Detecting axions seems to be extremely challenging. It is believed that the particle interacts very weakly with ordinary matter and has a very low mass — similar to a ghost particle. However, some spectacular experiments are currently being conducted to find this elusive particle. One experiment is attempting to “shine light through walls” by exposing a laser beam that contains photons to a 9 Tesla magnetic field. Another experiment is striving to find axions produced in the sun using a special kind of telescope. Moreover, another experiment is aiming to detect a postulated miniscule power source in a microwave cavity.

Some major challenges in the design of an experiment to detect axions are that no one knows the particle’s mass or how weakly it couples with ordinary particles. As a matter of fact, the range of predicted masses spans many orders of magnitude (anything from 1 μeV to 1 eV). Masses outside this range are considered less likely due to theoretical and astrophysical considerations. In addition, the so-called *coupling constant*, which determines how much the axion interacts with ordinary matter, is also unknown.

Several axion experiments are based on the prediction that axions and photons are converted into each other when subjected to a strong magnetic field. The wide range of predicted masses then translates into detecting electromagnetic radiation, or photons, in a span of frequencies. Each team is hoping to find the axion in a certain interval of possible masses and coupling constants, and they are constructing experimental equipment accordingly. Depending on the predicted axion properties, completely different techniques are used.

The Axion Dark Matter eXperiment (ADMX) takes place at the Center for Experimental Physics and Astrophysics at the University of Washington. The experiment uses a resonant cylindrical microwave cavity within a large superconducting magnet. If the axions turn out to have low masses, they may show up as microwaves detectable by the microwave cavity. The experiment can detect low-mass axions in the range of 1 μeV to 10 μeV. This mass is tiny compared to the electron, which weighs 0.5 MeV, or about a million million times more. The corresponding power spike that the ADMX needs to be able to detect is correspondingly puny, which sets almost impossible requirements for experimental equipment.

*Gravitational lensing of a galaxy cluster, which indicates the presence of dark matter.*

The innovative ADMX microwave receiver has an extremely low signal-to-noise ratio and uses a Superconducting Quantum Interference Device (SQUID) amplifier that’s cooled by liquid helium. The receiver is able to detect power spikes smaller than one percent of a yoctowatt (yoctowatt = 10^{-24} watt). This is an unbelievable level of sensitivity that’s close to the theoretical limit set by quantum mechanical fluctuations.

In a way, this experiment cannot fail. Finding the axion in the ADMX would be great, of course. However, proving that the axion is not within its search range would in itself count as a very important research result that would have implications on particle physics and astrophysics. There is also a chance that the ADMX could detect some even more exotic particles, such as chameleon particles or dark photons.

*The installation of the Tesla ADMX magnet. (“The 8.5 Tesla ADMX magnet being installed at the University of Washington, Seattle” by Lamestlamer — Own work. Licensed under Creative Commons Attribution Share-Alike 3.0, via Wikimedia Commons).*

Microwave cavities are used in many types of microwave applications such as radars, cell phone stations, and microwave ovens. They are also used as resonators because of their excellent ability to store electromagnetic energy.

In the world of electrical circuits, the “cousin” of the microwave cavity is the resonant *RLC-circuit*, which consists of resistive, inductive, and capacitive circuit elements. A resonant circuit can be made to resonate at a particular frequency by tuning its capacitors and inductors (the principle behind a radio tuner). In a similar way, a microwave cavity can be tuned to the frequency of the axion (or rather the photon that the axion is converted into).

Tuning a microwave cavity is made possible by using *tuning rods* — metallic or dielectric rods that protrude into the cavity. By simply changing the position of the tuning rods, you change the resonant frequency, or the “radio station”, of the cavity.

Predicting exactly what change in resonant frequency that a particular alteration in the tuning rod position will give is made easier using simulation. We simulated this scenario using the RF Module.

*The CAD geometry used in the COMSOL Multiphysics simulation. The cylindrical microwave cavity and the two metallic tuning rods are shown.*

We created both a 3D and 2D model of the cavity to compare results. The resonant mode that’s expected to couple to the axion-generated photon the strongest is the so-called *TM010 mode*. To find this particular mode, employing a 2D simulation is just as suitable as a 3D simulation.

The figure below shows the electric field distribution in the 3D model for the TM010 mode.

*The TM010 mode showing the electric field component aligned with the axis of the cylinder. The normalized field magnitude is plotted on three perpendicular slices.*

The superconducting magnet’s externally applied magnetic field has a strong magnetic flux component along the axis of the cylinder. If we assume that the cylinder axis is the *z*-axis, then we can represent the magnetic flux as approximately B = (0,0,Bz). Using this definition of the *z*-axis, the resonant TM010 mode is characterized by having a strong electric field component (Ez). So, with good approximation, we have E = (0,0,Ez). The field power is proportional to the dot product of E and B (Ez*Bz). The design of the experiment is meant to maximize this coupling and get the strongest possible signal if an axion-generated photon is created in the cavity.

The figure below shows the corresponding 2D simulation.

*The electric field in the 2D simulation of the cavity.*

The 2D and 3D simulations gave identical results. To understand the TM010 mode, it’s sufficient to use a 2D simulation, which is computationally much faster.

The figure below shows a simulation where the resonant frequency is plotted against the angular position of one of the tuning rods.

*Resonant frequency vs. rod position for the cylindrical cavity simulation.*

This simulation shows that this particular cavity design could be used for a search in the interval between ~500-700 MHz.

These results are similar to the results published by the AMDX team. One difference is that the cavity and tuning rods used in the simulation do not have the same properties and exact dimensions as in the ADMX.

Strictly speaking, the classification of a resonant mode being TM010 is only applicable for a cavity that does not contain tuning rods. As a matter of fact, the tuning rods introduce other modes similar to TM010. However, one doesn’t have to rely on the fundamental mode to get a coupling between the magnetic and electric fields. Other modes may also give a reasonable coupling.

The figure below shows a scan that includes an adjacent mode similar to TM010. Its sensitivity is in the range between ~740-800 MHz. The figure also includes a comparison to the 3D simulation.

*Resonant frequency vs. rod position for two adjacent modes. A comparison of the 3D simulation is included.*

It should be noted that even higher-order modes can be used. By using these, one could potentially scan a wider frequency range without having to change the overall cavity dimensions.

According to a recent presentation, the ADMX recently had major upgrades. The university’s experiment team is ready to begin what it refers to as “the definitive search for dark matter axions”. In the presentation, ADMX representative Gray Rybka confidently states that “if the axion is out there, we will find it”.

- Read this article: “Could dark matter be hiding in plain sight in existing experiments?“
- Watch this video on dark matter axions

Although it is possible to set up and solve a 3D model of a conical horn antenna, such a model would require a relatively large amount of computational resources to solve. We can solve for the electromagnetic fields much more quickly by exploiting the symmetry of the structure. Because we are dealing with a cone, our model is structurally symmetric around its axis, i.e., it’s *axisymmetric*.

Now, although the structure is axisymmetric, the electromagnetic fields will have some variation around the azimuth of the axis, that is, the fields have an *azimuthal variation*. The RF Module and the Wave Optics Module allow you to model axisymmetric structures with different azimuthal mode numbers.

We can exploit this feature; by building a 2D axisymmetric model and solving for several different azimuthal mode numbers, we can build a model that solves much quicker and uses less memory than a full 3D model. I like the sound of that. But first, a quick note on horn antennas.

There are various types of horn antennas in terms of both overall shape and pattern of the inside. These attributes determine the antenna’s beam profile, bandwidth, and cross-polarization.

Cross-polarization means that the electromagnetic fields are polarized opposite to what is intended. For example, we want the fields to be polarized vertically, but they are instead polarized horizontally.

The funnel part of the antenna is connected to a waveguide, which feeds electromagnetic waves into the antenna. The shape of the horn will dictate what application it’s suited for. For example, sectoral horns (labeled *b* and *c* in the image below) are typically used for wide search radar antennas.

*Various horn antenna shapes: a) pyramidal; b) sectoral, E-plane; c) sectoral, H-plane; d) conical; e) exponential. “Horn antenna types” by Chetvorno — Own work. Licensed under Creative Commons Zero, Public Domain Dedication via Wikimedia Commons.*

The antenna in our case is both shaped like a cone (labeled *d* in the image above) and has a corrugated surface inside; it’s a *corrugated conical horn antenna* fed by a circular waveguide. The waveguide passes the excited TE mode through the corrugated funnel, which, in turn, generates a TM mode. Due to the corrugated surface throughout the cone, the modes are mixed, leading to a lower cross-polarization at the aperture than the original excited TE mode.

*Conical horn antenna: A visualization in 3D based on a 2D axisymmetric model. The waveguide feeds the antenna with the TE _{1m} mode (*m

Above, I mentioned what cross-polarization is, but why would we want to reduce it? Well, if we have a lot of cross-polarization, our signal may interfere with other channels nearby, if they have alternating vertical and horizontal polarization. We wouldn’t want that.

To investigate the cross-polarization, we can use COMSOL Multiphysics and the RF Module to set up a model. As we learned earlier, we can save time by solving this as a 2D axisymmetric model instead of in 3D. We can do that by using the *Electromagnetic Waves, Frequency Domain* interface.

I will skip over the step-by-step model set-up and head straight to the fun stuff — the results. If you want to reproduce the plots shown here, feel free to download the model documentation and MPH-file from the Model Gallery.

First, we can see what the directive beam pattern of the antenna is:

*Far-field plot: The directive beam pattern of the antenna.*

Next, we can analyze the electric field at the antenna’s entrance and exit. By solving the model for both *m* = +1 and *m* = -1, we can compare the linear polarization in the *x*- and *y*-direction at the exit.

*Electric field at the entrance and exit of the antenna for the linear superposition of *m *= +1 and *m *= -1*.

At the waveguide feed, the field is mostly in the *x*-direction, but not linearly polarized. At the aperture, the field is very nearly linearly polarized. To quantify the polarization in both directions, we can evaluate the integral of the absolute value of each field component over the conical horn antenna’s entrance and exit. Doing so, we’ll find that the ratio is roughly 5:1 at the entrance and about 40:1 at the exit. In other words, we have reduced the cross-polarization by approximately a factor of 8.

Frequency selective surfaces (FSS) are periodic structures that function as filters for plane waves, such as microwave frequency waves. These structures can transmit, absorb, or reflect different amounts of radiation at varied frequencies. Typically, they have a bandstop or bandpass frequency response.

Frequency selective surfaces are used in a variety of applications. For example, the article “Picking the Pattern for a Stealth Antenna” from COMSOL News 2013 describes how engineers at Altran used FSS as RF filters to reduce the radar cross sections (RCS) of stealth antennas. In the article, designers employed FSS to reduce antenna gain in order to lower the RCS. There, the FSS were designed to absorb incident radiation, rather than scatter it. FSS surfaces are typically constructed with metallic patterns that are arranged periodically. Complementary split ring resonators can be used to build such structures.

As a type of planar resonator, complementary split ring resonators are primarily used to simulate metamaterial elements. When designing a bandpass structure, for example, they can be arranged periodically. Modeling these resonators in a periodic configuration can become quite complex and time-consuming. However, you can overcome these design challenges by implementing periodic boundary conditions into your model.

COMSOL Multiphysics, together with the RF Module, enables you to model a periodic complementary split ring resonator with ease by utilizing perfectly matched layers and periodic boundary conditions. As an example, we can refer to the Frequency Selective Surface, Periodic Complementary Split Ring Resonator model, which is available in our Model Gallery.

In this model, a split ring slot is patterned on a thin copper layer (which is modeled as a perfect electric conductor) that rests on a PTFE substrate that is 2 mm in thickness.

*A single unit cell of the complementary split ring resonator. The model is created with periodic boundary conditions.*

To simulate a 2D infinite array, as shown above, you can model just one unit cell of the complementary split ring resonator. This is done using Floquet-periodic boundary conditions on each of the four sides of the unit cell.

To learn how to do this, check out the model documentation, where step-by-step instructions are provided.

While this post mostly focuses on how you can save time modeling using periodic boundary conditions, this particular model’s documentation goes into further detail regarding the periodic structure’s bandpass frequency response in terms of S-parameters, as shown below.

*S-parameter plot showing the periodic structure functions as a bandpass filter near 4.6 GHz*.

Anechoic chambers are noise-canceling rooms that are designed to absorb sound or electromagnetic waves. *Acoustic anechoic chambers*, which typically have noise levels of around 10-20 dBA, can be used to test loudspeakers and the directivity of noise radiation, the sound quality of certain products (such as a Harley-Davidson), as well as to decrease the noise level produced by certain products (washing machines, computer fans, etc.).

First developed during World War II as a way to test aircraft that absorbed or scattered radar signals, *RF anechoic chambers* are still used today for a variety of different purposes. Like acoustic anechoic chambers, an RF anechoic chamber provides a space where no incident energy waves are present, allowing for devices to be tested without interference. Examples include satellites and the antenna performance of devices such as cell phones, RFID tags, and GPS. When testing on-board aircraft systems, these chambers can be large enough to house the entire aircraft itself.

The image below shows the cones, or *pyramidal absorbers*, that line the walls, ceiling, and floor of an RF anechoic chamber at the Surface Sensors and Combat Systems Facility at Naval Surface Warfare Center (NSWC) Dahlgren.

*Testing a maritime antenna in an anechoic chamber.*

You can easily model an RF anechoic chamber using COMSOL Multiphysics and the RF Module with perfectly matched layers and periodic boundary conditions. Let’s explore the one of our models from the Model Gallery: Using the Modeling of Pyramidal Absorbers for an Anechoic Chamber.

In our model example, pyramidal lossy structures (absorbers) are placed in an infinite array. When an incident wave strikes one of the pyramidal structures, many small reflections are created as the electromagnetic wave passes into the pyramid and is reflected into a second pyramid. The absorber is made of radiant-absorbent (RAM) material, which means that as the wave strikes the pyramid, the incident field is partially *reflected* and partially *transmitted* into the nearby absorber. Therefore, after many reflections and partial transmissions, the wave’s amplitude is drastically reduced by the time it reaches the base of the pyramid. In the model, we imitate the microwave absorption of a conductive carbon loaded-foam with a conductive material at σ = 0.5 S/m.

Below, you can see the infinite 2D array of the pyramidal structures. The structure can be modeled using *Floquet-periodic* boundary conditions on four sides, where one unit of the model contains the pyramidal structure as well as the block beneath it made of the same material. Above the pyramid is a perfectly matched layer (PML) and the space above and between the pyramid and PML is filled with air. At the bottom of the pyramid and block is a layer of highly conductive material that is used to block any noise from outside the chamber (shown in orange in the image above). This layer is modeled as a perfect electric conductor (PEC) in our model.

In the plot below on the left, the norm of the electric field and power flow in the model is shown for the case where the angle of incidence on the pyramid is 30°. As we can see, the strength of the incident wave is strong near the tip of the absorber and deceases toward the base of the pyramid. On the right, the graph displays the scattering parameter (S-parameter) for *y*-axis polarized waves as a function of incident angle.

- Download the Modeling of Pyramidal Absorbers for an Anechoic Chamber model to learn how to build it yourself

Behind discoveries such as the tau neutrino and bottom quark is a continually evolving chain of particle accelerators that has consistently provided high-intensity beams for particle physics experiments at Fermilab since the 1970s. At an intermediate stage of the accelerator chain is a cyclic particle accelerator (shown in the figure below), which is located about 20 feet below ground and is known as the Booster synchrotron. As its name implies, this synchrotron *boosts* the energy of incoming particles by a factor of 20 before they are transferred to the main injector or underground beam lines for experiments.

*The Fermilab accelerator chain showing the location of the Booster synchrotro*n.

Presently, there are 19 ferrite-tuned RF cavities spaced throughout the 474-meter Booster synchrotron tunnel. These RF cavities are the workhorses responsible for accelerating an incoming beam of positively charged protons, while additional bend magnets keep the particles moving in a circular trajectory. The RF cavities of the Booster currently support a repetition rate of 7 Hz, which indicates how often particles are generated and sent through the accelerator chain.

*A copper ferrite-tuned RF cavity removed from the Booster synchrotron (left) and one of the ferrite tuners (right).*

The Booster RF cavities, shown in more detail in the following diagram, are half-wave resonators designed with a size and shape that enables operation over frequencies from 37 MHz to 53 MHz.

*Front- and side-view drawings of a Booster RF cavity.*

Around the exterior of the cavity, you’ll find a tetrode power amplifier supplying the RF signal in addition to three coaxial ferrite tuners providing a variable load. By using three ferrite tuners spaced at 90-degree intervals around the cavity, the power loss density per tuner is reduced, thus aiding in keeping the operating temperature lower.

Protons cycling through the Booster enter each RF cavity in succession, where they are accelerated along the central beam pipe by oscillating electromagnetic fields. Protons will continue to cycle through until they reach their target energy of 8 GeV. To reach this target energy, the frequency of oscillation is rapidly increased by varying the bias on the ferrite tuners.

With upcoming experiments at the lab requiring higher intensity beams, the Proton Improvement Plan (PIP) is currently being enacted under the leadership of William Pellico and Robert Zwaska to ensure that the particle accelerators can meet these demands. Modernizing the particle accelerators, including the Booster, will enable them to produce and sustain particle beams at double the current intensity through 2025. For the Booster RF cavities, this means they will need to be upgraded to operate at a 15 Hz repetition rate — that’s double the current rate — and possibly at a higher accelerating voltage.

From the Accelerator Division at Fermilab, John Reid is coordinating the challenging process to refurbish, qualify, and test the unique ferrite-tuned RF cavities of the Booster in accordance with the PIP. Mohamed Awida Hassan and Timergali Khabiboulline, from the Superconductivity and Radiofrequency Development Department, are working together with Reid to ensure that electromagnetic losses do not lead to overheating of the RF cavities and tuners reducing their lifetime. In a recent interview and article about their work, Hassan explains that they “are using both multiphysics simulation and physical measurements, provided by colleagues in the Accelerator Division, to evaluate the RF, thermal, and mechanical properties of the Booster RF cavities.”

*The engineers featured in this article kindly gave up a few minutes of their time for a photograph in the Booster tunnel while the synchrotron was down for routine maintenance. From left to right they are: Robert Zwaska, William Pellico, Mohamed Hassan, and Timergali Khabiboulline. (John Reid was unavailable for the photograph that day.)*

Doubling the hourly proton yield from the Booster synchrotron places new demands on the 40-year-old ferrite-tuned RF cavities. Increasing the repetition rate and accelerating voltage is projected to increase the power dissipated in the RF cavities and tuners, potentially causing them to overheat. Physical measurements of temperature, although they are performed, are difficult to acquire and often inaccurate. Therefore, measurements are used in conjunction with thermal analysis in COMSOL Multiphysics® to estimate the additional cooling requirements of the upgraded RF cavities to ensure stable long-term proton production.

Hassan and Khabiboulline set up the RF cavity model as shown in the following screenshot of the COMSOL Desktop®.

*Screenshot showing the setup and analysis for a multiphysics model of a Booster RF cavity with ferrite tuners. A surface plot of temperature is shown in the Graphics window.*

In their model, RF analysis was initially performed to solve for the electric and magnetic fields that are used as heat sources in the thermal analysis. The results from their analysis confirm that a temperature increase of more than 40°C could occur when going from an accelerating voltage of 55 kV and repetition rate of 7 Hz to 60 kV and 15 Hz.

To help reduce the time, risks, and expense associated with upgrading the RF cavities, the simulation results are being used to facilitate design decisions with regard to the cooling mechanism. It is anticipated that an upgraded Booster will aid in the production of more protons for another decade of intense physics discovery.

Want more details about their simulations? Read “Doubling Beam Intensity Unlocks Rare Opportunities for Discovery at Fermi National Accelerator Laboratory” appearing on page 12 in the 2014 edition of *Multiphysics Simulation*, an *IEEE Spectrum* insert.

- Explore the RF Module
- Read “Multiphysics Analysis of the Fermilab Booster RF Cavity” presented at IPAC 2012 by Hassan et. al
- Key discoveries at Fermilab
- Particle accelerators at Fermilab

When energy from electromagnetic fields is transformed into thermal energy, RF heating occurs. There are two different types of RF heating methods: induction and dielectric. Induction heating takes place in materials with a high electrical conductivity, such as copper or other metals, for instance. Eddy currents are induced by the alternating electromagnetic field and the resistive losses heat up the material. Dielectric heating, on the other hand, happens in — you guessed it — nonconducting material, when it’s subjected to a high-frequency electromagnetic field. The alternating electromagnetic field causes the dielectric molecules to flip back and forth and the material to heat up due to internal friction.

The example we will go through here involves both induction and dielectric heating.

To redirect microwaves passing through a waveguide, you can add a bent section. This is appropriately referred to as a *waveguide bend*. When you have such a bend in between two straight, rectangular waveguides, it will look something like this:

*Schematic of an aluminum waveguide with a bend. The top is deliberately omitted in order to reveal copper coating and a dielectric block inside.*

As you can see, our particular waveguide not only includes a bend, but also a lossy dielectric block (an insulator). That may seem a bit odd; in reality, you might have a tuner or resonator or something in there instead. The block in our example is there to demonstrate in a simple way how we would model microwave heating.

We pass electromagnetic waves in through one end of the waveguide (the end farthest away from the dielectric block) via a power source of 100 watts. The waves oscillate at 10 GHz and move along the rectangular waveguide, around the bend, and then come in contact with the insulator before exiting our model through the other end.

Note that the waveguide in our model is assumed to continue ad infinitum.

We want to figure out how the waveguide and block heat up over time.

We can solve this in two stages using COMSOL Multiphysics and the RF Module:

- Electromagnetics
- Thermal

Let’s fast forward to the results.

First, we want to find out how the waveguide heats up after turning on the power source.

Next, we can study the electromagnetic fields and temperature for a steady-state solution after the assembly has reached thermal equilibrium. Via the “View” node in the Model Builder, we can hide geometric entities. By hiding the top two layers of the waveguide, we’ll get a much clearer picture of what’s going on inside:

*Inside the waveguide: The temperature of the dielectric block as well as the electric (red arrows) and magnetic fields (green arrows) and power flow (blue arrows) are displayed*.

Here’s a closer look at the magnetic fields and dielectric block:

We can also study the electromagnetic fields before and after heating up.

Before heating:

The block’s material properties are functions of temperature, so when it heats up, the electric properties change. Below, the loss tangent in the block is plotted for the steady-state solution. The nonuniform loss tangent distribution is the result of a nonuniform temperature distribution.

You can model RF and microwave heating with COMSOL Multiphysics and the RF Module. The model featured here can be downloaded either through the software (via the RF Model Library, under “Microwave Heating”) or the Model Gallery.

Note that the model can be opened with the geometry, mesh, materials, and such pre-loaded, but you will still need to go through the solution steps. In other words, if you’re following the model documentation, you can skip all the steps except for compute.

RF coils are electromagnetic (EM) coils that transmit and receive radio frequency (RF) signals. The transmitter coil generates EM fields, while the receiver coil, as the name suggests, receives them. RF coils can be found in various devices, such as magnetic resonance imaging (MRI) systems, for example. In MRI scanning equipment, radio waves and uniform, strong magnetic fields are used to form images of the patient’s anatomy. RF coils are also used for impedance matching of predominantly capacitive devices such as RF plasma reactors for semiconductor manufacturing.

*An RF coil.*

An important characteristic of an RF coil is its frequency response, especially the *resonance frequency*; the frequency at which the reactive part of the impedance between the input and output of a coil becomes zero.

One of our introduction tutorials for RF modeling involves an RF coil model. Using COMSOL Multiphysics and the RF Module, we can analyze the signal strength of an RF coil by solving for its resonance frequency and Q factor, among other properties.

Let’s have a look at a copper coil made of two turns.

First, we want to find the fundamental resonant frequency, i.e. the lowest frequency of the coil. We do so by performing an eigenfrequency analysis of the model geometry that is shown below.

*RF coil geometry.*

We will consider the coil to be a perfect electric conductor, so we will only need to solve the eigenfrequency equation for the electromagnetic waves in the surrounding air domain.

Next, we want to find the lowest eigenfrequency of the model. We use a lumped port to apply a time-harmonic driving port voltage between the two ends of the coil. The port is assigned a 50 Ω external cable impedance and a driving voltage of 1 volt (V).

*Mesh used in a driven version of the model. A slice is cut out and the air domain is made invisible to show the coil.*

Based on our results in Version 1, we run our model through a range of frequencies around the resonance frequency. After conducting our eigenfrequency analysis, we find that the lowest eigenfrequency is 180 MHz.

Then, we can obtain the distribution of electric and magnetic fields at this resonance:

*Simulation of the electric field (slice) and magnetic flux density (arrows) at the fundamental resonant frequency.*

Because this particular coil transmits or receives signals through resonance, it’s important to determine the *Q factor*, i.e. the quality factor. The Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is. It is determined by the ratio of peak frequency to the full width at half maximum for the frequency response of the coil impedance. A higher Q factor indicates that oscillations die out more slowly and the coil will be more efficient at the resonance frequency. However, it also indicates that the coil has a narrow bandwidth, so it will be more sensitive to any frequency mismatch.

The Q factor can be extracted by plotting the input impedance versus frequency. Electrical impedance describes the total resistance to an alternating current by an electric circuit. Using our previous results, we continue our analysis and solve for the impedance of the model across a range of frequencies around the resonant frequency of 180 MHz.

*Plot of input impedance magnitude versus frequency, where f _{0} is the peak or resonance frequency and Δf is the full width at half maximum.*

The Q factor for our model is around 300, which confirms that the system is not under-damped and will exhibit sufficient frequency bandwidth.

- The tutorial model shown here can be accessed from both the Model Gallery and the Model Library with step-by-step instructions and an MPH-file.

First, let’s look at one of the most common transmission line structures, the coaxial cable. An inner and outer conductor are separated by a dielectric, and a wave will travel along the length of the cable in a TEM mode, meaning that the electric and magnetic field are transverse to the direction of propagation.

If we can assume that the conductors and the dielectric are lossless (a good approximation for many cases), we can compute an impedance, as demonstrated in the Model Gallery benchmark example on Finding the Impedance of a Coaxial Cable (the model can also be found in the Model Library).

In that example, we draw a 2D cross section of the coax and specify the dielectric properties as well as an operating frequency below the cutoff frequency for any TE or TM modes. The COMSOL software will then solve an eigenvalue problem for the out-of-plane propagation constant as well as the fields, which can be used to compute the impedance of the cable. This approach is very efficient in terms of computation, but only works for TEM waveguides of uniform cross section.

Now, let’s consider a coaxial waveguide with a corrugated outer conductor. These are used when mechanical flexibility is desired.

*A corrugated coaxial cable. The slice plot is of the electric field and the arrow plot is the magnetic field.*

Such waveguides will not be operating in a purely TEM mode, meaning that there is some electric and magnetic field component in the direction of propagation. However, we will assume that these components are small and can, as a consequence, define the impedance as:

Z_0=\frac{\left|V\right|^2}{2P}

where V is the voltage that can be evaluated by taking the path integral of the electric field at any line between the inner and outer conductor, and P is the integral of the Poynting flux at any cross section. You can use *Integration Coupling Operators* to evaluate the fields at the edges and boundaries of the modeling domain to evaluate these quantities.

Rather than computing a long section of the waveguide, we can consider only one periodic section of the structure itself. But the effective wavelength of the signal traveling down the waveguide will be much longer than this, so we use the *Floquet periodic boundary condition* to specify that the wave traveling down the waveguide has a specified propagation constant.

*The Floquet periodic boundary condition interface.*

Via this approach, we can then solve an eigenvalue problem to compute the frequency of the wave that will have this propagation constant. When using a periodic boundary condition, we also need to ensure that the mesh on the boundaries is periodic.

*The Copy Face feature will ensure that the mesh on the periodic faces is identical. The interior is then meshed with free tetrahedral elements.*

Once the solution is computed for a single unit cell, we can evaluate the impedance at that frequency. We can also sweep over a range of effective wavelengths to compute the impedance over a range and observe that at higher frequencies, the impedance will go up. This means that we are approaching the frequency at which TE or TM modes will be present. At that point, we can no longer use this approach.

Here, we have shown that you can compute an impedance for a waveguide with periodic structure operating in the quasi-TEM regime. The *Floquet Periodic* boundary conditions and the *Copy Face* functionality are used to set up a unit cell model, which is solved to extract the impedance for a range of frequencies.

If you have questions about this type of modeling, please contact us.

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Branch line couplers are used for either splitting or combining power. This type of coupler is made up of two sets of coupled ports with a phase difference of 90° between them. The power enters through one input port and is then divided equally between two output ports. The fourth port is isolated.

*Geometry of a branch line coupler. In the image, transmission lines are horizontal and branch lines are vertical.*

Branch line couplers are designed to be symmetrical, as shown in the geometry above, which means that you should be able to select any of the ports for the input power and still achieve the same behavior. In the geometry depicted above, the bottom-left port is chosen as the input port, but we could just as well have swapped the input and isolated ports, or flipped it so the two outputs were on the left and the input and isolated ports on the right.

In the case of single-antenna transmitter/receiver systems, you have a transmitter at the input port (1), a 50 ohm termination at port (2), an antenna at one output (3), and a receiver at the isolated port (4). Here, the power moves in quadrature phases from the transmitter to the antenna, and then, the antenna sends a signal back to the receiver. The receiver and transmitter are unaffected by each other.

There are a few important criteria your branch line coupler design will need to meet in order to function:

- High isolation
- Should be as high as possible, but note that it can never actually be 100%

- Ability to set any ports as input, isolated, and outputs and still achieve the same frequency output
- Quadrature phase difference (S-parameters)

If you want to design an accurate branch line coupler, you can use COMSOL Multiphysics along with the RF Module to set up a model. To help you get started, we have included an example model in both the Model Gallery online and the Model Library in the RF Module (you can find it under “Passive Devices”).

Following the instructions in the model documentation, you will be able to compute the S-parameters and evaluate the phase difference to ensure it is 90 degrees. You will also be able to analyze the matching, isolation, and coupling at the ports around the operating frequency.

*Plot showing that the input power (red) is divided equally between the two output ports (yellow). The isolated port is blue.*

- Learn how to build the Branch Line Coupler model
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Sunlight is essentially incoherent light; it is composed of many wavelengths and varying polarizations. However, we can assume linearity of the electromagnetic fields, so any polarization of light can be treated as the sum of two orthogonal polarizations — one that has the *electric field* polarized parallel to the plane of the interface, and the other that has the *magnetic field* parallel to the plane of the interface.

When a ray of light (an electromagnetic wave) propagating through free space hits a dielectric medium, part of the light will be transmitted and part will be reflected. The fraction of the light that is reflected or transmitted is dependent upon the angle of incidence, the permittivity of the dielectric, and the polarization. This can also be described by the Fresnel equations, which are an analytic solution to Maxwell’s equations.

*Schematic showing light incident upon a dielectric interface. The angle of incidence is denoted by θ. Part of the light will be transmitted and part will be reflected.*

Instead of solving the Fresnel equations, we can build a COMSOL model to simulate an infinite plane wave of light incident upon a dielectric medium. Using either the RF Module or the Wave Optics Module, we can build a unit cell describing a small region around the dielectric interface. We solve the full Maxwell’s equations in the unit cell, with periodic boundary conditions and ports to truncate the modeling domain.

Let’s take a look at the results of our benchmark model, which solves for two orthogonal polarizations of light and computes the reflection and transmission coefficients with respect to incident angle.

*The electric field in the y-direction (surface slice plot) and the power flow (arrow plot).*

*The magnetic field in the y-direction and the power flow. Both are shown for θ = 70°.*

*Comparing COMSOL model results with analytic solution for reflectance and transmittance for electric field incidence (left) and magnetic field incidence (right).*

As you can see in the above plots, the benchmark model results agree with the analytic solution. We can also see that different polarizations of light will reflect differently off of an air-dielectric interface, and this tells us why polarized sunglasses are popular with boaters!

First, let’s consider a parallelepided volume of free space representing a periodically repeating unit cell with a plane wave passing through it at an angle, as shown below:

The incident wavevector, \bf{k}, has component magnitudes: k_x = k_0 \sin(\alpha_1) \cos(\alpha_2), k_y = k_0 \sin(\alpha_1) \sin(\alpha_2), and k_z = k_0 \cos(\alpha_1) in the global coordinate system. This problem can be modeled by using Periodic boundary conditions on the sides of the domain and Port boundary conditions at the top and bottom. The most complex part of the problem set-up is defining the direction and polarization of the incoming and outgoing wave.

Although the COMSOL software is flexible enough to allow any definition of base coordinate system, in this posting, we will pick one and use it throughout. The direction of the incident light is defined by two angles, \alpha_1 and \alpha_2; and two vectors, \bf{n}, the outward pointing normal of the modeling space and \bf{a_1}, a vector in the plane of incidence. The convention we choose here is to align \bf{a_1} to the global *x*-axis and align \bf{n} with the global *z*-axis. Thus, the angle between the wavevector of the incoming wave and the global *z*-axis is \alpha_1, the *elevation angle of incidence*, where -\pi/2 > \alpha_1 > \pi/2 with \alpha_1 = 0, meaning normal incidence. The angle between the incident wavevector and the global *x*-axis is the *azimuthal angle of incidence*, \alpha_2, which lies in the range, -\pi/2 > \alpha_2 \geq \pi/2. As a consequence of this definition, positive values of both \alpha_1 and \alpha_2 imply that the wave is traveling in the positive x- and y-direction.

To use the above definition of direction of incidence, we need to specify the \bf{a_1} vector. This is done by picking a *Periodic Port Reference Point*, which must be one of the corner points of the incident port. The software uses the in-plane edges coming out of this point to define two vectors, \bf{a_1} and \bf{a_2}, such that \bf{a_1 \times a_2 = n}. In the figure below, we can see the four cases of \bf{a_1} and \bf{a_2} that satisfy this condition. Thus, the *Periodic Port Reference Point* on the incoming side port should be the point at the bottom left of the *x-y* plane, when looking down the *z*-axis and the surface. By choosing this point, the \bf{a_1} vector becomes aligned with the global *x*-axis.

Now that \bf{a_1} and \bf{a_2} have been defined on the incident side due to the choice of the Periodic Port Reference Point, the port on the outgoing side of the modeling domain must also be defined. The normal vector, \bf{n}, points in the opposite direction, hence the choice of the Periodic Port Reference Point must be adjusted. None of the four corner points will give a set of \bf{a_1} and \bf{a_2} that align with the vectors on the incident side, so we must choose one of the four points and adjust our definitions of \alpha_1 and \alpha_2. By choosing a periodic port reference point on the output side that is diametrically opposite the point chosen on the input side and applying a \pi/2 rotation to \alpha_2, the direction of \bf{a_1} is rotated to \bf{a_1'}, which points in the opposite direction of \bf{a_1} on the incident side. As a consequence of this rotation, \alpha_1 and \alpha_2 are switched in sign on the output side of the modeling domain.

Next, consider a modeling domain representing a dielectric half-space with a refractive index contrast between the input and output port sides that causes the wave to change direction, as shown below. From Snell’s law, we know that the angle of refraction is \beta=\arcsin \left( n_A\sin(\alpha_1)/n_B \right). This lets us compute the direction of the wavevector at the output port. Also, note that this relationship holds even if there are additional layers of dielectric sandwiched between the two half-spaces.

In summary, to define the direction of a plane wave traveling through a unit cell, we first need to choose two points, the Periodic Port Reference Points, which are diametrically opposite on the input and output sides. These points define the vectors \bf{a_1} and \bf{a_2}. As a consequence, \alpha_1 and \alpha_2 on the input side can be defined with respect to the global coordinate system. On the output side, the direction angles become: \alpha_{1,out} = -\arcsin \left( n_A\sin(\alpha_1)/n_B \right) and \alpha_{2,out}=-\alpha_2 + \pi/2.

The incoming plane wave can be in one of two polarizations, with either the electric or the magnetic field parallel to the *x-y* plane. All other polarizations, such as circular or elliptical, can be constructed from a linear combination of these two. The figure below shows the case of \alpha_2 = 0, with the magnetic field parallel to the x-y plane. For the case of \alpha_2 = 0, the magnetic field amplitude at the input and output ports is (0,1,0) in the global coordinate system. As the beam is rotated such that \alpha_2 \ne 0, the magnetic field amplitude becomes (\sin(\alpha_2), \cos(\alpha_2),0). For the orthogonal polarization, the electric field magnitude at the input can be defined similarly. At the output port, the field components in the *x-y* plane can be defined in the same way.

So far, we’ve seen how to define the direction and polarization of a plane wave that is propagating through a unit cell around a dielectric interface. You can see an example model of this in the Model Gallery that demonstrates an agreement with the analytically derived Fresnel Equations.

Next, let’s examine what happens when we introduce a structure with periodicity into the modeling domain. Consider a plane wave with \alpha_1, \alpha_2 \ne 0 incident upon a periodic structure as shown below. If the wavelength is sufficiently short compared to the grating spacing, one or several diffraction orders can be present. To understand these diffraction orders, we must look at the plane defined by the \bf{n} and \bf{k} vectors as well as in the plane defined by the \bf{n} and \bf{k \times n} vectors.

First, looking normal to the plane defined by \bf{n} and \bf{k}, we see that there can be a transmitted 0^{th} order mode with direction defined by Snell’s law as described above. There is also a 0^{th} order reflected component. There also may be some absorption in the structure, but that is not pictured here. The figure below shows only the 0^{th} order transmitted mode. The spacing, d, is the periodicity in the plane defined by the \bf{n} and \bf{k} vectors.

For short enough wavelengths, there can also be higher-order diffracted modes. These are shown in the figure below, for the m=\pm1 cases.

The condition for the existence of these modes is that:

m\lambda_0 = d(n_B \sin \beta_m - n_A \sin \alpha_1)

for: m=0,\pm 1, \pm 2,…

For m=0 , this reduces to Snell’s law, as above. For \beta_{m\ne0}, if the difference in path lengths equals an integer number of wavelengths in vacuum, then there is constructive interference and a beam of order m is diffracted by angle \beta_{m}. Note that there need not be equal numbers of positive and negative m-orders.

Next, we look along the plane defined by the \bf{n} and \bf{k} vectors. That is, we rotate our viewpoint around the *z*-axis such that the incident wavevector appears to be coming in normally to the surface. The diffraction into this plane are indexed as the n-order beams. Note that the periodic spacing, w, will be different in this plane and that there will always be equal numbers of positive and negative n-orders.

COMSOL will automatically compute these m,n \ne 0 order modes during the set-up of a Periodic Port and define listener ports so that it is possible to evaluate how much energy gets diffracted into each mode.

Last, we must consider that the wave may experience a rotation of its polarization as it gets diffracted. Thus, each diffracted order consists of two orthogonal polarizations, the *In-plane vector* and *Out-of-plane vector* components. Looking at the plane defined by \bf{n} and the diffracted wavevector \bf{k_D}, the diffracted field can have two components. The Out-of-plane vector component is the diffracted beam that is polarized out of the plane of diffraction (the plane defined by \bf{n} and \bf{k}), while the In-plane vector component has the orthogonal polarization. Thus, if the In-plane vector component is non-zero for a particular diffraction order, this means that the incoming wave experiences a rotation of polarization as it is diffracted. Similar definitions hold for the n \ne 0 order modes.

Consider a periodic structure on a dielectric substrate. As the incident beam comes in at \alpha_1, \alpha_2 \ne 0 and there are higher diffracted orders, the visualization of all of the diffracted orders can become quite involved. In the figure below, the incoming plane wave direction is shown as a yellow vector. The n=0 diffracted orders are shown as blue arrows for diffraction in the positive z-direction and cyan arrows for diffraction into the negative z-direction. Diffraction into the n \ne 0 order modes are shown as red and magenta for the positive and negative directions. There can be diffraction into each of these directions and the diffracted wave can be polarized either in or out of the plane of diffraction. The plane of diffraction itself is visualized as a circular arc. Note that the plane of diffraction for the n \ne 0 modes is different in the positive and negative z-direction.

All of the ports are automatically set up when defining a periodic structure in 3D. They capture these various diffracted orders and can compute the fields and relative phase in each order. Understanding the meaning and interpretation of these ports is helpful when modeling periodic structures.

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