> **Note:** This idea has been superseded by our current design which uses **coherent beam combining**. See [[A realistic space-based prototype]] for more information. Recall that a Gaussian beam (the standard mathematical model of a laser beam) always has a diverging beam width $w(z)$ given by: $ w(z) = w_0 \sqrt{1 + \left(\dfrac{z}{z_R}\right)^2}, \quad z_R = \dfrac{n\pi w_0^2}{\lambda} $ For large $z$, we have $(z/z_R)^2 \gg 1$, so we have $1 + (z/z_R)^2 \approx (z/z_R)^2$. Thus, this expression effectively reduces to: $ w(z) \approx \dfrac{w_0 z}{z_R} = \dfrac{\lambda}{n\pi w_0}z $ Thus, the divergence of the Gaussian beam is not only highly dependent on the beam waist (laser aperture radius) $w_0$, but also on the wavelength $\lambda$. In fact, since $w(z) \propto \lambda$ for large $z$ (which is the far-field regime that we are interested in), the difference in the rate of divergence $\dfrac{dw}{dz}$ using an infrared laser versus a microwave laser may be a difference of nearly a million! So why not use an infrared laser? The first concern is **atmospheric attenuation**. The atmosphere is incredibly hostile to infrared light and blocks almost all infrared wavelengths. However, not all hope is lost. There exists a special wavelength range between $\pu{8\mu m}$ and $\pu{14 \mu m}$ known as the **infrared window**[^1]. Atmospheric attenuation in clear sky conditions is near-zero, so a far-infrared laser can be made incredibly accurate and have spot sizes of 30 meters or even less! Indeed, this may be a far more promising idea than using a microwave laser, which diverges rapidly to several kilometers.[^2] For Project Elara, a good idea is to choose the $\pu{12.1 - 12.5 \mu m}$ wavelength range in particular.[^3] This wavelength. Notice how in the below plot (generated from HITRAN simulation data)[^4] there is essentially no atmospheric attenuation of infrared radiation within this range: ![[spectraplot.png|500]] As further confirmation, consider the following (rather noisy) log graph generated from the [ATLAS web viewer](https://eodg.atm.ox.ac.uk/ATLAS/zenith-absorption). Within the $\pu{12-12.5\mu m}$ wavelength range, the optical thickness is close to one. The absorption $A$ can be calculated from the optical thickness with $A = 1 - e^{-\tau}$, where $\tau$ is the optical thickness. At $\lambda = \pu{12.165 \mu m}$ there is a particularly low optical thickness of just $\tau = 0.0237$ for water (all the other gases are even lower, so we can effectively ignore them); this corresponds to nearly 98% transmittance, which is near-perfect. ![[ATLAS-zenith-absorption.png|500]] Note that the $\pu{12-12.5 \mu m}$ range is not the only option. Another band at around $\pu{2-2.3\mu m}$ also offers exceptionally-low atmospheric attenuation at nearly 99.4%! This band comes with the additional advantage of having a shorter wavelength, meaning that the beam divergence is lower. ![[ATLAS-zenith-absorption-fullrange.png|500]] Unfortunately, all of this is assuming **clear skies**, which cannot always be guaranteed at all locations. During rain or stormy weather, atmospheric attenuation greatly increases for infrared wavelengths. Remember that the two plots we showed (and the calculations we performed) are for **atmospheric gases** (like water vapour) only, _not_ liquid water, fog, clouds, or dust (clouds and fog are just suspensions of _liquid_ water droplets in air). So an infrared laser would struggle to beam power during a hurricane or dust storm - hardly ideal for reliable power transmission. But all is not lost, because there _is_ a way to mitigate this.[^5] In good weather, we can directly transmit power from infrared lasers in space to our terrestrial receiving stations.[^6] However, in poor weather, we can use a two-stage backup power system. We can build large power receiving stations located in high mountain ranges with very dry weather (or alternatively, artificially-built towers in arid regions at high altitudes). When we have poor weather conditions in some area, we can point the infrared lasers at these backup receiving stations instead, which can receive power reliably thanks to the ideal atmospheric conditions at these sites. Then, we can use the power received from space to power phased arrays[^7] that beam power to the places where power is needed. Since phased arrays operate in the microwave range (1-2 GHz), they are unimpeded by the atmosphere and can easily penetrate storms or heavy weather to reach the places that the infrared lasers alone cannot reach. And since they are located at high altitudes, the microwaves can reach extremely far. Using the formula $D = \sqrt{2R_Eh + h^2}$ for the distance to the horizon $D$ (where $R_E = \pu{6,378 km}$ is Earth's radius and $h$ is the height above Earth's surface), we can compute some values for the transmission distance:[^8] | Height above ground | Microwave transmission distance | | --------------------------------------------- | ------------------------------- | | 100 m (typical skyscraper) | 35 km | | 200 m (radio tower) | 50 km | | 400 m (high radio tower) | 71 km | | 600 m (very high radio tower) | 87 km | | 1500 m (mountain/highland) | 138 km | | 3000 m (high mountains) | 195 km | | 4000 m (relatively high mountains e.g. Andes) | 225 km | | 6000 m (very high mountains) | 276 km | | 8000 m (highest mountains in the world) | 319 km | With these backup transmitters, we can bypass storms and ensure uninterrupted power (which, as an additional advantage, can transmit multiple directions at once due to the properties of phased arrays). In addition, if we want to transmit over longer distances, no problem[^9]: we can place additional microwave reflectors at elevated positions along the transmission path, bouncing microwaves to get power to wherever it's needed. Of course, there would be power losses this way, but transmitting via this reflector technique over oceans, which have nearly no (1-2 GHz microwave, <2.4 GHz) obstacles above 30 meters, can make sure that power is as minimally lost as possible. [^1]: For more details, see the [associated Wikipedia article](https://en.wikipedia.org/wiki/Infrared_window) [^2]: The inspiration for this idea came from [this LLNL paper](https://web.archive.org/web/20140919135937/https://e-reports-ext.llnl.gov/pdf/372187.pdf#expand) linked from the [DOE website](https://www.energy.gov/articles/space-based-solar-power), where the authors described using a $\pu{0.8 \mu m}$ infrared laser to achieve *incredibly* accurate beam sizes of up to $\pu{0.1 m}$. The webpage is now archived, hence the link points to a Wayback Machine snapshot of the page. [^3]: It is particularly advantageous that since we are now focusing our efforts on free electron lasers (see [[Prototype free-electron maser design]]) they can be tuned over a wide range of wavelengths, so if we find that one wavelength doesn't work, we can easily switch to another. [^4]: [HITRAN](https://hitran.org/) is a software package specialized for calculating spectral lines for a variety of molecular gas species and mixtures, and is often used for atmospheric modelling. The [Stanford SpectraPlot](https://www.spectraplot.com/absorption) can be used freely to view absorbance spectra from HITRAN, and was used to generate the referenced plot. [^5]: To our knowledge it is an original idea of ours, though it is likely that others may have come up with the idea independently. [^6]: Since infrared light is relatively short-wavelength, it would be ideally converted to electricity with solar thermal techniques, such as further concentrating the light with mirrors and using those extremely high temperatures to generate a turbine (or possibly heat a plasma to drive a dynamo/extract DC power from charged particles in the plasma). [^7]: We can use gyrotrons to be able to generate the powerful microwaves for these phased arrays using much the same technology as we already use to make sunlight-powered electron guns in our present design (see [[Prototype free-electron maser design]]). [^8]: In practice, due to terrain irregularities (for instance, other mountains, gorges, valleys, fjords, etc.) the actual distance may be lower, though this problem is less significant at higher transmission altitudes. [^9]: It is possible to do over-the-horizon transmission by other means, such as by exploiting the ability of long-wavelength microwaves to diffract around obstacles. [This wikipedia article](https://en.wikipedia.org/wiki/Over-the-horizon_radar) on the subject explains more. Note that it is (theoretically) also possible to place large microwave reflectors (parabolic reflectors) on blimps or ballooms high up in the atmosphere to be able to "bounce" microwaves from high up in the atmosphere and transmit over incredibly long distances. Also, yes, this idea is superficially similar to the fire beacons in Lord of the Rings.