Before we get to the rough surface, let’s start with something simple: a thin uniform layer of gold coating on top of optically flat glass, as shown in the image below. Such a model exhibits negligible structural variation in the plane of glass. In addition, it can be modeled quite simply in the COMSOL Multiphysics® software by considering a small two-dimensional unit cell that has a width much smaller than the wavelength.
This computational model is based on the Fresnel equation example, one of the verification models in the Application Gallery, but is modified to include a layer of gold with a wavelength-dependent refractive index. This type of index requires that we manually adjust the mesh size based on the minimum wavelength in each material as well as the skin depth, as described in a previous blog post.
Light incident on a metal coating on top of a glass substrate is reflected, transmitted, and absorbed.
The model includes Floquet periodic boundary conditions on the left and right sides of the modeling domain and a Port boundary condition at the top and bottom. The Port boundary condition at the top launches a plane wave at a specified angle of incidence and computes the reflected light, while the one at the bottom calculates the transmitted light. We can integrate the losses within the metal layer to compute the absorbance within the gold layer.
The computational model that calculates the optical properties of a metal film on glass.
If we are interested in computing incident light at off-normal incident angles, then we also have to concern ourselves with the height of the modeling domain — the distance between the material interfaces and the Port boundary conditions. This distance must be large enough such that any evanescent field drops off to approximately zero within the modeling domain.
The reason for this has to do with the Port boundary conditions, which can only consider the propagating component of the electromagnetic field. Any evanescent component of the field that reaches the Port boundary condition is artificially reflected, so we must place the port boundary far enough away from the material interfaces. In the most general cases, it is difficult to determine how far the evanescent field extends. A simple rule of thumb is to place the Port boundary conditions at least half a wavelength away from the material interfaces and to check if making the domain larger alters the results.
The sample results below show the transmitted, reflected, and absorbed light as well as their total — which should always add up to one. If these do not add up to one, then we must carefully check our model setup.
The transmittance, reflectance, and absorbance of light normally incident on a flat glass surface with a metal coating as a function of wavelength.
The transmittance, reflectance, and absorbance of 550-nm light at various angles of incidence.
Let’s now make things a little bit more complicated and introduce a periodic structural variation: a sinusoidal ripple. Clearly, we now need to consider a larger unit cell that considers a single ripple.
A surface with periodic variations reflects and transmits light into several different diffraction orders.
We can still apply the same domain properties and all of the same boundary conditions. However, if spacing is large enough, then we can have higher-order diffraction. In other words, light can be reflected and transmitted into several different directions. To properly compute the reflection and transmission, we need to add several diffraction order ports. The software computes the appropriate number of ports based on the domain width, material properties, and specified incident angle. If we are studying a range of incident angles, we must make sure to compute all of the diffraction orders present at the limits of the angular sweep.
There can be multiple diffraction orders present, depending on the ratio of wavelength to domain width, refractive index, and incident angles.
The conditions under which higher-order diffraction appears and the appropriate modeling procedure is presented in depth in the example of a plasmonic wire grating, so let’s not go into it at length here. In short, the wider the computational domain relative to the wavelengths in the materials above and below, the more diffraction orders can be present (the number of diffraction orders varies with the incident angle). The results shown below plot the total transmittance and reflectance; i.e., all of the light reflected into the different diffraction orders is added up, as is all of the transmitted light.
The transmittance, reflectance, and absorbance of light normally incident on a rippled glass surface with a metal coating.
The transmittance, reflectance, and absorbance of 550-nm light at various angles of incidence.
Let’s now move on to the most computationally difficult case: a surface with many random variations in the surface height. To model the randomness, we must model several different domains of increasing widths and different subsets of the rough profile. As the domain width increases — and as different subsets of the surface are sampled — the average behavior computed from these different models converges. That is, we generate a set of statistics by sampling the rough surface. Rather than going into detail on how to calculate these statistics, let’s focus on how to model one domain that approximates a rough surface by defining the height variation as the sum of different sinusoids with random height and phase, as described here.
A rough surface with random variations reflects and transmits light in random directions. The computational model must sample a statistically significant subset of the roughness profile.
Our computational domain must now be very wide, many times longer than the wavelength. As we still want to model a plane wave incident at various angles on the structure, we use the Floquet periodic boundary conditions, which require that we have an identical mesh on the periodic boundaries. Practically speaking, this means that we may need to slightly alter the geometry of our domain to ensure that the boundaries on the left and right side are identical. If we do use a sum of sine functions, as described here, then the profile will automatically be periodic.
We still want to launch the wave with a Port boundary condition. However, it is no longer practical to use diffraction order ports to monitor the reflected and transmitted light, as this can result in hundreds (or thousands) of diffraction orders. Furthermore, since this model represents a statistical sampling, the relative fraction of light scattered into these different orders is not of interest; we’re only interested in the sum of the total reflected and transmitted light. That is, this modeling approach computes the total integrated scatter plus the specular reflection and transmission of the surface.
The computational domain for a model of a rough surface. Light is launched from the interior port toward the material interface. Light reflected back toward this port passes though it and is absorbed in the PML, as is the transmitted light. Two additional boundaries are introduced to monitor the total reflectance and transmittance.
Thus, we introduce an alternative modeling strategy that does not use ports to compute reflection and transmission. Instead, we use a perfectly matched layer (PML) above and below to absorb all reflected and transmitted light as well as probes to compute reflection and transmission. PMLs absorb any fields incident upon them, as described in this blog post on using PMLs for wave electromagnetics problems.
The PML absorbs both propagating and evanescent components of the field, but we only want it to absorb the propagating component. Thus, we again need to ensure that we place the PMLs far enough away from the material interfaces. We use the same rule of thumb as before, placing the PML at least half a wavelength away from the material interfaces.
As we approach grazing angles of incidence, even the PML domain does not, by default, absorb all of the light. At nearly grazing angles, the effective wavelength in the absorbing direction is very long, and we need to modify the default wavelength in the PML settings (shown below). This change to the settings is only necessary if we are interested in angles of incidence greater than ~75°.
The PML settings modified to account for grazing angles of incidence.
Since our domain is now bounded by PMLs above and below, the port that launches the wave must now be placed within the modeling domain. To do this, we use the Slit Condition option to define an interior port that is backed by a domain. This means that the port now launches a wave in one direction, emanating from this interior boundary. Any light reflected back toward the boundary passes through unimpeded and then gets absorbed by the PML.
Although this is a good way to launch the wave, we will no longer use the Port boundary condition to compute how much light is reflected, since we would have to add hundreds of diffraction ports, and similarly, we’d need hundreds of ports to compute the total transmittance.
To monitor the total transmitted and reflected light, we instead introduce two additional interior boundaries to the model, placed just in front of the PML domains (shown in the schematic above). At these two boundaries, we integrate the power flux in the upward and downward directions, normalized by the incident power, which gives us the total reflectance and transmittance. To more accurately determine the integral of the power flux at these boundaries, we also introduce a boundary layer mesh composed of a single layer of elements much smaller than the wavelength.
On the incident side, we place this monitoring boundary above the interior port. The launching port introduces a plane wave propagating toward the material interface. The light reflected at the interface passes through this interior port, then moves through the boundary at which we monitor reflectance, and is absorbed in the PML.
The plots below show sample results of the transmittance, reflectance, and absorbance. They are notably different from the smooth surface and periodically varying surface results. Note that the sweep over the angle of incidence terminates at 85° off normal. Of course these plots will look slightly different for each different random geometry case that we run.
The transmittance, reflectance, and absorbance of light normally incident on a rough glass surface.
The transmittance, reflectance, and absorbance of 550-nm light at angles of incidence up to 85° off normal.
Here, we have introduced a modeling approach that is appropriate for computing the optical transmission and reflection from a rough surface. This method contrasts with the approach for modeling a uniform optically flat surface as well as the one for modeling surfaces with periodic variations. The modeling method for rough surfaces can also be used for the modeling of periodic structures that have a very long period, such as when the scattering into different diffraction orders is not of interest.
Modeling truly random surfaces does require some care, as the geometry needs to be altered to ensure that it is periodic. Furthermore, the domain size and number of different random geometries studied must be large enough to give statistically meaningful results. Since this requires solving many different variations of the same model and postprocessing the results, it is helpful to use the Application Builder, LiveLink™ for MATLAB®, or LiveLink™ for Excel® in our modeling workflow.
MATLAB is a registered trademark of The MathWorks, Inc. Microsoft and Excel are either registered trademarks or trademarks of Microsoft Corporation in the United States and/or other countries.
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Sensors, chemical imaging, and optoelectronics: These are just some of the applications in which midinfrared wavelengths are of significance. With their growing importance across various industries and technologies, the need for identifying optical fibers that can produce midinfrared light in a large wavelength range is increasing as well.
One approach is to fuse silica — a typical material for optical fibers — with infrared transparent semiconductors like germanium, zinc selenide, and silicon. To be more specific, silica makes up the cladding, or outer layer, of the optical fiber design and the selected semiconductor makes up its core. This combination provides opportunities for developing new types of midinfrared multimaterial optical fibers.
A multimaterial optical fiber design. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
In order for these optical fiber designs to be effective, an important step is to better understand their optical losses. While this can be time consuming and costly via experiments, simulation tools like the COMSOL Multiphysics® software provide a more efficient route for modeling this behavior and identifying means of optimization. Let’s look at an example from researchers at Pennsylvania State University and Pacific Lutheran University that involves the analysis and design of a germanium-based optical fiber.
For their analysis, the researchers used the RF Module, an add-on product to COMSOL Multiphysics. After defining their respective geometries, they applied refractive indices to both the core and cladding of the optical fiber at a particular wavelength. Along the outside of the cladding, the electric field is said to be zero. The plot below shows the electric field distribution for a characteristic HE_{11} mode that is confined and guided through a 6-μm core diameter made of germanium.
The electric field distribution of a characteristic HE_{11} mode within the core diameter. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
To identify the optimal fiber geometry and operative wavelength range, the research team simulated the mode’s propagation attenuation as a function of wavelength and core diameter. Between 2 and 4 μm, a window of low optical loss occurs. As the wavelength becomes longer, the loss begins to increase. This is due to the evanescent wave extending further into the cladding region and the increase in silica’s extinction coefficient at a longer wavelength. Note that the loss related to this effect is larger when the core diameter is smaller.
Left: The optical losses for the fibers. Right: The electric field distributions of the HE_{11} mode in the core diameter for varying wavelengths. Images by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
For these longer wavelengths, one strategy for reducing optical loss and achieving a wider window of high transmission is to add another layer of material between the germanium and silica. This material needs to have smaller refractive indices than germanium and a smaller extinction coefficient than silica over a large wavelength range. Two good candidates for this are silicon and zinc selenide. In the plot below, we can see the characteristic HE_{11} mode confined and guided through each of these new fiber structures.
The electric field distribution of a characteristic HE_{11} mode within the core diameter when an interfacial layer is added. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
Once again, the researchers simulated the optical loss as a function of the wavelength for a 6-μm core diameter. As the results indicate, introducing the additional layer significantly reduces the optical propagation losses, particularly at longer wavelengths. What’s more: It does so without having to sacrifice the size of the core diameter.
Left: The optical losses for the fibers that include an added layer. Right: The electric field distributions of the HE_{11} mode in the core diameter at a wavelength of 10 μm. Images by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
From the above results, it is clear that the reduction in optical loss is more pronounced in the zinc selenide case. This is because the refractive index between germanium and zinc selenide is greater than that between germanium and silicon, allowing light to be better confined. However, smaller refractive index differences often reduce the constraint of a small core diameter for single-mode guidance.
To account for this, the researchers calculated the single-mode guidance for each germanium-based core diameter configuration. The results show that the zinc selenide structure requires a smaller germanium core diameter to achieve single-mode guidance.
The germanium-based core diameter requirements for the zinc selenide (a) and silicon (b) structures. The gray areas represent conditions for single-mode guidance. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
With the flexibility of COMSOL Multiphysics, the research team was able to easily modify different parameters in their optical fiber design and analyze the impact on fiber performance. From there, they implemented further strategies to minimize optical losses and enable greater transmission. Their optimized fiber design has potential use in the medical field, specifically in endoscopes for spectroscopic imaging.
Implementing the Fourier transformation in a simulation can be useful in Fourier optics, signal processing (for use in frequency pattern extraction), and noise reduction and filtering via image processing. In Fourier optics, the Fresnel approximation is one of the approximation methods used for calculating the field near the diffracting aperture. Suppose a diffracting aperture is located in the plane at . The diffracted electric field in the plane at the distance from the diffracting aperture is calculated as
where, is the wavelength and account for the electric field at the plane and the plane, respectively. (See Ref. 1 for more details.)
In this approximation formula, the diffracted field is calculated by Fourier transforming the incident field multiplied by the quadratic phase function .
The sign convention of the phase function must follow the sign convention of the time dependence of the fields. In COMSOL Multiphysics, the time dependence of the electromagnetic fields is of the form . So, the sign of the quadratic phase function is negative.
Now, let’s take a look at an example of a Fresnel lens. A Fresnel lens is a regular plano-convex lens except for its curved surface, which is folded toward the flat side at every multiple of along the lens height, where m is an integer and n is the refractive index of the lens material. This is called an m^{th}-order Fresnel lens.
The shift of the surface by this particular height along the light propagation direction only changes the phase of the light by (roughly speaking and under the paraxial approximation). Because of this, the folded lens fundamentally reproduces the same wavefront in the far field and behaves like the original unfolded lens. The main difference is the diffraction effect. Regular lenses basically don’t show any diffraction (if there is no vignetting by a hard aperture), while Fresnel lenses always show small diffraction patterns around the main spot due to the surface discontinuities and internal reflections.
When a Fresnel lens is designed digitally, the lens surface is made up of discrete layers, giving it a staircase-like appearance. This is called a multilevel Fresnel lens. Due to the flat part of the steps, the diffraction pattern of a multilevel Fresnel lens typically includes a zeroth-order background in addition to the higher-order diffraction.
A Fresnel lens in a lighthouse in Boston. Image by Manfred Schmidt — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.
Why are we using a Fresnel lens as our example? The reason is similar to why lighthouses use Fresnel lenses in their operations. A Fresnel lens is folded into in height. It can be extremely thin and therefore of less weight and volume, which is beneficial for the optics of lighthouses compared to a large, heavy, and thick lens of the conventional refractive type. Likewise, for our purposes, Fresnel lenses can be easier to simulate in COMSOL Multiphysics and the add-on Wave Optics Module because the number of elements are manageable.
The figure below depicts the optics layout that we are trying to simulate to demonstrate how we can implement the Fourier transformation, applied to a computed solution solved for by the Wave Optics, Frequency Domain interface.
Focusing 16-level Fresnel lens model.
This is a first-order Fresnel lens with surfaces that are digitized in 16 levels. A plane wave is incident on the incidence plane. At the exit plane at , the field is diffracted by the Fresnel lens to be . This process can be easily modeled and simulated by the Wave Optics, Frequency Domain interface. Then, we calculate the field at the focal plane at by applying the Fourier transformation in the Fresnel approximation, as described above.
The figures below are the result of our computation, with the electric field component in the domains (top) and on the boundary corresponding to the exit plane (bottom). Note that the geometry is not drawn to scale in the vertical axis. We can clearly see the positively curved wavefront from the center and from every air gap between the saw teeth. Note that the reflection from the lens surfaces leads to some small interferences in the domain field result and ripples in the boundary field result. This is because there is no antireflective coating modeled here.
The computed electric field component in the Fresnel lens and surrounding air domains (vertical axis is not to scale).
The computed electric field component at the exit plane.
Let’s move on to the Fourier transformation. In the previous example of an analytical function, we prepared two data sets: one for the source space and one for the Fourier space. The parameter names that were defined in the Settings window of the data set were the spatial coordinates in the source plane and the spatial coordinates in the image plane.
In today’s example, the source space is already created in the computed data set, Study 1/Solution 1 (sol1){dset1}, with the computed solutions. All we need to do is create a one-dimensional data set, Grid1D {grid1}, with parameters for the Fourier space; i.e., the spatial coordinate in the focal plane. We then relate it to the source data set, as seen in the figure below. Then, we define an integration operator intop1
on the exit plane.
Settings for the data set for the transformation.
The intop1
operator defined on the exit plane (vertical axis is not to scale).
Finally, we define the Fourier transformation in a 1D plot, shown below. It’s important to specify the data set we previously created for the transformation and to let COMSOL Multiphysics know that is the destination independent variable by using the dest
operator.
Settings for the Fourier transformation in a 1D plot.
The end result is shown in the following plot. This is a typical image of the focused beam through a multilevel Fresnel lens in the focal plane (see Ref. 2). There is the main spot by the first-order diffraction in the center and a weaker background caused by the zeroth-order (nondiffracted) and higher-order diffractions.
Electric field norm plot of the focused beam through a 16-level Fresnel lens.
In this blog post, we learned how to implement the Fourier transformation for computed solutions. This functionality is useful for long-distance propagation calculation in COMSOL Multiphysics and extends electromagnetic simulation to Fourier optics.
Modern optical communication systems commonly use EO routers and macroscale MO devices. Each device, however, has its own drawbacks. EO routers require an electric field and often have operation voltages in the kilovolt range, while macroscale MO devices don’t allow for scalable solutions. The quest for small, low-power optic routing alternatives is thus an important focus for specialized researchers within this field. This search is complicated, as various specializations within physics and engineering are required to address the study and interaction of magnetism, magnetic materials, and light.
Scanning electron micrograph of an unpolished silicon-on-insulator rib waveguide used for optical routers on a chip. Image credit: J. Tioh, “Interferometric switches for transparent networks: development and integration,” 2012, Graduate Theses and Dissertations. Paper 12487.
One potential solution involves integrating optical components onto a silicon substrate to create an MO routing solution on a chip. This option reduces both the size and operation power of the device and can potentially enable new technologies like light processors. But before monolithically integrated MO routers become commonplace, there are still a few hurdles that this technology needs to overcome.
Standard industry practices, for instance, present a challenge when it comes to manufacturing monolithically integrated MO routers. To introduce new technology with minimal industry disruptions, standard practices must be used. In this case, silicon should be used as a base substrate and the combining materials must be compatible with silicon for successful monolithic integration. But bonding silicon and the suitable magneto-optic materials can be quite difficult using standard industry practices due to their crystal structures. As a result, the materials tend to become brittle and crack, significantly increasing optical losses.
Multiphysics simulation offers potential solutions to such challenges. These tools can help the research community identify optimal designs and manufacturing techniques for monolithically integrated MO routers. For his doctoral dissertation at Iowa State University, John Pritchard, an engineer who works within this field, turned to the COMSOL Multiphysics® software to provide new insight into the design and future of on-chip MO routers.
When analyzing an on-chip MO system, Pritchard chose to focus on a few key design elements. One point of focus was analyzing a codirectional coupler, a device that is commonly found in interferometer designs. The power coupling coefficients of a codirectional coupler vary based on the distance between the coupling section length and coupled waveguides. Through his simulation work, Pritchard determined how to generate an ideal power coupler coefficient by choosing a specific coupler length.
3D simulation results for the codirectional coupler. Copyright © John Pritchard.
Another point of analysis was an on-chip optical waveguide. The goal here was to design a rib waveguide that minimizes energy loss and maintains a sufficient beam profile throughout the device. To achieve this, Pritchard used silicon as the rib waveguide’s transmission medium, since it is suitably transparent to infrared light and useful for integration with electronic devices. Further, a low-index cladding model was placed between the substrate and waveguide to stop the optical mode from leaking out.
The optical mode of an SOI rib waveguide. Copyright © John Pritchard.
As for the waveguide’s silicon-on-insulator (SOI) platform, Pritchard used a buried oxide insulator on a silicon substrate. This waveguide configuration enabled him to confine relatively large optical modes in the waveguide and avoid harming the single-mode operation. Subsequent simulations of the configuration revealed that the optical mode is well confined within the waveguide and that this geometry can be used to design an interferometer. Pritchard also performed a frequency analysis of the top view of the design, as seen in the animation below and to the left. This waveguide design was deemed a success and is a significant step toward fully realizing MO routers on a chip.
Left: Wave propagation at the top of a dual waveguide and coupler at 1550 nm. Right: Mode profile of a coupler and dual waveguide. Copyright © John Pritchard.
The coupler and waveguide designs we’ve discussed thus far are ideal for the single-mode confinement of light at 1550 nm. Now, let’s see how adding a top layer of MO material to the SOI rib waveguide affects the device. Specifically, the goal is to find out the amount of light that is exposed to the Faraday rotation. This indicates when light with a rotated state of polarization interferes with nonrotated light.
Mode analysis of an SOI waveguide with a top layer of MO material. Copyright © John Pritchard.
The results, highlighted above, show that the MO material contains 3.9% of the light. Despite being a small percentage, previous research suggests that this creates sufficient Faraday rotation to observe interference at the electrical output. But for this to happen, the material needs to be magnetized with a permanent magnet or controlled magnetic field generator. Finding appropriate monolithically integrated magnetic field generators was therefore a final point of consideration.
While magnetic field generation techniques are important for creating on-chip MO modulators, the process itself is complex. The small size of MO systems makes it difficult to develop monolithically integrated magnetic field generators. To address this, Pritchard used simulation to validate the design of a novel dynamic magnetic field generator: a four-turn integrated coil.
Left: Geometry of an integrated magnetic field generator. Right: Geometry of the integrated magnetic field generator with the MO material highlighted in pink and the silicon waveguide shown in purple. Copyright © John Pritchard.
The results show that the coil generated 260 G near the center of the optical waveguide when energized with a 35 mA current. Within the tested waveguide dimensions, this magnetic field strength can magnetize Ce:YIG film on silicon.
It is possible to expand such research by investigating the field at the center of the MO material, which is part of the core of the four-turn integrated coil. Here, the simulation studies indicate that with a current of 35 mA, the magnetic field at the center of the MO material layer has a maximum field of about 210 G, a reduction possibly explained by the difference in properties of the material. Such findings speak to the potential of on-chip MO routing solutions and can be used as a resource in improving their design and manufacturing processes.
Simulation results for the magnetic field generator at the center of the coil. Copyright © John Pritchard.
Future on-chip optical network architects will have a variety of active switching and routing options, allowing them to make their networks more robust by using both EO and MO devices. While it’s important to note that the results mentioned here are preliminary and more research is needed, the simulations and design methodologies act as a proof of concept for on-chip magnetic field generators and silicon rib waveguides. They can serve as a useful foundation for continued studies on such devices, creating a path for furthering their optimization.
As John Pritchard notes: “Some of the most beautiful connections between light, magnetism, and quantum theory have led to breathtaking technologies ranging from superconductor imaging to gigawatt laser pulses. These have enabled revolutionary inventions in transportation; measurement instruments used to understand the birth of the universe; and, in the near future, optical integrated circuits.” Looking to the future, we are eager to see the continued role of multiphysics simulation in advancing optical research, a field with wide-reaching applications.
In Part 2 of the blog series, we used the Electromagnetic Waves, Frequency Domain interface, which we call a Full-Wave simulation, and a Far-Field Domain node to determine the electric field in the far field. We then coupled a Full-Wave simulation to the Electromagnetic Waves, Beam Envelopes interface (or a Beam-Envelopes simulation) in order to precisely calculate fields in any region, regardless of the distance from the source.
The Far-Field Domain and Beam-Envelopes solutions that we looked at in the previous blog post are effective, but they share one noteworthy restriction. In each case, we assumed that a homogeneous domain surrounded the antenna in all directions. For many situations, this information is sufficient. In other simulations, you may not have a homogeneous domain surrounding your antenna and you need to account for issues like atmospheric refraction or reflection off of nearby buildings. These simulations require a different approach.
A model of several hotels in Las Vegas. A directional antenna emits rays toward the ARIA® Resort & Casino.
The Geometrical Optics interface in the Ray Optics Module, an add-on product to the COMSOL Multiphysics® software, regards EM waves as rays. This interface can account for spatially varying refractive indices, reflection and refraction from complicated geometries, and long propagation distances. However, these features come with a tradeoff. Since waves are treated as rays, this approach neglects diffraction. In other words, we are assuming that the wavelength of light is much smaller than any geometric features in our environment. You can read a more thorough description of ray optics in a previous blog post.
As you may recall, we introduced an approach to coupling a radiating and receiving antenna in Part 3 of this series. When incorporating ray optics into our multiscale modeling, we are required to use a similar but more generalized approach. Before we show you how to set up a geometrical optics simulation in COMSOL Multiphysics, let’s first review this alternate method.
As a quick refresher, we are interested in calculating the fields at the location of the receiving antenna using the following equation:
We previously used an integration operator on a single point to calculate this along the line directly between the two antennas. We now wish to retain the angular dependence, so we need to recalculate this equation for each point in the receiving antenna’s domain. Since it is impractical to add numerous points and integration operators, we need to establish a more general technique.
To do so, we replace the integration operator with a General Extrusion operator. As before, we create a variable for the magnitude of r. We then use the General Extrusion operator to evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point in which we are actually interested.
To demonstrate this concept, we use a figure that is slightly more involved than that from the previous post. Note that the subscripts 1, 2, and r in represent a vector in component 1, a vector in component 2, and the offset between the antennas, respectively.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
As we previously outlined, the primary complication is determining where to calculate the scattering amplitude. We want the fields at the point , which requires calculating the scattering amplitude at . The complication, of course, is that each point in the domain around the receiving antenna (each vector ) will have its own evaluation location . We evaluate this by again rescaling the Cartesian coordinates, but instead of doing it for a single point, we define it inside of the general operator so that it can be called from any location. From the above figure, we know that this point is , with corresponding equations for y and z. The operator is defined in component 1, so the source will be defined in that component. It will be called from component 2, so the x, y, z in the following expressions refer to x_{2}, y_{2}, z_{2} in the above figure.
The General Extrusion operator used for the scattering amplitude calculation. Note that this is defined in component 1.
As a bookkeeping step, we store the calculated fields in a “dummy” variable. By a dummy variable, we mean that we add in an extra dependent variable that takes the value of a calculation determined elsewhere. We do this for two reasons.
The first reason is that most variables in COMSOL Multiphysics are calculated on demand from the dependent variables. In an RF simulation, for example, the dependent variables are the three Cartesian components of the electric field: Ex, Ey, and Ez. These are determined when computing the solution. In postprocessing, every other value (electric current, magnetic field, etc.) is calculated from the electric field when required. In most cases, this is a fast and seamless process. In our case, each field evaluation point requires a general extrusion of a scattering amplitude, and each scattering amplitude point requires a surface integration as defined in the Far-Field Domain node. This can take a while and we want to ensure that we perform this calculation only once.
The second reason why we do this has to do with the element order. The Scattered Field formulation requires a background electric field. COMSOL Multiphysics then calculates the magnetic field using the differential form of Faraday’s law (also known as the Maxwell-Faraday equation). This requires taking spatial derivatives of the electric field. There are no issues when taking the spatial derivatives of an analytical function like a plane wave or Gaussian beam, but it can cause a discretization issue when applied to a solved-for variable. This is a rather advanced topic, which you can find out more about in an archived webinar on equation-based modeling.
By using a cubic dummy variable to store the electric field, we can take a spatial derivative of the electric field and still obtain a well-resolved magnetic field for use in the Scattered Field formulation. Without the increased order of the dummy variable, the magnetic field used would be underresolved. Below, you can see what it looks like to put the General Extrusion operator together with the dummy variable setup. The variable r is identical to the one used in Part 3 of this blog series and is defined in component 2.
The dummy variable implementation. Notice that the dummy variable components are called Ebx, Eby, and Ebz.
The only remaining step is to use the dummy variables — Ebx, Eby, and Ebz — in a background field simulation of the half-wavelength dipole discussed in Part 1 and Part 3.
This technique isn’t actually very good for this particular problem. There may be situations where it is useful, but the technique from Part 3 is preferred in the vast majority of cases. The received power from the two simulations is extremely close, but this method takes much longer to calculate and the file size increases drastically. In the demo examples for this post, this method took several times longer than the previous simulation method. While you may conclude that this is not a terribly useful step overall, it is useful when we incorporate ray optics into our multiscale modeling, as discussed in the next section.
A geometrical optics simulation implicitly assumes that every ray is already in the far field. Earlier in the blog series, we saw that the Far-Field Domain feature correctly calculates the electric field at arbitrary points in the far field. Here, we use that information as the input for rays in a geometrical optics simulation. The simulation geometry, symmetry, and electric dipole point source used are the same as in Part 2.
The domain assignments for the simulation. The Full-Wave simulation is performed over the entire domain, with the outer region set as a perfectly matched layer (PML). The geometrical optics simulation is only performed in this outer region. Note that this image is not to scale.
With the domains assigned, we select the Geometrical Optics interface, change the Intensity computation to Compute intensity, and select the Compute phase check box. These steps are required to properly compute the amplitude and phase of the electric field along the ray trajectory.
Settings for the Geometrical Optics interface. The Intensity computation is set to Compute intensity and the Compute phase check box is selected.
We also apply an Inlet boundary condition to the boundary between the Full-Wave simulation domain and Geometrical Optics domain. The inlet settings can be seen in the image below, but let’s walk through them one at a time. First, the Ray Direction Vector section is configured. This will launch the rays normal to the curved surface we’ve selected for the inlet — in other words, radially outwards. The variables Etheta and Ephi are calculated from the scattering amplitude according to
with a similar assignment for Ephi.
This equation comes from our previous blog post about using the Far-Field Domain node to calculate the fields at an arbitrary location. These variables are used to specify the initial phase and polarization of the rays. The variable specifies the correct spatial intensity distribution for the rays (as antennas generally do not emit uniformly) and is calculated according to , where Z is the impedance of the medium.
The initial radius of curvature has two factors. The parameter is the radius of the spherical boundary that we are launching the rays from and will correctly initialize the curvature of the ray wavefront.
Finally, we use the Cartesian components of our spherical unit vector to specify the initial principal curvature direction. This ensures that the correct polarization orientation is imparted to the rays. The wavefront shape here must be set to Ellipsoid — even though the surface is technically a sphere — because we need to be able to specify a preferred direction for polarization. If we choose Spherical, then each orientation is degenerate and we cannot make that specification.
The settings for the Inlet boundary condition in the Geometrical Optics interface. Note that you can click the image to expand it.
Beyond setting the correct frequency, the only other setting here is the placement of a Freeze Wall condition on the exterior boundary to stop the rays. Let’s take a look at the results vs. theory. As before, we express the full solution for a point dipole as a sum of two contributions, which we have labeled near field (NF) and far field (FF).
The electric fields from a geometrical optics simulation compared against theory. Geometrical optics is always in the far field, so we see excellent agreement as the distance from the source increases. For reference, the far-field domain results from the previous post would overlap exactly with the ray optics and FF theory lines.
As mentioned before, the Geometrical Optics interface is necessarily in the far field, so we do not expect to be able to correctly capture the near-field information as we did in the Beam-Envelopes solution in Part 2. This can also be seen because we seeded the ray tracing simulation with data from the Far-Field Domain node calculation. It is therefore unsurprising that there is disagreement near the source, but we can clearly see that the results match with theory as the distance from the source increases.
From looking solely at the above plot, we have to ask ourselves: “What have we actually gained here?”
This is a fair question, because the plot shown above could have been constructed directly from any of the techniques covered in the series so far. To make this clear, let’s review each of them.
Multiscale Technique | Regime of Validity | Modules Used | Notes |
---|---|---|---|
Far-Field Domain node | Far field | RF or Wave Optics | Requires the antenna to be completely surrounded by a homogeneous domain. |
Beam-Envelopes | Any field | Wave Optics | Requires specification of the phase function or wave vector. |
Geometrical Optics | Far field | Ray Optics | Can account for a spatially varying index as well as reflection and refraction from complex geometries. Diffraction is neglected. |
A summary of the multiscale modeling techniques we have covered in this blog series.
Note that any of these techniques will require a Full-Wave simulation of the radiation source. This generally requires the RF Module, although there is a subset of radiation sources that can be modeled using the Wave Optics Module instead. The Far-Field Domain node is available in both the RF and Wave Optics modules.
We originally motivated this discussion by talking about signal transmission from one antenna to another, and solved that simulation using the Far-Field Domain node in the last post. In the next blog post in this series, we’ll redo that simulation using the Geometrical Optics interface introduced here.
Access the model discussed in this blog post and any of the model examples highlighted throughout this blog series by clicking on the button above.
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In the simulation of our receiving antenna, we will use the Scattered Field formulation. This formulation is extremely useful when you have an object in the presence of a known field, such as in radar cross section (RCS) simulations. Since there are a number of scattered field simulations in the Application Gallery, and it has been discussed in a previous blog post, we will assume a familiarity with this technique and encourage you to review those resources if the Scattered Field formulation is new to you.
The Scattered Field formulation is useful for computing a radar cross section.
When comparing the implementation we will use here with the scattering examples in the Application Gallery, there are two differences that need to be referenced explicitly. The first is that, unlike the scattering examples, we will use a receiving antenna with a Lumped Port. With the Lumped Port excitation set to Off, it will receive power from the background field. This is automatically calculated in a predefined variable, and since the power is going into the lumped power, the value will be negative. The second difference, which we will spend more time discussing, is that the receiving antenna will be in a separate component than the emitting antenna and we will have to reference the results of one component in the other to link them.
What does it mean when we have two or more components in a model? The defining feature of a component is that it has its own geometry and spatial dimension. If you would like to have a 2D axisymmetric geometry and a 3D geometry in the same simulation, then they would each require their own component. If you would like to do two 3D simulations in the same model, you only need one component, although in some situations it can be beneficial to separate them anyways.
Let’s say, for example, that you have two devices with relatively complicated geometries. If they are in the same component, then anytime you make a geometric change to one, they both need to be rebuilt (and remeshed). In separate components this would not be the case. Another common use of multiple components is submodeling, where the macroscopic structure is analyzed first and then a more detailed analysis is performed on a smaller region of the model. When we split into components, however, we then need to link the results between the simulations.
In our case, we have two antennas at a distance of 1000 λ. Separating them into distinct components is not strictly required, but we are going to do it anyways to keep things general. We will add in ray tracing later in this series and some users may find this multiple component method useful with an arbitrarily complex ray tracing geometry.
While we go through the details, it’s important that we have a clear image of the big picture. The main idea that we are pursuing in this post is that we first simulate an emitting antenna and calculate the radiated fields in a specific direction. Specifically, this is the direction of the receiving antenna. We then account for the distance between the antennas and use the calculated fields as the background field in a Scattered Field formulation for the receiving antenna. The emitting antenna is centered at the origin in component 1 and the receiving antenna is centered at the origin in component 2. Everything we will discuss here is simply the technical details of determining the emitted fields from the first simulation and using them as a background field in a second simulation.
Note: The overwhelming majority of the COMSOL Multiphysics® software models only have one component and only should have one component. Ensure that you have a sufficient need for multiple components in your model before implementing them, as there is a very real possibility of causing yourself extra work without benefit.
There are a number of coupling operators, also known as component couplings, available in COMSOL Multiphysics. Generally speaking, these operators map the results from one spatial location to another. Said in another way, you can call for results in one location (the destination), but have the results evaluated at a separate location (the source). While this may seem trivial at first glance, it is an incredibly powerful and general technique. Let’s look at a few specific examples:
As mentioned above, we want to simulate the emitting antenna (just like we did in Part 2 of the series) and calculate the radiated fields at a distance of 1000 λ. We then use a component coupling to map the fields to being centered about the origin in component 2.
If we look at the far-field evaluation discussed in Part 2, we know that the x-component of the far field at a specific location is
The only complication is determining where to calculate the scattering amplitude. This is because component couplings need the source and destination to be locations that exist in the geometry. We don’t want to define a sphere in component 1 at the actual location of the receiving antenna, since that defeats the entire purpose of splitting the two antennas into two components. What we will do instead is create a variable for the magnitude of r, and then evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point we are actually interested in. In the image below, we show the point where we would like to evaluate the scattering amplitude.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
We add a point to the geometry using the rescaling of the Cartesian coordinates shown in the above figure. Only x is shown in the figure, but the same scaling is also applied to y and z. For the COMSOL Multiphysics implementation, shown below, we have assumed that the receiving antenna is centered at a location of (1000 λ, 0, 0), and the two parameters used are ant_dist = and sim_r = .
The required point for the correct scattering amplitude evaluation.
Note that we create a selection group from this point. This is so that it can be referenced without ambiguity. We then use this selection for an integration operator. Since we are integrating only over a single point, we simply return the value of the integrand at that point similar to using a Dirac delta function.
The integration operator is defined using the selection group for the evaluation point.
The above discussion was all about how to evaluate the scattering amplitude at the correct location. The only remaining step is to use this in a background field simulation of the half-wavelength dipole discussed in Part 1. When we add in the known distance between the antennas, we get the following:
The variable definition for r. Note that this is defined in component 2.
The background field settings.
In the settings, we see that the expression used for the background field in x is comp1.intop1(emw.Efarx)*exp(-j*k*r)/(r/1[m]), which matches the equation cited above. Also note that r is defined in component 2, while intop1() is defined in component 1. Since we are calling this from within component 2, we need to include the correct scope for the coupling operator, comp1.intop1(). The remainder of the receiving antenna simulation is functionally equivalent to other Scattered Field simulations in the Application Gallery, so we will not delve into the specifics here.
It is interesting to note that running either the emission or background field simulations by themselves is quite straightforward. All of the complication in this procedure is in correctly calculating the fields from component 1 and using them in component 2. All of this heavy lifting has paid off in that we can now fully simulate the received power in an antenna-to-antenna simulation, and the agreement between the simulated power and the Friis transmission equation is excellent. We can also obtain much more information from our simulation than we can purely from the Friis equation, since we have full knowledge of the electromagnetic fields at every point in space.
It is worth mentioning one final point before we conclude. We have only evaluated the far field at an individual point, so there is no angular dependence in the field at the receiving antenna. Because we are interested in antennas that are generally far apart, this is a valid approximation, although we will discuss a more general implementation in Part 4.
We have now reached a major benchmark in this blog series. After discussing terminology in Part 1 and emission in Part 2, we can now link a radiating antenna to a receiving antenna and verify our results against a known reference. The method we have implemented here can also be more useful than the Friis equation, as we have fully solved for the electromagnetic fields and any polarization mismatch is automatically accounted for.
There is one remaining issue, however, that we have not discussed. The method used here is only applicable to line-of-sight transmission through a homogeneous medium. If we had an inhomogeneous medium between the antennas or multipath transmission, that would not be appropriately accounted for either by this technique or the Friis equation. To solve that issue, we will need to use ray tracing to link the emitting and receiving antennas. In Part 4 of this blog series, we will show you how we can link a radiating source to a ray optics simulation.
Let’s begin by discussing a traditional antenna simulation using COMSOL Multiphysics and the RF Module. When we simulate a radiating antenna, we have a local source and are interested in the subsequent electromagnetic fields, both nearby and outgoing from the antenna. This is fundamentally what an antenna does. It converts local information (e.g., voltage or current) into propagating information (e.g., outgoing radiation). A receiving antenna inverts this operation and changes incident radiation into local information. Many devices, such as a cellphone, act as both receiving and emitting antennas, which is what enables you to make a phone call or browse the web.
Antennas of the Atacama Large Millimeter Array (ALMA) in Chile. ALMA detects signals from space to help scientists study the formation of stars, planets, and galaxies. Needless to say, the distance these signals travel is much greater than the size of an antenna. Image licensed under CC BY 4.0, via ESO/C. Malin.
In order to keep the required computational resources reasonable, we model only a small region of space around the antenna. We then truncate this small simulation domain with an absorbing boundary, such as a perfectly matched layer (PML), which absorbs the outgoing radiation. Since this will solve for the complex electric field everywhere in our simulation domain, we will refer to this as a Full-Wave simulation.
We then extract information about the antenna’s emission pattern using a Far-Field Domain node, which performs a near-to-far-field transformation. This approach gives us information about the electromagnetic field in two regions: the fields in the immediate vicinity of the antenna, which are computed directly, and the fields far away, which are calculated using the Far-Field Domain node. This is demonstrated in a number of RF models in the Application Gallery, such as the Dipole Antenna tutorial model, so we will not comment further on the practical implementation here.
One question that occasionally comes up in technical support is: “How do I use the Far-Field Domain node to calculate the radiated field at a specific location?” This is an excellent question. As stated in the RF Module User’s Guide, the Far-Field Domain node calculates the scattering amplitude, and so determining the complex field at a specific location requires a modification for distance and phase. The expression for the x-component of the electric field in the far field is:
and similar expressions apply to the y- and z-component, where r is the radial distance in spherical coordinates, k is the wave vector for the medium, and emw.Efarx is the scattering amplitude. It is worth pointing out that emw.Efarx is the scattering amplitude in a particular direction, and so it depends on angular position , but not radial position. The decrease in field strength is solely governed by the 1/r term. There are also variables emw.Efarphi and emw.Efartheta, which are for the scattering amplitude in spherical coordinates.
To verify this result, we simulate a perfect electric dipole and compare the simulation results with the analytical solution, which we covered in the previous blog post. As we stated in that post, we split the full results into two terms, which we call the near- and far-field terms. We briefly restate those results here.
where is the dipole moment of the radiation source and is the unit vector in spherical coordinates.
Below, we can see the electric fields vs. distance calculated using the Far-Field Domain node for a dipole at the origin with . For comparison, we have included the Far-Field Domain node, the full theory, as well as the near- and far-field terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the Far-Field Domain node and the far-field theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the Far-Field Domain node will only account for radiation that goes like 1/r, and so the agreement improves with increasing distance as the contribution of the 1/r^{2} and 1/r^{3} terms go to zero. In other words, the Far-Field Domain node is correct in the far field, which you probably would have guessed from the name.
A comparison of the Far-Field Domain node vs. theory for a point dipole source.
For most simulations, the near-field and far-field information is sufficient and no further work is necessary. In some cases, however, we also want to know the fields in the intermediate region, also known as the induction or transition zone. One option is to simply increase the simulation size until you explicitly calculate this information as part of the simulation. The drawback of this technique is that the increased simulation size requires more computational resources. We recommend a maximum mesh element size of for 3D electromagnetic simulations. As the simulation size increases, the number of mesh elements increases, and so do the computational requirements.
Another option is to use the Electromagnetic Waves, Beam Envelopes interface, which here we will simply refer to as Beam-Envelopes. As discussed in a previous blog post, Beam-Envelopes is an excellent choice when the simulation solution will have either one or two directions of propagation, and will allow us to use a much coarser mesh. Since the phase of the emission from an antenna will look like an outgoing spherical wave, this is a perfect solution for determining these fields. We perform a Full-Wave simulation of the fields near the source, as before, and then use Beam-Envelopes to simulate the fields out to an arbitrary distance, as required.
The simulation domain assignments. If the outer region is assigned to PML, then a Full-Wave simulation is performed everywhere. It is also possible to solve the inner region using a Full-Wave simulation and the outer region using Beam-Envelopes, as we will discuss below. Note that this image is not to scale, and we have only modeled 1/8 of the spherical domain due to symmetry.
How do we couple the Beam-Envelopes simulation to our Full-Wave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the Full-Wave and Beam-Envelopes domains. First, we set the exterior boundary of the Full-Wave simulation to PMC, which is the natural boundary condition for that simulation. The second step is to set that same boundary to an Electric Field boundary condition for Beam-Envelopes. We then specify the field values in the Beam-Envelopes Electric Field boundary condition according to the fields computed from the Full-Wave simulation, as shown here.
The Electric Field boundary condition in Beam-Envelopes. Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the Beam-Envelopes domain to absorb the outgoing spherical wave. The remaining boundaries are set to PEC and PMC according to symmetry. We must also set the solver to Fully Coupled, which is described in more detail in two blog posts on solving multiphysics models and improving convergence from a previous blog series on solvers.
If we again examine the comparison between simulation and theory, we see excellent agreement over the entire simulation range. This shows that the PMC and Electric Field boundary conditions have enforced continuity between the two interfaces and they have fully reproduced the analytical solution. You can download the model file in the Application Gallery.
A comparison of the electric field of the Full-Wave and Beam-Envelopes simulations vs. the full theory.
In today’s blog post, we examined two ways of computing the electric field at points far away from the source antenna and verified the results using the analytical solution for an electric point dipole. These two techniques are using the Far-Field Domain node from a Full-Wave simulation and linking a Full-Wave simulation to a Beam-Envelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using Beam-Envelopes has the additional advantage in that it also computes fields in the intermediate region. In the next post in the series, we will combine the calculated far-field radiation with a simulation of a receiving antenna and determine the received power. Stay tuned!
Multiscale modeling is a challenging issue in modern simulation that occurs when there are vastly different scales in the same model. For example, your cellphone is approximately 15 cm, yet it receives GPS information from satellites 20,000 km away. Handling both of these lengths in the same simulation is not always straightforward. Similar issues show up in applications such as weather simulations, chemistry, and many other areas.
While multiscale modeling can be a general topic, we will focus our attention on the practical example of antennas and wireless communication. When we wirelessly transmit data via antennas, we can break the operation down into three main stages:
Modern communications require long-distance wireless data transfer via antennas.
The two length scales that we will consider for this process are the wavelength of the radiation and the distance between the antennas. To use a specific example, FM radio has a wavelength of approximately three meters. When you listen to the radio in your car, you are often ten km or more away from the radio tower. Because many antennas, such as dipole antennas, are similar in size to a wavelength, we will not consider this to be another distinct length scale. As a result, we have one length scale for the emitting antenna, a different length scale for the signal propagation from source to destination, and then the original length scale again for the receiving antenna.
Let’s go over some of the most important equations, terms, and considerations when working with multiple scales in the same high-frequency electromagnetics model.
The Friis transmission equation calculates the received power for line-of-sight communication between two antennas separated by a lossless medium. The equation is
where the subscripts r and t discriminate between the transmission antenna and the receiving antenna, G is the antenna gain, P is the power, is the reflection coefficient for impedance mismatch between antenna and transmission line, p is the polarization mismatch factor, λ is the wavelength, r is the distance between the antennas and is associated with the so-called free-space path loss, and and are the angular spherical coordinates for the two antennas.
Note that we have explicitly included two impedance mismatch terms, and so:
The Friis transmission equation is derived in many texts, so we will not do so again here.
A visualization of the gain for a transmitting and receiving antenna. When using the Friis transmission equation, we require the orientation of each antenna for correct gain specification. The distance between the antennas is r.
Let’s now discuss spherical coordinates , since they are incredibly useful for antenna radiation and we will use them repeatedly. Starting from the Cartesian coordinates (x, y, z), we can easily express these as follows.
For convenience, we have used the actual COMSOL Multiphysics commands — sqrt(), acos(), and atan2(,) — instead of their mathematical symbols. In our simulation setup, we will also make use of the Cartesian components of the spherical unit vector .
Similar assignments can be made for the Cartesian components of and , but is the most important for our purposes. This will be discussed later in this blog series when we cover ray optics.
A given point shown in both Cartesian (x, y, z) and spherical coordinates. The unit vectors for the spherical coordinates are also included. Note that the spherical unit vectors are functions of location.
We are generally interested in the radiated power from antennas. The power flux in W/m^{2} is represented by the complex Poynting vector .
Many antenna texts also use radiation intensity, which is defined as the power radiated per solid angle and measured in W/steradian. Mathematically speaking, this is . For clarity, we have included two conventions here, as it is common to use in electrical engineering, while physicists will generally be more familiar with . We can then calculate the radiated power by integrating this quantity over all angles.
Gain and directivity are similar in that they both quantify the radiated power in a given direction. The difference is that gain relates this radiated power to the input power, whereas directivity relates this to the overall radiated power. Put more simply, gain accounts for dielectric and conductive losses and directivity does not. Mathematically, this reads as and for gain and directivity, respectively. P_{in} is the power accepted by the antenna and P_{rad} is the total radiated power. While both quantities can be of interest, gain tends to be the more practical of these two as it accounts for material loss in the antenna. Because of its prevalence and usefulness, we also include the definition of gain (in a given direction) from “IEEE Standard Definitions of Terms for Antennas”, which is: “The ratio of the radiation intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically.”
IEEE also includes three notes about gain in their definition:
In practice, an actual antenna will be connected to a transmission line. Because the antenna and the transmission line may not have the same impedance, there can be a loss factor due to impedance mismatch. The realized gain is simply the gain when accounting for impedance mismatch. Mathematically, this is , where is the reflection coefficient from transmission line theory, Z_{c} is the characteristic impedance of the transmission line, and Z is the impedance of the antenna.
When using a lumped port with a characteristic impedance in COMSOL Multiphysics, the far-field gain that is calculated corresponds to the IEEE realized gain. This is important to mention explicitly, since various definitions of gain have changed over the last few decades. Starting with COMSOL Multiphysics version 5.3, which will be released in 2017, the variable names in the COMSOL software will be changed to match the IEEE definitions.
The realized gain and electric field from a Vivaldi antenna, simulated using COMSOL Multiphysics and the RF Module. You can find the Vivaldi Antenna tutorial model in the Application Gallery.
The terms we have discussed so far have referred to antennas emitting radiation, but they are also generally applicable to receiving antennas. The reason we have put more emphasis on emission thus far is because antennas generally obey reciprocity (the Lorentz reciprocity theorem is a fixture in most antenna textbooks). Reciprocity means that an antenna’s gain in a specific direction is the same regardless of whether it is emitting in that direction or receiving a signal from that direction. Practically speaking, you can calculate the gain in any direction from a single simulation of an emitting antenna, which is easier than simulating the inverse process for each desired direction.
When we talk about receiving antennas, we are often interested in calculating the received power for an incoming signal. This can be done by multiplying the effective area, , of the antenna by the incident power flux and accounting for impedance mismatch in the line, yielding . As you may expect, this bears a striking similarity to several terms of the Friis transmission equation.
Today, we will talk about one type of emitter: the perfect electric point dipole. Depending on the literature, you may have seen this referred to as a perfect, ideal, or infinitesimal dipole. This emitter is a common representation of radiation for electrically small antennas. The solution for the field is
where is the dipole moment of the radiation source (not to be confused with the polarization mismatch) and k is the wave vector for the medium.
One breakdown of the various regions for the electromagnetic field generated from an electrically small antenna.
In this equation, there are three factors of 1/r^{n}. The 1/r^{2} and 1/r^{3} terms will be more significant near the source, while the 1/r term will dominate at large distances. While the electromagnetic field will be continuous, it is common to refer to different regions of the field based on the distance from the source. One such distribution for an electrically small antenna is shown above, although there are other conventions that refer to the magnitude of kr.
Later, we will see how to calculate the fields at any distance from a given source, but the most important region for antenna communications is the far field or radiation zone, which is the region farthest away from the source. In this region, the fields take the form of spherical waves, , a fact that we will take advantage of.
We will now split up the E-field equation above into two terms. For simplicity, we will call the 1/r term the far field (FF) and the 1/r^{2} and 1/r^{3} terms the near field (NF).
As mentioned before, we can calculate the radiated power in watts by integrating over all angles. Note that only the far-field term will contribute to this integral, which is a primary reason why the far field is of practical interest to antenna engineers. The total power radiated from a point dipole is , where Z_{0} is the impedance of free space and c is the speed of light. The maximum gain is 1.5 and is isotropic in the plane normal to the dipole moment (e.g., the xy-plane for a dipole in ).
A note on units: The equations above are given with the traditional definition of the dipole moment in Coulomb*meters (Cm). In antenna and engineering texts, it is common to specify an infinitesimal current dipole in Ampere*meters (Am). COMSOL Multiphysics follows the engineering convention. The two definitions are related by a time derivative, so for a COMSOL software implementation, the dipole moment should be multiplied by a factor of to obtain the infinitesimal current dipole.
We will use a perfectly conducting half-wavelength dipole as our receiving antenna.
A visual representation of radiation incident on a half-wavelength dipole antenna.
Many texts cover an infinitely thin wire, which has an impedance of and a directivity of . It is worth mentioning that the antenna impedance will change from these values for an antenna of finite radius. The receiving antenna we use here has a length of 0.47 λ and a length-to-diameter ratio of 100. With these values, we simulate an impedance of , which is close to the infinitely thin value and also agrees reasonably well with experimental values. Regrettably, there is no theoretical value to compare to this number, but this highlights the need for numerical simulation in antenna design.
The comparison between the directivity of the infinitely thin dipole and our simulated dipole antenna is shown below. Because the antenna is lossless, this is equivalent to the antenna gain. You can download the dipole antenna model here.
A comparison of the directivity for two half-wavelength antennas (oriented in z) as a function of theta. The COMSOL Multiphysics® simulation is of a finite radius cylinder and the theory is for an infinitely thin antenna.
We can now use the Friis transmission equation to calculate the power that is emitted from a perfect point dipole and received by a half-wave dipole antenna. To use this equation, we simply need to know the gain and impedance mismatch (or realized gain), wavelength, distance between the antennas, and input power. Since we are using a point electric dipole, we have a dipole moment instead of input power and impedance mismatch. We can account for this by removing the impedance mismatch term and replacing the input power by the radiated power of the perfect electric dipole from above — effectively saying that power in equals power out.
If we assume that our emitter and detector are both located in the xy-plane, are polarization matched, and are separated by 1000 λ, as well as that the dipole moment of the emitter is 1 Am in , the Friis equation yields a received power of 380 μW. We will simulate this value in part 3 of this series for verification of our simulation technique. We can then use our simulation to confidently extract results and introduce complexity that the Friis equation cannot account for.
In this blog post, we have introduced the idea of multiscale modeling and discussed all of the relevant terms, definitions, and theory that we will need moving forward. For those of you with a strong background in electromagnetics and antenna design, this has likely been a quick review. If the concepts presented here are new to you, we strongly recommend further reading in a book on classical electromagnetics or antenna theory.
In the following blog posts, we will focus primarily on practical implementation of multiscale modeling in COMSOL Multiphysics and we will repeatedly refer to concepts discussed today.
Stay tuned for more installments in our multiscale modeling blog series:
Because they can be focused to the smallest spot size of all electromagnetic beams, Gaussian beams can deliver the highest resolution for imaging, as well as the highest power density for a fixed incident power, which can be important in fields such as material processing. These qualities are why lasers are such attractive light sources. To obtain the tightest possible focus, most commercial lasers are designed to operate in the lowest transverse mode, called the Gaussian beam.
As such, it would be reasonable to want to simulate a Gaussian beam with the smallest spot size. There is a formula that predicts real Gaussian beams in experiments very well and is convenient to apply in simulation studies. However, there is a limitation attributed to using this formula. The limitation appears when you are trying to describe a Gaussian beam with a spot size near its wavelength. In other words, the formula becomes less accurate when trying to observe the most beneficial feature of the Gaussian beam in simulation. In a future blog post, we will discuss ways to simulate Gaussian beams more accurately; for the remainder of this post, we will focus exclusively on the paraxial Gaussian beam.
A schematic illustrating the converging, focusing, and diverging of a Gaussian beam.
Note: The term “Gaussian beam” can sometimes be used to describe a beam with a “Gaussian profile” or “Gaussian distribution”. When we use the term “Gaussian beam” here, it always means a “focusing” or “propagating” Gaussian beam, which includes the amplitude and the phase.
The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwell’s equations. This is the first important element to note, while the other portions of our discussion will focus on how the formula is derived and what types of assumptions are made from it.
Because the laser beam is an electromagnetic beam, it satisfies the Maxwell equations. The time-harmonic assumption (the wave oscillates at a single frequency in time) changes the Maxwell equations to the frequency domain from the time domain, resulting in the monochromatic (single wavelength) Helmholtz equation. Assuming a certain polarization, it further reduces to a scalar Helmholtz equation, which is written in 2D for the out-of-plane electric field for simplicity:
where for wavelength in vacuum.
The original idea of the paraxial Gaussian beam starts with approximating the scalar Helmholtz equation by factoring out the propagating factor and leaving the slowly varying function, i.e., , where the propagation axis is in and is the slowly varying function. This will yield an identity
This factorization is reasonable for a wave in a laser cavity propagating along the optical axis. The next assumption is that , which means that the envelope of the propagating wave is slow along the optical axis, and , which means that the variation of the wave in the optical axis is slower than that in the transverse axis. These assumptions derive an approximation to the Helmholtz equation, which is called the paraxial Helmholtz equation, i.e.,
The special solution to this paraxial Helmholtz equation gives the paraxial Gaussian beam formula. For a given waist radius at the focus point, the slowly varying function is given by
where , , and are the beam radius as a function of , the radius of curvature of the wavefront, and the Gouy phase, respectively. The following definitions apply: , , , and .
Here, is referred to as the Rayleigh range. Outside of the Rayleigh range, the Gaussian beam size becomes proportional to the distance from the focal point and the intensity position diverges at an approximate divergence angle of .
Definition of the paraxial Gaussian beam.
Note: It is important to be clear about which quantities are given and which ones are being calculated. To specify a paraxial Gaussian beam, either the waist radius or the far-field divergence angle must be given. These two quantities are dependent on each other through the approximate divergence angle equation. All other quantities and functions are derived from and defined by these quantities.
In COMSOL Multiphysics, the paraxial Gaussian beam formula is included as a built-in background field in the Electromagnetic Waves, Frequency Domain interface in the RF and Wave Optics modules. The interface features a formulation option for solving electromagnetic scattering problems, which are the Full field and the Scattered field formulations.
The paraxial Gaussian beam option will be available if the scattered field formulation is chosen, as illustrated in the screenshot below. By using this feature, you can use the paraxial Gaussian beam formula in COMSOL Multiphysics without having to type out the relatively complicated formula. Instead, you simply need to specify the waist radius, focus position, polarization, and the wave number.
Screenshot of the settings for the Gaussian beam background field.
Plots showing the electric field norm of paraxial Gaussian beams with different waist radii. Note that the variable name for the background field is ewfd.Ebz
.
In the scattered field formulation, the total field is linearly decomposed into the background field and the scattered field as . Since the total field must satisfy the Helmholtz equation, it follows that , where is the Laplace operator. This is the full field formulation, where COMSOL Multiphysics solves for the total field. On the other hand, this formulation can be rewritten in the form of an inhomogeneous Helmholtz equation as
The above equation is the scattered field formulation, where COMSOL Multiphysics solves for the scattered field. This formulation can be viewed as a scattering problem with a scattering potential, which appears in the right-hand side. It is easy to understand that the scattered field will be zero if the background field satisfies the Helmholtz equation (under an approximate Sommerfeld radiation condition, such as an absorbing boundary condition) because the right-hand side is zero, aside from the numerical errors. If the background field doesn’t satisfy the Helmholtz equation, the right-hand side may leave some nonzero value, in which case the scattered field may be nonzero. This field can be regarded as an error of the background field. In other words, under certain conditions, you can qualify and quantify exactly how and by how much your background field satisfies the Helmholtz equation. Let’s now take a look at the scattered field for the example shown in the previous simulations.
Plots showing the electric field norm of the scattered field. Note that the variable name for the scattered field is ewfd.relEz
. Also note that the numerical error is contained in this error field as well as the formula’s error.
The results shown above clearly indicate that the paraxial Gaussian beam formula starts failing to be consistent with the Helmholtz equation as it’s focused more tightly. Quantitatively, the plot below may illustrate the trend more clearly. Here, the relative L2 error is defined by , where stands for the computational domain, which is compared to the mesh size. As this plot suggests, we can’t expect that the paraxial Gaussian beam formula for spot sizes near or smaller than the wavelength is representative of what really happens in experiments or the behavior of real electromagnetic Gaussian beams. In the settings of the paraxial Gaussian beam formula in COMSOL Multiphysics, the default waist radius is ten times the wavelength, which is safe enough to be consistent with the Helmholtz equation. It is, however, not a “cut-off” number, as the approximation assumption is continuous. It’s up to you to decide when you need to be cautious in your use of this approximate formula.
Semi-log plot comparing the relative L2 error of the scattered field with the waist size in the units of wavelength.
In the above plot, we saw the relationship between the waist size and the accuracy of the paraxial approximation. Now we can check the assumptions that were discussed earlier. One of the assumptions to derive the paraxial Helmholtz equation is that the envelope function varies relatively slowly in the propagation axis, i.e., . Let’s check this condition on the x-axis. To that end, we can calculate a quantity representing the paraxiality. As the paraxial Helmholtz equation is a complex equation, let’s take a look at the real part of this quantity, .
The following plot is the result of the calculation as a function of x normalized by the wavelength. (You can type it in the plot settings by using the derivative operand like d(d(A,x),x)
and d(A,x)
, and so on.) We can see that the paraxiality condition breaks down as the waist size gets close to the wavelength. This plot indicates that the beam envelope is no longer a slowly varying one around the focus as the beam becomes fast. A different approach for seeing the same trend is shown in our Suggested Reading section.
Real part of the paraxiality along the x-axis for paraxial Gaussian beams with different waist sizes.
Today’s blog post has covered the fundamentals related to the paraxial Gaussian beam formula. Understanding how to effectively utilize this useful formulation requires knowledge of its limitation as well as how to determine its accuracy, both of which are elements that we have highlighted here.
There are additional approaches available for simulating the Gaussian beam in a more rigorous manner, allowing you to push through the limit of the smallest spot size. We will discuss this topic in a future blog post. Stay tuned!
When it comes to the design of a polarizing beam splitter, the most common configuration comes in the form of a cube. This cube design is valued as a viable alternative to the plate design for many reasons. Because there is only one reflecting surface in the cube configuration, it avoids producing ghost images. Further, as compared to the input beam, the translation of the transmitted output beam is quite small, which simplifies the process of aligning optical systems.
Let’s take a closer look at such a design. Polarizing beam splitter cubes are comprised of two prisms positioned at right angles. One of these prisms includes a dielectric coating evaporated on the intermediate hypotenuse surface. When a light wave enters the cube, the coating transmits the portion of the incident wave with the electric field that is polarized in the plane of incidence and reflects the portion of the incident wave with the electric field that is orthogonal to the plane of incidence. These parts of the incident wave are represented by p-polarization and s-polarization, respectively, in the schematic shown below.
Polarizing beam splitter cube schematic.
Polarizing beam splitter devices such as this are useful for broadband or tunable sources as well as selected laser lines, since the dielectric coating can be designed as either spectrally broadband or narrowband. Additionally, these coatings can be tailored for use in high-power laser applications that feature very large damage thresholds.
To ensure that these devices perform properly within their respective system, it is important to study their design and make modifications as needed to achieve optimal performance. Numerical modeling apps, as we’ll highlight here, help to make this process much more efficient.
The basis of our Polarizing Beam Splitter app is the simple MacNeille design. In this configuration, there are a pair of layers that feature a consecutively high and low refractive index. The light waves interact with the layer boundary at the Brewster angle, thus reflecting the s-polarization and transmitting the p-polarization at every internal layer boundary.
Now that we’ve reviewed the underlying design, let’s take a look at our app’s user interface (UI). Note that when creating your own apps, it is up to you to decide on its layout and structure, including the parameters that are made available for modification. This example, and the others that we share within our Application Gallery, are designed to serve as both a source of inspiration and guidance within your own app-building processes.
The Polarizing Beam Splitter app’s UI.
In the app’s Design section, users can enter their own refractive indices for the prisms as well as the layers within the dielectric stack or select a material from the available list of options. Here, they can also define the number of layers within the dielectric stack. Selecting the Sweep type, either Wavelength or Spot radius, is possible via the Simulation Parameters section. For each of these sweep types, users have the option to choose the polarization for the simulation that will be performed.
Moving over to the Graphics window, you will notice a series of displayed tabs. The Geometry and Mesh tabs display the current geometry and mesh, respectively. When a solution does exist, the Electric Field tab shows the following for a specific Wavelength or Spot radius and Polarization: the norm of the electric field, the first wave’s electric field, or the second wave’s electric field. The Reflectance and Transmittance tab, meanwhile, highlights the reflectance and transmittance of the polarizations of the performed simulation. Lastly, there is the Refractive Index tab. In the case of a Wavelength sweep, this tab shows either the refractive index as compared to each material’s wavelength or the spatial refractive index profile across a cut-line over the prisms and the dielectric stack. In the case of a Spot radius sweep, only a spatial refractive index profile is shown.
After obtaining their simulation findings, users can choose to create a customized report via the Report button. This will generate a Microsoft® Word® report that contains the respective input data and results from their analyses. They can use this report to communicate their results to others in a clear, simplified format.
Every design workflow encounters its own set of challenges. With their customization capabilities and ease of use, numerical modeling apps serve as a powerful tool for meeting the specific needs of individuals and organizations, all while balancing efficiency with accuracy.
Our Polarizer Beam Splitter app is just one example of how you can use the Application Builder to create an easy-to-use tool to advance simulation analyses. We encourage you to start building apps of your own and experience the many benefits that come with deploying them to others.
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The ability to implement the Fourier transformation in a simulation is a useful functionality for a variety of applications. Besides Fourier optics, we use Fourier transformation in Fraunhofer diffraction theory, signal processing for frequency pattern extraction, and image processing for noise reduction and filtering.
In this example, we calculate an image of the light from a traffic light passing through a mesh curtain, shown below. To simplify the model, we assume the electric field of the lights is a plane wave of uniform intensity; for instance, 1 V/m. Let the mesh geometry be measured by the local coordinates and in a plane perpendicular to the direction of the light propagation, and let the image pattern be measured by the local coordinates and near the eye in a plane parallel to the mesh plane.
A Fraunhofer diffraction pattern as a Fourier transform of a square aperture in a mesh curtain.
According to the Fraunhofer diffraction theory, then, we can calculate the image above simply by Fourier transforming the light transmission function, which is a periodic rectangular function if the mesh is square. Let’s consider a simplified case of a single mesh whose transmission function is a single rectangular function. We will discuss the case of a periodic transmission function later on.
We are interested in the light hitting one square of the mesh and getting diffracted by the sharp edges of the fabric while transmitting in the center of the mesh. In this case, the light transmission function is described by a 2D rectangular function. By implementing a Fourier transformation into a COMSOL Multiphysics simulation, we can more fully understand this process.
In order to learn how to implement Fourier transformation, let’s first discuss the concept of data sets, or multidimensional matrices that store numbers. There are two possible types of data sets in COMSOL Multiphysics: Solution and Grid. For any computation, the COMSOL software creates a data set, which is placed under the Results > Data Sets node.
The Solution data set consists of an unstructured grid and is used to store solution data. To make use of this data set, we specify the data to which each column and row corresponds. If we specify Solution 1 (sol1), the matrix dimension corresponds to that of the model in Study 1. If it is a time-dependent problem, for example, the data set has a three-dimensional array, which may be written as with . Here, is the number of stored time steps, is the number of nodes, and is the number of the space dimension. Similarly, the data set for a time-dependent parametric study consists of a 4D array. Again, note that the spatial data (other than the time and parameter data) links with the nodal position on the mesh, not necessarily on the regular grid.
On the other hand, the Grid data set is equipped with a regular grid and is provided for functions and all other general purpose uses. All numbers stored in the Grid data set link to the grid defined in the Settings window. This data set is automatically created when a function is defined in the Definition node and by clicking on Create Plot. This creates a 1D Grid data set in the Data Sets node.
You also need to specify the range and the resolution of your independent variables. By default, the resolution for a 1D Grid data set is set to 1000. If the independent variable (i.e., x) ranges from 0 to 1, the Grid data set prepares data series of 0, 0.001, 0.002, …, 0.999, and 1. The default resolution is 100 for 2D and 30 for 3D. For Fourier transformation, we use the Grid data set. We can also use this data set as an independent tool for our calculation, as it does not point to a solution.
To begin our simulation, let’s define the built-in 1D rectangular function, as shown in the image below.
Defining the built-in 1D rectangular function.
Then, we click on the Create Plot button in the Settings window to create a separate 1D plot group in the Results node.
A plot of the built-in 1D rectangular function.
Let’s look at the Settings window of the plot. We expand the 1D Plot Group 1 node and click on Line Graph 1 to see the data set pointing to Grid 1D. In the Grid 1D node settings, we see that the data set is associated with a function rect1
.
Settings for the built-in 1D rectangular function.
Settings for the 1D Grid data set.
We can create a 2D rectangular function by defining an analytic function in the Definitions node as rect1(x)*rect1(y)
. For learning purposes, we will create and define a 2D Grid data set and plot it manually instead of automatically. The results are shown in the following series of images.
In the Grid 2D settings, we choose All for Function because the 2D rectangular function uses another function, rect1
. We also assign and as independent variables, which we previously defined as the curtain’s local coordinates, and set the resolution to 64 for quicker testing. To plot our results, we choose the 2D grid data, renamed to Grid 2D (source space), for the data set in the Plot Group settings window.
Defining the function in the Grid 2D settings.
Creating and defining a 2D data set.
Setting the 2D plot group for the 2D rectangular function.
A 2D plot of the 2D rectangular function.
Now, let’s implement a Fourier transform of this function by calculating:
Here, and represent the destination space (Fourier/frequency space) independent variables, as we previously discussed.
Since we already created a 2D data set for and , now we can create a Grid 2D data set, renamed to Grid 2D (Destination space), for and (shown below). We choose Function from Source and All from Function because the rect
function calls the rect1
function as well. We can change the resolution to 64 here, as we did for the 2D data set, for quicker calculation.
Settings for the Grid 2D data set for the Fourier space.
Now, we are at the stage in our simulation where we can type in the equations by using the integrate
operator.
Entering the equation for the Fourier transform of the 2D rectangular function.
We finally obtain the resulting Fourier transform, as shown in the figure below. Compare this (more accurately, the square of this) to each twinkling colored light in the photograph of the mesh curtain. In practice, this image hasn’t been truly seen yet. To calculate the image on its final destination, the retina of the eye, we would need to implement the Fourier transformation one more time.
The Fourier transform of the 2D rectangular function.
In COMSOL Multiphysics, you can use the data set feature and integrate
operator as a convenient standalone calculation tool and a preprocessing and postprocessing tool before or after your main computation. Note that the Fourier transformation discussed here is not the discrete Fourier transformation (FFT). We still use discrete math, but we carry out the integration numerically by using Simpson’s rule. This function is used in the integration operator in COMSOL Multiphysics, while the discrete Fourier transform is formed by the operation of number sequences. As a result, we don’t need to be concerned with the aliasing problem, Fourier space resolution issue, or Fourier space shift issue.
There is more to discuss on this subject, but let’s comment on the two cases that we simplified earlier. We calculated for a single mesh. In practice, the mesh curtain is made of a finite number of periodic square openings. It sounds like we have to redo our calculation for the periodic case, but fortunately, the end result differs only by an envelope function of the periodicity. For details, Hecht’s Optics outlines this topic very well.
The second simplification was that we assumed a sharp rectangular function for the mesh transmission function. In COMSOL Multiphysics, all functions other than the user-defined functions are smoothed to some extent for numerical stability and accuracy reasons. You may have noticed that our rectangular function had small slopes. This may be a complication rather than a simplification because the simplest case is a rectangular function with no slopes and we used a smoothed rectangular function instead of a sharp one.
The Fourier transforms of the two extreme cases are known; i.e., a rectangular function with no slopes is transformed to a sinc function (sin(x)/x
) and a Gaussian function to another Gaussian function. A sinc function has ripples around the center representing a diffraction effect, while a Gaussian function decays without any ripples. Our smoothed rectangular function is somewhere between these two extremes, so its Fourier transform is also somewhere between a sinc function and a Gaussian function. As we previously mentioned, the curtain fabric can’t have sharp edges, so our results may be more accurate for this example case anyway.