color dispersions, several approaches were proposed, which include depth-division multiplexing (DDM) [21], multiplexing encoding (ME) [22], space-division multiplexing (SDM) [23] and frequency-division multiplexing (FDM) [24] methods.

Thus, in this paper, we propose a new approach for color dispersion-free single SLM full-color holographic 3-D display based on sampling and selective frequency-filtering (SFF) methods. Three sampled color holograms corresponding to each of the *R*, *G* and *B*-color images are calculated with the NLUT (novel-look-up-table) method based on its unique property of shift-invariance [26–28]. Those sampled color holograms can provide periodic *n × n* arrays of their frequency spectrums on the Fourier domain. For color separation, lower-order 3 × 3 spectrum locations are selected from them and three groups of three spectrum locations are uniquely allocated to each color hologram. Then, by selectively filtering out those three groups of three spectrums with their own spectrum filtering masks (SFMs) and inversely Fourier-transforming them, frequency-filtered (FF) versions of those sampled R, G and B-holograms can be obtained. These FF-*R*, *G* and *B*-holograms are then multiplexed into a single hologram, which is called a FF-color multiplexed hologram (FF-CMH). After then, this FF-CMH is optically reconstructed into a full-color 3-D object image on the 4-*f* lens system. That is, the FF-CMH loaded on the SLM is transformed into the frequency domain and illuminated with a multi-wavelength light source which is composed of *R*, *G* and *B*-lasers, where additional color dispersion due to the pixelated structure of the SLM can be removed just by using the optical versions of SFMs, and with which each group of three original *R*, *G* and *B*-frequency spectrums is selectively filtered out. Then, from these optically-filtered three color holograms, a color distortion-free full-color 3-D object image can be reconstructed on the 4-*f* lens system.

To confirm the feasibility of the proposed method, Fourier-optical analysis and optical experiments with 3-D color objects in motion are carried out, and the results are comparatively discussed with those of the conventional methods.

## 2. Proposed system

Figure 1 shows a flowchart of the proposed system, which is composed of four-step processes. In the 1st step, *R*, *G* and *B*-color and depth images are extracted from the input 3-D object and three sampled color holograms are generated with the NLUT [29,30]. In the 2nd step, 3 × 3 periodic frequency-spectrums provided by the sampled color holograms are divided into three groups of three spectrums, which are then allocated to each color hologram. Just by filtering out those spectrums with their own SFMs, frequency-filtered (FF) versions of the sampled *R*, *G* and *B*-holograms are then obtained. In the 3rd step, these FF-color holograms are multiplexed into a so-called FF-CMH. In the 4th step, the FF-CMH is reconstructed into a color distortion-free full-color 3-D image on the optical 4-*f* lens system by being illuminated with a multi-wavelength light source, where color dispersion due to the SLM can be removed with a pinhole-type optical SFM.

#### 2.1 Generation of R, G and B-holograms from the sampled color images

Three *R*, *G* and *B*-color images extracted from the input 3-D object are sampled with specific spatial periods, and then hologram patterns for those three color images are calculated using the NLUT (novel-look-up-table) method. Here it must be noted that unlike the conventional computer-generated hologram (CGH) algorithms, the NLUT method can simply calculate the hologram pattern of a 3-D object just by shifting and adding operations of pre-calculated principal fringe patterns (PFPs) based on its unique property of shift-invariance [31–33]. Thus, the sampling of color images and calculation of their hologram patterns can be done in a single step with the NLUT. For the NLUT-based hologram generation, the NLUT should be constructed in advance with pre-calculated three sets of PFPs for each color and for each depth plane of the 3-D object, which are called sets of *R*-PFPs, *G*-PFPs and *B*-PFPs, respectively.

### 2.1.1 Operational principle of the NLUT method

Figure 2 shows an example for showing a hologram generation process of the NLUT method for three object points of *A*(*-x _{1}, y_{1}, z_{1}*),

*B*(

*-x*) and

_{2}, -y_{2}, z_{1}*C*(

*x*) on a depth plane of

_{3}, -y_{3}, z_{1}*z*based on its shift invariance property [29–33]. Here, the object point

_{1}*A*(

*-x*) in Fig. 2(a) is located at the distances of

_{1}, y_{1}, z_{1}*-x*and

_{1}*y*along the

_{1}*x*and

*y*-directions, respectively, from the center point

*O*(

*0, 0, z*), so the hologram pattern for this point can be obtained just by shifting the PFP for the center-located object point, which is shown in Fig. 2(b), with amounts of

_{1}*-x*and

_{1}*y*along the

_{1}*x*and

*y*-directions, respectively, as seen in Fig. 2(c). Likewise, hologram patterns for other object points of

*B*(

*-x*) and

_{2}, -y_{2}, z_{1}*C*(

*x*) in Fig. 2(a) can be also obtained by shifting the same PFP with amounts of

_{3}, -y_{3}, z_{1}*-x*and

_{2}*y*, and

_{2}*x*and

_{3}*-y*along the

_{3}*x*and

*y*-directions, respectively, as show in Fig. 2(c). Then, by adding them together and tailoring them with the pre-determined size on the overlapped regions with all those shifted versions of PFPs, the hologram pattern for those three object points on the depth plane of

*z*can be obtained, which is shown in Fig. 2(d). Therefore, the hologram pattern for a 3-D object can be generated just by carrying out those processes to all object points on all depth planes of the object.

_{1}Figure 3 shows a conceptual diagram of the shift-invariance property of the NLUT method. As seen Fig. 3(a), the hologram pattern for the object point *A*(*0, 0, z _{1}*), which is represented by

*‘PFP-A’*is recorded as a form of the Fresnel-zone-plate (FZP) in the NLUT method. Since the center of

*‘PFP-A’*is positioned at the origin of (

*0, 0, 0*) at the hologram plane as shown in Fig. 3(b), the object point

*A'*reconstructed from this hologram pattern of

*‘PFP-A’*is also located at (

*0, 0, z*). When this hologram pattern of

_{1}*‘PFP-A’*is shifted to the location of (

*x*) along the

_{1}, y_{1}, 0*x*and

*y*-directions, then the object point

*A”*reconstructed from the shifted version of

*‘PFP-A”’*is also moved to the new location of (

*x*). Likewise, when the

_{1}, y_{1}, z_{1}*‘PFP-A’*is shifted to the location of (

*x*) along the

_{2}, -y_{2}, 0*x*and

*y*-directions, the object point

*A”'*reconstructed from the shifted version of

*‘PFP- A”'’*is also moved to the location of (

*x*).

_{2},-y_{2}, z_{1}In other words, in the NLUT method, if two object points are in the same depth plane, hologram patterns for those two object points can be directly obtained just by shifting the corresponding PFP to the locations of two object points and adding them together without any additional calculations, which is referred to the shift-invariance property of the NLUT method [32,33].

### 2.1.2 NLUT-based calculation of sampled color holograms for the color images

Three *R*, *G* and *B*-holograms for each color image are generated using the NLUT method. As mentioned above, three sampled color holograms can be simply calculated just by shifting and adding processes of their corresponding color-PFPs, which is the reason why the NLUT method is employed here rather than other CGH algorithms. That is, in the NLUT method, *R*, *G* and *B*-hologram patterns for each color image of the 3-D object can be calculated with their corresponding color sets of PFPs as follows [26].

Where *a _{Rp}*,

*a*and

_{Gp}*a*represent the

_{Bp}*R*,

*G*and

*B-*intensity values of the

*p*object point, respectively, as well as

^{th}*k*and

_{0}= 2π/λ*λ*mean the wave number and the wavelength of the light source. Each resolution of the

*R*-PFP,

*G*-PFP and

*B*-PFP should be set to be large enough for their lateral shifting, adding and cropping operations in the hologram generation processes. That is, the minimum resolution of the PFP can be given by Eqs. (5)-(6) [26].

In Eqs. (5)-(6), *h _{x}* and

*h*represent the horizontal and vertical resolution of the hologram pattern, and

_{y}*s*,

*O*and

_{x}*O*denote the sampling step and horizontal and vertical image sizes, respectively. When

_{y}*H*is calculated for the

_{R}(x, y)*p*object point

^{th}*(x*), the center of the corresponding

_{Rp}, y_{Rp}, z_{p}*R*-PFP has to be shifted by

*s × x*and

_{Rp}*s × y*along the

_{Rp}*x*and

*y*-directions from its origin location of (

*0, 0*), respectively, and cropped as the same resolution size as of the final hologram. Then, the

*R*-PFP(

*x, y; s∙x*) represents the finally-shifted and cropped

_{Rp}, s∙y_{Rp}, z_{p}*R*-PFP for the

*p*object point, where (

^{th}*x, y*) denotes the coordinate of the finally cropped

*R*-PFP, and (

*s∙x*) denotes the coordinate of the

_{Rp}, s∙y_{Rp}, z_{p}*p*

^{th}object point [26]. In the NLUT method, a 3-D object is assumed to be composed of

*n*depth-sliced object planes with a step of

*Δz*. Here, the PFP is shifted depending on the sampling step of s along the

*x*and

*y*-directions, so that the depth step along the

*z*-direction has to be also shifted with an amount of

*s∙Δz*.

Figure 4 shows two kinds of *R*-holograms generated with the original and sampling NLUT methods based on the shifting and adding operations of the corresponding *R*-PFP on the hologram plane. As seen in Fig. 4(a), three object points of *A, B* and *C* are assumed to be on the depth plane of *z _{m}*, where the discrete pixel pitch along the

*x*and

*y-*directions are set to be

*p*and

_{x}*p*respectively. Figure 4(b) shows the original

_{y},*R*-hologram generated for those object points with the corresponding

*R*-PFP, where three points such as

*R*-PFP-

*A*,

*R*-PFP

*-B*and

*R*-PFP

*-C*represent the center locations of those

*R*-PFPs. Thus, the relative distances between the

*R*-PFP-

*A*and

*R*-PFP-

*B*on the hologram plane are given by

*Δx*and

*Δy*, respectively, along the

*x*and

*y*-directions, which are same with the relative distances between the object points of

*A*and

*B*on the object plane. In addition, Fig. 4(c) shows the frequency spectrum of the original

*R*-hologram, where

*X*,

_{c}*Y*

_{c}, and

*X*,

_{s}= 1/p_{x}*Y*represent the cut-off frequencies and sampling frequencies along the

_{s}= 1/p_{y}*X*and

*Y-*directions, respectively. Here, the object image reconstructed from the original hologram of Fig. 4(b), turns out to be the same original object image of Fig. 4(a) as shown in Fig. 4(d).

On the other hand, Fig. 4(b’) shows the sampled *R*-hologram generated with the corresponding *R*-PFP for those three object points with the spatial periods of *x*_{s} = 3*Δx =* (1/*X _{s}*) and

*y*

_{s}= 3

*Δy =*(1/

*Y*), respectively, along the

_{s}*x*and

*y*-directions. As seen in Fig. 4(b’), the relative distances between the

*R*-PFP-

*A*′ and

*R*-PFP-

*B*′ are given by

*s × x*and

_{Rp =}3 × Δx*s × y*, respectively, along the

_{Rp}= 3 × Δy*x*and

*y*-directions, which are same with those between the object points of

*A*′ and

*B*′ on the object plane. Thus, the shifting distances between the

*R*-PFP-

*A*′ and PFP-

*B*′ for the sampled hologram case of Fig. 4(b’) gets increased by 3-fold compared to those for the original hologram case of Fig. 4(b). In addition, Fig. 4(c’) shows the frequency spectrum of the sampled

*R*-hologram, where

*X*′

_{c},

*Y*′

*and*

_{c}*X*′

*,*

_{s}= 1/3p_{x}*Y*′

*represent the sampled cut-off frequencies and sampling frequencies along the*

_{s}= 1/3p_{y}*X*and

*Y*-directions, respectively. Here, relationships of

*X*′

*and*

_{s}= X_{s}/3*Y*′

*can be derived, which means that on the same frequency spectrum of the original hologram, a periodic 3 × 3 array of frequency spectrums can be obtained in case of the sampled hologram. Thus, for the sampled*

_{s}= Y_{s}/3*R*-hologram with the spatial periods of

*Δp*′

_{x}*= 3Δp*and

_{x}*Δp*′

_{y}*= 3Δp*along the

_{y}*x*and

*y*-directions, a periodic 3 × 3 array of frequency spectrums of the original

*R*-hologram can be obtained in the same frequency spectrum as seen in Fig. 4(c’). Likewise, for the

*G*and

*B*-holograms, the same periodic 3 × 3 arrays of their frequency spectrums can be also obtained. Thus, by assigning three groups of three frequency spectrums, which are uniquely selected from 9(3 × 3) frequency spectrums, to each color hologram, three color holograms can be uniquely separated based on color-dependent spatial filtering of those frequency spectrums on the Fourier domain.

#### 2.2 Generation of frequency-filtered three color holograms

For color separation, three sampled *R*, *G* and *B*-holograms are selectively band-pass filtered on the Fourier plane of a computational 4-*f* lens system using the fast Fourier-transform (FFT) and inverse FFT (IFFT) algorithms as shown in Fig. 5.

As mentioned above, sampled color holograms can provide periodic arrays of their frequency spectrums at the discrete *n* × *n* spectrum locations on the Fourier domain, which may give us a chance to effectively separate those three-color holograms. In fact, three kinds of arrays of spatial-frequency spectrums for those sampled color holograms are overlapped on the same periodic spectrum locations on the Fourier plane. Thus, for separating those color holograms, lower-order 3 × 3 periodic spectrum locations are selected as seen in Fig. 5(b), and three groups of three spectrum locations, which are exclusively selected from nine (3 × 3) spectrum locations based on permutation, are uniquely allocated to each color. As seen in Fig. 5(c), three groups of three spectrum locations of (1,1) (2,3) (3,2), (1,3) (2,2) (3,1) and (1,2) (2,1) (1,3), are assigned to *R*, *G* and *B*-holograms, respectively. Thus, with three kinds of spectrum filtering masks (SFMs), which are called *R*, *G* and *B*-SFMs, respectively, we can generate frequency-filtered (FF) versions of those original sampled color holograms, which are called FF-*R*, *G* and *B*-holograms, respectively, as seen in Fig. 4(e).

Thus, frequency spectrums of those FF-*R*, *G* and *B-*holograms may exist only on their allocated locations. In other words, with these computational SFMs, three color holograms can be made clearly separable in the process of full-color display, which means that the CMH-CD problem can be solved in the proposed system.

#### 2.3 Generation of the frequency-filtered color-multiplexed hologram

As mentioned above, there have been many attempts to digitally multiplex three *R*, *G* and *B*-holograms into a single hologram for the single SLM-based full-color holographic display [21–24]. In this paper, three kinds of FF-*R*, *G* and *B*-holograms are multiplexed into a single hologram, called a frequency-filtered color-multiplexed hologram (FF-CMH) as seen in Fig. 5(f). Moreover, the FF-CMH is generated as a form of the simple amplitude-type hologram. The employed SLM can provide an 8-bit data value for each pixel, thus the intensity levels for the amplitude-type hologram may be limited by the range of 0~255 [6]. For multiplexing those FF-color holograms into a single FF-CMH, the range of the intensity must be shared together with three *R*, *G* and *B*-holograms. Thus, each pixel-amplitude of those FF-color holograms needs to be properly scaled before being multiplexed together. The independent controllability of light intensities of three lasers enables the *R*, *G*, and *B*-intensity values of the final 3-D image to be adjusted without recalculation processes of the FF-CMH. Thus, the amplitude scaling factor of the *R*, *G* and *B*-components can be simply set to be same. Moreover, to avoid the value overflow of the FF-CMH, the amplitude levels of all those hologram pixels are adjusted by one third, which means that they are made exist within the range of 0~85. Three amplitude-scaled FF-*R*, *G* and *B*-holograms are finally added up together to generate the FF-CMH as seen in Fig. 5(f).

#### 2.4 Optical frequency-filtering and reconstruction of the FF-CMH

Figure 6 shows a simple optical 4-*f* lens system for reconstructing the FF-CMH into a color distortion-free full-color 3-D object image. The FF-CMH is loaded on the SLM locating at the input plane of the 4-*f* lens system, and illuminated with a collimated light source composed of three *R*, *G* and *B*-lasers. The FF-CMH is then optically transformed into the Fourier domain, where 3 × 3 spatial-frequency spectrums of the FF-CMH are obtained. That is, as seen in Fig. 6, *R, G* and *B*-frequency spectrums appear on each set of three locations of (1,1) (2,3) (3,2), (1,3) (2,2) (3,1) and (1,2) (2,1) (3,3), respectively, which correspond to the allocated locations of the computational SFMs of Fig. 5(c). However, as seen in Fig. 6(a), in each location, not only one of the originally-allocated *R*, *G* and *B*-frequency spectrums, but also two other color-dispersed spectrums are simultaneously generated due to the color-dispersion resulted from the pixelated structure of the SLM, called a SLM-CD. Thus, by employing optical versions of the computational SFMs on the Fourier plane of the 4-*f* lens system, which is called optical SFMs and shown in Fig. 6(b), all those color-dispersed spectrums can be blocked, while only nine original *R*, *G* and *B*-frequency spectrums are passed as seen in Fig. 6(c). Thus, with this optical band-pass filtering process with the optical SFMs, the SLM-CD problem can be also solved in the proposed system.

Here, the optical SFMs corresponding to the computational SFMs, can be made as a form of the pinhole-type film mask, and with which not only the zero and higher-order diffraction terms, but also the color-dispersed terms can be blocked, while allowing only nine terms of original *R*, *G* and *B*-frequency spectrums to be passed. The optical SFM can be fabricated based on the Abbe image formation theory [25, 34]. From these optically frequency-filtered three color holograms, a color distortion-free full-color 3-D object image can be reconstructed at the output plane of the 4-*f* lens system as seen in Fig. 6. Here, in this paper, the Fresnel hologram type, which can provide a 3-D reconstruction with depth information, is adopted for the holographic display.

## 3. Experiments and the results

#### 3.1 Overall configuration of the experimental setup

Figure 7 shows an overall experiment setup of the proposed system, which is largely composed of digital and optical systems. In the experiments, two kinds of volumetric 3-D color objects such as ‘Cube’ and ‘Airplane’ with the sizes of 10*mm* × 10*mm* × 10*mm* and 10*mm* × 10*mm* × 4*mm*, respectively, are computationally generated with the 3Ds Max and used for the test objects. Here, a cubic object of ‘Cube’ with six faces colored with ‘red’, ’green’, ‘blue’, ‘purple’, ‘yellow’ and ‘cyan’, is set to be located at the distance of 40*cm* from the hologram plane.

As seen in the digital system of Fig. 7(a), *R*, *G* and *B*-intensity and depth image data are extracted from the test object of ‘Cube’. Sampled color holograms whose resolution are 1920 × 1200 pixels, are calculated from each set of the *R*, *G* and *B*-intensity and depth data using the sampling NLUT method. With color-dependent SFMs, three kinds of FF-*R*, *G* and *B*-holograms are generated and multiplexed into the FF-CMH.

In the optical system of Fig. 7(b), the FF-CMH is reconstructed on the optical 4-*f* lens system, where two dichromatic convex lenses (Model: #63-564 Edmond Optics) whose focal lengths are 15*cm,* are used. As seen in Fig. 7, a multi-wavelength light source, which is composed of three *R*, *G* and *B*-lasers (Model: STRADUS^{TM} 633, 532 and 473, VORTRAN Laser Technology) lasers whose wavelengths are 630*nm*, 532*nm* and 472*nm*, respectively, is collimated and expanded with the beam expander (Model: HB-4XAR.14, Newport), and then illuminated into the SLM where the FF-CMH is loaded. Here, the color-distorted images generated due to the SLM-CD can be removed with a fabricated optical SFM. In the experiment, a reflection-type SLM (Model: HOLOEYE LC-R-1080) with the resolution of 1920 × 1200 pixels, 8.1*μm* pixel-pitch and 60% diffraction efficiency, is used. This SLM is operated in the amplitude-modulation mode to realize the light modulation of the Fresnel hologram pattern.

#### 3.2 Calculation of sampled color holograms for each color image

Figures 8(a)-8(c) show the calculated three color holograms for each color image of ‘Cube’ with the sampling NLUT method. Here, the test object of ‘Cube’ whose dimension is 10*mm* × 10*mm* × 10*mm* is modeled as a collection of 256 depth-sliced plane object images with a step size of *Δz =* 0.04*mm*. Thus, three sets of 256 *R*, G and *B*-PFPs for each depth plane are pre-calculated and stored in the NLUT. Then, each of the 256 sampled color images is sequentially selected, and hologram patterns for those object points locating on each selected object plane are then calculated just by shifting and adding processes of the corresponding color PFPs. Here, the test object of ‘Cube’ located at the distance of 40*cm* from the hologram plane can be reconstructed within a display space of 15.6*mm* × 11.7*mm* × 12.4*mm*. With no sampling process, the resolution of each color image is given by 960 × 720 pixels with the spatial pixel periods of *Δp _{x} = Δp_{y} =* 2 × 8.1

*µm*. Thus, the shifting amounts of the PFPs corresponding to each image pixel become same with those of the spatial pixel periods such as

*Δs*and

_{x}= Δp_{x}*Δs*. However, with the sampling process, the resolution of each color image becomes 320 × 240 when the sampling spatial periods are given by

_{y}= Δp_{y}*Δp*′

*′*

_{x}= Δp*3 ×*

_{y}=*Δp*3 ×

_{x}=*Δp*3 × 2 × 8.1

_{y}=*µm*. The depth distance between two PFPs is also increased by 3-fold compared to the original depth distance of 2 × 8.1

*µm*. Since the cut-off frequencies are related with the spatial periods such as

*X*and

_{c}= 1/Δx*Y*, the spatial periods are also changed from

_{c}= 1/Δy*Δx*and

*Δy*to 3 ×

*Δx*and 3 ×

*Δy*, while their cut-off frequencies are changed from

*X*and

_{c}*Y*to

_{c}*X*× 1/3 and

_{c}*Y*× 1/3, respectively, along the

_{c}*x*and

*y*-directions. Thus, the sampling process causes an extension of the single frequency spectrum into the 3 × 3 periodic array of them in the same frequency bandwidth as shown in Fig. 8(d).

#### 3.3 SFMs-based generation of FF-color holograms

Since the resolution of the sampled hologram is set to be 1920 × 1200 pixels, the resolution of its frequency spectrum obtained on the Fourier domain is also given by 1920 × 1200 pixels. As seen in Fig. 8(d), there exist 3 × 3 periodic frequency spectrums and their conjugate terms on the Fourier domain. Figures 8(d)-8(h) show the SFM-based selective frequency-filtering processes of the sampled color holograms. Three groups of three spectrums uniquely assigned to each color hologram are filtered out with their own SFMs, and then FF-versions of those sampled three-color holograms such as FF-*R*, *G* and *B*-holograms can be generated. As mentioned above, three groups of three spectrum locations of (1, 1) (2, 3) (3, 2), (1, 3) (2, 2) (3, 1) and (1, 2) (2, 1) (3, 3) are allocated to the *R*, *G* and *B*-holograms, respectively. Those FF-color holograms are then equally scaled by one third in amplitude and multiplexed into the FF-CMH as seen in Fig. 8(h).

#### 3.4 Optical reconstruction of the FF-CMH on the 4-f lens system

Figure 9 shows a conceptual configuration of the reconstruction process of the FF-CMH on the optical 4-*f* lens system using the pinhole-type optical SFM. As seen in Fig. 9, the FF-CHM of Fig. 8(h) is loaded on the SLM, and then a multi-wavelength light source composed of *R*, *G* and *B*-lasers whose wavelengths are 630*nm*, 532*nm* and 472*nm*, respectively, is illuminated into the SLM. Figure 9(a) shows twenty-seven spectrums of the FF-CMH which are optically generated on the Fourier domain, where original nine (3 × 3) spectrums uniquely allocated to each color are consistent with those of the FF-*R*, *G* and *B*-holograms. However, due to the pixelated structure of the SLM, each of those nine spectrums is dispersed into two other color spectrums. Thus, twenty-seven color spectrums in total would be generated on the optical Fourier domain. That is, eighteen points of color-dispersed spectrums along with nine points of original spectrums are simultaneously generated as seen in Fig. 9(a).

For filtering out those nine original spectrums (three for *R*-color spectrums, three for *G*-color spectrums and three for *B*-color spectrums), a pinhole-type filtering mask called an optical SFM, which corresponds to the computational SFMs, is fabricated and inserted on the optical Fourier domain of the 4-*f* lens system. This optical SFM then blocks the zero-order and undesired eighteen color-dispersed terms, while passing nine original spectrums. Here, the optical SFM acting as an optical band-pass filter, can be designed based on the Abbe image formation theory [34]. That is, actual positions of those nine spectrums can be calculated from the relationship between the frequency spectrums and frequency positions such as *x _{f} = X∙f∙λ* and

*y*[25, 34], where

_{f}= Y∙f∙λ*x*,

_{f}*y*and

_{f}*X*,

*Y*represent the

*x*,

*y-*coordinates of the spatial-frequency position and frequency, as well as

*f*and

*λ*denotes the focal length of the lens and wavelength of the laser

Now, the spatial-frequency *X* is equal to the cut-off frequency spectrum, which is calculated to be *X _{m} =* 1

*/Δx =*1/(6 × 8.1

*µm*). Then, the frequency distances between two adjacent frequency spectrums can be calculated by the relation of

*Δx*, where the focal length of the lens and wavelengths of the

_{f}= X_{m}∙f∙λ*R*,

*G*and

*B*-lasers are given by 15

*cm*, and 630

*nm*, 532

*nm*, 472

*nm*, respectively. Table 1 shows the calculated center locations and diameters of those nine pinholes on the optical SFM for filtering out those original nine color spectrums.

Here, the spatial interval of those 3 × 3 periodic frequency spectrums for the case of the *R*-laser illumination, *Δx _{R}* is calculated to be 1.9

*mm*as seen in Fig. 9(a). When the origin coordinate of the frequency spectrums is set to be

*P*(0, 0), the (

_{0}*n, m*)

*position of 3 × 3*

^{th}*R*-color spectrum points can be given by

*P*(

_{m,n}*m*× 1.9

*mm, n*× 1.9

*mm*), where

*m*and

*n*denote positive integers. Likewise, the (

*n, m*)

*positions of 3 × 3*

^{th}*G*and

*B*-color spectrum points can be also given by

*P*(

_{m,n}*m*× 1.6

*mm, n*× 1.6

*mm*) and

*P*(

_{m,n}*m*× 1.5

*mm, n*× 1.5

*mm*), respectively. For example, the actual location coordinate of the

*B*-color spectrum point of the (2, 1) position can be given by

*P*(1.5

_{2,1}*mm,*3.0

*mm*), while those of the

*G*-color spectrum points of the (2, 2) and (2, 3) positions are also given by

*P*(3.2

_{2,2}*mm,*3.2

*mm*) and

*P*(5.7

_{2,3}*mm,*3.8

*mm*), respectively, as seen in Table 1 and Fig. 9. All those nine pinholes on the optical SFM have been fabricated to have the same diameters of 0.5

*mm*on their calculated center positions with a CNC machine as seen in Fig. 9(c).

#### 3.5 Optically reconstructed 3-D color object images

Figure 10 shows two kinds of 3-D color objects of ‘Cube’ and ‘Airplane’ and their optically reconstructed images on the proposed system. For the comparative performance evaluation of the proposed method, four kinds of experiments have been carried out. As mentioned above, in the proposed system, two kinds of digital and optical frequency-filtering operations based on 2-D spatial sampling are newly employed for solving both of the CMH-CD and SLM-CD problems.

Now, in the 1st experiment, object images have been optically reconstructed without performing both of the digital and optical frequency-filtering operations, which mean that color-distortions due to the CMH-CD and SLM-CD inevitably occur in the reconstructed object images as seen in Figs. 10(b) and (b’). In this case, not only the original *R*-color image diffracted from the *R*-hologram, but also two other *R*-color images diffracted from the *G* and *B*-holograms are reconstructed with very small angular deviations when the *R*-laser is illuminated into the CMH, which is referred to the CMH-CD. In addition, when the SLM loaded with the CMH is illuminated with three *R*, *G* and *B*-lasers simultaneously, not only the *R*, *G* and *B*-color images, but also three sets of two other color images are also diffracted from each of the *R*, *G* and *B*-holograms of the CMH due to the pixelated structure of the SLM, which is referred to the SLM-CD. Thus, one color image (*R*, *G* and *B* single color images overlapped) and 6 other single color images reconstructed from each of the *R*, *G* and *B*-holograms, must coexist in Figs. 10(b) and (b’). In addition, the zero-order term also occurs together with nine color images as seen at the top and left corner areas of Figs. 10(b) and (b’).

In the 2nd experiment, object images have been tried to be reconstructed with the optical frequency-filtering process, where the color-distortion due to the CMH-CD happens to occur in the reconstructed object images as seen in Figs. 10(c) and (c’). In this case, three *R*, *G* and *B*-frequency spectrums are overlapped together at all 3 × 3 spectrum areas of the CMH. Thus, even though the optical SFM can filter out nine original color spectrums while blocking the other color-dispersed spectrums, all those optically-filtered color spectrums contains three-color components. It means that from each of the optically-filtered color spectrums three kinds of same color images have been reconstructed simultaneously. Thus, Figs. 10(c) and (c’) contain one color image (*R*, *G* and *B* single color images overlapped) and 6 other single color images reconstructed from each of the *R*, *G* and *B*-holograms. However, in this experiment, the zero-order term has been removed with the optical SFM as seen in the top and left corner areas of Figs. 10(c) and (c’) unlike the cases of Figs. 10(b) and (b’).

Meanwhile, in the 3rd experiment, object images have been tried to be reconstructed with the digital frequency-filtering process, where the color-distortion due to the SLM-CD then must occur in the reconstructed object images as seen in Figs. 10(d) and (d’). In this case, each of the 3 × 3 spectrums of the CMH contains only one color data, thus *R*-color spectrum in the (1, 1), (2, 3) and (3, 2) positions of Fig. 9(b) can be reconstructed into the corresponding *R*-color image. But, since no optical SFM is employed in this experiment, *G* and *B*-color images are also reconstructed from the *R*-color hologram. Thus, 6 color-distorted single color images along with original one color image (*R*, *G* and *B* single color images overlapped) are simultaneously reconstructed due to the pixelated structure of the SLM, which cause the reconstructed 3-D image to be severely distorted in color as seen in Figs. 10(d) and (d’). Moreover, the zero-order term also co-exist together with those reconstructed color images since the optical frequency-filtering process has not been adapted here.

On the other hand, Figs. 10(e) and (e’) finally show the object images reconstructed from the proposed system, where both of the digital and optical frequency-filtering processes have been employed, so that color-distortions due to the CMH-CD and SLM-CD have been totally removed from the reconstructed object images as seen in Figs. 10(e) and (e’). In this case, *R*, *G* and *B*-color spectrums are set to be separately allocated to each group of three spectrums areas and then synthesized into the CMH as seen in the digital frequency-filtering operation of Fig. 7. Furthermore, each group of three original *R*, *G* and *B*-color spectrums is selectively filtered out in the corresponding optical frequency-filtering operation of Fig. 9. Then, from these optically-filtered three *R*, *G* and *B*-color spectrums, a color distortion-free full-color 3-D object image without the zero-order term can be finally reconstructed on the proposed 4-*f* lens system as seen in Figs. 10(e) and (e’). However, there exist small amounts of color defects on the reconstructed images as seen in Figs. 10(e) and (e’). It may happen to occur due the inaccurate spectrum filtering operation of the optical mask since the optical pinhole mask was manually fabricated in this paper. Thus, those color defects expect to be removed by employing a precisely-manufactured optical pinhole mask on a thin metal with the computer-controlled milling machine.

#### 3.6 Optically reconstructed 3-D color video images for the objects in motion

For the test scenarios, two kinds of full-color 3-D video images for the self-revolving ‘Cube’ and flying ‘Airplane’, are generated with the 3Ds Max. The first test scenario is composed of 100 frames of video images of a continuously self-revolving 3-D ‘Cube’ from a ‘shuffled’ state to the ‘solved’ state with a speed of 24 frames per second as shown in Fig. 11(a). In addition, the second test scenario is composed of 50 frames of video images of a 3-D ‘Airplane’ moving from the far right to the near left direction with a speed of 16 frames per second as seen in Fig. 11(b).

Two kinds of FF-CMHs for each of the test scenarios of Fig. 11 are generated and optically reconstructed on the proposed 4-*f* lens system. Those reconstructed video object images are directly captured with the digital camera (Nikon D7000) at the output plane of the optical 4-*f* lens system, which are shown in Fig. 12. As seen in five sample frames of Fig. 12(a) and (b), all those full-color 3-D video images have been successfully reconstructed on the proposed system. The total video frames for each of the first and second video scenarios have been compressed into the video files of Visualization 1[Fig. 12(a)], and Visualization 2[Fig. 12(b)], respectively, and attached to Fig. 12. All those successful experimental results show that the proposed system can reconstruct full-color 3-D holographic videos on a simple 4-*f* lens system based on the sampling and selective frequency-filtering methods with a single amplitude-type SLM, which finally confirms the feasibility of the proposed system in the practical application fields.

## 4. Conclusions

In this paper, a new type of the single SLM full-color holographic 3-D display system based on the sampling and selective frequency-filtering methods has been proposed. Just by removing the CMH and SLM-based color distortions with digital and optical SFM generated based on the 2-D sampling theorem, a color distortion-free full-color 3-D image has been optically reconstructed from the FF-CMH on the simple 4-*f* lens system employing a single SLM. Theoretical analysis and successful experiments with 3-D color objects in motion confirm the feasibility of the proposed system in the practical application fields.

## Funding

National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2011-0030079).

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