We will have to accept that our ground-based receiver stations will not be able to capture most of the energy that comes from orbit, at least not initially. This arises from two fundamental limitations of our system: - From theory, the diffraction limit means that the minimum spot size from geostationary orbit is $w_\text{earth} = \sqrt{2D\lambda/\pi}$, which we derived in [[Ideal laser beam divergence]]. We also showed in [[Computing the theoretical minimum divergence beam]] that this is a fundamental limit that cannot be overcome, and even in this theoretically-ideal circumstance, a maximum of only $1-1/e^2 \approx 86\%$ of the beam's intensity can be concentrated in this region (though increasing the ground receiver size to 1.52x the beam width can increase this figure to 99%, see [Wikipedia](https://en.wikipedia.org/wiki/Gaussian_beam#Power_through_an_aperture)). - From practical considerations, $w_\text{earth} \approx \pu{2.6 km}$ for 1 GHz waves and $w_\text{earth} \approx \pu{755m}$ for 12 GHz waves in the _best_ case scenario (see [[Ideal laser beam divergence]]). This is notwithstanding the fact that any frequencies gt;\pu{2GHz}$ will likely get attenuated greatly in adverse weather conditions and incur huge power losses, so our best-case spot size (beam radius) for the beam at Earth's surface is likely to be over a kilometer. The spot size (beam diameter) of a 2 GHz beam upon reaching the Earth would be around 3.7km wide, which is around the size of a big solar thermal power station's land area[^4]. This is why we want our stations to built off the coast, where the space-to-Earth beam would be located away from population centers for maximum safety. However, assuming we can cover the entire area of the beam, this amounts to a massive amount of power (though the average power density will be very low and should be as safe as a microwave). For a solar power satellite of our design with 10km radius mirrors, not accounting for inefficiencies, we can generate 427 GW of power, enough to satisfy the power demands of a country. In reality, however, losses would probably mean an efficiency of only around 10-25%; however, this is still quite substantial, and a constellation of power satellites would have an impressive power generation capability. Larger mirrors (up to 30 km) could provide up to 3.8 TW of power each (in ideal circumstances), making these satellites the most powerful energy generators in the world, and in the vacuum and freefall environment of space, we can go nearly as big as we want. It is certainly not unfeasible, from a purely theoretical standpoint, to achieve something extraordinary: for these solar satellites to power the entire world, and launch us to other planets and even the stars. ## Concept design: sparse-array beam-combined system The very first concept design (see the archived sections at the bottom of the page) benefits from using a variety of relatively proven technologies, such as parabolic antennas, waveguides and beam combining. However, it has the disadvantage of being extremely large (power satellites kilometers across) and extremely impractical to build. It is also somewhat wasteful, due to the fact that it requires hundreds of parabolic dishes to be steered for the beam-combining to work. For these reasons, we also propose another design that shares some commonalities but is overall very different. This newer design relies on an idea very similar to conventional beam combining, but uses an array of spatially-separated power satellites with individual masers rather than parabolic antennas. By physically separating the power satellites in space, their individual maser beams can together create a **synthesized beam** with a much larger _effective aperture size_ than their individual apertures. The divergence of a synthesized beam made by coherently-combining beams separated by a distance $D_b$ (called the _baseline distance_) is given by[^5]: $ \theta = \frac{1}{z} \sqrt{ \frac{\lambda}{2} z } $ Where $z$ is the distance from the source (in our case, we have $z = D = \text{35,786 km}$, the distance from Earth to geostationary orbit). Note how this expression is _completely independent_ of the size of the individual masers, meaning that we can have a very large synthesized beam even with (relatively) small masers! Meanwhile, the effective spot size $D_b$ of the beam on Earth (i.e. its width by the time it reaches the Earth's surface) is given by[^6]: $ D_{b} = \sqrt{ 2 \lambda z } \approx \begin{cases} \pu{4.6 km} & (\text{1 GHz}) \\ \pu{3.3 km} & (\text{2 GHz}) \\ \pu{2.9 km} & (\text{2.5 GHz}) \\ \pu{2.7 km} & (\text{3 GHz}) \\ \end{cases} $ The theoretical basis behind synthesized beams comes from the fact that all electromagnetic waves exhibit **interference**. Thus, by combining several light (or microwave) sources that are in-phase, constructive interference creates a powerful "peak" of power density in the center of the combined beam. This means that while the system is still bound by the diffraction limit, it can concentrate power more effectively within a smaller radius, which has the same effect as having a larger single aperture. This is why, for instance, existing astronomical radio telescopes such as the [Very Large Telescope](https://en.wikipedia.org/wiki/Very_Large_Array) use a very similar technique by spacing out individual radio antennas over a span of over 20 kilometers to capture as much light (technically, radio waves) as possible. > **Note:** More detail on the theory of synthesized beams can be found in [[Principles of coherent beam synthesis]]. Thus, in this design, we use separate power satellites, each with their own solar mirror and maser, which orbit together at geostationary orbit at a fixed separation of $D_b$ with respect to each other. Each maser is based off the same basic design as our passively-driven free-electron maser design shown below: ![[prototype-free-electron-maser.excalidraw.svg]] The main difference, however, is that their output coupler is a simple microwave horn, which helps keep the maser beams as close as possible to the ideal Gaussian beam (conventional waveguides, by contrast, can introduce distortions[^7]). The maser beams are then coherently combined to be able to form a large synthesized beam multiple kilometers across, thus achieving the ideal $w_0$ for minimizing the divergence of the beam. Since the power satellites are spatially-separated and there is nothing physically holding them together, we call this arrangement a **sparse array**. We show the design in the diagram below: ![[sparse-array-coherent-beam-combining.excalidraw.svg]] This system relies on precision masers that are all exactly **in-phase** and aimed at the same direction so that constructive and destructive interference can take place. For the masers to be in phase, they must be **monochromatic** (producing microwaves at exactly **one** frequency) and perfectly-synchronized (so there are no phase shifts). This means that the distances between the power satellites must be kept constant to incredible precision throughout each orbit, in a similar fashion as space-based interferometry satellites like the [LISA space observatory](https://lisa.nasa.gov/). For this reason, the power satellites will likely need to adopt a [zero-drag satellite](https://en.wikipedia.org/wiki/Zero-drag_satellite) design so that the masers can be kept in as controlled an environment as possible. They must do so while keeping their mirrors focused perfectly towards the Sun (as shown in the diagram below), to maintain the maximum solar coverage for power collection. ![[space-satellite-orbital-arrangement.excalidraw.svg|500]] In addition, since the beams are spatially-separated by kilometers and thus arrive at an angle to the ground, it is also necessary to slightly correct for the phase-shift caused by this discrepancy in path length, even though this discrepancy is _extremely_ small. If we let $D_b$ be the synthesized beam diameter (as with before) and $D = \pu{35,786 km}$ be the distance to geostationary orbit, the path-length difference $\delta s$ caused due to the angled approach of the beam is given by: $ \delta s = \sqrt{D^2 +\left(\dfrac{D_b}{2}\right)^2} - D \approx \pu{4.7 cm} $ While it may seem like this path-length difference is very small, it is still around 40% of a wavelength (assuming $\pu{2.5GHz}$ microwaves), meaning that active phase correction may still be necessary to make sure that the different waves all arrive exactly in-phase with each other. However, using a sparse array power system means that instead of building huge power satellites, we can build much more moderately-sized ones (30-meter mirrors, 1-meter horns, and weighing a few tons) that are well within our current capabilities for satellite construction and can (each) be launched with a single rocket. By simply spacing the individual power satellites far apart from each other, we can create a huge synthesized beam that creates a tightly-focused beam, as opposed to needing ridiculously-sized power satellites armed with a mind-boggling number of antennas. > **Note:** It is possible to connect all the individual masers to a single large mirror and simply position them at the rims of the mirror, rather than spacing them out in an ring formation (and consequently needing to carefully align their orbits). However, this would only be efficient if the mirror itself was several kilometers across, which would require future technologies to make possible. ### Concept 2 prototype design Since Concept 2 requires no large-scale space engineering and only relies on highly-accurate satellite positioning, the prototype design would mostly be limited (on an engineering level) by the size and weight of the maser and altitude control systems (as well as onboard electronics), _not_ the heavy parabolic antennas and support structures of Concept 1. While we do not yet know the specifics for how heavy our (fairly unique) free-electron masers will be, we can make some estimates. As our work is currently on building compact and highly-efficient free-electron masers, assuming that we can create a tabletop-sized free-electron maser, it is possible (although not guaranteed) that the maser mass could be brought down to a hundred kilograms or less. Assuming a 5-meter radius and very thin solar mirror made of a roll-able aluminium-coated plastic, it is likely that the mirror mass can be brought down very low - perhaps just a few kilograms. The microwave horn would also be very light. Thus, the majority of the mass would be the altitude control systems, which must be made as light as reasonable. To ensure a reliable spacecraft, we want to have as few moving parts as possible. The spacecraft would be able to power itself since it acts as an energy converter, and energy recovery systems in the free electron maser can also be used to power the spacecraft. Backup solar panels could provide emergency power in case of solar storms or system failures. For the altitude control system, traditional thrusters and reaction wheels can easily malfunction and have many moving parts. Instead, passive systems can be used, like [electrodynamic tethers](https://en.wikipedia.org/wiki/Electrodynamic_tether), which use a conducting rod with DC current to raise (or lower) a spacecraft relative to Earth's magnetic field, and [magnetorquers](https://en.wikipedia.org/wiki/Electrodynamic_tether) that work using a similar principle, but for changing orientation rather than altitude. We can also use small secondary mirrors that can be twisted, relying on the _radiation pressure gradient_ from sunlight to help rotate the spacecraft, although this would require mechanical components as well. So, for backup, we would still want some reaction wheels, though technologies like magnetic-bearing reaction wheels could eliminate the friction that can cause mechanical wear and tear on traditional bearings. For our electronics, we can use cheap consumer-grade devices, much like NASA's [PhoneSat](https://en.wikipedia.org/wiki/PhoneSat) used commercial phones, and of course we would be using open-source technologies like NASA's [F prime](https://github.com/nasa/fprime) open-source satellite control system and our own Project Elara software and libraries. Communications would likely use smaller monopole RF antennas to avoid conflicting with the microwave frequency range used by the maser. ## Terrestrial receivers Once our power beams arrive on Earth's surface, we still need to collect the power and turn it into useful electricity that we can then distribute. For our terrestrial receiver stations, we plan to convert old semi-submersible oil rigs that can be towed by tugboats, and retrofit them with massive collections of parabolic antennas that collect the microwaves from space and convert it back to electricity. This electricity can then be transmitted back to shore with undersea cables. As a bonus, by using old oil rigs as our base, we can make these receivers semi-mobile and able to be towed around the world to supply electricity everywhere. However, owing to the scale of our collectors at 6km in diameter, we will need to combine a large number of old oil rig platforms and will need powerful tugboats to pull the receivers to sea. Additionally, to shelter our delicate parabolic antennas onboard from corrosion and storms, we will need to surround them with [radomes](https://en.wikipedia.org/wiki/Radome). The engineering challenges will be no doubt massive, but this is a concept that could in theory work. ## Mitigating divergence and secondary arrays The primary power satellites, as described above, would be located in **geostationary orbit**. However, this is a problem, since the power satellites will need very large terrestrial collectors to be able to capture most of the beam as it diverges over the incredibly long distance from geostationary orbit. Even with the beam-combining strategies we outlined above, which does increase the effective aperture and decreases the divergence, the ground-based stations would be still incredibly large. We do partially mitigate this issue by letting our collector stations be offshore and semi-mobile, although this limits access to purely coastal regions. To mitigate this issue, the plan is to implement an extensive microwave power relay system. By mounting microwave antennas on high, prefabricated towers, we can send power wirelessly from our primary receiver stations at sea to inland regions, while not needing (as much) substantial infrastructure as typical electrical power lines. Since microwaves between $\pu{1-2.5 GHz}$ barely attenuate in the atmosphere, it is also more robust and not as easily damaged by weather or needing maintenance. A relay of these power transmission towers can bring power far inland, making it much more widely-available. Furthermore, with economies of scale, we can greatly reduce the cost of these towers, reducing the cost of construction as well. But the problem, of course, is in areas without electrical grids or indeed without much infrastructure at all. While we can subsidize the construction of our power relay system so that these communities do not need to pay the majority of the upfront costs (again, we are not-for-profit after all), building high power transmission towers may take too long, especially when these regions are very difficult for construction crews to reach. These remote and underdeveloped regions need power directly from space. For this, we plan to create a **secondary array** of power satellites, only instead of being built in geostationary orbit, they are placed in low-Earth orbit only $\pu{600-800km}$ from Earth's surface in **Sun-synchronous orbits** that keep them always in full view of the Sun. The much, much shorter transmission distance (compared to geostationary orbit) means that the divergence is much less, so smaller, potentially air-droppable or ship-bound power receivers can be adequate, although the extremely-fast orbital velocities of these satellites in lower orbit relative to the ground means that they need to use sophisticated phased arrays or (as in Concept 2) tightly-synchronized formations to track the ground. In addition, a constellation of satellites would be necessary, as each satellite comes into and out of view of the ground receivers. Needless to say, this is a tremendous technological challenge. This also allows us to cover for our other use case: bringing power to **disaster regions and warzones**. A combination of air/helicopter-dropped and ship-bound power receivers would allow much improved access to power infrastructure, which can power medical equipment and other equipment used by humanitarian workers. These can specifically be used to receive power from the power satellites in low Earth orbit, which have much tighter beams and therefore do not need as much infrastructure. ## Archived portions ### Discontinued concept design: single large power satellite system Our original first concept design is shown in the diagram below: ![[beam-combining-design.excalidraw.svg|500]] The design, in general, consists of a very large composite solar mirror, made of many smaller segments joined together. The segments are extremely thin and are coated with a very thin layer of a pure, highly-reflective metal. The composite solar mirror is parabolic in shape, and reflects the sunlight onto a secondary mirror (which is also parabolic in shape). The struts used to support the secondary mirror are extremely thin and lightweight to minimize interference. A suitable material could be titanium, although it would be incredibly expensive. The secondary mirror directs the now-concentrated sunlight into a small hole in the primary mirror, where it enters the laser cavity (not shown). The laser is a semi-tunable free-electron maser (or similar) that uses the concentrated heat and UV radiation in the sunlight to cause the thermionic emission and photoemission of electrons from a tungsten (or specialized ceramic) cathode, chosen due to its ability to withstand very high temperatures and its conductive nature. The cathode is negatively-charged, repelling electrons off its surface, which are attracted by a series of anodes and focused with magnetic lenses. The basic design of the electron gun and undulator for the laser would be a variation of the passively-driven free-electron maser design shown below: ![[prototype-free-electron-maser.excalidraw.svg]] The maser produces an intensive microwave beam with a power of several hundred MW (megawatts), and this beam is fed via cylindrical waveguides to circular-shaped transmitters on the sides of the solar mirror. These transmitters are several hundred meters to over a kilometer across, and contain hundreds of small parabolic reflectors (dishes). The maser beam travelling in the waveguide is split among the reflectors, which each have a primary and secondary reflector. This creates a very focused beam. By using reflectors of different sizes in a confocal arrangement and different phases, the smaller parabolic reflectors can effectively combine their microwave beams into a much larger microwave beam, allowing the circular transmitters to have an effective aperture of over 1 km, which, as we know from [[Ideal laser beam divergence]], leads to a highly-collimated beam. ![[beam-combining-design-closeup.excalidraw.svg|500]] > **Note:** Only a few of the "stripes" of the parabolic reflectors are shown, and their sizes are exaggerated, but in reality, the entire surface of the transmitters would be covered by criss-crossed stripes. The circular transmitters are attached on all sides of a central hub[^1], allowing the mirrors to continuously face the Sun while the transmitters always face the Earth. As the spacecraft's shifts from the night to the day side, the parabolic reflectors rotate to track the ground-based receiving stations. They are mounted on magnetic bearings to minimize friction and avoid moving parts. And since the position of the receiver stations with respect to the spacecraft changes throughout the course of the day, the microwave beam splitter redirects power from the circular transmitters on one side to the next, depending on which is facing the Earth at the moment. This enables the beam to stay locked on the same station(s) throughout the day. Additionally, having transmitters on each side means that the power satellite can send power to multiple stations at once, which, again, is one of our goals for making this a global, international power source. [^1]: This is distantly inspired by the AN/FPS-132 Solid State Phased Array Radar System, for instance, at the [RAF Fylingales military radar station](https://en.wikipedia.org/wiki/RAF_Fylingdales). In addition, it was also inspired by [this concept of a solar-powered space tug](https://commons.wikimedia.org/wiki/File:Solardisk.jpg), originally from [Wikipedia's article on space-based solar power](https://en.wikipedia.org/wiki/Space-based_solar_power). [^2]: This is just an illustration, and is not an optimal arrangement of the parabolic reflectors. See the [Wikipedia entry on optimal circle packing](https://en.wikipedia.org/wiki/Circle_packing_in_a_circle) for the optimal packings. [^3]: This can be established by a quick calculation. The proportion of the beam power that is received is the ratio of the receiver (combined) size versus the beam spot size. Let $D_r \approx \pu{200m}$ be the effective diameter of the terrestrial receiving station, and let $D_b \approx \pu{20 km}$ be the beam diameter on Earth. The proportion of beam power receiver $\eta$ is given by $\eta = \pi r_1^2/\pi r_2^2 = \pi (D_r/2)^2)/(\pi (D_b/2)^2) = (D_r/D_b)^2$. [^4]: The [Ivanpah solar power facility](https://en.wikipedia.org/wiki/Ivanpah_Solar_Power_Facility) located in the Mojave Desert has a total land area of 1420 hectares. Since its base is roughly circular, its diameter can be calculated from $D = 2\sqrt{A/\pi}$ (from $A = \pi (D/2)^2$, the formula for the area of a circle in terms of its diameter), which evaluates to roughly 4.25 km. [^5]: This formula is [[Procedure for testing coherent beam synthesis]]. [^6]: This is assuming that the masers are arranged in a circular ring but along the same plane. Formula is from [[Principles of coherent beam synthesis]]. [^7]: An ideal laser/maser has the beam profile of a [Gaussian beam](https://en.wikipedia.org/wiki/Gaussian_beam) and is thus the most tightly-focused beam possible, since the Gaussian beam diverges by the diffraction limit (the theoretical minimum divergence). However, a waveguide's boundary conditions give rise to modes that disrupt the Gaussian nature of the beam. As the [Wikipedia article on electromagnetic modes](https://en.wikipedia.org/wiki/Transverse_mode#Waveguides) explains, _"hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode"_; since TEM (transverse electromagnetic) waves are non-Gaussian, the beam quality degrades after passing through a waveguide.