# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.2 Under Illumination  
The output of many experiments [40], [PERSON], [PERSON], et al. [39] shows that the tunneling diodes behave differently under illumination as compared to their behavior without illumination. This difference in behavior can be expressed in quantum terms such that illumination can be considered as an oscillating potential or perturbation in the Hamiltonian function. Many approaches to solve this complicated Schrodinger problem have been proposed where each of them tries to approach the problem in a different manner, and the most frequent ones may be briefly reviewed here. Tien-Gordon:  
This effect was first introduced in [65] and is also referred to as photon-assisted tunneling (PAT) quite frequently in the literature [39; 66]. In a similar style to [54], it assumes that the time-dependent wave function can be described as shown in (13). \[\Psi(\mathbf{r},t)=f(\mathbf{r}).e^{\frac{-iEt}{\hbar}}, \tag{13}\]  
where \(\Psi(\mathbf{r},t)\) is the modeled wave function, \(f(\mathbf{r})\) is a spatial function, \(E\) is the quasi-particle energy, \(i\) is the imaginary number and \(\hbar\) is the reduced Planck's constant. The introduction of an oscillating perturbation into the Hamiltonian results in the following Equation (14). \[\hat{H}=\hat{H}_{0}+eVcos\omega t, \tag{14}\]  
[PERSON] and [PERSON] [65] support their space-independent perturbation approach by the assumption that the wave function drops very sharply at both ends of the tunnel, so the interaction with the electric field is quite small. Based on that, they put the following form for the wave equation as shown in (15). \[\Psi(\mathbf{r},t)=f(\mathbf{r}).e^{\frac{-iEt}{\hbar}}\left(\sum_{n=-\infty} ^{n=+\infty}B_{n}e^{in\omega t}\right), \tag{15}\]  
Working out some math, they find the coefficients \(B_{n}\) to be \(B_{n}=J_{n}\left(\frac{eV}{\hbar\omega}\right)\) where \(J_{n}\) represents the \(n\)-th order Bessel function of the first kind. Without delving too much into this math, the current value for a tunneling device under illumination is found to be dependent on both the photon energy represented by its frequency \(\omega\) and the oscillation amplitude \(V\). The following current equation is found from the [PERSON] and Gordon [65] model and is frequently used to explain the experiments' outputs up to a few THz [39; 66]:  
\[I_{illum}=\sum_{n=-\infty}^{n=+\infty}J_{n}^{2}\left(\frac{qV_{w}}{\hbar\omega} \right)\times I_{dark}\left(V_{D}+n\frac{\hbar\omega}{q}\right), \tag{16}\]  
where \(I_{illum}\) is the resultant illumination current, \(q\) is the particle charge, the electron charge in this case, and \(V_{D}\) is the DC voltage across the tunnel. The terms are summed from \(-\infty\) to \(+\infty\) to scan all the possible wave function modes. In physical terms, the illumination current is the composite result of emitting and/or absorbing photons with frequency \(\omega\) such that the \(n<0\) terms represent the absorption of photons and \(n>0\) represents their emission. The amplitude of the Bessel function for each term represents the likelihood of the corresponding occurrence (i.e., a photon gets absorbed/emitted). Another issue is how many terms shall be included in a calculation? This by far depends on the value of the \(n\)-th Bessel function with respect to the fundamental one where the voltage amplitude has a major impact. Empirically, [PERSON] and [PERSON] [66] use values \(n=-1,0,1\) in their model. Therefore, it makes sense to deploy more Bessel terms for higher oscillation amplitudes. Contrary to the easier approaches of tunneling under no illumination, this concept results in a nonlocal functionality of current to voltage where the notation \(I(V)\) becomes no longer accurate because it represents an instantaneous dependency; whereas the current \(I\) depends on the current and previous values of voltages in this model.