## Abstract

Mesh-based computer generated hologram enables realistic and efficient representation of three-dimensional scene. However, the dark line artifacts on the boundary between neighboring meshes are frequently observed, degrading the quality of the reconstruction. In this paper, we propose a simple technique to remove the dark line artifacts by matching the phase on the boundary of neighboring meshes. The feasibility of the proposed method is confirmed by the numerical and optical reconstruction of the generated hologram.

© 2015 Optical Society of America

## 1. Introduction

Computer generated hologram (CGH) is a technique to synthesize the hologram or the complex optical field corresponding to three-dimensional (3D) scene. The 3D scene is modeled by a collection of primitives and the CGH is synthesized by adding elementary optical field of each primitive. The types of the primitives representing the 3D scene include a point [1], a light ray [2], and a triangular mesh [3–12]. The triangular-mesh based CGH has advantages in the point that it enables efficient calculation, and direct consideration of the texture and diffusiveness of the 3D object surfaces. Recently, the fully analytic triangular-mesh-based CGH methods which use an analytic mathematical formula of the angular spectrum of the reference triangle have drawn increasing attention as they enable exact calculation of the optical field without resampling or interpolation [9–12].

One problem of the fully analytic triangular-mesh-based CGH methods is dark line artifacts on the mesh boundary. In the fully analytic methods, each triangular mesh is modeled by an aperture of the triangular shape. By a carrier wave, the object surface is reconstructed with piecewise continuous phase distribution and the abrupt change of the phase on the mesh boundaries causes the interference between the optical fields of neighboring meshes, creating dark lines on the boundary [10–12].

Two researches have been reported regarding the dark line artifacts. Y.-Z. Liu et al. adjusted the center-of-mass of each mesh at a distance of the integer multiple of the wavelength from the hologram plane [11]. However, the method can only be applied when the optical carrier is a plane wave whose direction is parallel to the optical axis. D. Im et al. moved the hologram synthesis plane to the retinal plane of the observer in order to deal with converging carrier wave case [12]. However, it requires two-step calculation, i.e. initial hologram synthesis on the retinal plane and its numerical propagation to the original plane through the crystal lens, increasing the computational burden.

In this paper, we propose a simple technique to remove the dark line artifacts. The proposed method matches the phase on the mesh boundary by giving proper phase bias to each mesh according to the plane carrier wave. For converging or diverging carrier wave case, the initial CGH with the plane carrier wave is numerically multiplied by a convex or concave lens function. The proposed method is more general than previous ones [11, 12] in the point that it can deal with all of the planar, converging, and diverging carrier waves for off-axis holograms, viewing window based holographic displays [13], and holographic projectors [14], respectively. Unlike Y.-Z. Liu et al.’s work [11] which deals with the planar carrier wave, the direction of the planar carrier wave in the proposed method is not restricted to the optical axis. Unlike D. Im et al.’s work [12] which deals with the converging carrier wave, the proposed method is much simpler, only requiring a multiplication with the lens function after the initial CGH synthesis with a plane wave.

## 2. Principle

Figure 1 shows the concept of the fully analytic triangular-mesh-based CGH synthesis [9, 10]. In the mesh-based analytic CGH synthesis, the angular spectrum G(*f _{x}*,

*f*) of the optical wave in the hologram plane corresponding to a single mesh is first represented using the angular spectrum G

_{y}_{l}(

*f*,

_{xl}*f*) in the local plane containing the mesh;

_{yl}**f**= [

_{x,y}*f*]

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

**f**= [

_{xl,yl}*f*]

_{xl}f_{yl}^{T},

**f**= [

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

_{xl}f_{yl}f_{zl}^{T}=

**Rf**,

_{x,y,z}*f*= {(1/λ)

_{z}^{2}-

*f*

_{x}^{2}-

*f*

_{y}^{2}}

^{0.5}, and

*f*= {(1/λ)

_{zl}^{2}-

*f*

_{xl}^{2}-

*f*

_{yl}^{2}}

^{0.5}.

**R**and

**c**are 3 × 3 rotation matrix and 3 × 1 shift vector which satisfy

**r**=

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

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

**r**and

_{x,y,z}**r**are 3 × 1 position vectors in the global and local coordinates. Then the G

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

*f*) is calculated using the analytic mathematical formula of the angular spectrum G

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

*f*,

_{xr}*f*) of a reference triangular mesh and the affine transform between the given and the reference triangular meshes;

_{yr}**A**and

**b**are 2 × 2 affine matrix and 2 × 1 shift vector between the given triangle in the local plane and the reference triangle. The carrier wave is used to shift the angular spectrum of the meshes so that the meshes are visible from the opposite direction of the carrier wave as shown in Fig. 1. For a general plane carrier wave whose direction is represented by a unit 3 × 1 vector

**u**, the G

_{c}_{l}(

*f*,

_{xl}*f*) in Eq. (1) is modified by

_{yl}**u**and

_{xl}**u**are unit vectors of

_{yl}*x*and

_{l}*y*axes represented in the global coordinates.

_{l}The frequency shift in Eq. (3) by the carrier wave makes all meshes visible from the observation direction, but it also causes the phase mismatch between the neighboring meshes, resulting in the dark line defects. By Eq. (3), the phase value on each vertex of the mesh is dependent on the selection of the local coordinates for the mesh. Therefore, the vertex shared by neighboring meshes like vertex V_{2} in Fig. 2 has different phase values for neighboring meshes likes mesh 1 and mesh 2 in Fig. 2. This phase mismatch happens not only on the shared vertex but also on the boundary between the meshes, creating the dark line defect.

The proposed method removes the dark line defect by matching the phase on the boundary between the meshes. The phase matching can be simply done by giving proper phase bias to each mesh considering the carrier wave direction. In the proposed method, the G_{l}(*f _{xl}*,

*f*) in Eq. (1) is modified to

_{yl}**r**is the 2D position vector of any of three vertices of the mesh represented in the (

_{v,xlyl}*x*,

_{l}*y*) local coordinates, and

_{l}*d*is the distance between the vertex and the carrier wavefront that passes through a reference point in the hologram plane as shown in Fig. 2. The first term in Eq. (4) is given by Eq. (3) and shifts the angular spectrum to make the mesh visible from the observation direction as before. The second term in Eq. (4) initializes the phase of the selected vertex of the mesh to be zero. Then the third term sets the phase of the vertex according to the distance from the carrier wavefront. Note that even though only a single vertex out of three vertices for each mesh is arbitrarily selected and used for

_{v,vf}**r**and

_{v,xlyl}*d*in Eq. (4), the second and third terms of Eq. (4) give the phase bias to the mesh so that every point in the mesh has the phase according to its distance from the carrier wavefront. The process is repeated for all meshes while sharing the reference point in the hologram plane to give the phase to all meshes consistently.

_{v,vf}In case of the converging or diverging carrier wave as shown in Fig. 3, the CGH for a plane carrier wave is first synthesized using the proposed phase bias, then the CGH is simply multiplied by the lens transmittance function exp[ ± *j*{π/(λ*f*)}(*x*^{2} + *y*^{2})] where *f* determines the curvature of the converging or diverging carrier wave. The distortion of the 3D object by the lens function can easily be pre-compensated in the initial CGH synthesis stage by considering the image of the target 3D object by the lens.

Note that the term which is equivalent to the second and third terms in Eq. (4) giving the proper phase bias can be found in H. Kim et al.’s work [9]. In ref [9], however, random phases are assigned to the meshes in the simulation and experiment, and thus the effect of the term on the dark line artifact is not analyzed. Note also that the significant demagnification of the 3D object by the lens is not required in the proposed method unlike D. Im et al.’s work [12]. The proper phase bias in the initial CGH creates continuous phase distribution of the plane carrier wave on the object surface. After the multiplication with the lens function, the continuity is still maintained over the object surface while the carrier wave is transformed to the converging or diverging wave. Finally note that for the converging or diverging carrier wave, it may be considered to assign different plane carrier waves to the corresponding meshes in a way that the directions of the plane carrier waves are converging or diverging. However, in this case, the phase distribution on the object surface is only piecewise continuous with discontinuities on the mesh boundary, resulting in the dark line artifact.

## 3. Simulation and experiment

In numerical simulation, a CGH for two objects, i.e. a box at 2cm and a teapot at + 4cm from the hologram plane was synthesized. The pixel pitch of the hologram was 8um and the carrier wave was set to be a plane wave whose propagation direction is given by 0° for azimuthal and 0.5° for polar angles. The resolution of the synthesized CGH is 499 × 499. Figure 4 shows the simulation result. It can be confirmed that the dark line artifacts is completely removed in the proposed phase matching method. In the magnified view of the phase distribution on the reconstructed box in Fig. 4(a), it can be observed that there are many phase discontinuities at the mesh boundary which causes the dark line artifacts. These phase discontinuities are eliminated in the proposed method as shown in Fig. 4(b). The observed phase distribution inside the box area in Fig. 4(b) is the phase distribution of the plane carrier wave.

Figure 5 shows the simulation result for plane, converging and diverging carrier wave cases. In order to see the effect of the lens function, the CGH in Fig. 4 was multiplied with a convex or concave lens function of the focal length of 20cm without pre-compensation of the 3D object. Therefore it is expected that the size and the location of the reconstructed objects will be changed. In Fig. 5 it is observed that the proposed method eliminates the dark line defect completely in both of the converging and diverging carrier wave cases as well. The magnified views of the phase on the reconstructed box also show the continuous phase distribution of the corresponding carrier waves. It is also observed that the size of the reconstructed objects are reduced in the converging case and increased in the diverging case as expected.

Figure 6 shows another simulation result in plane carrier wave case. In Fig. 6, the size of the box object was extended in axial direction in order to see subtle focus changes inside the box. The axially extended box of the size 1.7mm(W) × 2.0mm(H) × 14.2mm(T) was located around the hologram plane and 499 × 499 resolution CGH of 8um pitch was used. From Fig. 6, it can be observed that the rear, middle, and the front parts of the box object are focused separately, confirming that the proposed method preserves the depth information while eliminating the dark line defects effectively.

In optical experiment, a CGH for a teapot at 50cm from the spatial light modulator (SLM) was synthesized for plane, converging, and diverging carrier wave cases. The pixel pitch of the SLM is 8 um and the wavelength of the laser is 532 nm. The polarizers before and the after the SLM were adjusted to achieve the amplitude modulation. For the synthesis of the CGH, the complex optical field of the teapot was first calculated with a plane carrier wave of 90° for azimuthal and 1° for polar angles so that the reconstructed image is separated vertically from the DC part. For converging and diverging carrier wave cases, the virtual convex or concave lens function of the focal length 100cm was multiplied to the calculated complex optical field without pre-compensation of the object. After the complex optical fields for plane, converging, and diverging carrier waves are obtained, the interference pattern between the calculated complex optical field and a plane reference wave was calculated and loaded to the SLM. The resolution of the final CGH was 1019 × 1019.

Figure 7 shows the optical reconstruction results. For the plane carrier wave case, the reconstructed image was captured directly by using a usual digital camera. Note that the plane carrier wave is converted into a converging wave around the camera sensor plane by the imaging lens of the camera. For the converging wave case, the reconstruction was also directly captured by the camera, but in this case, the camera was located at the focal point of the convex lens function, i.e. 100cm from the SLM, like usual viewing window based holographic display. For the diverging carrier wave case, a diffuser was placed at the reconstructed distance and the diffused image was captured by the digital camera. The weak ghost images observed above and below the reconstruction are believed to come from the high order diffraction from the SLM. The horizontal lines visible in the reconstruction comes from the amplitude modulation of the SLM with the vertically inclined carrier wave. In spite of the ghost image and horizontal lines, it can be observed that the dark line artifacts is effectively removed in the proposed method as expected.

Figure 8 shows another optical reconstruction result of the proposed method with two teapots whose depths have 12 cm difference. For more visibility, the shading of the mesh face is adjusted in the CGH synthesis. The resolution of the synthesized CGH was 1979 × 1019 and the direction of the carrier wave was set to 0° for azimuthal and 1° for polar angles, so that the reconstruction is horizontally separated from the DC. From Fig. 8, it can be seen that the left and right teapots are focused separately, showing the 3D nature of the optical reconstruction. It is also observed that the dark line artifact is removed not only in the focused object but also in the defocused object.

## 4. Conclusion

In this paper, we have proposed a simple method to remove the dark line artifacts on the mesh boundary in the fully analytic triangular-mesh-based CGH synthesis. The proposed method gives proper phase bias to each triangular mesh so that the boundary between the neighboring meshes shares the same phase distribution according to the carrier wave direction in all cases of the plane, converging, and diverging carrier wave. The numerical and optical reconstruction results support the feasibility of the proposed method.

## Acknowledgments

This research was supported by 'The Cross-Ministry Giga KOREA Project' of The Ministry of Science, ICT and Future Planning, Korea. [GK14D0100, Development of Telecommunications Terminal with Digital Holographic Table-top Display]

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