Diffraction
Notes on diffraction for holograms
Theory
Rayleigh-Sommerfeld diffraction theory
Approximations
Fresnel diffraction
Fresnel diffraction can be used for intermediate distances (near-field), when \(\displaystyle F = \frac{a^2}{L \lambda} \geq 1\).
The Fresnel approximation is: \[ E(x,y,z) = \frac{e^{ikz}}{i \lambda z} \int \int E(x', y', 0) e^{\frac{ik}{2z} \left[ (x - x')^2 + (y - y')^2 \right]} dx' dy' \] where
- \(\displaystyle k=2 \pi / \lambda\)
Expanding out terms, we get: \[ \begin{align} E(x,y,z) &= \frac{e^{ikz}}{i \lambda z} \int \int E(x', y', 0) e^{\frac{ik}{2z} \left[ (x - x')^2 + (y - y')^2 \right]} dx' dy'\\ &= \frac{e^{ikz}}{i \lambda z} \int \int E(x', y', 0) e^{\frac{i\pi}{\lambda z} \left[ x^2 - 2x x' + x'^2 + y^2 - 2yy' + y'^2 \right]} dx' dy'\\ &= \frac{e^{ikz}}{i \lambda z} e^{\frac{-i\pi}{\lambda z} \left[ x'^2+ y'^2 \right]} \int \int E(x', y', 0) e^{\frac{i\pi}{\lambda z} \left[ x^2 + y^2 \right] } e^{\frac{-i2\pi}{\lambda z} \left[ x x' + yy' \right]} dx' dy' \\ &= \frac{e^{ikz}}{i \lambda z} e^{\frac{-i\pi}{\lambda z} \left[ x'^2+ y'^2 \right]} \mathcal{F} \left\{ E(x', y', 0) e^{\frac{i\pi}{\lambda z} \left[ x^2 + y^2 \right] } \right\}_{p=\frac{x}{\lambda z}, q=\frac{y}{\lambda z}} \end{align} \]
Fraunhofer diffraction
Fraunhofer diffraction is used for far-field holograms.
Resources
- Digital holography and wavefront sensing by Ulf Schnars, Claas Falldorf, John Watson, and Werner Jüptner, Springer-verlag Berlin an, 2016