Terahertz Electromagnetic Wave Detection

Members

Luo Shizhuo E1353445@u.nus.edu Zhang Bohan E1349227@u.nus.edu

Project Overview

Terahertz (THz) waves are rather useful in communication (6G communication: 0.3~3 THz), astronomy (millimetre telescope: 0.22 THz) and solid state material characterisation. However, the generation and detection of THz wave so difficult that it is named as the "THz Gap".

In this project, we aim to generate and detect a THz pulse with a traditional method (Electro-Optical Sampling) and then calibrate a commercial VO2 thermal detector which is reported to be able to detect THz wave. (Specifically, we will use LiNbO3 (LN) crystal, which is generally applied in intense THz pulse generation, to generate a common THz pulse.)

This project can provide a cheaper option compared to specific THz camera (like THz fluorescence camera), and enable us to observe the pattern of THz wave in a more cost-effective way.

Theory Basis

LN-based THz pulse generation

Phase Match Condition

We assume that an electromagnetic wave goes in the direction with an angle $ϵ$ between z axis. The electric field is given as:

$E(t)=\int E(\omega )e^{i\omega t-ikz\cdot \cos ϵ-ikx\cdot \sin ϵ}d\omega =\int E(\omega )e^{i\omega t-i\phi }d\omega$

where $\phi =kz\cdot \cos ϵ+kx\cdot \sin ϵ$, and \textit{k} is the wave vector. The phase term $\phi$ indicates the direction of the motion. When an angular dispersion by frequency $\omega$ is introduce in deviation angle $ϵ$, the phase term $\phi$ becomes:

$\phi \approx \phi ({\omega }_0)+\frac{\partial \phi }{\partial \omega }(\omega -{\omega }_0)$

Here, ${\omega }_0$ is the central frequency. Substituting $v_g=\frac{c}{n_g}$ and $\tan \gamma$ for $\frac{d\omega }{dk}$ and $\frac{n({\omega }_0)}{n_g}{\omega }_0\frac{dϵ}{d\omega }$, respectively, the formula above can be simplified to:

$\phi \approx \frac{n({\omega }_0){\omega }_0}{c}(z\cos ϵ({\omega }_0)+x\sin ϵ({\omega }_0))+\frac{\omega -{\omega }_0}{v_g\cos \gamma }(z\cos (ϵ({\omega }_0)+\gamma )+x\sin (ϵ({\omega }_0)+\gamma ))$

Hence, the speed of the wave packet (term with a low frequency $(\omega -{\omega }_0)$) is modified to be $v_g\cos \gamma$. The phase match condition becomes:

$v_{\text{THz}}=v_g\cos \gamma$

This shows that the phase match condition can be modulated by the parameter $\gamma$.

• Exact Calculation

The THz field is generated by Difference Frequency Generation (DFG) described by:

$E_{\text{THz}}(t)=\iint \chi ({\omega }_1)E^{*}({\omega }_1)e^{-i{\omega }_{1}t+i{k}_{1}z\cos {ϵ}_1+i{k}_{1}x\sin {ϵ}_1}E({\omega }_2)e^{i{\omega }_{2}t-i{k}_{2}z\cos {ϵ}_2-i{k}_{2}x\sin {ϵ}_2}d{\omega }_{1}d{\omega }_2$

When we substitute $\omega$ and $\omega +\mathrm{\Omega }$ for ${\omega }_1$ and ${\omega }_2$, the equation above becomes:

$E_{\text{THz}}(t)=\iint \chi (\omega )E^{*}(\omega )E(\omega +\mathrm{\Omega })e^{i\mathrm{\Omega }t+i\frac{\mathrm{\Omega }}{v_g\cos \gamma }(z\cos \gamma +x\sin \gamma )}d\omega d\mathrm{\Omega }$

Hence, the THz wave by frequency $\mathrm{\Omega }$ is:

$E_{\text{THz}}(\mathrm{\Omega })=e^{i\mathrm{\Omega }t-i\frac{\mathrm{\Omega }}{v_{\text{THz}}}z}\int \chi (\omega )E^{*}(\omega )E(\omega +\mathrm{\Omega })e^{i\frac{\mathrm{\Omega }}{v_g\cos \gamma }z-i\frac{\mathrm{\Omega }}{v_{\text{THz}}}z}d\omega$

where a rotation of $\gamma$ in $x-z$ plane is imposed. The term $e^{i\frac{\mathrm{\Omega }}{v_g\cos \gamma }z-i\frac{\mathrm{\Omega }}{v_{\text{THz}}}z}$ represents the phase mismatch.

Optimization

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