## Abstract

We propose a method that eliminates speckle artifact in the reconstruction of triangular mesh based computer generated holograms. The proposed method generates a number of holograms with different sets of interleaved plane carrier waves. The generated holograms are displayed sequentially with time-multiplexing, achieving speckle-free reconstruction without sacrificing the viewing angle or depth of focus. The proposed method is verified experimentally using viewing-window type holographic display setup. The experimental results indicate that the proposed method can achieve speckle-free reconstruction with smaller number of the time-multiplexing than conventional temporal speckle averaging techniques.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Speckle is one of the main causes that degrade the reconstruction quality in holographic three-dimensional (3D) displays using coherent light source. The random phase distribution of high spatial frequency on the surface of the reconstructed 3D image interferes in the observer’s eye retina plane or camera’s sensor plane, creating grain shaped noise in the reconstructed images [1].

One traditional method to reduce the speckle noise in holographic displays is spectral averaging [2]. Instead of monochromatic laser, light source of larger bandwidth like light emitting diode (LED) is used in the reconstruction. Each monochromatic component of the LED light generates holographic reconstruction with different speckle pattern. These different speckle patterns are averaged in the observation, resulting in the reduction of the speckle contrast. Each monochromatic component of the LED light, however, generates the holographic reconstruction not only with different speckle pattern but also different magnification and shift, and thus the observed image is not crisp but blurred.

Another traditional method to reduce the speckle noise is temporal averaging. The reconstructions with different phase distributions are presented to observer sequentially such that different speckle patterns are averaged by after-image effect [3–5]. The reduction ratio of the speckle contrast, however, is low, i.e. *N*^{-1/2} for *N* time-multiplexing [1], and thus large number of the frames in the time-multiplexing are required to achieve satisfactory result.

Recently, a spatial interleaving method has been proposed for speckle-less reconstruction of point cloud based computer generated holograms (CGHs) [6-7]. In conventional point cloud based CGHs, a single hologram is synthesized by accumulating spherical waves from all object points in the cloud. In the spatial interleaving method, multiple holograms are synthesized with different sets of interleaved object points and presented to the observer sequentially with time-multiplexing. The spatial interval between the object points in a single hologram is set to be larger than the circle of confusion of the observer so that the interference in the retina plane is avoided. Therefore each hologram with interleaved object points has speckle-free reconstruction, unlike conventional temporal averaging methods. By time-multiplexing multiple holograms with different sets of the interleaved object points, the entire 3D objects can be filled without speckle noise. Although this method uses time-multiplexing like the temporal averaging methods, the required number of the time-multiplexing can be much lower. This spatial interleaving method has also been applied to light ray field based CGHs, where the light rays corresponding to the interleaved object points are selected for each hologram synthesis [8].

In this paper, we propose a novel method to eliminate the speckle in triangular mesh based CGH. The triangular mesh based CGH synthesizes the hologram by accumulating the wave from each triangular mesh constituting 3D objects [9–16]. As the triangular mesh representation of 3D objects is widely used in computer graphics, the triangular mesh based CGH is highly compatible with existing computer graphics objects. Since the hologram synthesis is performed per each triangular mesh, it is more computationally efficient than the point cloud based CGHs where waves from individual points in the object surface should be calculated separately. One difficulty of the triangular mesh based CGH is that the wave from each triangular mesh may have different form according to the shape, size, and orientation of the triangle, unlike point cloud based CGHs where the wave from each object point is always a spherical wave. The fully-analytic triangular mesh based method uses geometric relationship between the arbitrarily given triangle and a reference triangle whose angular spectrum can be obtained in analytic formula, ensuring exact wave calculation for the given sampling grid in the hologram plane. Due to these advantages, the fully-analytic triangular mesh based CGH attracts growing attention recently. To the authors’ best knowledge, however, the speckle reduction in the triangular mesh based CGH has not been addressed yet. Also, the spatial interleaving method cannot be applied to the triangular mesh based CGH because the hologram of interleaved meshes still has a speckle noise inside each triangle.

The proposed method in this paper eliminates the speckle by using angular spectrum interleaving. In the proposed method, multiple holograms are synthesized using analytic triangular mesh based CGH with different sets of interleaved plane carrier waves. For each hologram, the angular spacing between interleaved plane carrier waves is set such that the holographic reconstruction associated with only one plane carrier wave is captured by the observer’s eye pupil. The observed reconstruction of each hologram has linear phase distribution on the triangular mesh surface corresponding to the captured plane carrier wave, and thus it is an entire 3D object without speckle noise. By presenting the synthesized multiple holograms sequentially in time-multiplexing manner, the speckle-free reconstruction is visible in the full viewing angle of the hologram. Unlike the conventional temporal averaging methods where speckle patterns of each frame in the time-multiplexing are averaged, each frame in the proposed method is free from speckle noise. Therefore, the number of the frames in the time-multiplexing does not affect the speckle contrast, but it is only related to the angular range and density of the reconstruction to be covered. Unlike the spatial interleaving methods where the object points are spatially interleaved, the proposed method interleaves the plane carrier waves, achieving angular interleaving. Since the control of the carrier wave is natural and exact in the analytic triangular mesh based CGH, the proposed method is well suited to it.

In the next sections, we review the carrier wave control in the analytic triangular mesh based CGH. Then we explain the principle of the proposed angular spectrum interleaving method with experimental verification results.

## 2. Carrier wave control in analytic triangular mesh based CGHs

In analytic triangular mesh based CGHs, each triangular mesh constituting a 3D object is modelled as a triangular shaped amplitude transparency illuminated by a carrier wave. The corresponding angular spectrum in the hologram plane is calculated from the geometric relationship between two planes, i.e. the hologram plane and the local plane containing the triangular mesh, and between two triangles, i.e. the given triangle and a reference triangle whose angular spectrum is given by an analytic formula.

Suppose that the hologram plane is *z* = 0 plane in global (*x*,*y*,*z*) coordinates and the triangular mesh lies in *z _{l}* = 0 plane in local (

*x*,

_{l}*y*,

_{l}*z*) coordinates with one of its vertices located at the local coordinates origin (

_{l}*x*= 0,

_{l}*y*= 0,

_{l}*z*= 0). The global and local coordinates are related using a 3 × 3 rotation matrix R and 3 × 1 shift vector

_{l}**c**by

**r**= R

_{xl,yl,zl}**r**+

_{x,y,z}**c**where

**r**= [

_{xl,yl,zl}*x*,

_{l}*y*,

_{l}*z*]

_{l}^{T}and

**r**= [

_{x,y,z}*x*,

*y*,

*z*]

^{T}. The carrier wave is a plane wave represented by u(

**r**) =

_{x,y,z}*a*(

**ν**)exp[

_{x,y}*j*2π

**ν**

_{x,y,z}^{T}

**r**] where

_{x,y,z}**ν**= [

_{x,y,z}*ν*,

_{x}*ν*,

_{y}*ν*]

_{z}^{T}and

**ν**= [

_{x,y}*ν*,

_{x}*ν*]

_{y}^{T}are 3 × 1 and 2 × 1 spatial frequency vectors with

*ν*= (1/λ

_{z}^{2}-

*ν*

_{x}^{2}-

*ν*

_{y}^{2})

^{1/2}for wavelength λ. The angular spectrum of the mesh in the hologram plane G(

**f**;

_{x,y}**ν**;

_{x,y}*a*(

**ν**)) is given by [9]

_{x,y}**f**= [

_{x,y,z}*f*,

_{x}*f*,

_{y}*f*]

_{z}^{T}and the local spatial frequency

**f**= [

_{xl,yl,zl}*f*,

_{xl}*f*,

_{yl}*f*]

_{zl}^{T}are related by

**f**= R

_{xl,yl,zl}**f**and

_{x,y,z}**f**and

_{x,y}**f**are defined by [

_{xl,yl}*f*,

_{x}*f*]

_{y}^{T}and [

*f*,

_{xl}*f*]

_{yl}^{T}, respectively. The

*f*and

_{z}*f*satisfy

_{zl}*f*= (1/

_{z}*λ*

^{2}-

*f*

_{x}^{2}-

*f*

_{y}^{2})

^{1/2}and

*f*= (1/

_{zl}*λ*

^{2}-

*f*

_{xl}^{2}-

*f*

_{yl}^{2})

^{1/2}. The

**r**is a 3 × 1 global position vector of the local coordinate origin. A is the 2 × 2 matrix relating the triangular mesh in the local plane g

^{o}_{x,y,z}_{l}(

**r**) = g

_{xl,yl}_{l}([

*x*,

_{l}*y*]

_{l}^{T}) with the reference triangle g

_{r}by g

_{l}(

**r**) = g

_{xl,yl}_{r}(A

**r**) assuming that the one of the vertices of the reference triangle is located at the origin. The term B(

_{xl,yl}**f**;

_{x,y}**ν**) is given by

_{x,y}_{r}is the angular spectrum of the reference triangle whose analytic formula is known.

Using Eqs. (1) and (2), the angular spectrum for a specific carrier wave u(**r _{x,y,z}**) =

*a*(

**ν**)exp[

_{x,y}*j*2π

**ν**

_{x,y,z}^{T}

**r**] is obtained. For multiple carrier waves, the angular spectrum is given by

_{x,y,z}**f**and D(

_{x,y}**f**) is given by

_{x,y}**f**) can be obtained by Eq. (4) efficiently.

_{x,y}## 3. Proposed method

The proposed method removes the speckle noise by interleaving and scanning the carrier waves. The motivation of the proposed method is that the hologram synthesized with a single plane carrier wave; i.e. D(**f _{x,y}**) =

*δ*(

**f**-

_{x,y}**ν**) does not show speckle noise in its reconstruction. With a single carrier wave, every mesh surface has linear phase distribution. Therefore their addition within the circle of confusion of the user’s eye produces uniform intensity without random fluctuations, resulting in speckle-free reconstruction. In spite of speckle-less reconstruction, the hologram with a single plane carrier wave alone is not highly adequate for holographic 3D displays due to limited viewing angle of the reconstruction. Figure 1 shows an example of the angular spectrum G(

_{x,y}**f**;

_{x,y}**ν**;

_{x,y}*a*) which corresponds to a single carrier wave of the spatial frequency

**ν**. As can be seen in Fig. 1, most of the signal energy is concentrated around the carrier wave’s spatial frequency

_{x,y}**ν**. Since the extent of the angular spectrum defines the angular range of the reconstruction, i.e. viewing angle of the hologram, the hologram with a single plane carrier wave has limited viewing angle, much smaller than the maximum viewing angle determined by the sampling pitch of the hologram.

_{x,y}In order to exploit the speckle-free reconstruction of the hologram with a single plane carrier wave while preserving the full viewing angle, the proposed method uses interleaving and scanning of the plane carrier waves. Figure 2 illustrates the concept of the proposed method. The proposed method synthesizes multiple holograms G* _{n}*(

**f**),

_{x,y}*n*= 1,2,…,

*N*with different sets of interleaved plane carrier waves, i.e.

*n*-th hologram |

**s**| is smaller than the angular spacing between the plane carrier waves Δ

_{x,y}^{(n)}**ν**, i.e. |

_{x,y}**s**|<|Δ

_{x,y}^{(n)}**ν**|. In each hologram, the angular spacing between the plane carrier waves Δ

_{x,y}**ν**is determined such that their lateral separation in the user’s eye pupil plane is larger than the eye pupil. Since the wave field captured by the eye pupil is associated only with a single plane carrier wave, the observed image is speckle-free reconstruction of the hologram. The enlarged depth of focus (DOF) of the observed image which is caused by the limited angular spectrum extent smaller than the eye pupil can be made shallow by time-multiplexing multiple holograms G

_{x,y}*(*

_{n}**f**) with plane carrier wave shifts

_{x,y}**s**within the eye pupil as illustrated in Fig. 2. Therefore the proposed method achieves the speckle-free reconstruction without sacrificing the large viewing angle and shallow DOF.

_{x,y}^{(n)}In the proposed method, the number of the time multiplexing *N* determines the number of the plane carrier waves within an eye pupil. If we increase *N*, more plane carrier waves are sequentially captured by the eye pupil with small spacings, achieving natural continuous blur when the reconstruction is not focused. With small *N*, the defocus blur can be discrete, but we can still maintain the same amount of the blur, i.e. DOF, by spanning the full eye pupil sparsely.

Note that in the proposed method, each frame G* _{n}*(

**f**) of the time multiplexed holograms gives speckle-free reconstruction of the 3D objects unlike the temporal averaging methods. Therefore the number of the time-multiplexing

_{x,y}*N*affects not the speckle contrast but the smoothness of the defocus blur. This feature makes it possible to realize the speckle-free shallow DOF reconstructions with the number of the time multiplexing much smaller than the conventional temporal averaging methods. In the experimental results described in the next section, we found that the number of the time-multiplexing

*N*can be reduced to 5.

Note also that although the proposed method has been explained in the context of the triangular mesh CGHs, it can be applied to other CGH techniques as well, by assigning proper phase to individual element of 3D object, for instance a 3D point in point cloud CGHs, with consideration of the plane carrier wave.

Finally, note that the angular spectrum extent of the individual hologram in the proposed method is given by the bandwidth of the 3D scene as shown in Fig. 2 and it should be smaller than the eye pupil for effective speckle-free reconstructions. Therefore fine surface texture that corresponds to an angular spectrum range similar to or larger than the eye pupil may not be reconstructed with proper contrast in the proposed method.

## 4. Experimental verification

For the verification of the proposed method, optical experiments were performed. Figure 3 shows the schematics of the experimental setup. The holographic 3D images created by the SLM is filtered by an aperture in the Fourier plane of the 4-*f* system to pass only the desired angular spectrum range. The field lens in the image plane of the SLM forms a viewing window at its focal distance where the eye or the camera is located.

In the first experiment, the speckle-less reconstruction with a single hologram having interleaved plane carrier waves was verified. A phase only liquid crystal on silicone (LCoS) device having 8um pixel pitch, 1920 × 1080 resolution, and 8 bit depth (model name: PLUTO, manufacturer: HOLOEYE Photonics AG) was used as the SLM and the wavelength of the laser was 532nm. A 3D scene having two teapot mesh models at different depths is used as the target scene. For this target 3D scene, a single hologram was synthesized using two plane carrier waves, i.e. D(**f _{x,y}**) =

*δ*(

**f**-

_{x,y}**ν**) +

_{x,y1}*δ*(

**f**-

_{x,y}**ν**). In the proposed method, the separation |

_{x,y2}**ν**-

_{x,y2}**ν**| between the interleaved plane carrier waves should be larger than the eye or camera pupil. In the experiment, however, it was set to be deliberately small in order to increase the period of the interference patterns and thus enhance its visibility when both plane carrier waves fall inside the camera pupil.

_{x,y1}Figure 4 shows the experimental results. In Fig. 4, upper parts show the angular spectrum captured in the Fourier plane of the 4-*f* system or equivalently in the viewing window plane. Since two plane carrier waves are used in the synthesis of the hologram, we can see two bright spots centered at the corresponding spatial frequencies of the plane carrier waves. Figure 4(a) shows the reconstruction when the aperture was set to pass both spots. Since the reconstruction contains two plane carrier wave components which are separated horizontally, the strong vertical interference pattern is observed. Note that if we use a hologram with large number of the plane carrier waves, this interference pattern becomes complicated, resulting in usual grain-shaped speckle pattern. Figure 4(b) is the result when the aperture was set to pass only a single spot in the Fourier plane, which is equivalent to the situation when only the single plane carrier wave component falls within the camera pupil. As expected, the vertical interference pattern disappears, showing clear reconstruction. Therefore it can be confirmed that speckle-free reconstruction can be observed even though the hologram itself has multiple plane carrier waves if only a single plane carrier wave component falls within the camera or eye pupil by interleaving the plane carrier waves with sufficient spacing.

In the second experiment, the speckle-free reconstruction with shallow DOF by presenting multiple holograms sequentially with shifted single plane carrier wave within the eye pupil was examined. We performed two experiments; non-real-time operation using low frame rate LCoS SLM and real-time operation using high frame rate DMD SLM. The 3D scene is the same as previous experiment, i.e. two teapots with different depths. In both experiments, two types of holograms were prepared. In one type which is the case of the conventional temporal averaging, the convolution kernel having constant amplitude with random phase, D* _{n}*(

**f**) = exp[j

_{x,y}*ϕ*(

_{n}**f**)] was used in the synthesis of each hologram where

_{x,y}*ϕ*(

_{n}**f**) is uniformly distributed over [0,2π). In the other type, the convolution kernel of each hologram is given by D

_{x,y}*(*

_{n}**f**) = δ(

_{x,y}**f**-

_{x,y}**s**) as the proposed method. In the non-real-time experiment using LCoS SLM, the reconstruction of each hologram was captured separately, and the captured images were added digitally to emulate the time-multiplexing.

_{x,y}^{(n)}Figure 5 shows the experimental results using the LCoS SLM. The time-multiplexed reconstructions with different number of holograms *N* are shown in two types of the holograms. In each of Fig. 5(a) and 5(b), upper row shows the captured reconstructions when the camera is focused at the left teapot and lower row shows the captured images when the camera focus is at the right teapot. The white insets in the upper row indicate the angular spectrum accumulated sequentially by the time multiplexing. In the conventional temporal averaging method shown in Fig. 5(a), each hologram has extended angular spectrum range as shown in the leftmost part which is captured in the Fourier plane of the 4-*f* system. Its reconstruction has severe speckle noise as expected when *N* = 1, i.e. no time-multiplexing case. As the number of the holograms in the time-multiplexing increases from *N* = 1 to *N* = 81, the speckle noise decreases due to averaging of speckle patterns. However, the speckle noise is severe when *N* = 9 and it is still observable when *N* = 81.

In the proposed method shown in Fig. 5(b), the angular spectrum of each hologram is concentrated around the plane carrier wave **s _{x,y}^{(n)}** as shown in the leftmost part of Fig. 5(b) which was captured in the Fourier plane of the 4-

*f*system. By time-multiplexing, the angular spectrum can span the extent of the angular spectrum of the conventional temporal averaging method shown in Fig. 5(a). In Fig. 5(b), it can be observed that even with

*N*= 1, the reconstruction is clean without the speckle noise as expected. Comparing the amount of blur of the defocused object between Fig. 5(a) and 5(b), however, it can be seen that the defocus blur of the proposed method with

*N*= 1 is much smaller than that of the conventional temporal averaging method due to limited extent of angular spectrum. However, by increasing the number of the holograms from

*N*= 1 to

*N*= 81 as shown in Fig. 5(b), the amount of the defocus blur quickly saturates to that of the temporal averaging method from

*N*= 5. Therefore it can be claimed that the time multiplexing of 5 holograms are sufficient to reconstruct the speckle-free images while maintaining the shallow DOF in the proposed method.

In the experiment using the DMD SLM, the real-time operation was conducted using high frame rate of the SLM. The DMD SLM used in the experiment is a binary amplitude modulating SLM with 13.86um pixel pitch and 1024 × 768 resolution (model name: DMD Discovery 3000, manufacturer: Texas Instruments Inc.). The multiple holograms were displayed with a frame rate 6 KHz and the reconstruction was captured by a camera with an exposure time 1/10 sec. The wavelength of the laser source used in this experiment is 633nm. Before loading the hologram data to the DMD SLM, each original 8 bit hologram was transformed to a binary hologram by thresholding operation, to fit the binary modulation property of the SLM. The experimental results are shown in Fig. 6. Although the reconstruction quality is slightly worse than the case of the LCoS SLM experiment because we reduced the bit depth of the hologram to binary in the DMD SLM experiment, it is sufficient to verify the speckle-free reconstruction with shallow DOF of the proposed method. Similar to the experimental results of the LCoS SLM, it can be seen in Fig. 6 that the time-multiplexing of 5 holograms, *N* = 5, is sufficient to display clear 3D images without sacrificing the DOF.

Figure 7 shows another real-time experimental result using the DMD SLM. In this experiment, a planar scene with uniform amplitude was used as the target object to measure the speckle contrast quantitatively. The speckle contrast was measured within the large uniform tilted square at the center of the pattern. Figure 8 shows the measured speckle contrast. Figure 8(a) is the speckle contrast measured from Fig. 7 and Fig. 8(b) is the speckle contrast measured from the non-real-time LCoS SLM experimental result in Fig. 5. In case of Fig. 8(b), the speckle contrast was measured from the most homogeneous section of the teapot. As can be seen in Fig. 8, the proposed method shows lower speckle contrast in all time-multiplexing cases and it reaches the minimum speckle contrast level quickly at *N* = 5 and maintain the similar value with larger *N*. Therefore, it can be concluded that the number of the time-multiplexing required in the proposed method can be as small as 5 which is much smaller than the number of the time-multiplexing required in the conventional temporal averaging methods.

## 5. Conclusion

In this paper, we proposed a novel speckle reduction technique using carrier wave interleaving in analytic mesh based CGHs. By time-multiplexing holograms with interleaved plane carrier waves, speckle-free reconstructions can be achieved without sacrificing the shallow DOF and the viewing angle of the hologram. The principle of the proposed method was verified by optical experiments using LCoS SLM and DMD SLM and it was found that the number of time multiplexing can be reduced to 5 using the proposed method which is much smaller than the conventional temporal speckle averaging techniques at the same speckle contrast and the DOF level.

## Funding

Cross-Ministry Giga KOREA Project, Ministry of Science and ICT (MSIT), Korea (GK17D0100, Development of Telecommunications Terminal with Digital Holographic Table-top Display); Information Technology Research Center (ITRC), Ministry of Science and ICT (MSIT), Korea (IITP-2017-2015-0-00448). INHA University Research Grant, Korea (INHA-49293).

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