# Holographic light transport¶

Odak contains essential ingredients for research and development targeting Computer-Generated Holography. We consult the beginners in this matter to Goodman's Introduction to Fourier Optics book (ISBN-13: 978-0974707723) and Principles of optics: electromagnetic theory of propagation, interference and diffraction of light from Max Born and Emil Wolf (ISBN 0-08-26482-4). This engineering note will provide a crash course on how light travels from a phase-only hologram to an image plane.

Holographic image reconstruction. A collimated beam with a homogenous amplitude distribution (A=1) illuminates a phase-only hologram $$u_0(x,y)$$. Light from this hologram diffracts and arrive at an image plane $$u(x,y)$$ at a distance of z. Diffracted beams from each hologram pixel interfere at the image plane and, finally, reconstruct a target image.

As depicted in above figure, when such holograms are illuminated with a collimated coherent light (e.g. laser), these holograms can reconstruct an intended optical field at target depth levels. How light travels from a hologram to a parallel image plane is commonly described using Rayleigh-Sommerfeld diffraction integrals (For more, consult Heurtley, J. C. (1973). Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points. JOSA, 63(8), 1003-1008.). The first solution of the Rayleigh-Sommerfeld integral, also known as the Huygens-Fresnel principle, is expressed as follows:

$$u(x,y)=\frac{1}{j\lambda} \int\!\!\!\!\int u_0(x,y)\frac{e^{jkr}}{r}cos(\theta)dxdy,$$

where field at a target image plane, $$u(x,y)$$, is calculated by integrating over every point of hologram's field, $$u_0(x,y)$$. Note that, for the above equation, $$r$$ represents the optical path between a selected point over a hologram and a selected point in the image plane, theta represents the angle between these two points, k represents the wavenumber ($$\frac{2\pi}{\lambda}$$) and $$\lambda$$ represents the wavelength of light. In this described light transport model, optical fields, $$u_0(x,y)$$ and $$u(x,y)$$, are represented with a complex value,

$$u_0(x,y)=A(x,y)e^{j\phi(x,y)},$$

where A represents the spatial distribution of amplitude and $$\phi$$ represents the spatial distribution of phase across a hologram plane. The described holographic light transport model is often simplified into a single convolution with a fixed spatially invariant complex kernel, $$h(x,y)$$ (Sypek, Maciej. "Light propagation in the Fresnel region. New numerical approach." Optics communications 116.1-3 (1995): 43-48.).

$$u(x,y)=u_0(x,y) * h(x,y) =\mathcal{F}^{-1}(\mathcal{F}(u_0(x,y)) \mathcal{F}(h(x,y)))$$

There are multiple variants of this simplified approach:

• Matsushima, Kyoji, and Tomoyoshi Shimobaba. "Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields." Optics express 17.22 (2009): 19662-19673.,
• Zhang, Wenhui, Hao Zhang, and Guofan Jin. "Band-extended angular spectrum method for accurate diffraction calculation in a wide propagation range." Optics letters 45.6 (2020): 1543-1546.,
• Zhang, Wenhui, Hao Zhang, and Guofan Jin. "Adaptive-sampling angular spectrum method with full utilization of space-bandwidth product." Optics Letters 45.16 (2020): 4416-4419.

In many cases, people choose to use the most common form of h described as

$$h(x,y)=\frac{e^{jkz}}{j\lambda z} e^{\frac{jk}{2z} (x^2+y^2)},$$

where z represents the distance between a hologram plane and a target image plane. Note that beam propagation can also be learned for physical setups to avoid imperfections in a setup and to improve the image quality at an image plane:

• Peng, Yifan, et al. "Neural holography with camera-in-the-loop training." ACM Transactions on Graphics (TOG) 39.6 (2020): 1-14.,
• Chakravarthula, Praneeth, et al. "Learned hardware-in-the-loop phase retrieval for holographic near-eye displays." ACM Transactions on Graphics (TOG) 39.6 (2020): 1-18.,
• Kavaklı, Koray, Hakan Urey, and Kaan Akşit. "Learned holographic light transport." Applied Optics (2021)..