# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.2 Under Illumination
The functionality represented in (16) is in the frequency domain such that the current harmonics are found instead of its time-dependent value, which is attempted by [PERSON] and [PERSON] [39] and takes a more complicated form. Another notable issue in this model is that a single frequency oscillation generates multi-mode wave functions. This is connected to the findings of other more sophisticated models of the wave equation. It should be noted, however, that [PERSON] and [PERSON] [39] state that more than single tone currents can be found out from this model, i.e., current components for \(\omega,2\omega,3\omega,...\). However, the only distinguished current component is the fundamental with frequency \(\omega\) for practical reasons, as they explain. Equally important, the most prominent reason is that the cut off frequency is much lower than the second harmonic such that the diode capacitance appears as a short circuit to these higher-order current components. In other words, although the diode is considered a nonlinear device, only the first harmonic should be accounted for in any nonlinear simulation as the higher order harmonics have more complicated notation and much less impact. In the end, this model has been in use and quoted many times due to its simplicity and the good correspondence it gives with experiments up to a few THz frequency. Having said that, the assumption of (13) has been disputed especially for higher frequencies and more complicated structures. Other models:  
To reach a more precise model of the time-dependent wave equation, there have been many introduced models in the literature. The following just reviews the most frequently-cited ones. For example, the Floquet model [67] exploits the case when the oscillation is a single frequency such that the Hamiltonian is periodically perturbed \(\hat{H}(t)=\hat{H}(t+T)\) where \(T\) is the perturbation period, then it can be proven that the wave equation solution of the Schrodinger equation is as follows (17). \[\Psi(\mathbf{r},t)=e^{-i\varepsilon_{r}t},\Phi_{\alpha}(\mathbf{r},t), \tag{17}\]  
Whereas the \(\varepsilon_{r}\) is the Floquet characteristic exponent or quasi-energy [68], \(\Phi_{\alpha}(\mathbf{r},t)\) are the Floquet mode obeying functions such that \(\Phi_{\alpha}(\mathbf{r},t)=\Phi_{\alpha}(\mathbf{r},t+T)\). Another noticeable attempt is that of [PERSON] [69] who introduces a space-time coordinate transformation to solve for the wave equation and decompose its terms as shown in [69] (2). This transformation is claimed to improve accuracy over the models that adopt a spatially-uniform time-dependent field approach. Similar to [PERSON] and [PERSON] [63], the algorithm can account for phase-change oscillating barriers with linear and parabolic relationships. The interesting output of this model is that \"the onset of frequency dependence is not a general property of the barrier\" as opposed to the Buttiker-Landauer [70] approach, which states that the maximum tunneling frequency \(\omega\) is limited by the tunneling time \(\tau\) through the relation \(\omega.\tau=1\). This means that tunneling at higher frequencies is still expected to happen with more efficiency than expected by the Buttiker-Landauer approach. Indeed, [PERSON]'s [71] experiment supports this finding where he states that \"the tunneling time consideration is only an estimate.\"  
In the end, it is worth noting that the recent models are less frequently used in the engineering field, and most of them are presented by physics research groups.