## Abstract

We propose a novel method that synthesizes computer-generated holograms from light field data. Light field, or ray space, is the spatio-angular distribution of light rays coming from three-dimensional scene, and it can also be represented using a large number of views from different observation directions. The proposed method synthesizes a hologram by applying the complex field recovery technique from its Wigner distribution function to the light field data. Unlike conventional approaches, the proposed method synthesizes holograms without hogel configuration, generating a converging parabolic wave for each object point with continuous wavefront. The proposed method does not trade the spatial resolution with angular resolution like conventional hogel-based approaches. Moreover, the proposed method works not only for random phase light field like conventional approaches, but also for arbitrary phase distribution with corresponding carrier waves. Therefore, the proposed method is useful in synthesizing holographic contents for a wide range of applications. The proposed method is verified by simulations and optical experiments, showing successful reconstruction of three-dimensional objects.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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### Equations (16)

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(1)
$$L\left(x,y,u,v\right)={\displaystyle \sum _{m}{a}_{m}^{2}\delta \left(u+\frac{x-{x}_{m}}{\lambda {z}_{m}},v+\frac{y-{y}_{m}}{\lambda {z}_{m}}\right)},$$
(2)
$$H(x,y)={\displaystyle \iint H\left(x,y;{x}_{c},{y}_{c}\right)W({x}_{c},{y}_{c})d{x}_{c}d{y}_{c}},$$
(3)
$$H\left(x,y;{x}_{c},{y}_{c}\right)={\displaystyle \iint L\left(\frac{x+{x}_{c}}{2},\frac{y+{y}_{c}}{2},u,v\right){e}^{j2\pi \left\{u\left(x-{x}_{c}\right)+v\left(y-{y}_{c}\right)\right\}}dudv}.$$
(4)
$$\begin{array}{l}H\left(x,y;{x}_{c},{y}_{c}\right)={\displaystyle \iint {\displaystyle \sum _{m}{a}_{m}^{2}\delta \left(u+\frac{\frac{x+{x}_{c}}{2}-{x}_{m}}{\lambda {z}_{m}},v+\frac{\frac{y+{y}_{c}}{2}-{y}_{m}}{\lambda {z}_{m}}\right)}{e}^{j2\pi \left\{u\left(x-{x}_{c}\right)+v\left(y-{y}_{c}\right)\right\}}dudv}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\displaystyle \sum _{m}{a}_{m}^{2}}\mathrm{exp}\left[-j\frac{\pi}{\lambda {z}_{m}}\left\{{\left(x-{x}_{m}\right)}^{2}+{\left(y-{y}_{m}\right)}^{2}\right\}\right]\mathrm{exp}\left[j\frac{\pi}{\lambda {z}_{m}}\left\{{\left({x}_{c}-{x}_{m}\right)}^{2}+{\left({y}_{c}-{y}_{m}\right)}^{2}\right\}\right].\end{array}$$
(5)
$$x={t}_{x}+\frac{{\tau}_{x}}{2},\text{\hspace{1em}}y={t}_{y}+\frac{{\tau}_{y}}{2},\text{\hspace{1em}}{x}_{c}={t}_{x}-\frac{{\tau}_{x}}{2},\text{\hspace{1em}}{y}_{c}={t}_{y}-\frac{{\tau}_{y}}{2}.$$
(6)
$$\begin{array}{l}H\left(x,y;{x}_{c},{y}_{c}\right)=H\left({t}_{x}+\frac{{\tau}_{x}}{2},{t}_{y}+\frac{{\tau}_{y}}{2};{t}_{x}-\frac{{\tau}_{x}}{2},{t}_{y}-\frac{{\tau}_{y}}{2}\right)\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}={\displaystyle \iint L\left({t}_{x},{t}_{y},u,v\right)\mathrm{exp}\left[j2\pi \left(u{\tau}_{x}+v{\tau}_{y}\right)\right]}dudv\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=\tilde{L}({t}_{x},{t}_{y},{\tau}_{x},{\tau}_{y})=\tilde{L}\left(\frac{x+{x}_{c}}{2},\frac{y+{y}_{c}}{2},x-{x}_{c},y-{y}_{c}\right),\end{array}$$
(7)
$$H(x,y)={\displaystyle \iint \tilde{L}\left(\frac{x+{x}_{c}}{2},\frac{y+{y}_{c}}{2},x-{x}_{c},y-{y}_{c}\right)W({x}_{c},{y}_{c})}d{x}_{c}d{y}_{c},$$
(8)
$$U(x)={\displaystyle \sum _{m}{U}_{m}(x)}={\displaystyle \sum _{m}{a}_{m}{e}^{j{\varphi}_{m}}\mathrm{exp}\left[-j\frac{\pi}{\lambda {z}_{m}}{\left(x-{x}_{m}\right)}^{2}\right]},$$
(9)
$$\begin{array}{l}WDF({t}_{x},u)={\displaystyle \int U\left({t}_{x}+\frac{{\tau}_{x}}{2}\right){U}^{*}\left({t}_{x}-\frac{{\tau}_{x}}{2}\right){e}^{-j2\pi u{\tau}_{x}}d{\tau}_{x}}\\ ={\displaystyle \int {\displaystyle \sum _{m}{U}_{m}\left({t}_{x}+\frac{{\tau}_{x}}{2}\right){U}_{m}{}^{*}\left({t}_{x}-\frac{{\tau}_{x}}{2}\right){e}^{-j2\pi u{\tau}_{x}}}}d{\tau}_{x}+{\displaystyle \int {\displaystyle \sum _{m,n(\ne m)}{U}_{m}\left({t}_{x}+\frac{{\tau}_{x}}{2}\right){U}_{n}{}^{*}\left({t}_{x}-\frac{{\tau}_{x}}{2}\right){e}^{-j2\pi u{\tau}_{x}}}}d{\tau}_{x}\\ ={\displaystyle \sum _{m}{a}_{m}^{2}\delta \left(u+\frac{{t}_{x}-{x}_{m}}{\lambda {z}_{m}}\right)}+{\displaystyle \int {\displaystyle \sum _{m,n(\ne m)}{U}_{m}\left({t}_{x}+\frac{{\tau}_{x}}{2}\right){U}_{n}{}^{*}\left({t}_{x}-\frac{{\tau}_{x}}{2}\right){e}^{-j2\pi u{\tau}_{x}}}}d{\tau}_{x}\\ =L({t}_{x},u)+{\displaystyle \int {\displaystyle \sum _{m,n(\ne m)}{U}_{m}\left({t}_{x}+\frac{{\tau}_{x}}{2}\right){U}_{n}{}^{*}\left({t}_{x}-\frac{{\tau}_{x}}{2}\right){e}^{-j2\pi u{\tau}_{x}}}}d{\tau}_{x},\end{array}$$
(10)
$${{\displaystyle \int WDF({t}_{x},u){e}^{j2\pi u{\tau}_{x}}du}|}_{{t}_{x}=\frac{x+{x}_{c}}{2},{\tau}_{x}=x-{x}_{c}}=U\left(x\right){U}^{*}({x}_{c})={\displaystyle \sum _{m}{U}_{m}(x)}{\displaystyle \sum _{n}{U}_{n}^{*}({x}_{c})},$$
(11)
$$\begin{array}{l}{{\displaystyle \int L({t}_{x},u){e}^{j2\pi u{\tau}_{x}}du}|}_{{t}_{x}=\frac{x+{x}_{c}}{2},{\tau}_{x}=x-{x}_{c}}={\tilde{L}({t}_{x},{\tau}_{x})|}_{{t}_{x}=\frac{x+{x}_{c}}{2},{\tau}_{x}=x-{x}_{c}}={\displaystyle \sum _{m}{U}_{m}(x){U}_{m}^{*}({x}_{c})}\\ ={\displaystyle \sum _{m}{a}_{m}^{2}\mathrm{exp}\left[-j\frac{\pi}{\lambda {z}_{m}}{\left(x-{x}_{m}\right)}^{2}\right]}\mathrm{exp}\left[j\frac{\pi}{\lambda {z}_{m}}{\left({x}_{c}-{x}_{m}\right)}^{2}\right]=H(x;{x}_{c}),\end{array}$$
(12)
$$\Delta {t}_{x}=\Delta x,\text{\hspace{1em}}{N}_{tx}={N}_{x}.$$
(13)
$$\Delta u{N}_{u}=\frac{1}{\Delta {\tau}_{x}}=\frac{1}{\Delta x}=\frac{\mathrm{sin}{\theta}_{full}}{\lambda},$$
(14)
$${B}_{tx}={B}_{x},\text{\hspace{1em}}{B}_{u}=\lambda \left|z\right|{B}_{x},$$
(15)
$$\Delta {t}_{x}\le \frac{1}{{B}_{x}},\text{\hspace{1em}}\Delta u\le \frac{1}{\lambda \left|z\right|{B}_{x}}.$$
(16)
$${N}_{u}\ge \lambda \left|z\right|{B}_{x}^{2}.$$