Diffraction: Difference between revisions

Latest revision as of 15:16, 14 April 2023

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, when \(\displaystyle F = \frac{a^2}{L \lambda} \lt \lt 1\).

Resources

Digital holography and wavefront sensing by Ulf Schnars, Claas Falldorf, John Watson, and Werner Jüptner, Springer-verlag Berlin an, 2016
- See https://github.com/OptoManishK/Digital_Holography

@@ Line 7: / Line 7: @@
 ===Fresnel diffraction===
 {{main | Wikipedia: Fresnel diffraction}}
-Fresnel diffraction can be used for intermediate distances.
+Fresnel diffraction can be used for intermediate distances (near-field), when <math>F = \frac{a^2}{L \lambda} \geq 1</math>.
+The Fresnel approximation is:
+<math display="block">
+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'
+</math>
+where
+* <math>k=2 \pi / \lambda</math>
+Expanding out terms, we get:
+<math display="block">
+\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}
+</math>
 ===Fraunhofer diffraction===
 {{main | Wikipedia: Fraunhofer diffraction}}
-Fraunhofer diffraction is used for far-field holograms.
+Fraunhofer diffraction is used for far-field holograms, when <math>F = \frac{a^2}{L \lambda} << 1</math>.
 ==Resources==
 * Digital holography and wavefront sensing by Ulf Schnars, Claas Falldorf, John Watson, and Werner Jüptner, Springer-verlag Berlin an, 2016
 ** See https://github.com/OptoManishK/Digital_Holography