PC5271 wiki - User contributions [en]

Terahertz Electromagnetic Wave Detection

2025-04-24T03:50:30Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile Sensors.jpeg|thumb|center|500px|Thermopile Sensors]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:49:03Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile_Sensors.png|thumb|center|500px|Thermopile Sensors]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:47:59Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile Sensors.png|thumb|center|500px|Thermopile Sensors]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:47:16Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile Sensors.png|thumb|center|500px|Thermopile Sensors.]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:46:50Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile Sensors.png|thumb|center|500px|]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:46:14Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

Thermopiles are used to provide an output in response to temperature as part of a temperature measuring device, such as the infrared thermometers widely used by medical professionals to measure body temperature, or in thermal accelerometers to measure the temperature profile inside the sealed cavity of the sensor. They are also used widely in heat flux sensors and pyrheliometers and gas burner safety controls. The output of a thermopile is usually in the range of tens or hundreds of millivolts. As well as increasing the signal level, the device may be used to provide spatial temperature averaging. Thermopiles are also used to generate electrical energy from, for instance, heat from electrical components, solar wind, radioactive materials, laser radiation or combustion. The process is also an example of the Peltier effect (electric current transferring heat energy) as the process transfers heat from the hot to the cold junctions.

[[File:Thermopile Sensors.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

File:Thermopile Sensors.jpeg

2025-04-24T03:44:24Z

Bohan: Thermopile Sensors

== Summary ==
Thermopile Sensors

Terahertz Electromagnetic Wave Detection

2025-04-24T03:40:37Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Differential Temperature Thermopile.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:36:21Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.
[[File:Thermal_Resistance_Layer.png|thumb|center|500px|Schematic of a thermopile structure showing heat flow and voltage generation due to temperature gradient.]]

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

Terahertz Electromagnetic Wave Detection

2025-04-24T03:25:31Z

Bohan: /* Thermopile Detector */

==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 via a commercial thermopile sensor. (Specifically, we will use LiNbO3 (LN) crystal, which is reported to be capable of generating THz pulses easily.)

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 <math>\epsilon</math> between z axis. The electric field is given as:

<math>E(t)=\int E(\omega) e^{i\omega t-ikz\cdot\cos\epsilon-ikx\cdot\sin\epsilon} d\omega=\int E(\omega) e^{i\omega t-i\varphi} d\omega</math>

where <math>\varphi=kz\cdot\cos\epsilon+kx\cdot\sin\epsilon</math>, and <math>k</math> is the wave vector. The phase term <math>\varphi</math> indicates the direction of the motion. When an angular dispersion by frequency <math>\omega</math> is introduce in deviation angle <math>\epsilon</math>, the phase term <math>\varphi</math> becomes:

<math>\varphi\approx\varphi(\omega_0)+\frac{\partial\varphi}{\partial\omega}(\omega-\omega_0)</math>

Here, <math>\omega_0</math> is the central frequency. Substituting <math>v_g=\frac{c}{n_g}</math> and <math>\tan \gamma</math> for <math>\frac{d\omega}{dk}</math> and <math>\frac{n(\omega_0)}{n_g}\omega_0\frac{d\epsilon}{d\omega}</math>, respectively, the formula above can be simplified to:

<math>\varphi\approx \frac{n(\omega_0)\omega_0}{c}(z\cos\epsilon(\omega_0)+x\sin\epsilon(\omega_0)) + \frac{\omega-\omega_0}{v_g\cos\gamma}(z\cos(\epsilon(\omega_0)+\gamma)+x\sin(\epsilon(\omega_0)+\gamma))</math>

Hence, the speed of the wave packet (term with a low frequency <math>(\omega-\omega_0)</math>) is modified to be <math>v_g\cos\gamma</math>. The phase match condition becomes:

<math>v_\text{THz}=v_g\cos\gamma</math>

This shows that the phase match condition can be modulated by the parameter <math>\gamma</math>.

*'''Exact Calculation'''

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

<math>E_\text{THz}(t)=\iint\chi(\omega_1)E^*(\omega_1)e^{-i\omega_1t+ik_1z\cos\epsilon_1+ik_1x\sin\epsilon_1}E(\omega_2)e^{i\omega_2 t-ik_2z\cos\epsilon_2-ik_2x\sin\epsilon_2}d\omega_1d\omega_2</math>

When we substitute <math>\omega</math> and <math>\omega+\Omega</math> for <math>\omega_1</math> and <math>\omega_2</math>, the equation above becomes:

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

Hence, the THz wave by frequency <math>\Omega</math> is:

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

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

===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: <math>\varepsilon_{ij}=\chi^{(1)}_{ij}(\omega)+\sum_k\chi^{(2)}_{ijk}(0,\omega;\omega)E_{k,\text{THz}}</math>. To give the refractive indices, we shall use the refractive ellipse. We reverse the dielectric tensor as <math>\varepsilon^{(-1)}</math>. When applied Taylor expansion in the order of <math>E_{k,THz}</math>, we have <math>\varepsilon^{-1}\approx\varepsilon^{-1}_{ij,0}+\sum_kr_{ijk}E_{k,\text{THz}}</math>. Here, <math>\varepsilon^{-1}_{ij,0}</math> is the dielectric tensor when there is no outer electric field, and <math>r_{ijk}</math> is the EO coefficient

According to the conservation of energy density <math>w=\frac{1}{2}\sum_{i,j}D_i\epsilon^{-1}_{ij}D_j</math> and electric displacement <math>D_i</math> with and without the medium, we can normalise the conversation relation as:

<math>
1=\sum_{i,j}\frac{u_iu_j}{n_{ij}^2}=u_iu_j\varepsilon^{-1}_{ij}\approx \sum_{i,j}u_iu_j(\varepsilon^{-1}_{ij,0}+\sum_k r_{ijk}E_{k,\text{THz}})
</math>

where <math>u_i=\frac{D_i}{\sqrt{2w}}</math> represents the direction of the outer optical field. The above quadric equation can be definitely simplified in the form of <math>\sum_i \frac{u_i\prime^2}{n_i^2}=1</math> with a transformation <math>u_i\prime=\sum_{j}R_ij u_j</math> via diagonalzing the tensor <math>\varepsilon^{-1}_{ij}</math>. Then, we can simply gives the Jones matrix to describe the change of the polarisation of the incident laser. The Jones matrix can be:
<math>\mathbf{J} = \begin{bmatrix}
e^{ikn_1 d} & 0 \\
0 & e^{ikn_2d} \\
\end{bmatrix}</math>
, where we assume that the incident direction is in the z-direction (index 3).

===Thermopile Detector===
To detect the generated THz waves, we employ a thermopile detector, which offers a cost-effective and sensitive solution for broadband THz detection. A thermopile is a thermal detector that consists of a series of thermocouples connected in series or parallel, producing a voltage in response to temperature differences across its junctions.

When the THz radiation is absorbed by the blackened surface of the thermopile, it causes a temperature rise, generating a measurable voltage signal due to the Seebeck effect. Unlike semiconductor photodetectors, thermopiles do not rely on photon energy exceeding the material bandgap. This makes them especially suitable for detecting low-energy THz photons (∼0.003 eV), which are otherwise difficult to detect using conventional photodiodes.

In our setup, the thermopile detector is positioned to capture the emitted THz beam after passing through the optical setup. Its broadband response allows us to record the relative intensity of the THz signal under different generation or modulation conditions. Though it does not provide time-resolved measurements, its simplicity and compatibility with room-temperature operation make it a practical tool for evaluating THz pulse generation in a laboratory environment.

==Measurements and Setups==

=== EO Sampling with ZnTe===

We use Zinc Telluride (ZnTe) for the EO Sampling. As the theory mentioned above, we shall first give out the electro-optical tensor <math>r_{ijk}</math>. As reference says that the tensor can be simplified to be <math>r_{ijk}=r_{231}|\epsilon_{ijk}|</math>, where <math>\epsilon_{ijk}</math> is the Levi-Civita symbol. In most references, <math>r_{231}</math> is simplified to be <math>r_{41}</math>.

Meanwhile, though ZnTe is isotropic, the measurement using its (110) plane. Thus, we shall first rotate the frame to fit the incident plane. Now we assume that the basis is <math>\mathbf{e}_1=\mathbf{e}_x=\frac{1}{\sqrt{2}}(-1,1,0)</math>, <math>\mathbf{e}_2=\mathbf{e}_y=(0,0,1)</math> and <math>\mathbf{e}_3=\mathbf{e}_z=\frac{1}{\sqrt{2}}(1,1,0)</math>.

Now, we assume that the incident Terahertz wave has a polarisation in (110) plane with an angle <math>\alpha</math> between <math>\mathbf{e}_x</math>. The Terahertz wave vector should be <math>\mathbf{E}_\text{THz}=E_\text{THz}\left(-\frac{\cos\alpha}{\sqrt{2}}, \frac{\cos\alpha}{\sqrt{2}},\sin\alpha\right)</math>. Hence, we can write out the refractive index as:

<math>
\mathbf{\varepsilon}^{-1}=\begin{bmatrix}
\frac{1}{n_0^2} & r_{41} E_\text{THz}\sin\alpha & r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} E_\text{THz}\sin\alpha & \frac{1}{n_0^2} & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha \\
r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & -r_{41} \frac{E_\text{THz}}{\sqrt{2}}\cos\alpha & \frac{1}{n_0^2}\\
\end{bmatrix}
</math>

which gives Eigenvalues:

<math>
n_1^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2^{-2}=\frac{1}{n_0^2}-\frac{r_{41}E_\text{THz}}{2}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3^{-2}=\frac{1}{n_0^2}+r_{41}E_\text{THz}\sin\alpha
</math>

and the eigenvectors:

<math>
u_1\prime\propto\left(-1, 1,b_1(\alpha)\right)
</math>

<math>
u_2\prime\propto\left(1, -1,b_2(\alpha)\right)
</math>

<math>
u_3=\frac{1}{\sqrt{2}}(-1,-1,0)
</math>

where <math>b_{1,2}(\alpha)=\frac{\sqrt{7\cos^2\alpha+1}\pm3\sin\alpha}{\cos^2\alpha+1\pm\sin\alpha\sqrt{7\cos^2\alpha+1}}\cos\alpha</math>. Notice that <math>r_{41}\approx4pm/V</math>. Hence, when <math>E_{THz}<1 MV/cm</math>, we have <math>r_{41}E_\text{THz}\leq4\times10^{-4}\ll1</math>. The refractive indices are approximately:

<math>
n_1=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha+\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_2=n_0-\frac{n_0^3r_{41}E_\text{THz}}{4}(\sin\alpha-\sqrt{1+3\cos^2\alpha})
</math>

<math>
n_3=n_0+\frac{n_0^3r_{41}E_\text{THz}\sin\alpha}{2}
</math>

Hence, we find that the birefringent coefficient <math>\Delta n=n_1-n_2=\frac{n_0^3r_{41}E_\text{THz}}{2}(\sqrt{1+3\cos^2\alpha})\leq n_0^3r_{41}E_\text{THz}</math> has a maximum at <math>\alpha=0</math>. Here, the eigenvectors, i.e., the principle axis of ZnTe, are <math>
u_1\prime=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y, u_2\prime=\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\mathbf{e}_x+\frac{1}{\sqrt{2}}\mathbf{e}_y</math>. It shows that the angle between the new axis and the old ones is strictly 45 degree. Therefore, when the normally incident beam has a polarisation along <math>\mathbf{e}_x</math>, it would becoming elliptically polarised. Specifically, in the experiment, the linearly polarised beam would pass through a Quarter Wave Plate (QWP), then be analysed by Wollaston prism and finally collected by balance detector after passing ZnTe. The balance detector here actually consists of 2 detectors same in type.

We can thereby show the optical field after the Wollaston prism as:

<math>
\left(E_1\prime, E_2\prime\right)^T=R\mathbf{Q}\mathbf{J}R(E_\text{optical},0)^T
</math>

where the QWP matrix is
<math>
\mathbf{Q}=\begin{bmatrix}
1&0\\
0&i\\
\end{bmatrix}
</math> and the rotation matrix representing a rotation of 45 deg is
<math>
\mathbf{R}=\begin{bmatrix}
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\end{bmatrix}
</math>. Hence, the formula can be simplified as:

<math>
\left(E_1\prime, E_2\prime\right)^T=\frac{e^{ik\frac{n_1+n_2}{2}d}E_\text{optical}}{2}\left(ie^{ki\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d},-ie^{ik\frac{n_1-n_2}{2}d}+e^{-ik\frac{n_1-n_2}{2}d}\right)^T
</math>

In the experiment, we extract the difference between the signals from the 2 detectors via lock-in amplifier. As the intensity <math>I_0\propto E_\text{optical}^2</math>, the final measurement would gives:

<math>
I_\text{signal}=I_0\sin\left(k(n_1-n_2)d\right)=I_0\sin\left(kr_{41} E_\text{THz}n_0^3d\right)\approx I_0kr_{41} E_\text{THz}n_0^3d
</math>

Therefore, we can inversely give the electric field of Terahertz wave as:
<math>
E_\text{THz}=\frac{I_\text{signal}}{I_0kr_{41}E_\text{THz}n_0^3d}
</math>

===Modification given finite delay in the circuit ===

In the measurement, we use a source of 2 kHz laser and the frequency of chopper is set to be 1 kHz. However, due to the delay in the circuit, we saw periodic exponential-like signal other than Dirac delta signal as expected. Here, we analyse how the exponential data would affect the result. Assume that the detector is not saturated. Hence, in 1 period, the signal would be the sum of all past exponentially-decaying signals:

<math>
s_p(t)=\sum_{n=0}^{+\infty}s_0e^{-\frac{nT+t}{\tau}}=\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}
</math>

where <math>\tau</math> is the time constant of the detector and <math>T</math> is the period of the signal. Given a periodic continuation of the signal, we can give the Fourier components by:

<math>
\tilde{s}(f)=\frac{1}{T}\int_{0}^T\frac{s_0e^{-\frac{t}{\tau}}}{1-e^{T/\tau}}e^{i2\pi ft}dt=\frac{\tau}{T}\frac{1}{1+i\frac{2\pi}{f\tau}}s_0
</math>

where frequency <math>f=\frac{n}{T}</math> is a discrete variable. In the experiment, we measure the Terahertz wave in frequency <math>f_0</math> and the optical intensity in frequency <math>f_1=2f_0</math>. Therefore, the lock-in outputs <math>U_{1,2}</math> and the real values <math>s_{1,2}</math> should have a relation as:

<math>
\frac{s_1}{s_2}=\sqrt{\frac{1+\left(\frac{2\pi}{f_0\tau}\right)^2}{1+\left(\frac{2\pi}{2f_0\tau}\right)^2}}\frac{U_1}{U_2}
</math>

This relation shows that when <math>\tau\rightarrow0</math>, the real value might be 2 times of the outputs, verified by our program. Here, we measured the

===EO Sampling Setup===
[[file:schem_eosampling.png|600px|thumb|right|Schematics of Setup with Electro-Optical Sampling]]

Firstly, we try to detect the Terahertz wave with the most traditional method, electro-optical sampling, to demonstrate that we are truely measuring the terahertz wave. This is because the the EO sampling actually measures the wave shape of the Terahertz pulse with a resolution of 0.07 ps which is much smaller than the period of the terahertz wave (2 ps). Hence, we expect to see a oscillation signal representing the electric field of the terahertz wave. Then, we can compare the wave to others result, thereby demonstrating that this pulse is terahertz wave.

The setup (figure in the right) mainly consist of 3 parts, pump path for generating terahertz wave (red one), terahertz wave path (blue one) and the probe path for EO sampling (green one). All pump and probe beam are 2 kHz, 1030 nm Yb:YAG laser with a duration of about 200 fs.

'''Pump Path'''

The delay line is used to control the time delay between the THz pulse and the probe pulse. Then, a telescope with a cylindrical convex lens (f=100 mm) and a conjugate concave lens (f=-50 mm) compresses the beam in the direction perpendicular to the table. The grating then introduces a tilt in the wavefront of the pulse. Subsequently, the pulse will be imaged into the Lithium Niobate (LN) crystal with another telescope. The second one consist of cylindrical convex lens with focal lengths of 50 mm and 100 mm. The Half-Wave Plate (HWP) before the grating is used to optimise the efficiency of the grating, while the one before LN crystal ensures that the polarisation of the beam aligns with the d33 axis to give a maximum THz generation efficiency.

'''Probe Path'''

The probe beam has an energy that is much smaller than the pump path to protect the EO crystal (Zinc Telluride, ZnTe) and the photodiode detectors. When attenuated by the HWP and Polarisation Beam Splitter (PBS), the probe beam would firstly go into a delay path to match the time delay in the pump path. Then, a convex lens with a focal length of 150 mm focuses the beam on the ZnTe to detect the change of birefringence due to Terahertz pulse. Then, the beam is collimated by a conjugated convex lens with a focal length of 50 mm and analysed by the Wollaston prism. The splitted beams would go into 2 same detectors. Experimentally, the linear range of the photodiode on the lock-in amplifier is about from 0~9 mV in our setup.

'''Terahertz Path'''

The Terahertz path is collected by the first Off-Axis Parabolic mirror (OAP) with a focal length of 76.2 mm. Then, it's collimated by conjugated OPA with focal lengths of 76.2 mm and 101.6 mm. In the end, it would be focused by the final OPA on the ZnTe. The OPA here has a tiny tunnel to allow the 1030 nm probe beam to pass.

===Thermopile Setup===

[[file:schem_thermopile.png|600px|thumb|right|Schematics of setup with thermopile sensor]]

In our second attempt, we try to detect the Terahertz pulse directly via a thermopile sensor.

==Source for Codes==
===Test Program for Signal Distortion===

===Source for Auto-measurement Program===

==Reference==

File:Differential Temperature Thermopile.png

2025-04-24T03:22:26Z

Bohan:

Terahertz Electromagnetic Wave Detection

2025-03-04T02:29:38Z

Bohan: /* Members */

Main Page

2025-03-04T02:28:59Z

Bohan: /* Terahertz Electromagnetic Wave Detection */

'''Welcome to the wiki page for the course PC5271: Physics of Sensors!'''

This is the repository where projects are documented. Creation of new accounts have now been blocked,and editing/creating pages is enabled. If you need an account, please contact Christian.

==Projects==
===[[Project 1 (Example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join. Once the project has stabilized, keep stuff in the project page linked by the headline.

===[[Laser Gyroscope]]===
Team members: Darren Koh, Chiew Wen Xin

Build a laser interferometer to detect rotation.

===[[Laser Distance Measurer]]===
Team members: Arya Chowdhury, Liu Sijin, Jonathan Wong

Description: To build a device that uses lasers to measure distances.

(CK: We should have fast laser diodes and fast photodiodes, mounted in optics bench kits)

===[[Alcohol Concentration Measurement]]===

Team members: Lim Gin Joe,Sun Weijia, Yan Chengrui, Zhu Junyi

This project aims to build a sensor to measure the concentration of alcohol by optical method

(CK: you can check Optics Letters <b>47</b>, 5076-5079 (2022) https://doi.org/10.1364/OL.472890 for some info)

===[[Ultrasonic Acoustic Remote Sensing]]===
Team member(s): Chua Rui Ming

How well can we use sound waves to survey the environment?

(CK: we have some ultrasonic transducers around 40kHz, see datasheets below)

===[[Blood Oxygen Sensor]]===
Team members: He Lingzi, Zhao Lubo, Zhang Ruoxi, Xu Yintong

This project aims to build a sensor to detect the oxygen concentration in the blood.

(CK: We have LEDs at 940nm and 660nm peak wavelenth emission, plus some Si photodiodes)

===[[Terahertz Electromagnetic Wave Detection]]===
Team members: Shizhuo Luo, Bohan Zhang

This project aims to detect Terahertz waves, especially terahertz pulses (This is because they are intense and controllable). We may try different ways like electro-optical sampling and VO2 detectors.

===[[Optical measurement of atmospheric carbon dioxide]]===
Team member(s): Ta Na, Cao Yuan, Qi Kaiyi, Gao Yihan, Chen Yiming

This project aims to make use of the optical properties of carbon dioxide gas to create a portable and accurate measurement device of carbon dioxide.

===[[Photodetector with wavelength @ 780nm and 1560nm]]===
Team members: Sunke Lan

To design photodetector as power monitor with power within 10mW.

(CK: Standard problem, we have already the respective photodiodes)

===[[Single photon double-slit interference]]===

Team members: Cai Shijie, Nie Huanxin, Yang Runzhi

1.Build a single photon detector using LED. The possible LED is gallium compounds based, emitting wavelength around 800nm(red light).

2. Other possible detector: photomultiplier or avalanche photon detector(do we have that?).

3.Do single double-slit interference experiment.

Other devices needed: use LED as single photon source (wavelength shorter than the emitting wavelength 800nm)

prove the detection is single photon: need optical fibre, counting module

===[[STM32-Based IMU Attitude Estimation]]===

Team members: Li Ding, Fan Xuting

This project utilizes an STM32 microcontroller and an MPU6050 IMU sensor to measure angular velocity and acceleration, enabling real-time attitude angle computation for motion tracking.

===Hall effect magnetometer===
Team members: Ni Xueqi

This project uses the Hall effect to quantify the magnitude of an external magnetic field.

===[[Light Sensing System Based on the Photoelectric Effect]]===
Team members: Xu Ruizhe, Wei Heyi, Li Zerui, Ma Shunyu

This project utilizes the principle of the photoelectric effect to design a smart light sensing system. The system can detect ambient light intensity and process the data using Arduino or Raspberry Pi. When the light intensity changes beyond a predefined threshold, the system can trigger responses such as lighting up an LED, activating a buzzer, or automatically adjusting curtains.

==Resources==
===Books and links===
* A good textbook on the Physics of Sensors is Jacob Fraden: Handbook of Mondern Sensors, Springer, ISBN 978-3-319-19302-1 or [https://link.springer.com/book/10.1007/978-3-319-19303-8 doi:10.1007/978-3-319-19303-8]. There shoud be an e-book available through the NUS library at https://linc.nus.edu.sg/record=b3554643

===Software===
* Various Python extensions. [https://www.python.org Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* [https://www.gnuplot.info Gnuplot]: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Matlab: Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask if interested.
* Mathematica: More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license. Ask if interested.

===Apps===
Common mobile phones these days are equipped with an amazing toolchest of sensors. There are a few apps that allow you to access them directly, and turn your phone into a powerful sensor. Here some suggestions:

* Physics Toolbox sensor suite on [https://play.google.com/store/apps/details?id=com.chrystianvieyra.physicstoolboxsuite&hl=en_SG Google play store] or [https://apps.apple.com/us/app/physics-toolbox-sensor-suite/id1128914250 Apple App store].

===Data sheets===
A number of components might be useful for several groups. Some common data sheets are here:
* Photodiodes:
** Generic Silicon pin Photodiode type [[Media:Bpw34.pdf|BPW34]]
** Fast photodiodes (Silicon PIN, small area): [[Media:S5971_etc_kpin1025e.pdf|S5971/S5972/S5973]]
* PT 100 Temperature sensors based on platinum wire: [[Media:|PT100_TABLA_R_T.pdfCalibration table]]
* Thermistor type [[Media:Thermistor B57861S.pdf|B57861S]] (R0=10kΩ, B=3988Kelvin). Search for [https://en.wikipedia.org/wiki/Steinhart-Hart_equation Steinhart-Hart equation]. See [[Thermistor]] page here as well.
* Thermopile detectors:
** [[Media:Thermopile_G-TPCO-035 TS418-1N426.pdf|G-TPCO-035 / TS418-1N426]]: Thermopile detector with a built-in optical bandpass filter for light around 4μm wavelength for CO<sub>2</sub> absorption
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]
* Ultrasonic detectors:
** plastic detctor, 40 kHz, -74dB: [[Media:MCUSD16P40B12RO.pdf|MCUSD16P40B12RO]]
** metal casing/waterproof, 48 kHz, -90dB, [[Media:MCUSD14A48S09RS-30C.pdf|MCUSD14A48S09RS-30C]]
** metal casing, 40 kHz, sensitivity unknown, [[Media:MCUST16A40S12RO.pdf|MCUST16A40S12RO]]
** metal casing/waterproof, 300kHz, may need high voltage: [[Media:MCUSD13A300B09RS.pdf|MCUSD13A300B09RS]]

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

== Old Wiki ==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2]