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.
ARIA is a registered trademark of CityCenter Land, LLC.
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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 5 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.
Both of these interfaces solve the frequency-domain form of Maxwell’s equations, but they do it in slightly different ways. The Electromagnetic Waves, Frequency Domain interface, which is available in both the RF and Wave Optics modules, solves directly for the complex electric field everywhere in the simulation. The Electromagnetic Waves, Beam Envelopes interface, which is available solely in the Wave Optics Module, will solve for the complex envelope of the electric field for a given wave vector. For the remainder of this post, we will refer to the Electromagnetic Waves, Frequency Domain interface as a Full-Wave simulation and the Electromagnetic Waves, Beam Envelopes interface as a Beam-Envelope simulation.
To see why the distinction between Full-Wave and Beam-Envelope is important, we will begin by discussing the trivial example of a plane wave propagating in free space, as shown in the image below. We will then apply the lessons learned to the dielectric slab.
A graphical representation of a plane wave propagating in free space, where the red, green, and blue lines represent the electric field, magnetic field, and Poynting vector, respectively.
To properly resolve the harmonic nature of the solution in a Full-Wave simulation, we need to mesh finer than the oscillations in the field. This is discussed further in these previous blog posts on tools for solving wave electromagnetics problems and modeling their materials. To simulate a plane wave propagating in free space, the number of mesh elements will then scale with the size of the free space domain in which we are interested. But what about the Beam-Envelopes simulation?
The Beam-Envelopes method is particularly well-suited for models where we have good prior knowledge of the wave vector, . Practically speaking, this means that we are solving for the fields using the ansatz . Notice that the only unknown in the ansatz is the envelope function . This is the quantity that needs to be meshed to obtain a full solution, hence the mention of beam envelopes in the name of the interface. In the case of a plane wave in free space, the form of the ansatz matches exactly with the analytical solution. We know that the envelope function will be a constant, as shown by the green line in the figure below, so how many mesh elements do we need to resolve the solution? Just one.
The electric field and phase of a plane wave propagating in free space. In the field plot (left), the blue and green lines show the real part and absolute value of E(r), which are and , respectively. The phase plot (right) shows the argument of E(r). In both plots, the x-axis is normalized to a wavelength, so this represents one full oscillation of the wave.
In practice, Beam-Envelopes simulations are more flexible than the ansatz we just used. This is for two reasons. First, instead of specifying a wave vector, we can specify a user-defined phase function, . Second, there is also a bidirectional option that allows for a second propagating wave and a full ansatz of . This is the functionality that we will take advantage of in modeling the dielectric slab (also called a Fabry-Pérot etalon).
The points discussed here will come up again in the dielectric slab example, and so we highlight them again for clarity. The size of mesh elements in a Full-Wave simulation is proportional to the wavelength because we are solving directly for the full field, while the mesh element size in a Beam-Envelopes simulation can be independent of the wavelength because we are solving for the envelope function of a given phase/wave vector. You can greatly reduce the number of mesh elements for large structures if a Beam-Envelopes simulation can be performed instead of a Full-Wave simulation, but this is only possible if you have prior knowledge of the wave vector (or phase function) everywhere in the simulation. Since the degrees of freedom, memory used, and simulation time all depend on the number of mesh elements, this can have a large influence on the computational requirements of your simulation.
Using the 2D geometry shown below, we can clearly see the different waves that need to be accounted for in a simulation of a dielectric slab illuminated by a plane wave. On the left of the slab, we have to account for the incoming wave traveling to the right, as well as the reflected wave traveling to the left. Because of internal reflections inside the slab itself, we have to account for both left- and right-traveling waves in the slab, and finally, the transmitted waves on the right. We also choose a specific example so that we can use concrete numbers.
Let’s make the dielectric slab an undoped silicon (Si) wafer that is 525 µm thick. We will simulate the response to terahertz (THz) radiation (i.e., submillimeter waves), which encompasses wavelengths of approximately 1 mm to 100 µm and is increasingly used for classifying semiconductor properties. The refractive index of undoped Si in this range is a constant n = 3.42. We choose the domain length to be 15 mm in the direction of propagation.
The simulation geometry. Red arrows indicate incident and reflected waves. The left and right regions are air with n = 1 and the Si slab in the center has a refractive index n = 3.42. The x_{i}s on the bottom denote the spatial location of the planes. The slab is centered in the simulation domain, such that x_{1} = (15 mm – 525 µm)/2. Note that this image is not to scale.
For a 2D Full-Wave simulation, we set a maximum element size of to ensure the solution is well resolved. The simulation is invariant in the y direction and so we choose our simulation height to be . Because we have constrained the wave to travel along the x-axis, we choose a mapped mesh to generate rectangular elements. The mesh will then be one mesh element thick in the y direction, with a mesh element size in the x direction of , where n depends on whether it is air or Si. Again, note that this is a wavelength-dependent mesh.
Before setting up the mesh for a Beam-Envelopes simulation, we first need to specify our user-defined phase function. The Gaussian Beam Incident at the Brewster Angle example in the Application Gallery demonstrates how to define a user-defined phase function for each domain through the use of variables, and we will use the same technique here. Referring to x_{0}, x_{1}, and x_{2} in the geometry figure above, we define the phase function for a plane wave traveling left to right in the three domains as
where n = 3.42 and the first line corresponds to in the leftmost domain, the second line is in the Si slab, and the bottom line is in the rightmost domain. We then use this variable for the phase of the first wave, and its negative for the phase of the second wave. Because we have completely captured the full phase variation of the solution in the ansatz, this allows a mapped mesh of only three elements for the entire model — one for each domain. Let’s examine what the mesh looks like in the Si slab for these two interfaces at two different wavelengths, corresponding to 1 mm and 250 µm.
The mesh in the Si (dielectric) slab. From left to right, we have the Full-Wave mesh at 1 mm, the Full-Wave mesh at 250 µm, and the Beam-Envelopes mesh at any wavelength. Note that the Full-Wave mesh density clearly increases with decreasing wavelength, while the Beam-Envelopes mesh is a single rectangular element at any wavelength.
Yes, that is the correct mesh for the Si slab in the Beam-Envelopes simulation. Because the ansatz matches the solution exactly, we only need three total elements for the entire simulation: one for the Si slab and one each for the two air domains on either side of it. This is independent of wavelength. On the other hand, the mesh for the Full-Wave simulation is approximately four times more dense at = 250 µm than at = 1 mm. Let’s look at this in concrete numbers for the degrees of freedom (DOF) solved for in these simulations.
Wavelength Simulated |
Full-Wave Simulation DOF Used |
Beam-Envelopes Simulation DOF Used |
---|---|---|
1 mm | 4,134 | 74 |
250 µm | 16,444 | 74 |
The number of degrees of freedom (DOF) used at two different wavelengths for the Full-Wave and Beam-Envelopes simulations.
Again, it is important to point out that this does not mean that one interface is better or worse than another. They are different techniques and choosing the appropriate option is an important simulation decision. However, it is fair to say that a Full-Wave simulation is more general, since we did not need to supply it with a wave vector or phase function. It can solve a wider class of problems than Beam-Envelopes simulations, but Beam-Envelopes simulations can greatly reduce the DOF when the wave vector is known. As we have seen in a previous blog post, memory usage in a simulation strongly depends on the number of DOF. Do not blindly use a Beam-Envelopes simulation everywhere though! Let’s take a look at another example where we intentionally make a bad choice for the wave vector and see what happens.
In the hypothetical free space example above, we chose a unidirectional wave vector. Here, we will do the same for the Si slab. It is important to emphasize that choosing a single wave vector where we know that the solution will be a superposition of left- and right-traveling waves is an exceptionally bad choice, and we do this here solely for demonstration purposes. Instead of using the bidirectional formulation with a user-defined phase function, let’s naively choose a single “guess” wave vector of and see what the damage is. Using our ansatz, inside of the dielectric slab we have
where the left-hand side is the solution we are computing and the right-hand side is exact. Now, we manipulate the equation slightly to examine the spatial variation in the solution.
We intentionally chose the case where , which means we can simplify to
Since and are constants determined by the Fresnel relations at the boundaries of the dielectric slab, this means that the only spatial variation in the computed solution will come from . The minimum mesh requirement in the slab is then determined by the “effective” wavelength of this oscillating term
which is half of the original wavelength. Not only have we made the Beam-Envelopes mesh wavelength dependent, but the required mesh in the dielectric slab for this choice of wave vector needs to be twice as dense as the mesh for a Full-Wave simulation. We have actually made the situation worse with the poor choice of a single wave vector for a simulation with multiple reflections. We could, of course, simply double the mesh density and obtain the correct solution, but that would defeat the purpose of choosing the Beam-Envelopes simulation in the first place. Make smart choices!
Another practical question is how do the results of a Full-Wave and Beam-Envelopes simulation compare? They are both solving Maxwell’s equations on the same geometry with the same material properties, and so the various results (transmission, reflection, field values) agree as you would expect. There are slight differences though.
If you want to evaluate the electric field of the right-propagating wave in the dielectric slab, you can do that in the Beam-Envelopes simulation. This is, of course, because we solved for both right- and left-propagating waves and obtained the total field by summing these two contributions. This could be extracted from the Full-Wave simulation in this case as well, but it would require additional user-defined postprocessing and may not be possible in all cases. It may seem counterintuitive in that we actually have more information readily available from a Beam-Envelopes simulation, even though it is computationally less expensive. We must remember, however, that this is simply the result of solving the model using the ansatz we specified initially.
We have examined the simple case of a dielectric slab in free space using both the Electromagnetic Waves, Frequency Domain and Electromagnetic Waves, Beam Envelopes interfaces. In comparing Full-Wave and Beam-Envelopes simulations, we showed that a Beam-Envelopes simulation can handle much larger simulations, but only in cases where we have good knowledge of the wave vector (or phase function) everywhere in the simulation. This knowledge is not required for a Full-Wave simulation, but the simulation must then be meshed on the order of a wavelength, as opposed to meshing the change in the envelope function in a Beam-Envelopes simulation. It is also worth mentioning that most Beam-Envelopes meshes will need more than the three elements shown here. This was only possible here because we chose a textbook example with an analytical solution to use as a teaching model. For more realistic simulations, you can refer to the Mach-Zehnder Modulator or Self-Focusing Gaussian Beam examples in the Application Gallery.
Note that the Electromagnetic Waves, Frequency Domain interface is available in both the RF and Wave Optics modules, although with slightly different features. The Full-Wave simulation discussed in this post could be performed in either module, although the Beam-Envelopes simulation requires the Wave Optics Module. For a full list of differences between the RF and Wave Optics modules, you can refer to this specification chart for COMSOL Multiphysics products.
In a previous blog post discussing bit-by-bit hologram simulation, we introduced holographic data storage, its applications in consumer electronics, and how to simulate a bit-by-bit hologram. Now, we’ll discuss the other form of holographic data storage: page data storage. A page is a block of data represented by a spatial light modulator (SLM) that is either transmissive or reflective by using microelectromechanical systems (MEMS) or liquid crystal on silicon (LCoS).
As mentioned in the previous blog post, simulation for holographic data storage has traditionally been performed by the beam propagation method, which can handle very large computational domains, but cannot correctly handle a large focusing angle. COMSOL Multiphysics, on the other hand, uses a full-wave method, which can handle any kind of beam, but uses relatively more memory. With COMSOL Multiphysics, we can simulate a page (multibyte) data storage system in a small domain. To demonstrate, let’s consider a rectangular domain similar to that used in the previous study. This time, we will cipher one-byte (or eight bits) of data.
A typical optical layout of page-type holographic data storage (the character code of my name is encoded in binary data in the SLM).
For this simulation, we will use the binary data converted from the character code of a part of my own name in its native language. 01001101
, which means “water”, can be seen in the fifth row in the SLM in the image above. To be more realistic, we’ll use a set of Fourier lenses to focus the object beam into the holographic material to record, expand, and visualize the retrieved object beam onto the detector in the retrieval process. Of course, we won’t model a lens, but instead make a focused beam by Fourier transforming the electric field amplitude after the SLM and providing it as the incident field in the scattering boundary condition on the incident boundary.
To image the retrieved object beam on the detector, we again Fourier transform the retrieved electric field amplitude and square the norm to get an intensity that a charge-coupled device (CCD) or complementary metal-oxide semiconductor (CMOS) sensor detects as a signal. More signal processing takes place afterward to create a cleaner signal and lessen the bit error rate to a significantly smaller level, but we will not go into this process here.
A holographic page data storage system, carrying one-byte data.
In our previous discussion, we used a slightly diverging super Gaussian beam. For this simulation, the domain size will be inevitably wider along the direction of the reference beam propagation, which we will discuss later. So, if we use a diverging beam, the beam will eventually touch the boundaries, which needs to be avoided. Instead of launching a 10 um beam with a flat phase on the left boundary, we will add the following quadratic phase function so the beam slightly focuses in the middle of the domain, assuming the out-of-plane electric field solved for
where is the waist radius of the reference beam, is the refractive index of the holographic material, is the wave number in the vacuum, and is the wavefront curvature at distance from the beam waist (focal plane) position defined by
Here, is the Rayleigh range in which the beam is almost straight.
For = 10 um, = 1 um, and = 1.35, it gives = 424 um. We will see later that this number is far larger than our domain size, which means that the beam is almost collimated in the computational domain. To define the wavefront curvature, we have borrowed the paraxial Gaussian beam formula. We ignored a constant phase and the Gouy phase, which are not necessary here. The image below shows how to enter the incident field with a right curvature at the left boundary ().
Defining the reference beam with a wavefront curvature.
As we are using a 10 um beam radius, the vertical domain size, , of 30 um is large enough. The biggest obstacle here is how to determine the horizontal domain size, , for the object beam entrance. Now, the aperture through which the object beam transmits is a 1 x 8 SLM with 8 pixels. The SLM behaves like a diffraction grating with a period of . When the object beam transmits through the SLM and is focused, the zeroth-order beam is focused into a circle of the so-called Airy ring radius and the diffracted beams of higher orders will spread out at angles corresponding to the diffraction orders.
To get sufficient information from the SLM and store correct data in the holographic material, we want to capture up to at least the first-order beams (0^{th} and ±1^{st}). Otherwise, we may get some retrieved signal, but the signal might not fully restore the original data. Another reason why we only take up to the first orders is because all other higher orders will be too weak in intensity to be recorded in the holographic material.
The first requirement is that the zeroth-order beam radius, , must be 10 um, which determines the numerical aperture (NA) of the lens system. The Airy ring radius, , is given by the Airy ring radius formula
where is the wavelength in the air.
We want the Airy ring radius to be 10 um. From this requirement, we get the NA for a given and as
On the other hand, the NA is originally defined as
where is the focusing angle, is the number of SLM pixels, is the half size of the SML pixel, and is the focal length of the Fourier lens.
From this equation, a ratio, , is derived as
We apply the grating equation for the first order
where is the diffraction angle of the first-order beams.
We get the deviation of the beam position of the first-order beams from the zeroth-order beam at a distance as
Inserting the known numbers, = 8 and = 10 um, we get = 65.6 um. Adding some margin to capture the “whole” first-order beams, half of may be 80 um; that is, = 160 um. It’s worth mentioning that this particular figure is one of the key elements of holographic technology.
Other than this number, , , and are undetermined. Now that we know all of the domain sizes, we can estimate the number of meshes needed from the maximum mesh size, , where is the refractive index of the holographic material and is the intersecting angle between the object and reference beams. With the RAM capacity of my own computer, = 1 um seems to be the shortest wavelength. Then, we get = 131.1, of which the numbers and are dependent. For now, let be 40 um, followed by = 5.2 mm. We now have all of the simulation parameters.
To prepare the 1 x 8 pixel data, we can define the primitive built-in rectangular function to represent a single pixel. To make pixel data, the rectangular function is shifted and added up. 01001101
is defined as an analytic function, as shown in the figure below. The open subapertures stand for “1″.
An SLM aperture opacity function, representing the eight-bit data of 01001101
.
Next, we focus the object beam. In Fourier optics, the image of the input electric field that is focused by a Fourier lens in the focal plane is the Fourier transform of the input field. The complex electric field amplitude in the image plane focused by a Fourier lens with the focal length is calculated by
where is the spatial coordinate in the Fourier/image space and represents the spatial frequency.
Do we need to use additional software to implement the Fourier transformation? No. By using COMSOL Multiphysics, all of the required capabilities are included in one package. You can also use COMSOL Multiphysics as a convenient scientific computational software in the GUI of the same platform as other finite element computations.
The Settings window is shown in the figure below, followed by the result of the Fourier transformation of the page data 01001101
, calculated by the COMSOL software.
The settings for the incident object beam, which is the Fourier transform of the electric field amplitude after the SLM.
The computed incident object beam as the Fourier transform of the binary data 01001101
.
The center beam is the zeroth-order beam and the two side beams with the opposite phase are the first-order beams. This is a typical Fraunhofer diffraction pattern of a grating. As we calculated before, our computational domain fits these three beams exactly. This electric field amplitude is given as the Electric Field boundary condition for the object beam. The following figures are the result of the page data recording.
The electric field amplitude (top) and intensity (bottom) for the page data recording.
Our hologram simulation is starting to look more interesting thanks to our encoding and ciphering work. The data for my name has been encoded by an industrial standard and then converted to a binary code. Then, it was Fourier transformed by a Fourier lens, which can be thought of as another ciphering process. Finally, the code was ciphered in a hologram. Of course, you can’t crack the code by simply looking at any of the images above.
Next, we move on to the data retrieval step. To retrieve the data, as was described in the previous blog post, we can use the same COMSOL Multiphysics feature to turn the functionalities on and off. We do this by adding the Wave Equation, Electric 2 node with a user-defined refractive index, which specifies the modulated index.
The Settings window for the modulated refractive index.
The modulated refractive index. The modulation amplitude corresponds to the position where the electric field intensity exceeds the threshold.
By turning the object beam off and keeping the reference beam on, as well as having the modulated index, we get the result of the retrieval simulation.
The electric field amplitude (top) and intensity (bottom) for the page data retrieval.
The electric field amplitude at the bottom edge during page data retrieval (cross section).
Now, we want to image this retrieved data onto the CCD surface by using the other Fourier lens. To do so, we will Fourier transform the retrieved electric field amplitude again and take the square of this amount. The following figure is the final result. The CCD detects the 1
positions in the original code, 01001101
. We finally see the code again!
The retrieved data on the CCD surface. The dashed line represents the position of 1
in the original code.
We have implemented a holographic page data storage model using the wave optics capabilities of COMSOL Multiphysics. Though the rigorous Maxwell solver persuades us to pay more attention to some specific restrictions, we were able to catch a glimpse of the holography created by the design calculation we performed prior to the simulation. We also went over some helpful and convenient uses of COMSOL Multiphysics as a scientific calculator. As we learned, the COMSOL software can perform all of these tasks in one environment, with sequential finite element computations and other scientific calculations performed simultaneously.
What if airplane walls could appear transparent, offering an expansive view while flying high above the clouds? Now, imagine if these same lightweight windows could also double as interactive entertainment screens. Such advancements could translate into greater fuel and cost savings, while providing further space and comfort for passengers. With the help of an emerging technology — organic light-emitting diodes (OLEDs) — these ideas are becoming a potential reality.
A flexible OLED device. Image by meharris. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
OLEDs function similarly to LED lights, except that they use organic molecules to produce light. This newer technology is valued for its many favorable attributes, including being thin, flexible, lightweight, and bright. In general, OLEDs also feature a low operating voltage as well as low power consumption. Significant light loss, however, is an important concern, with only 20% of emitted light leaving OLED devices. This translates into a low outcoupling efficiency and low energy efficiency.
So what, you might wonder, is the cause of such light loss? Several factors can contribute. For instance, mismatches in the refractive index between the different OLED layers can result in total internal reflections. Another potential source is light coupling to surface plasmons at the metal cathode.
As a leader in the development of OLED lighting panels, Konica Minolta Laboratory noticed a lack of research behind the latter of these two cases — the plasmon effect. Using the RF Module in COMSOL Multiphysics, the team sought to analyze how plasmon coupling and structure impact the efficiency of OLEDs, presenting their findings at the COMSOL Conference 2015 Boston.
To begin, let’s take a closer look at the inner workings of an OLED. Such devices typically consist of two or more layers of organic material placed between two electrodes, namely the anode and cathode. All of these components are deposited on a substrate, which is often made of glass or plastic.
The diagram below provides an overview of the different individual layers. They include a metal (Ag) cathode; three organic layers: the electron transport layer (ETL), emitting layer (EML), and hole transport layer (HTL); a transparent anode (commonly made of an indium tin oxide, or ITO); and a substrate.
The structure of an OLED. Image by Leiming Wang, Jun Amano, and Po-Chieh Hung and taken from their COMSOL Conference 2015 Boston presentation.
The metal cathode, referred to as a metal electrode in the diagram above, is an important point of focus in plasmon loss. In fact, around 40% of the total emitted light ends up coupling to surface plasmons at this point — a significant percentage of the total emission. Reducing plasmon loss at the metal cathode is therefore an essential step when designing OLEDs.
Looking to do just that, the research team at Konica Minolta Laboratory used simulation to test the impact of incorporating a nanostructured or nanograting cathode structure into their OLED design. Here’s an overview of what they found…
When beginning their research studies, the team’s initial step was to analyze mode distribution and plasmon coupling in real space. To do so, they used a 2D simulation of a multilayer bottom-emitting OLED. This made it possible to easily identify the coupling of dipole emission into various light modes.
The initial set of results indicates that the waveguide mode does not contribute to light emission, as it essentially propagates toward the sides. With that in mind, the researchers shifted their attention to a wave featuring SPP wave characteristics, which you can see highlighted in the following figure. A surface plasmon polarization (SPP) wave is a surface wave that is confined to a narrow region at the boundary between the metal cathode and the neighboring electron transport layer.
The studies show that the excitation of the SPP wave at the cathode interface, and thus the coupling of dipole emissions into SPP, appears to be the main reason for plasmon loss. The findings ultimately confirmed the team’s decision to focus on evaluating plasmon loss and designing an alternate cathode structure.
The simulation domain in 2D (top) and the field distribution of a multilayer OLED structure’s dipole emission (bottom). Images by Leiming Wang, Jun Amano, and Po-Chieh Hung and taken from their COMSOL Conference 2015 Boston paper.
The next item on the list was to measure the plasmon coupling effect for both flat and nanograting cathode structures. Creating electromagnetic models of the plasmon coupling effect at the metal cathode was a required step for the analysis. In an effort to focus specifically on the plasmon effect, the team used a simple model representing an Ag/EML structure featuring two layers. The finite element method (FEM) model enabled the researchers to simulate optical effects resulting from arbitrary subwavelength structures, which can be rather difficult to achieve through analytical simulations.
From the results, it is possible to draw a comparison between the dipole emission for a flat interface and a nanograting interface. The flat interface model (shown in the image below on the top) illustrates that the dipole emission is primarily coupled to the SPP wave, with just a small amount radiated out as usable light. On the other hand, SPP coupling is greatly suppressed when using a nanograting interface (shown in the image below on the bottom). Such findings suggest that using a nanostructured cathode can help significantly reduce plasmon loss. Before drawing any final conclusions, however, the team wanted to compare the two structures in a few other ways.
A field distribution simulation of a dipole emission for the two-layer OLED structure with a flat (top) and nanograting (bottom) interface. The insert, located in the bottom right-hand corner, depicts the structural parameters of the nanograting cathode. Images by Leiming Wang, Jun Amano, and Po-Chieh Hung and taken from their COMSOL Conference 2015 Boston paper.
For further insight into the structures, a power flow analysis was performed. The researchers were able to use the results found here to calculate the partition of total emission power into the light mode and plasmon mode. The results from this study refined the team’s earlier research by suggesting that to significantly reduce plasmon loss when using a nanograting structure, the cathode and emission layer must be less than 100 nm apart from one another.
The simulation studies up until this point involved the use of 2D models. 3D models, however, are superior for characterizing the isotropic nature of OLED light. The researchers therefore opted to add 3D simulations of OLEDs into the mix. As depicted by their results, strong field intensity exists in the cross-sectional xy-plane at the flat interface, confirming that strong SPP excitation occurs in the flat structure. The findings also reiterate that coupling to SPP is negligible for the nanograting structure.
3D field distribution simulations of a dipole emission in an OLED model with a flat (top) and nanograting (bottom) interface. Images by Leiming Wang, Jun Amano, and Po-Chieh Hung and taken from their COMSOL Conference 2015 Boston paper.
Building off their initial research studies, the team additionally sought to analyze the influence of size, shape, and nanograting period on the plasmon loss reduction. This translated into running parametric studies to optimize the nanograting cathode structure and see how structural changes affect plasmon loss. Here, we’ll focus on one such study, which looks at the grating structure’s effect on the overall plasmon reduction.
Left: The average relative plasmon loss (the plasmon loss with the grating relative to the plasmon loss with the flat surface) as a function of two different grating geometrical parameters: pitch height (on the x-axis) and pitch duty ratio (on the y-axis). Here, the pitch duty ratio is the quotient of the grating post width and the grating period. Right: Plotting the corresponding standard deviation of the wavelength averaging. Images by Leiming Wang, Jun Amano, and Po-Chieh Hung and taken from their COMSOL Conference 2015 Boston presentation.
The studies show that smaller pitch duty ratios lead to larger reductions in plasmon loss (represented by the darker colors in the figure above on the left). The dark colors in the right figure represent parameter combinations with a small wavelength variation. Therefore, the encircled common darker cells in the bottom-right corners of the figures indicate the optimal structure configuration for both reducing plasmon loss and having broadband performance. In fact, the circled cell generates an approximate 50% plasmon loss reduction over a broadband emission. This serves as additional proof that an optimized nanograting cathode structure can improve OLED efficiency.
The simulation studies highlighted here mark a pivotal point in OLED research, with the mode distribution and plasmon coupling of OLEDs visualized in real space. The research findings provide opportunities for further innovative research into the design and optimization of the technology. As the efficiency of OLEDs continues to improve, their widespread commercial use will increase.