Test build of a free-electron maser

This article details the basic steps for building and testing a free-electron maser. The key components of a free-electron maser are:

An electron source that provides an electron beam
An undulator composed of alternating magnets
An optical cavity sealing the undulator, composed of two mirrors at each end of the undulator

Notably all components have to be kept in vacuum or the maser will not work. What follows is a very rudimentary setup for creating a free-electron maser, and hasn’t considered the broad literature on the topic. Consider this a proof-of-concept engineering sketch. What follows is a discussion on the experimental and engineering aspects of each of the components of the maser. The diagram below is available to serve as a reference for the general design of the maser:

Simplified diagram of general maser design

Note: For more diagrams and information, please see Prototype free-electron maser design.

Suggested shopping list

Vacuum chamber - note, however, that we need a very good vacuum to make this work, and this is not really something we’d “buy” as much as getting permission to use
Wires/fasteners (while in theory it would be best to solder the wires, it may be too much work and risky to do in a closed lab environment)
Zinc plates (for anode/cathode, must be scoured (that is, scraped and polished) or leached with acid prior to experiment and placed in vacuum)
Circular metal disks used as microwave reflectors
UV-C germicidal lamp
N52 neodymium magnets (and we’ll need lots of them)
Small dish radio antenna with digital interface
Optical table or at least a table with small holes that allows fastening items to it
RF-absorbing foam
Iris diaphragm

Experimental objectives

The point of the concept is to (a) demonstrate that we can indeed build a functional (if impractical) free-electron maser and (b) test our our theoretical models with actual results, so we can check discrepancies, identify inefficiencies, and improve our understanding.

Experimental design

The experiment will consist of three parts:

Working prototype of an electron source producing an electron beam
Working prototype of an undulator
Demonstration of lasing

Part 1: Electron beam

To create the electron beam, we’ll be making something somewhat similar to a vacuum tube. However, instead of relying on thermionic emission (electrons “boil off” the cathode when it gets really hot), we will use photoelectric emission by letting UV light be incident on a metal (likely Zinc) plate. By the photoelectric effect, the UV photons knock off electrons in the cathode, setting them free. By applying a voltage between the anode and cathode with the voltage source, we can create a potential difference between the two electrodes, so electrons fly towards the anode. With a magnet lens (see sextupole magnet), we can collimate the flying electrons into a beam that exits out from a hole in the anode.

Note: Among other reasons, to make sure that the scoured Zinc plate does not rapidly oxidize, everything must be placed in an oxygen-free environment. To more clearly see the electron beam, it might be useful to temporarily pass it through a chamber filled with some sort of noble gas so that the electron beam becomes clearly visible, although the same process causes the electrons to lose energy, so it should only be used for alignment and test purposes.

A diagram of a general free-electron maser

Depending on need, we’ll then use other sets of magnets to be able to bend the electron beam so it enters the undulator at our desired angle of incidence, and another set to let the electron beam escape after it passes through the undulator (shown in the above diagram). These should be relatively straightforward to set up, especially because our maser will have no moving parts and (in theory) a minimum of electrical wiring required.

Part 2: Undulator

The undulator will be the most important part of the design, but probably the hardest to actually work right. It consists of a set of alternating magnets in two rows (one upper, one lower) that causes the electron beam to “wiggle” and therefore emit radiation. From Calculation of free electron maser parameters we know that:

λ_{r} = \frac{L _{u}}{2 γ ^{2}} (1 + \frac{1}{2} K^{2}), K \approx (93.4 A / N) B_{0} L_{u}

Where $L_{u}$ is the spatial period of the magnets and $K$ is the undulator parameter. Note that for coherent emission of radiation, we must have $K ≪ 1$ , and thus:

λ_{r} \approx \frac{L _{u}}{2 γ ^{2}}

We recall that the magnetic field $B_{0} = B_{y} = - \frac{B _{r} V}{2 π d ^{3}}$ between a pairs of magnets, derived in Free-electron maser physics reduces to $B_{y} \approx - B_{r} /2 π$ when the magnets are brought within a distance $d \leq 3 V$ of each other, and $B_{y} \approx - B_{r}$ for $d \leq 3 V /2 π$ . In our case, as we are buying magnets of dimensions $60 \times 10 \times 5 mm$ , the following parameters should be fairly suitable:

Parameter	Description	Suggested value
$B_{0}$	Undulator magnetic field strength	$\leq 0.1 T$
$d$	Separation between upper and lower rows of magnets	$\leq 1.4 cm$
$L_{u}$	Double of the separation of two adjacent magnets (see Calculation of free electron maser parameters for diagram)	$2.5 - 4.5 cm$
$Δ V$	Applied voltage between anode and cathode	$\geq 1 kV$ at least
$f_{r} = c / λ_{r}$	Frequency of maser output radiation	Dependent on parameters

The undulator field strength decreases with the cube of distance, so raising or lowering the magnets (that is, changing $d$ ) into the undulator chamber can be used to change the magnetic field strength. However, the rapid decay of the magnetic field with distance also means that the magnets should generally be brought within $d \leq 3 V \approx 1.4 cm$ of each other, considering the dimensions of our magnets. In addition, increasing $L_{u}$ in general increases the wavelength of the output radiation of the maser, and thus lowers the frequency. Thus, a maser with large dimensions (large $L_{u}$ ) is required for emitting long-wavelength microwaves in the $1 - 3 GHz$ range. However, for a prototype, a small $L_{u}$ can be used to make the maser more compact, at the cost of only being able to produce shorter-wavelength microwaves.

Photocathode characteristics

For simplicity and ease of handling, the prototype uses a photocathode rather than the thermocathode more common among free-electron laser designs (future designs will use a thermocathode, see Thermocathode design for electron gun). The photocurrent $I$ can be roughly computed using the following expression:

I = η (\frac{e P}{h f _{c}}) = η (\frac{e P λ _{c}}{h c}) \approx (8.07 \times 1 0^{5} C \cdot J^{- 1} m^{- 1}) \times η λ_{c} P

Where $η$ is the quantum efficiency (QE) of the photocathode, $e$ is the elementary charge, $P$ is the power of the light source used to illuminate the photocathode (in this case, a tin cathode), $c$ is the speed of light, $h$ is Planck’s constant, and $f_{c}, λ_{c}$ are the central frequency and wavelength of the light source respectively, which is found as follows:

⟨ λ ⟩ = \frac{1}{λ _{ma x} - λ _{min}} \int_{λ_{min}}^{λ_{ma x}} ρ (λ) d λ

Where $λ_{min}, λ_{ma x}$ are the maximum and minimum wavelengths of light produced by the light source, and $ρ (λ)$ is the spectral power density (power per unit wavelength) of the light source with bandwidth $λ_{ma x} - λ_{min}$ . Note that $ρ (λ)$ , $η$ , and $b$ can all be found in the literature¹. Some values are listed below for some commercially-available pure metals in vacuum² (listed from highest to lowest):

Material	Max. QE ( $\times 1 0^{- 4}$ )	Literature reference
Aluminium	2250 (@58.4 nm), 300 (@121.6 nm), 0.1 - 1 (@355 nm)	Samson & Cairns (1955) (first two values*), Pimpec et. al. (2024) (last value)
Lead	1400 (@80 nm) 300 (120 nm), 5.6 (@213 nm), 12.8 (@193 nm)	Samson & Cairns (1966) (first two values), Smedley et. al. (2008) (last two values)
Zinc	1600 (@80 nm), 1000 (@100 nm), 2.89 (@226 nm), 0.18 (@302 nm)	Samson & Cairns (1966) (first two values), DeVoe (1936) (last two values)
Copper	1250 (@800 nm), 1000 (@100 nm), 0.2-1.1, averaging around 1.0 (@262 nm)	Samson & Cairns (1966) (first two values), Prat et. al. (2015) (last value)
Nickel	1250 (@79 nm), 200 (@120 nm)	Samson & Cairns (1966)
Tin	1100 (@100 nm), 400 (@120 nm)	Samson & Cairns (1966)
Magnesium	1000+ (@100 nm), 20 (@258 nm), 10-20 (@262 nm), 7.6 (@355 nm)	First value is postulated**, last three values are respectively from Teichert et. al. (2021), Nakajyo et. al. (2002), Perrone et. al. (2025)
Gold	800 (@100 nm), 300 (@120 nm)	Samson & Cairns (1966)
Silver	750 (@100 nm), 300 (@120 nm)	Samson & Cairns (1966)
Tungsten	380 (@121.6 nm)	Samson & Cairns (1966)

_{*: The first two values were actually of an aluminium-magnesium alloy, containing 94% Al and 4% Mg.} _{**: The postulated value is based on the value of aluminium (from Samson & Cairns, 1966) at around the same frequency, since aluminium’s work function of 4.08 eV is close to magnesium’s 3.68 eV}

Note: Several of the older articles are combined in this NASA technical report, which is essentially a compilation of papers, and was the original source that main of the measurements were obtained from. Also, the determination of the values for zinc from DeVoe (1936) was found by taking the y-values in Fig. 3 (which were in units of $3.9 \times 1 0^{- 13} A \cdot er g^{- 1} \cdot s$ ), converting them into SI units of $A / W$ , then multiplying them by a factor of $h c / (e λ)$ to get the QE.

Upon considering these options, zinc may prove to be our preferred option, as it is generally inexpensive and safe to handle, and is a common choice in classroom demonstrations of the photoelectric effect. However, the oxide layer that forms on pure zinc exposed to air reduces its quantum efficiency dramatically. This layer can be removed by simply shaving it off (in vacuum), or it can be removed by immersion in sulfuric acid or hydrochloric acid; the acid reacts with the oxide layer and forms a precipitate, leaving pure zinc metal behind. Magnesium appears to form the best photocathodes of any pure metal² and also forms an oxide layer on its surface, but it is more dangerous to handle as it is an alkali (earth) metal. For all cases, it is useful to first clean the metal surface with isopropyl alcohol to remove contaminants, a general technique that is widely-used.

The UV source for the photocathode can, for our purposes, just be a germicidal UV-C lamp, which can reach wavelengths of close to $100 nm$ (the lower, the better) and is reasonably affordable. Industrial- and research-grade free electron lasers typically use a photoinjector, which essentially uses a laser as the light source for the photocathode, and can achieve much higher electron beam intensities, but this is out of scope for the time being.³

Electron beam characteristics

Any free-electron maser requires a powerful and usually relativistic electron beam to operate efficiently. The approximate reason for this is that generating coherent radiation requires that the synchrotron radiation in the chamber produces standing waves that are strong enough to exert a “recoil force” back on the electrons (see Free-electron maser physics for details). The power of synchrotron radiation is proportional to the fourth power of the Lorentz factor⁴, so it is important to accelerate the electrons to (mildly) relativistic velocities (indeed producing such high-energy electron beams is a major reason why typical free-electron lasers are the size of a room⁵). Since we are using purely electrostatic acceleration, the (total) potential difference $Δ V$ required (combination of all the cathode-anode voltages) is quite high, particularly for continuous operation (as opposed to pulsed operation). We assume a potential difference in the order of $1 - 2.2 keV$ , where a higher potential difference may be used, but nothing lower.

Undulator strength

For an undulator to produce coherent radiation, another important requirement is that the undulator strength parameter $K \approx (93.4 A / N) B_{0} L_{u}$ must be small⁶. Since $K$ is linear in $B_{0}$ and $L_{u}$ , the magnetic field strength must be carefully tailored to maintain $K ≪ 1$ . In practice this means that $B_{0}$ must be kept relatively small, particularly for operation at long wavelengths where $L_{u}$ must be large (as we discussed earlier).

RF chamber and output coupling

The undulator takes the form of a resonant cavity (RF cavity) with a mirror on each end; this amplifies the radiation emitted by the rapidly oscillating electrons and through complex physical effects creates a highly collimated beam. Since microwaves have relatively long wavelengths, we do not need to worry much about surface imperfections for the mirrors⁷. Additionally, the type of mirror does not especially matter, and essentially any type of metal (or metal-coated surface) will be sufficient at the wavelengths we are considering.⁸. We have put aluminium disks in the suggested shopping list to serve as mirrors, since aluminium is relatively lightweight (compared to steel). Shielding, however, will be important, because it is near-certain that this first experimental prototype will be extremely low-efficiency and thus produce a weak microwave beam (on the order of a watt or even lower; note, however, that this is still already more than a million times that of typical gas masers, and if focused, can cause burns). Moreover, our target microwave range is close to typical frequencies for telecommunications equipment and Wi-Fi. Thus, it goes without saying that the entire experimental setup must be completely shielded and we cannot use phones, computers, or other similar electronic devices anywhere close to the experimental apparatus, other than specially-shielded instrumentation and electronics.

To be able to transmit part of the microwaves, we’ll need to make a very small aperture (hole, in plain language) in one of the two mirrors.⁹ This is the output coupler of the maser, and transfers power from the interior of the resonant cavity to the outside. An iris diaphragm is used to be able to adjust the size of this resonant cavity and thus change the amount of power transmitted in the beam. To avoid Fresnel reflections, an anti-reflective RF coating will be applied¹⁰, although for practical purposes, this may be a RF-absorbent foam instead of a coating. It is not possible to calculate (in exact terms) how much of the microwaves will be (or indeed, should be) transmitted through the aperture since the problem has no analytical solution. We have done numerical simulations that do give approximate results - see RF cavity simulation with aperture.

Note: A heuristic, however, is to presume that the transmission coefficient is proportional the the ratio of the aperture area to the mirror area, and thus proportional to the square of the ratio between the aperture radius $r_{a p}$ and the mirror radius $R_{m}$ , that is, $T \sim \frac{A _{a p}}{A _{mi rror}} \sim (\frac{r _{a p}}{R _{m}})^{2}$ , though the effectiveness of this approximation is not known.

The resonant (RF) cavity can be 3D-printed and should be painted with an EM-absorbing coating everywhere except for the mirrors to absorb any microwaves that are not along the optical axis. This means that only the microwaves along the axis are reflected, thus producing a directional beam; note that in our initial prototype, however, we can omit the EM coating as most 3D print filaments are made of microwave-transparent materials and it is expected that the need for a coating is not yet a substantial area of concern. It will also need slots to be able to put in our neodymium magnets for the undulator, as well as the aforementioned. Other than that, there should be two sets of beam-bending magnets to guide the electron beam into (and out of) the optical cavity. Once the electron beam exits the cavity, it should be directed into an electron dump, which, for our purposes, can just be a concrete block.

Precautions

It is important to observe the following safety and isolation measures when building and testing the free-electron maser:

Electron beams can be dangerous as they consist of very fast charged particles (beta particles) which are ionizing. No one should be in the way of the electron beam when it is turned on. It must be operated remotely.
For the maser to work, it must be placed inside a vacuum chamber at extremely-low pressures. Operating a vacuum chamber is also hazardous and should be done with care.
The entire maser and all its components (sensors, wires, microcomputers, etc.) must be electromagnetically-shielded (e.g. with Faraday cages) to prevent unwanted EM interference, or otherwise the readings from the antenna will be unreliable. All wires must be properly shielded for the same reason.

Note that these precautions are just a few of the safety measures that should be observed. Other lab safety measures and an experienced research mentor/technician must also be present at all times. Please see Project Elara Safety Review and Precautions for more information.

Data collection and results to measure

Experimental setup

For the best results, the test should be performed in a shielded anechoic chamber to obtain accurate measurements, as is standard for testing RF antennas. However, with even small anechoic chambers costing several hundred or even thousands of dollars, this is not practical for our purposes. A slightly more affordable option is to buy slabs of pre-manufactured RF absorbing panels, which are typically carbon-loaded polyurethane foam cut into pyramidal wedges, and are used to coat the walls of anechoic chambers. This, however, is also expensive, though slightly less so. A last option is to use a RF-absorbent material like carbon-loaded foams to line the test chamber, such as this activated carbon foam sponge sheet. While the efficacy of such materials is generally unknown, they are the only option within our price range.

Note: It may be possible to try to calibrate for the imperfect reflections caused by the RF absorber and use that to reduce the noise caused, but there is no guarantee this would succeed.

Meanwhile, outside the chamber, a Faraday cage (or equivalent) can be used to block external EM fields and minimize interference. Note that a common rule of hand is the one-tenth rule; that is to say, a mesh made of a conducting material with a gap of $λ /10$ effectively blocks electromagnetic fields just as well as a solid conductor. Using metal wiring (which can be bought quite cheaply in wire rolls), we can therefore create a mesh that surrounds the testing chamber and shields it from external interference. Nevertheless, background readings of the EM field with the maser turned off must be conducted to measure any remaining background noise, and ensure that noise does not overwhelm the readings. After the experiment, the noise may be removed from the experimental data to obtain cleaner results.

For our maser design, the key parameters to measure are the following:

Parameter	Measurement method	Ideal result
Radiation pattern	Test antenna	Gaussian intensity profile
Frequency range	Test antenna	Narrow range, centered at the lasing wavelength
Electron beam leakage	External Geiger counters	Very low; if there is any ionizing radiation significantly greater than background, then we must locate the source of the leakage, reinforce the relevant section(s) around the source, and add more shielding
Magnetostatic fields inside undulator	Built-in gaussmeter in the undulator	Close to the computed analytical and numerical solutions; see Numerical simulations for magnetostatics and Numerical modelling of electron beams for the specific solutions
Electron gun anode-cathode potential	External voltmeter	Close to the operating potential of the beam; for instance, for a 2.2 keV beam, this would be 2.2 kV
Beam current	Built-in current sensor	Close to the theoretically-predicted value of the photocurrent
Temperature of wiring, electron guns, and microcontroller(s)	Built-in temperature sensors	Generally low; if detected temperature is overly high, then the electrical components may overheat and short-circuit
Optical noise (variation in power output of the maser)	Test antenna	Generally low, ensuring the beam is stable
Operation time	Timer	As long as possible; the maser should be capable of continuous operation over extended durations

Note: For obtaining the radiation pattern, the test antenna would be moved at equidistant intervals in a rectangular or polar sampling grid to find the radiation pattern.

Other parameters can then be experimentally determined from the results, as follows:

Parameter	Determination method	Ideal result
Central wavelength	Center of frequency range	Equal to $λ_{r}$ , the lasing wavelength, and $λ_{0}$ , the wavelength of the fundamental mode of the laser cavity
Bandwidth	Width of frequency range	Very small, since a free-electron maser is a coherent light source
Gain	Gain formula combined with measured radiation pattern	Very high gain on the optical axis, with a maximum gain of $> 20 dB$
Total output power	Integrating the radiation pattern across all solid angles	Equal to the input power
Efficiency	Dividing the total output power by the input power (power supplied by the UV light source)	>50% efficiency (not realistically possible)

To collect our data, we will need to rely on an antenna placed in front of the maser that can pick up the maser’s microwave beam and therefore measure both its wavelength and signal power. This is the test antenna, and can take the form of a small dish antenna (similar to those used for receiving satellite TV or radio) that is reasonably mobile and has a known gain. Ideally, the antenna would be connected via a coaxial cable to computers located outside of the shielded vacuum chamber used to test the maser; however, if a physical link cannot be established without compromising the quality of the vacuum inside the chamber, then the antenna would need to be linked to a microcomputer for storing data until after the maser is turned off. In addition, if it is not possible to use a transparent vacuum chamber, miniature TV cameras may be used to obtain a live video feed. However, note that wireless remote control or data links must be avoided as their radio waves/microwaves may cause interference with the microwaves from the maser.

For the measurement pattern of the radiation pattern in particular, it would be necessary to move the test antenna to different grid points along a polar grid (from left to right). This is a non-trivial measurement to perform; consult this video and these notes for more information. Needless to say, we will also need to connect the test antenna by a well-shielded coaxial cable to a device outside the vacuum chamber and shielding, even as we move it to different locations around the maser. The specifics will be determined at a later time.

Maser control

The primary way we will be controlling the maser’s beam characteristics is by varying the separation $L_{u}$ between the pairs of magnets, which yields a change in the wavelength approximately proportional to $L_{u}$ .

We can also vary the distance $d$ between the upper and lower rows of magnets in the undulator, which can be used to adjust the magnetic field strength. The dependence of the magnetic field on the wavelength is not a direct relationship, but a stronger field in general increases the wavelength (slowly).

In addition, we can vary the voltage applied between the anode and the cathode. Increasing the voltage increases the Lorentz factor $γ$ of the electrons (as they move faster) and should yield a (very small) change proportional to $γ^{- 2}$ in the wavelength of the emitted radiation by the equation - see Calculation of free electron maser parameters for more on this.

Finally, the iris aperture can be adjusted to increase the amount of the beam that exits the resonant cavity. We again will restate that the dependence is highly complex and cannot be easily modelled analytically, although it is expected that

As a summary of the expected response of the maser to variable changes in its parameters, we have:

Parameter	Symbol	Effect on wavelength	Effect on maser power output
Spatial period of magnets	$L_{u}$	Increase leads to longer wavelength	No effect (in theory)
Distance of undulator magnets from beam centerline	$d$	Increase leads to stronger magnetic field	Increase leads to higher microwave beam power
Cathode-anode potential difference	$Δ V$	Increase leads to shorter wavelength (due to higher Lorentz factor)	Increase leads to higher microwave beam power
Iris aperture radius	$a$	No effect (in theory)	Increase leads to higher microwave beam power
Number of magnets used (per row)	$N$	Increase leads to more uniform magnetic field	Increase leads to more stable beam

Last but not least, the maser must have a SCRAM button for quickly shutting it off in case something goes wrong. There must be a physical switch as well as a digital switch for redundancy, in case one does not work. Before the operation of the maser, a safety briefing must be conducted, and the maser must be run with care and due caution.

Magnetic fields

We would ideally want to determine the actual magnetic fields of the magnets, which may deviate significantly from our theoretical result of $B \approx B_{r} /2 π$ . For this, a rather crude way is to use iron filings and photographs to be able to trace the field lines, although this is only useful for a visual inspection.

More realistically, we would take readings from the built-in gaussmeters in the maser at different positions and fit the measured values against theoretical models (as given in Numerical simulations for magnetostatics).

Resources

Tables of magnetic remanence (permanent magnets):

Other tables:

Neodynium N52 specs

Elara Research Docs

Explorer

Test build of a free-electron maser

Suggested shopping list

Experimental objectives

Experimental design

Part 1: Electron beam

Part 2: Undulator

Photocathode characteristics

Electron beam characteristics

Undulator strength

RF chamber and output coupling

Precautions

Data collection and results to measure

Experimental setup

Maser control

Magnetic fields

Resources

Graph View

Table of Contents

Backlinks

Elara Research Docs

Explorer

Test build of a free-electron maser

Suggested shopping list

Experimental objectives

Experimental design

Part 1: Electron beam

Part 2: Undulator

Photocathode characteristics

Electron beam characteristics

Undulator strength

RF chamber and output coupling

Precautions

Data collection and results to measure

Experimental setup

Maser control

Magnetic fields

Resources

Footnotes

Graph View

Table of Contents

Backlinks