# QPSK MMW Wireless Communication System Based On p-i-n InGaAs Photomixer
## 2 CW THz Wave Generation by Photomixing: General Approach  
The generation of CW THz radiation by photomixing is based on the THz-periodic generation of electrons and holes in semiconductors by absorption of two interfering laser beams of frequencies \(\
u_{1}\) and \(\
u_{2}\). The interference results in the intensity modulation of the laser beams with the difference carrier frequency of \(\
u_{THz}\). The two laser beams with electric field amplitudes of \(E_{1}\) and \(E_{2}\) focused onto an area of the semiconductor with dimensions smaller than the THz wavelength. The total electric field \(E(t)\) yields  
\[\mathrm{E}(t)=\mathrm{E}_{1}\cdot\exp(\mathrm{i}2\
u\
u_{1}t)+\mathrm{E}_{2} \cdot\exp(\mathrm{i}2\
u\
u_{2}t) \tag{1}\]  
Assuming the same polarization and amplitude of electrical field, \(E_{1}=E_{2}=E_{0}\), the electrical field superposition on the photomixer can be rewritten as [7]  
\[\mathrm{E}(t)=\mathrm{E}_{0}[\exp(\mathrm{i}\pi(\
u_{1}+\
u_{2})t)\cdot\cos( \pi(\
u_{2}-\
u_{1})t)] \tag{2}\]  
where \(\
u_{2}-\
u_{1}\) is the difference THz frequency, \(\
u_{THz}\). The frequency component of \(\
u_{1}+\
u_{2}\) cannot pass the photomixer low-pass filter (LPF). The total laser power, \(P_{L}\), where \(P_{L}=P_{1}+P_{2}\), becomes  
\[\begin{split} P_{L}(t)&=E^{2}(t)=2E_{0}{}^{2}[cos(2 \pi\
u_{THz}t)+1]\\ &=P_{0}[1+cos(2\pi\
u_{THz}t)]\end{split} \tag{3}\]  
where \(P_{1}\) and \(P_{2}\) are the laser sources output power. At photon energies above the band gap energy, the optical absorption coefficient of semiconductors is very large. As a result, the electron-hole pair generation takes place within a very thin layer compared to the THz wavelength. Electrons and holes are separated by a built-in field or an external field due to applied DC bias voltage. The resulting current contains a THz component due to the periodic modulation of the carrier generation rate.  
The AC current is converted into THz radiation due to the photoconductive antenna, in which the device is connected. The ideal THz output power of antenna-based photomixer is given by [6; 7]  
\[P_{THz}=\frac{1}{2}R_{A}{I_{THz}}^{2}=\frac{1}{2}R_{A}\left[\frac{eP_{L}}{hv} \right]^{2} \tag{4}\]  
where \(I_{THz}\) is the photogenerated current, \(R_{A}\) is the radiation resistance of the antenna, \(hv\) is the photon energy, and \(e\) is electron charge. The radiation resistance is proportional to the wave impedance, \(Z=\sqrt{\mu_{0}\mu_{eff}/\epsilon_{0}\epsilon_{eff}}\), where \(Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}=377\)\(\Omega\) is the vacuum impedance and \(\mu_{eff}\) & \(\epsilon_{eff}\) are magnetic and dielectric constants, respectively. For non-dielectric materials, \(\mu_{eff}=1\), the radiation resistance is calculated as  
\[R_{A}=\frac{60\pi}{\sqrt{\epsilon_{eff}}}=\frac{60\pi}{\sqrt{\frac{1+\epsilon _{A}}{2}}} \tag{5}\]  
here, \(\epsilon_{s}\) is the dielectric constant. As shown in Equation (4), the quadratic relation between laser power and THz power is \(P_{THz}=\gamma{P_{L}}^{2}\), where \(\gamma\) is the scaling factor. The expression given in Equation (4) is a general form of the THz power emitted by an antenna with radiation resistance \(R_{A}\) without including the limitations of the Resistor-Capacitor (RC) time constant and transport time of carriers. The THz power including the 3 dB transit time roll-off frequency, \(v_{tr}\) and RC 3 dB roll-off frequency, \(v_{RC}\) is given in Equation (6) [6]:  
\[P_{THz}=\frac{1}{2}R_{A}I_{ph}\left[\frac{1}{1+\left(\frac{v_{THz}}{v_{tr}} \right)^{2}}\right]\left[\frac{1}{1+\left(\frac{v_{THz}}{v_{RC}}\right)^{2}}\right] \tag{6}\]  
The two frequencies \(v_{tr}\) and \(v_{RC}\) are calculated from \(\frac{1}{2\tau_{tr}}\) and \(\frac{1}{2\pi\tau_{RC}}\), respectively, where \(\tau_{tr}\) is the transit time and \(\tau_{RC}\) is the RC time constant of the device [6].