Rectenna and antenna physics

For Project Elara’s power system to be useful, it is not enough to just collect solar energy from space and transmit it by means of a microwave laser (maser); in addition, it must be received on the Earth’s surface by various types of receivers, from small, easily deployable receivers pressed into flexible sheets for powering remote regions, to city-sized power stations in the ocean, to specialized types of receivers for emergency workers and humanitarian aid.

Parabolic antennas

Parabolic antennas are the first method we will use in power receiving. They are useful for high-intensity direct beaming, either in locations with pre-existing telecommunications antennas (e.g. decommissioned TV/satellite radio dishes, which can be modified to become ground-based power receivers), or in specialized locations (e.g. on ships or seaborne platforms).

The mathematical basis for parabolic antennas is the same as that of those used for the parabolic mirrors. As described in Solar mirror engineering, the electric field of the parabolic antenna is given (in time-independent form) by:

$$E(\theta )=\frac{2\lambda }{\pi D}\frac{J_1[(\pi D/\lambda )\sin \theta ]}{\sin \theta }$$

Where $J_1(\theta )$ is a Bessel function of the 1st kind, $D$ is the diameter of the mirror, $E(\theta )=|\mathbf{E}(\theta )|$ is the magnitude of the electric field, and $\lambda$ is the wavelength. Note that the full electric field is simply $E(\theta )$ multiplied by a factor of $e^{i\omega t}=e^{i\frac{2\pi c}{\lambda }t}$ and some polarization vector $\hat{\mathbf{n}}$, that is:

$$\mathbf{E}=E_0\hat{\mathbf{n}}\frac{J_1[(\pi D/\lambda )\sin \theta ]}{\sin \theta }\exp \left(i\frac{2\pi c}{\lambda }t\right),E_0\propto \frac{2\lambda }{\pi D}$$

Note: We assume that the diameter $D$ of the antenna satisfies $D>\lambda$. Otherwise, the parabolic antenna would not be able to efficiently receive microwaves.

Electrical transmission with a parabolic antenna

The more complex challenge for parabolic antennas is that they are not simply reflectors (like the mirrors); rather, they must also convert the received power into useful electricity. The theoretical basis for how this is possible at all comes from Maxwell’s equations: since they treat electricity and magnetism as unified rather than separate phenomena, an electromagnetic wave can produce an electrical current through a conductor, and likewise a conductor with an oscillating current can produce an electromagnetic wave.

The precise mathematical statement of this physical phenomenon follows from Ampere’s law, which says that (in integral form) that the magnetic field $\mathbf{B}$ is related to the (total) current $I$ as follows:

$$\oint \mathbf{B}\cdot d\mathbf{r}={\mu }_{0}I$$

Note that in the formulation of Maxwell’s equations in matter (where we primarily use the $\mathbf{H}$ field rather than the $\mathbf{B}$ field), we may write this more succinctly as:

$$I=\oint \mathbf{H}\cdot d\mathbf{r}$$

Therefore, rearranging, one may obtain the (free) current in a conductor by integrating in a closed path around the conductor (specifically, along the boundary of a cross-section of the conductor). One may also define the impedance $Z$, which is in general given by the formula:

$$Z=\frac{E_j}{H_k}$$

Where $E_j,H_k$ are particular components of the electric and magnetic fields that are transverse (perpendicular) to each other and nonzero. Now, while the magnetic field (more precisely, the $\mathbf{H}$ field) is related to the current, the electric field also has an important effect, since it produces a potential difference (often called an electromotive force or EMF, but we will generally avoid such terminology when possible since it is a confusing name). By the integral form of Faraday’s law, the potential difference, which we denote here by $V$, is the effect of a changing magnetic flux through some surface $\Sigma$:

$$V=-\frac{d{\Phi }_B}{dt}=-\frac{\partial }{\partial t}{\iint }_{\Sigma }\mathbf{B}\cdot d\mathbf{A}$$

However, the changing magnetic flux itself is the result of a changing electric field, due to Ampere’s law, whose differential form is given by:

$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

By combining Ampere’s law and Faraday’s law in integral form, we obtain:

$$\begin{array}{rl}V& =-\frac{\partial }{\partial t}{\iint }_{\Sigma }\mathbf{B}\cdot d\mathbf{A}\\ & =-{\iint }_{\Sigma }\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{A}\\ & =-{\iint }_{\Sigma }(\nabla \times \mathbf{E})\cdot d\mathbf{A}\end{array}$$

Where we can use Stoke’s theorem to rewrite the above as:

$$V=-{\iint }_{\Sigma }(\nabla \times \mathbf{E})\cdot d\mathbf{A}=-{\oint }_{\partial \Sigma }\mathbf{E}\cdot d\mathbf{r}$$

Where the line integral, as previously, is over the boundary of a cross-section of the conductor. In general, the current and potential difference are time-dependent due to the oscillating electric and magnetic fields, so it is implied that $I=I(t)$ and $V=V(t)$, where by Ohm’s law we have:

$$V(t)=I(t)Z$$

We will now apply, on a basic level, a mathematical analysis to a basic parabolic receiver antenna, and then discuss the modifications necessary to ensure such an antenna can handle kW-level power beams from space.

Antenna feed

For a parabolic antenna to function, several essential components must be present:

1. The parabolic dish itself, to capture and focus the incident microwaves, along with any secondary dish (if present)
2. A feed antenna that receives microwaves and guides them into to a transmission line
3. A transmission line that delivers AC current from the beam

Construction and repurposing of parabolic antennas

The construction of a large parabolic antenna to collect microwave beams with wide cross-sections, as is essential for space-based transmission, comes with substantial difficulties. The first difficulty is due to the fact that the sizes of these parabolic antennas make them very labor-intensive and material-intensive to build, as well as a technical challenge to ensure structural stability. The second difficulty is due to the fact that the complex and sensitive electronics of the feed and transmission line(s) are mounted high up on the antenna

Note that in general, light of a wavelength $\lambda$ can be reflected by a reflector that is smooth up to $\lambda /10$. Which means that 1cm microwaves can be reflected by a 0.1cm (i.e. 1mm) mesh rather than needing one solid parabolic reflector, saving material, cost, and weight. This is very relevant for terrestrial use.

It is also possible to repurpose existing telecommunications antennas as power receivers.

Phased arrays

On the largest scales, phased arrays the size of skyscrapers can yield the best possible transmission efficiencies.

Rectennas