Simultaneous Color Holographic Display

Introduction

A holographic display is a type of display system that produces 3D or 2D images by manipulating the wavefront of light. With the help of a spatial light modulator (SLM), a holographic display can manipulate the phase of a coherent wavefront at the pixel level. This allows it to reshape the wavefront precisely as it would originally emanate from a real object, creating an image with genuine depth cues.

Holographic displays typically use a laser as a light source, resulting in monochromatic holograms. To achieve color holograms, color holographic displays sequentially switch between RGB lasers at a high rate, leveraging the persistence of vision property of the human eye. This allows the human eye to fuse sequential monochromatic holograms into a perceived color hologram. However, this color scheme sacrifices the refresh rate of the SLM, as displaying one frame of a color target requires displaying three phase patterns, one for each RGB channel.

One potential solution to fully utilize the SLM’s refresh rate is to simultaneously activate the three primary laser lights and have the SLM modulate these three wavefronts with different wavelengths at the same time, using the same phase pattern. This approach could potentially allow for full utilization of the SLM’s refresh rate.

This project aims to investigate the effectiveness of traditional phase retrieval pipelines in this setup and explore potential improvements in reconstruction quality through the use of different loss functions.

Background

How to get light phase from intensity

Unlike conventional displays that directly control light intensity, holographic displays use a Spatial Light Modulator (SLM) to modulate the phase of light on a per-pixel basis. The modulated wavefront then propagates through free space from a starting plane $0$ to the image plane $z$. Our goal is to determine the phase pattern on the SLM such that, at the image plane $z$, the resulting intensity distribution matches a desired target intensity pattern.

Angular Spectrum Method

The Angular Spectrum Method is a computational technique used to model how wavefronts propagate through free space. It can be expressed in the following form:

$u(x,y,z)={ℱ}^{-1}\{ℱ\{u(x,y,0)\}\cdot \mathscr{H}(k_x,k_y,z)\}$

$\mathscr{H}(k_x,k_y,z)=\{\begin{array}{cc}{e}^{-\frac{i2\pi }{\lambda }\sqrt{1-(\lambda k_x{)}^2-(\lambda k_y{)}^2}z}& \text{if }\sqrt{k_x^2+k_y^2}<\frac{1}{\lambda },\\ 0& \text{otherwise}.\end{array}$

$u(x,y,0)$ is the wavefield at the plane $z=0$. By applying the 2D Fourier transform to it, $ℱ\{u(x,y,0)\}$, we decompose the wavefield into a superposition of plane waves traveling in various directions. This continuous distribution of plane waves is known as the angular spectrum. The spatial frequencies $k_x$ and $k_y$ determine the propagation direction of each plane wave component.

As each plane wave propagates through free space, it accumulates a distance-dependent phase shift. This phase shift is described by the transfer function $\mathscr{H}(k_x,k_y,z)$. In the Fourier domain, by multiplying the angular spectrum $ℱ\{u(x,y,0)\}$ by $\mathscr{H}(k_x,k_y,z)$, we effectively propagate all plane wave components over the distance $z$.

Finally, to reconstruct the propagated wavefield at $z$, we apply the inverse Fourier transform: ${ℱ}^{-1}\{ℱ\{u(x,y,0)\}\cdot \mathscr{H}(k_x,k_y,z)\}$.

Image Formation Model

In our setup of the holographic display, the coherent light source that illuminates the SLM has a source field $u_{src}(x,y,\lambda )$.

The phase phase-only SLM can apply a spatially-varying delay $\varphi (x,y,\lambda )$ on the phase of the field $u_{src}(x,y,\lambda )$, so the wavefield at the SLM becomes to:

$u_{SLM}(x,y,\lambda )=e^{iq(\varphi (x,y,\lambda ))}{u}_{src}(x,y,\lambda )$

The SLM is at plane $z=0$, we can use the Angular Spectrum Method to model what the wavefield will looks like at image plane $z$.

$u_z(x,y,\lambda )=ASM(u_{SLM}(x,y,\lambda ),z)$

At the image plane, what people see is the intensity of light not the wavefield. We can get light intensity by squaring the wavefield.

$I_z(x,y,\lambda )=|u_z(x,y,\lambda ){|}^2$

In combination, the final light intensity distribution is $I_z(x,y,\lambda )=|ASM(e^{iq(\varphi (x,y,\lambda ))}{u}_{src}(x,y,\lambda ),z){|}^2$

For notational convenience, we can write the intensity pattern at the image plane $z$ as:

${𝑰}_{𝒛}\mathit{(}𝒙,𝒚,𝝀\mathit{)}\mathit{=}\mathit{|}𝑨𝑺𝑴\mathit{(}{𝒆}^{𝒊𝒒\mathit{(}𝝓\mathit{)}},𝒛\mathit{)}{\mathit{|}}^2$

This is how we can get the light intensity at the image plane $z$, by showing a phase pattern $\varphi$ at the SLM.

Iterative Method To Get Light Phase from Intensity

Now we know how to calculate light intensity from the phase pattern. However in order to show some images on the holographic display, we need to find a way to calculate phase pattern from the light intensity. People usually use gradient descent to solve this kind of inverse problem.

At iteration 0 we can generate some phase pattern