Free-electron maser efficiency calculations

A critical area of importance for any power system, especially those used for long-distance power transfer, is the efficiency of the system. As detailed in Free-electron maser physics, efficiency comes in two areas:

• First, the power efficiency of the maser. This is a measure of how much energy is lost due to mechanical, thermal, and radiative effects that come from the maser mechanism itself, and can be calculating by finding the difference between the power output of the maser and the input power supplied to it (by sunlight or an artificial light source)
• Second, the directivity of the maser. This is a measure of how focused the maser beam is, and how well it stays focused. For instance, we might want to know how much of the power density of the beam stays within 80% of the maser aperture, and how that changes with increasing distance.

It is important to establish theoretical baselines (as well as upper limits) for the efficiency of a free-electron maser. This will be the focus of this page.

Electron gun efficiency

Assuming an otherwise ideal system, the theoretical maximum efficiency of a thermocathode comes from the Carnot efficiency equation, given by:

$${\Gamma }_{ac}=1-\frac{T_c}{T_h}$$

Where ${\Gamma }_{ac}$ is the efficiency of the cathode-anode combination, $T_h$ is the temperature of the thermocathode, and $T_c$ is the temperature of the anode at the far end of the electron gun. Since the anode will be in space, we can use the temperature of the Moon’s far side (which is about 100 K¹) as an approximation for $T_c$. Focused sunlight can heat up materials to over 2,700K², but as a conservative estimate we can let $T_h=2000\mathrm{K}$. Using these values, we find that the maximum theoretical efficiency is about 95%.

However, in practice, there are other sources of inefficiencies. The current state-of-the-art for thermionic technologies is approximately 25% efficiency in the lab, with top efficiency (based on materials science calculations) calculated at 35%.³ Thus, while it is theoretically possible to reach the Carnot limit, current limitations in materials science mean that the practical limit is under 40% efficiency. It may be possible to further increase the efficiency by coupling thermionic emission with photoelectric emission, since we are using optical heating. This approach, known as photon-enhanced thermionic emission (PETE). Photoelectric emission can (in theory) can reach up to 35% efficiency⁴, although again, this is theoretical and our ability to actually reach it depends on our materials science capabilities. Ignoring the difficulty of making a material that has both extremely high thermionic and photoelectric efficiencies and can withstand the harsh, cold conditions of deep space, with a temperature differential of well over 2000 K, the combined efficiency could reach up to 70%⁵, although to date this has never been reached, either in the lab or from theoretical calculations. A figure of around 40-50% would be most likely the highest efficiency that could be reached in a practical setting, although this may change with developments in materials science.

Undulator losses

The next source of losses comes from the undulator in a free-electron laser/maser. Research has indicated that longer wavelengths in general lead to higher efficiencies of 30% experimentally and 40% in theory⁶, although this relies on extremely-advanced techniques such as electron beam energy recovery. The theoretical limit is quite complex, and different papers give drastically different values with different techniques and different designs, though it seems to be around 30-40%, although one paper⁷ gives an astoundingly-high theoretical value of 70% (though this is only in pulsed operation).

Directivity calculations

The directivity of the maser can be measured using the gain, notation as $G(\theta ,\varphi )$, which is a measure of the power output of the maser beam compared to an isotropic radiator as a function of angle. The higher the gain, the higher the directivity, which is why we want to increase the gain as much as possible. Using a pure maser beam, the theoretical highest directivity within the spot radius would be given by $1-1/e^2\approx 86.5\%$ (this is derived from the Gaussian beam model and comes from the diffraction limit, a fundamental limit of all electromagnetic waves, see A realistic space-based prototype), although this increases to 99% for a circular region 1.52x the spot radius⁸. Furthermore, Gaussian beams are a mathematical idealization and creating an actual maser beam that closely approximates a Gaussian beam is all we can do in practice.

Combined efficiency

Combining the efficiency values we have previously mentioned, the following table gives the calculated efficiency of the free-electron laser/maser:

Case	Electron gun efficiency	Undulator efficiency	Total laser/maser efficiency
Theoretical best-case scenario	70%	70%	49%
Theoretical optimistic scenario	50%	40%	20%
Realistic scenario	25%	30%	7.5%

The highest efficiency ever recorded for a millimeter-wave free-electron maser (which is close to our maser operating wavelength) is at 35%⁹, although shorter-wavelength free-electron masers and gyrotrons (a related technology, see the very bottom of the Free-electron maser physics page) can achieve higher efficiencies. While this is a matter of speculation, efficiencies of around 50% may be achievable with (major) advances in technology. Note that even 35% is already quite high for any laser, since carbon dioxide lasers (already considered a high-efficiency laser) can reach only up to around 20% efficiency¹⁰, though some solid-state and diode lasers can achieve much higher efficiencies of over 50%¹¹.

Footnotes

1. Source: NASA. Note that while they give the temperature as -173 degrees Celsius, we have converted to the equivalent in Kelvin ($100.15\mathrm{K}$) for ease of calculations. ↩

2. This comes from the Uzbekistan solar furnace, which can reach 3000 degrees Celsius. ↩

3. These figures come from the paper Thermo-electric characteristics analysis of thermionic energy conversion in space nuclear reactors (Applied Thermal Engineering, 2025) by Zhao et. al. (https://doi.org/10.1016/j.applthermaleng.2024.124997) ↩

4. The figure comes from the paper Excellent visible light absorption and ultra-high photoelectric conversion efficiency of two-dimensional (MoSe2)x(MoSTe)1-x mosaic heterostructures (Chemical Physics, 2024) by Deng et. al. (https://doi.org/10.1016/j.chemphys.2024.112299) ↩

5. The figure comes from the paper Limit of efficiency for photon-enhanced thermionic emission vs. photovoltaic and thermal conversion (Solar Energy Materials and Solar Cells, 2015) by Segev et. al. (https://doi.org/10.1016/j.solmat.2015.05.001) ↩

6. The figures come from the paper High efficiency tapered free-electron lasers with a prebunched electron beam (Phys. Rev. Accel. Beams, 2017) by Emma et. al. (https://doi.org/10.1103/PhysRevAccelBeams.20.110701), which references another paper that gave the efficiency values. ↩

7. S. Benson, High power free-electron lasers, Proceedings of the 1999 Particle Accelerator Conference (Cat. No.99CH36366), New York, NY, USA, 1999, pp. 212-216 vol.1 (https://doi.org/10.1109/PAC.1999.795667) ↩

8. See Wikipedia’s article on Gaussian beams in the power through an aperture section ↩

9. Ginzburg NS, Kaminsky AA, Kaminsky AK, Peskov NY, Sedykh SN, Sergeev AP, Sergeev AS. High-efficiency single-mode free-electron maser oscillator based on a bragg resonator with step of phase of corrugation. Phys Rev Lett. 2000 Apr 17;84(16):3574-7. PMID: 11019149. (https://doi.org/10.1103/PhysRevLett.84.3574) ↩

10. See the Wikipedia article on carbon dioxide lasers. ↩

11. See this IPG photonics article ↩