One critically-important part of the entire design is the shape, material composition, and engineering of the the solar mirrors used. The key engineering challenges are numerous, but we'll start by focusing on one particular area: as the power satellites orbit in geostationary orbit, the solar mirror must be designed to reflect sunlight regardless of its orientation with respect to the Earth or the Sun. This means that the orientation of the solar mirror with respect to Earth varies across the day. Addressing this issue is a major challenge, compounded by the fact that, for optimal solar collection, we want to make sure that the mirror is always oriented towards the Sun, regardless of where it is during its orbit. ## A naive solution A "simple" solution to the issue can be for the mirror to have a "hole" at its center so that sunlight can travel right through to an off-axis light inlet on the transmitting satellites, as shown in the diagram below: ![[off-angle-laser-diagram.excalidraw.svg|600]] However, this approach is wasteful, because it means that the mirrors will always be at an angle with respect to the Sun. Recall that the irradiance $I$ - that is, the power density of light - is given by[^1]: $ I = \langle \mathbf{S}\rangle \cos \alpha $ Where $\mathbf{S}$ is the Poynting vector and $\langle \mathbf{S}\rangle$ denotes its time-averaged value. Since the axial tilt $\theta_a$ of the Earth is about $\theta_a = 23.4 \degree$ and $\cos(\theta_a) \approx 0.92$, we lose about 8% of solar irradiance (and thus 8% of the power we could collect) using this method, which is obviously not ideal. There is also the problem of using an opening in the mirror itself, which further restricts available space, especially with the transmitter sunlight already blocking part of the light. All said, this design could be much-improved, so we have independently come up with an alternate solution, which we discuss in the following section. ## A more sophisticated solution A better solution, then, is to consider a more advanced mirror configuration, where, instead of just one mirror, we have a primary mirror (the big parabolic reflector) and a secondary mirror, where the transmitter satellite is located at the root of the primary mirror. This is a common configuration in large reflecting telescopes on Earth. In addition, the mirror is continually turned so that it always faces the Sun. This approach is much more challenging, since turning the mirror requires more complex station-keeping and perhaps reaction wheels to turn the mirrors. (Alternative turning approaches, such as pre-spinning the mirrors on launch, may be a viable alternative, since reaction wheel failures are a common cause of spacecraft malfunctions). Additionally, the transmitter satellites will need _two_ gimbal-mounted output couplers, one behind and one in front of the satellite; they will use their front output coupler during points in the orbit when the mirror is *behind* the Earth, and switch to their back output coupler during points in the orbit when the mirror is _in front of_ the Earth. We illustrate this in the diagram below: ![[sun-facing design.excalidraw.svg|800]] While this approach is technologically-complex (since it requires an additional secondary mirror as well as two gimbal-mounted output couplers and a mechanism to continuously turn the mirror very precisely), it allows the primary mirror to be facing the Sun and oriented parallel to the Sun's rays, making the best use of the solar irradiance. **This is the design we are currently going for**. ### Other design considerations We spoke previously about the difficulties involved with reaction wheels and similar mechanisms. One promising technology that may solve many issues with reaction wheels is to use [magnetically-suspended reaction wheels](https://ntrs.nasa.gov/citations/19760012100), which, to quote the linked NASA paper, offers _"low drag torque, wearfree, unlubricated, vacuum-compatible operation, and unlimited life"_. Additionally, to protect the sensitive electronics in gimbal-mounted output couplers from being damaged by cosmic rays and micrometeorite impacts, it may be a good idea to use a [radome](https://en.wikipedia.org/wiki/Radome). Radomes are transparent to long-wavelength electromagnetic radiation (presumably including microwaves) but can provide much-needed shielding for the maser output couplers; otherwise, micrometeorites, which can have the kinetic energy of a bullet (if not more), can seriously damage the output couplers. ## Construction and building See [[Prototype solar mirror design]] for more details. ## Solar mirror modelling An approximate mathematical model of a parabolic mirror is just a paraboloid, which can be described as the surface of revolution obtained by rotating the parametric curve $z = t^2/2R, y = t$ about the $z$ axis, where $R$ is the radius of the mirror. We may find the electric field and radiation pattern of the mirror using the Helmholtz equation, the fundamental partial differential equation that governs the propagation of electromagnetic waves. The Helmholtz equation is given by: $ \nabla^2 \mathbf{E} + k^2 \mathbf{E} = 0 $ Where $\mathbf{E}(\mathbf{x}, t)$ is the **electric field** of the electromagnetic wave and $k$ is the wavevector, related to the wavelength $\lambda$ of the wave by $k = 2\pi/\lambda$. One can then solve the Helmholtz equation with Neumann (perfectly-reflecting) boundary conditions to model sunlight reflecting off the mirror. There exists an analytical solution to this boundary-value problem, which may be written in polar coordinates as: $ E(\theta) = \dfrac{2\lambda}{\pi D} \dfrac{J_1[(\pi D/\lambda)\sin \theta]}{\sin \theta} $ Where $J_1(\theta)$ is a [Bessel function of the 1st kind](https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind:_J%CE%B1), $D$ is the diameter of the mirror, and of course $\lambda$ is the wavelength. This is an idealized solution that doesn't account for the non-idealized reflectivity of the mirror, or scattering effects due to the (albeit small) protrusions of the gimbal-mounted output coupler and light inlet of the transmitter satellite, see [[sun-facing design.excalidraw.svg]]. It also doesn't account for the material properties of the mirror, which plays an essential role in how the mirror reflects (and scatters) light. [This photonics paper](https://www.mdpi.com/2304-6732/10/7/848) works out the physics for a better theoretical model. > **Note:** A [Desmos interactive plot](https://www.desmos.com/calculator/yx2yniraj6) is also available to play around with different parameters of a parabolic reflector and view the radiation pattern. However, to incoporate all the effects, analytical techniques will be limited, and we must use numerical simulations to solve the Helmholtz equation on a computer. This is especially difficult when we are working with actual CAD models rather than idealized mathematical models, where it is impossible calculate an exact solution by hand; analytical solutions are better-purposed as a way to sanity-check our numerical solutions. We can use both finite-element and finite-difference software to do our numerical simulations, although in the finite-difference case we will need to use the cylindrical form of the Helmholtz equation, and we are limited to simpler geometries than the more powerful (but complicated) finite-element method. Currently, we use the [FinDiff](https://findiff.readthedocs.io/en/latest/) and [FreeFEM++](https://freefem.org/) software packages, although we are working on switching to the pure-Python [FEniCS](https://fenicsproject.org/) library. An idea that may be worth exploring is using machine-learning solvers to solve PDEs; this may be beneficial, or it may not be. [^1]: See https://en.wikipedia.org/wiki/Irradiance [^2]: See https://en.wikipedia.org/wiki/Parabolic_antenna#Radiation_pattern_formula