# THz Rectennas and Their Design Rules
## 4 Theory of Operation
### Tunneling
#### 4.1.1 Without Illumination
Another drawback that this model suffers from is that the model is primarily a single insulator layer model as opposed to multi-insulator structures where more than one insulating layer is laminated in the diode structure. It requires a complicated mathematical work to account for multiple insulators with little output expected from this model. Although the model has many drawbacks and is kind of dated, it is still quite frequently cited in the literature. It is usually used as a reference model to compare to other more complicated models. [PERSON] model was proposed in the late 1970s [45] with a more systematic approach to rectennas. In this model, [PERSON] et al. focus on the best practice to optimize the power transfer between the antenna and the diode. It has been presented for infrared detection, as noted in their abstract, such that the diode is assumed nonlinear, the current of which has a third order polynomial functionality with respect to the voltage drop across the diode. Calculations are then made to optimize the power transfer between the antenna and the diode. The aim of the math is to find the relationship between incident power level and the physical and geometric characteristics to achieve maximum power transfer; again, using the same WKB model shown in (4) and used by [PERSON]. As the model is presented for detection, special attention is given to its noise performance with the introduction of the noise equivalent power (NEP) figure of merit. To compare between their model expectations and the results of an experiment they performed using a \(\lambda=377\,\upmu\)m laser, they measured the dependence of the antenna efficiency represented by \(\eta_{0}\) on the focused laser beam incidence angle \(\theta_{f}\) and found that \(\eta_{0}=f(\theta_{f}^{2})\), which is consistent with the effective aperture concept of the antenna theory. Lastly, the model is quite frequently quoted in the literature and represents a more realistic approach to a systematic rectenna despite the fact that it is structurally limited to single-layer insulators. Moreover, the model indirectly suggests not to apply it to an insulator thickness more than \(10\,\mathrm{\SIUnitSymbolAngstrom}\) as mentioned in [45] (Section III), and so, it has fewer application fields. Furthermore, the best achievable coupling efficiency was found to be 1% as shown in [45] (Figure 8) using a 377 \(\upmu\)m laser. It is worth noting that a \"weak bolometric response\" was observed. Transfer matrix method:  
This is also referred to in the literature as TMM. Technically, the transfer matrix method has been applied in other fields such as in the optics, acoustics and electromagnetic fields [52]. [PERSON] and [PERSON] [53] applied it to [PERSON]'s equation deploying [PERSON]'s approximation [54]. The concept is to use numerical modeling on a physical problem such that the problem structure is divided into smaller parts within which the wave equation, the potential energy, as well as the particle effective mass are assumed constant. Then, any change in any of these values is accounted for gradually when moving from one cell to another. Figure 7a shows the potential barrier value such that the whole curve is divided into many parts, and each part has a constant potential value throughout the containing cell. To simplify the problem, it is assumed that:  
1. The modeled structure is a one-dimensional structure such that the modeled diode metal insulator interface infinitely extends and that the potential value changes only in one direction normal to the interface. Figure 7: Energy-band diagram of a silicon p-n junction solar cell (Reproduced with permission from [55], IEEE, 2009). The Schrodinger wave equation time is approximated using [PERSON]'s assumption [PERSON] (1968) in which case all the rapid changes of the wave function with respect to time are neglected, and the wave function becomes as shown in (5). This simplification allows one to treat the wave function as a time-independent parameter.