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

Electro-Optical Sampling

The Terahertz photons has so low energy (~0.003 eV) that it can hardly be detected via photodiode made up of common semiconductor (band gap: >0.5eV). Even narrow bandgap semiconductor has a bandgap in the magnitude of 0.01 eV. Hence, conventional detectors cannot directly detect Terahertz photons. Nevertheless, the variance rate of terahertz frequency is in the magnitude of picoseconds. This hints us to use pulses with a duration of 0.1 picosecond magnitude as a "shutter" to capture the shape of terahertz wave.

Optical pulses can detect the transient change of the optical property. Once the electric field of the terahertz wave alters the optical properties of the material, the probe pulse, i.e. the optical pulse, can instantaneously detect this change. This measurement technique is "Electro-Optical (EO) Sampling". When modifying the time delay between the optical pulse and terahertz wave, the optical pulse can depict the electric field of terahertz wave point by point.

In the experiment, birefringence is the most popular optical property used in EO sampling. This effect can be seen as a special sum-frequency generation . Since the frequency of terahertz wave is much lower than the optical pulse, it can be deemed as zero-frequency term in the special sum-frequency generation. Hence, the dielectric tensor becomes: ${\epsilon }_{ij}={\chi }_{ij}^{(1)}(\omega )+{\chi }_{ijk}^{(2)}(0,\omega ;\omega )E_{k,\text{THz}}$. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as ${\epsilon }^{(-1)}$. When applied Taylor expansion in the order of $E_{k,THz}$, we have ${\epsilon }^{-1}\approx {\epsilon }_{ij,0}^{-1}+r_{ijk}{E}_{k,\text{THz}}$. Here, ${\epsilon }_{ij,0}^{-1}$ is the dielectric tensor when there is no outer electric field, and $r_{ijk}$ is the EO coefficient

According to the conservation of energy density $w=\frac{1}{2}{D}_i{ϵ}_{ij}^{-1}{D}_j$ and electric displacement $D_i$ with and without the medium, we can normalise the conversation relation as:

$1=\frac{u_i{u}_j}{n_{ij}^2}=u_i{u}_j{\epsilon }_{ij}^{-1}\approx u_i{u}_j({\epsilon }_{ij,0}^{-1}+r_{ijk}{E}_{k,\text{THz}})$

where $u_i=\frac{D_i}{\sqrt{2w}}$ represents the direction of the outer optical field. When diagnosing the tensor, we can simply give out the birefringent refractive indices.