When approaching the question of what a metal is, we can do so from the point of view of the governing Maxwell’s equations that are solved for electromagnetic wave problems. Consider the frequency-domain form of Maxwell’s equations:
The above equation is solved in the Electromagnetic Waves, Frequency Domain interface available in the RF Module and the Wave Optics Module. This equation solves for the electric field, \mathbf{E}, at the operating (angular) frequency \omega = 2 \pi f. The other inputs are the material properties: \mu_r is the relative permeability, \epsilon_r is the relative permittivity, and \sigma is the electrical conductivity.
For the purposes of this discussion, we will say that a metal is any material that is both lossy and has a relatively small skin depth. A lossy material is any material that has a complex-valued permittivity or permeability or a non-zero conductivity. That is, a lossy material introduces an imaginary-valued term into the governing equation. This will lead to electric currents within the material, and the skin depth is a measure of the distance into the material over which this current flows.
At any non-zero operating frequency, inductive effects will drive any current flowing in a lossy material towards the boundary. The skin depth is the distance into the material within which approximately 63% of the current flows. It is given by:
where both \mu_r and \epsilon_r can be complex-valued.
At very high frequencies, approaching the optical regime, we are near the material plasma resonance and do in fact represent metals via a complex-valued permittivity. But when modeling metals below these frequencies, we can say that the permittivity is unity, the permeability is real-valued, and the electrical conductivity is very high. So the above equation reduces to:
Before you even begin your modeling in COMSOL Multiphysics, you should compute or have some rough estimate of the skin depth of all of the materials you are modeling. The skin depth, along with your knowledge of the dimensions of the part, will determine if it is possible to use the Impedance boundary condition or the Transition boundary condition.
Now that we have the skin depth, we will want to compare this to the characteristic size, L_c, of the object we are simulating. There are different ways of defining L_c. Depending on the situation, the characteristic size can be defined as the ratio of volume to surface area or as the thickness of the thinnest part of the object being simulated.
Let’s consider an object in which L_c \gg \delta. That is, the object is much larger than the skin depth. Although there are currents flowing inside of the object, the skin effect drives these currents to the surface. So, from a modeling point of view, we can treat the currents as flowing on the surface. In this situation, it is appropriate to use the Impedance boundary condition, which treats any material “behind” the boundary as being infinitely large. From the point of view of the electromagnetic wave, this is true, since L_c \gg \delta means that the wave does not penetrate through the object.
The Impedance boundary condition is appropriate if the skin depth is much smaller than the object.
With the Impedance boundary condition (IBC), we are able to avoid modeling Maxwell’s equations in the interior of any of the model’s metal domains by assuming that the currents flow entirely on the surface. Thus, we can avoid meshing the interior of these domains and save significant computational effort. Additionally, the IBC computes losses due to the finite conductivity. For an example of the appropriate usage of the IBC and a comparison with analytic results, please see the Computing Q-Factors and Resonant Frequencies of Cavity Resonators tutorial.
The IBC becomes increasingly accurate as L_c / \delta \rightarrow \infty; however, it is accurate even when L_c / \delta \gt > 10 for smooth objects like spheres. Sharp-edged objects such as wedges will have some inaccuracy at the corners, but this is a local effect and also an inherent issue whenever a sharp corner is introduced into the model, as discussed in this previous blog post.
Now, what if we are dealing with an object that has one dimension that is much smaller than the others, perhaps a thin film of material like aluminum foil? In that case, the skin depth in one direction may actually be comparable to the thickness, so the electromagnetic fields will partially penetrate through the material. Here, the IBC is not appropriate. We will instead want to use the Transition boundary condition.
The Transition boundary condition (TBC) is appropriate for a layer of conductive material with a thickness relatively smaller than the characteristic size, and curvature, of the objects being modeled. The TBC can be used even if the thickness is many times greater than the skin depth.
The TBC takes the material properties as well as the thickness of the film as inputs, computing an impedance through the thickness of the film as well as a tangential impedance. These are used to relate the current flowing on the surface of either side of the film. That is, the TBC will lead to a drop in the transmitted electric field.
From a computational point of view, the number of degrees of freedom on the boundary is doubled to compute the electric field on either surface of the TBC, as shown below. Additionally, the total losses through the thickness of the film are computed. For an example of using this boundary condition, see the Beam Splitter tutorial, which models a thin layer of silver via a complex-valued permittivity.
The Transition boundary condition computes a surface current on either side of the boundary.
So far, with both the TBC and the IBC, we have assumed that the surfaces are perfect. A planar boundary is assumed to be geometrically perfect. Curved boundaries will be resolved to within the accuracy of the finite element mesh used, the geometric discretization error, as discussed here.
Rough surfaces impede current flow compared to smooth surfaces.
All real surfaces, however, have some roughness, which may be significant. Imperfections in the surface prevent the current from flowing purely tangentially and effectively reduce the conductivity of the surface (illustrated in the figure above). With COMSOL Multiphysics version 5.1, this effect can be accounted for with the Surface Roughness feature that can be added to the IBC and TBC conditions.
For the IBC, the input is the Root Mean Square (RMS) roughness of the surface height. For the TBC, the input is instead given in terms of the RMS of the thickness variation of the film. The magnitude of this roughness should be greater than the skin depth, but much smaller than the characteristic size of the part. The effective conductivity of the surface is decreased as the roughness increases, as described in “Accurate Models for Microstrip Computer-Aided Design” by E. Hammerstad and O. Jensen. There is a second roughness model available, known as the Snowball model, which uses the relationships described in The Foundation of Signal Integrity by P. G. Huray.
It is also worth looking at the idealized situation — the Perfect Electric Conductor (PEC) boundary condition. For many applications in the radio and microwave regime, the losses at metallic boundaries are quite small relative to the other losses within the system. In microwave circuits, for example, the losses in the dielectric substrate typically far exceed the losses at any metallization.
The PEC boundary condition is a surface without loss; it will reflect 100% of any incident wave. This boundary condition is good enough for many modeling purposes and can be used early in your model-building process. It is also sometimes interesting to see how well your device would perform without any material losses.
Additionally, the PEC boundary condition can be used as a symmetry condition to simplify your modeling. Depending on your foreknowledge of the fields, you can use the PEC boundary condition, as well as its complement — the Perfect Magnetic Conductor (PMC) boundary condition — to enforce symmetry of the electric fields. The Computing the Radar Cross Section of a Perfectly Conducting Sphere tutorial illustrates the use of the PEC and PMC boundary conditions as symmetry conditions.
Lastly, COMSOL Multiphysics also includes Surface Current, Magnetic Field, and Electric Field boundary conditions. These conditions are provided primarily for mathematical completeness, since the currents and fields at a surface are almost never known ahead of time.
In this blog post, we have highlighted how the Impedance, Transition, and Perfect Electric Conductor boundary conditions can be used for modeling metallic surfaces, helping to identify situations in which each should be used. But, what if you cannot use any of these boundary conditions? What if the characteristic size of the parts you are simulating are similar to the skin depth? In that case, you cannot use a boundary condition. You will have to model the metal domain explicitly, just as you would for any other material. This will be the next topic we focus on in this series, so stay tuned.
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When modeling electromagnetic structures (e.g., antennas, waveguides, cavities, filters, and transmission lines), we can often limit our analysis to one small part of the entire system. Consider, for example, a coaxial splitter as shown here, which splits the signal from one coaxial cable (coax) equally into two. We know that the electromagnetic fields in the incoming and outgoing cables will have a certain form and that the energy is propagating in the direction normal to the cross section of the coax.
There are many other such cases where we know the form (but not the magnitude or phase) of the electromagnetic fields at some boundaries of our modeling domain. These situations call for the use of the Lumped Port and the Port boundary conditions. Let us look at what these boundary conditions mean and when they should be used.
We can begin our discussion of the Lumped Port boundary condition by looking at the fields in a coaxial cable. A coaxial cable is a waveguide composed of an inner and outer conductor with a dielectric in between. Over its range of operating frequencies, a coax operates in Tranverse Electro-Magnetic (TEM) mode, meaning that the electric and the magnetic field vectors have no component in the direction of wave propagation along the cable. That is, the electric and magnetic fields both lie entirely in the cross-sectional plane. Within COMSOL Multiphysics, we can compute these fields and the impedance of a coax, as illustrated here.
However, there also exists an analytic solution for this problem. This solution shows that the electric field drops off proportional to 1/r between the inner and outer conductor. So, since we know the shape of the electric field at the cross section of a coax, we can apply this as a boundary condition using the Lumped Port, Coaxial boundary condition. The excitation options for this condition are that the excitation can be specified in terms of a cable impedance along with an applied voltage and phase, in terms of the applied current, or as a connection to an externally defined circuit. Regardless of these three options, the electric field will always vary as 1/r times a complex-valued number that represents the sum of the (user-specified) incoming and the (unknown) outgoing wave.
The electric field in a coaxial cable.
For a coaxial cable, we always need to apply the boundary condition at an annular face, but we can also use the Lumped Port boundary condition in other cases. There are also a Uniform and a User-Defined option for the Lumped Port condition. The Uniform option can be used if you have a geometry as shown below: a surface bridging the gap between two electrically conductive faces. The electric field is assumed to be uniform in magnitude between the bounding faces, and the software automatically computes the height and width of the Lumped Port face, which should always be much smaller than the wavelength in the surrounding material. Uniform Lumped Ports are commonly used to excite striplines and coplanar waveguides, as discussed in detail here.
A typical Uniform Lumped Port geometry.
The User-Defined option allows you to manually enter the height and width of the feed, as well as the direction of the electric field vector. This option is appropriate if you need to manually enter these settings, like in the geometry shown below and as demonstrated in this example of a dipole antenna.
An example of a User-Defined Lumped Port geometry.
Another use of the Lumped Port condition is to model a small electrical element such as a resistor, capacitor, or inductor bonded onto a microwave circuit. The Lumped Port can be used to specify an effective impedance between two conductive boundaries within the modeling domain. There is an additional Lumped Element boundary condition that is identical in formulation to the Lumped Port, but has a customized user interface and different postprocessing options. The example of a Wilkinson power divider demonstrates this functionality.
Once the solution of a model using Lumped Ports is computed, COMSOL Multiphysics will also automatically postprocess the S-parameters, as well as the impedance at each Lumped Port in the model. The impedance can be computed for TEM mode waveguides only. It is also possible to compute an approximate impedance for a structure that is very nearly TEM, as shown here. But once there is a significant electric or magnetic field in the direction of propagation, then we can no longer use the Lumped Port condition. Instead, we must use the Port boundary condition.
To begin discussing the Port boundary condition, let’s examine the fields within a rectangular waveguide. Again, there are analytic solutions for propagating fields in waveguide. These solutions are classified as either Transverse Electric (TE) or Transverse Magnetic (TM), meaning there is no electric or magnetic field in the direction of propagation, respectively.
Let’s examine a waveguide with TE modes only, which can be modeled in the 2D plane. The geometry we will consider is of two straight sections of different cross-sectional area. At the operating frequency, the wider section supports both TE10 and TE20 modes, while the narrower supports only the TE10 mode. The waveguide is excited with a TE10 mode on the wider section. As the wave propagates down the waveguide and hits the junction, part of the wave will be reflected back towards the source as a TE10 mode, part will continue along into the narrower section again as a TE10 mode, and part will be converted to a TE20 mode, and then propagate back towards the source boundary. We want to properly model this and compute the split into these various modes.
The Port boundary conditions are formulated slightly differently from the Lumped Port boundary conditions in that you can add multiple types of ports to the same boundary. That is, the Port boundary conditions each contribute to (as opposed to the Lumped Ports, which override) other boundary conditions. The Port boundary conditions also specify the magnitude of the incoming wave in terms of the power in each mode.
Sketch of the waveguide system being considered.
The image below shows the solution to the above model with three Port boundary conditions, along with the analytic solution for the TE10 and TE20 modes for the electric field shape. Computing the correct solution to this problem does require adding all three of these ports. After computing the solution, the software also makes the S-parameters available for postprocessing, which indicates the relative split and phase shift of the incoming to outgoing signals.
Solution showing the different port modes and the computed electric field.
The Port boundary condition also supports Circular and Coaxial waveguide shapes, since these cases have analytic solutions. However, most waveguide cross sections do not. In such cases, the Numeric Port boundary condition must be used. This condition can be applied to an arbitrary waveguide cross section. When solving a model with a Numeric Port, it is also necessary to first solve for the fields at the boundary. For examples of this modeling technique, please see this example first, which compares against a semi-analytic case, followed by this example, which can only be solved by numerically computing the field shape at the ports.
Rectangular, Coaxial, and Circular Ports are predefined.
Numeric Ports can be used to define arbitrary waveguide cross sections.
The last case, when using the Port boundary condition, is appropriate for the modeling of plane waves incident upon quasi-infinite periodic structures such as diffraction gratings. In this case, we know that any incoming and outgoing waves must be plane waves. The outgoing plane waves will be going in many different directions (different diffraction orders) and we can determine ahead of time the directions, albeit not the relative magnitudes. In such instances, you can use the Periodic Port boundary condition, which allows you to specify the incoming plane wave polarization and direction. The software will then automatically compute all the directions of the various diffracted orders and how much power goes into each diffracted order.
For an extensive discussion of the Periodic Port boundary condition, please read this previous blog post on periodic structures. For a quick introduction to the use of these boundary conditions, please see this model of plasmonic wire grating.
We have introduced the Lumped Port and the Port boundary conditions for modeling boundaries at which an electromagnetic wave can pass without reflection and where we know something about the shape of the fields. Alternative options for the modeling of boundaries that are non-reflecting in cases where we do not know the shape of the fields can be found here.
The Lumped Port boundary condition is available solely in the RF Module, while the Port boundary condition is available in the Electromagnetic Waves interface in the RF Module and the Wave Optics Module as well as the Beam Envelopes formulation in the Wave Optics Module. This previous blog post provides an extensive description of the differences between these modules.
But what about those boundaries that are not transparent, such as the conductive walls of the waveguide we have looked at today? These boundaries will reflect almost all of the wave and require a different set of boundary conditions, which we will look at next.
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The modulator controls the amplitude of an optical wave as it passes through the device. Its name stems from the use of a Mach-Zehnder interferometer located between two 50/50 directional couplers. By applying a voltage across one of the two interferometer arms, we can alter the refractive index of the waveguide material and trigger a phase shift of the propagating electromagnetic wave. Then, the two waves combine again in a second directional coupler, and thanks to the phase difference created by the voltage, we get an amplitude modulation.
Alternatively, if the modulator’s input and output ports are all connected to other waveguides, the device can act as a spatial switch instead of an amplitude modulator. In that case, we can tune the voltage so the light switches between the two output ports.
A Mach-Zehnder modulator with an applied voltage on one of the interferometer arms.
Suppose we want to design a Mach-Zehnder modulator. We need it to produce low loss, give us a 50/50 split of power through the two output arms, and be used as a spatial switch.
To determine the ideal design of the device, we turn to COMSOL Multiphysics and the Wave Optics Module.
In order to meet the general requirement of keeping the overall size of the device small, we need to find the smallest possible bend radius that also provides low loss. To figure this out, we can plot the total modal transmission over the increasing bend radius of the curvature. Doing so shows us that a minimum bend radius of 2.5 millimeters gives us an acceptable 2% loss (the plot depicts a total modal transmission of 98%). We can also confirm our results by generating an electric field norm plot.
Plotting the total modal transmission over the bend radius of curvature in meters.
Next, we have to find how long the coupler needs to be in order to give us our desired 50/50 split of incident power through the two output arms of the Mach-Zehnder interferometer. This can be achieved by monitoring the power difference in the two arms and sweeping the length of the coupler. If we plot the results of the parameter sweep, we will see that a coupler length of 380 μm will ensure a 50/50 split of power between the arms. Again, we can confirm our results with an electric field norm plot.
Here’s what the plot will look like:
Electric field norm plot confirming that the power is close to equal in the two interferometer waveguide arms for a 380 micrometer long directional coupler. Gemetry is scaled by a factor of 80 in the y-direction.
Finally, we want to confirm that we can use the device as a spatial switch if we have a scenario where all the input and output ports are connected to other fibers or waveguides. In other words, we need to check that the wave can be switched between two output ports by applying a voltage across the waveguide in one of the arms and then tuning it. The below plot shows that we can, indeed, switch the output port by tuning the voltage:
Transmission (y-axis) to the upper (blue line) and lower (green line) output waveguides versus the applied voltage (x-axis).
Note that if only one input and one output port are active, the device will act as an amplitude modulator instead of a spatial switch.
Sunlight is essentially incoherent light; it is composed of many wavelengths and varying polarizations. However, we can assume linearity of the electromagnetic fields, so any polarization of light can be treated as the sum of two orthogonal polarizations — one that has the electric field polarized parallel to the plane of the interface, and the other that has the magnetic field parallel to the plane of the interface.
When a ray of light (an electromagnetic wave) propagating through free space hits a dielectric medium, part of the light will be transmitted and part will be reflected. The fraction of the light that is reflected or transmitted is dependent upon the angle of incidence, the permittivity of the dielectric, and the polarization. This can also be described by the Fresnel equations, which are an analytic solution to Maxwell’s equations.
Schematic showing light incident upon a dielectric interface. The angle of incidence is denoted by θ. Part of the light will be transmitted and part will be reflected.
Instead of solving the Fresnel equations, we can build a COMSOL model to simulate an infinite plane wave of light incident upon a dielectric medium. Using either the RF Module or the Wave Optics Module, we can build a unit cell describing a small region around the dielectric interface. We solve the full Maxwell’s equations in the unit cell, with periodic boundary conditions and ports to truncate the modeling domain.
Let’s take a look at the results of our benchmark model, which solves for two orthogonal polarizations of light and computes the reflection and transmission coefficients with respect to incident angle.
The electric field in the y-direction (surface slice plot) and the power flow (arrow plot).
The magnetic field in the y-direction and the power flow. Both are shown for θ = 70°.
Comparing COMSOL model results with analytic solution for reflectance and transmittance for electric field incidence (left) and magnetic field incidence (right).
As you can see in the above plots, the benchmark model results agree with the analytic solution. We can also see that different polarizations of light will reflect differently off of an air-dielectric interface, and this tells us why polarized sunglasses are popular with boaters!
First, let’s consider a parallelepided volume of free space representing a periodically repeating unit cell with a plane wave passing through it at an angle, as shown below:
The incident wavevector, \bf{k}, has component magnitudes: k_x = k_0 \sin(\alpha_1) \cos(\alpha_2), k_y = k_0 \sin(\alpha_1) \sin(\alpha_2), and k_z = k_0 \cos(\alpha_1) in the global coordinate system. This problem can be modeled by using Periodic boundary conditions on the sides of the domain and Port boundary conditions at the top and bottom. The most complex part of the problem set-up is defining the direction and polarization of the incoming and outgoing wave.
Although the COMSOL software is flexible enough to allow any definition of base coordinate system, in this posting, we will pick one and use it throughout. The direction of the incident light is defined by two angles, \alpha_1 and \alpha_2; and two vectors, \bf{n}, the outward pointing normal of the modeling space and \bf{a_1}, a vector in the plane of incidence. The convention we choose here is to align \bf{a_1} to the global x-axis and align \bf{n} with the global z-axis. Thus, the angle between the wavevector of the incoming wave and the global z-axis is \alpha_1, the elevation angle of incidence, where -\pi/2 > \alpha_1 > \pi/2 with \alpha_1 = 0, meaning normal incidence. The angle between the incident wavevector and the global x-axis is the azimuthal angle of incidence, \alpha_2, which lies in the range, -\pi/2 > \alpha_2 \geq \pi/2. As a consequence of this definition, positive values of both \alpha_1 and \alpha_2 imply that the wave is traveling in the positive x- and y-direction.
To use the above definition of direction of incidence, we need to specify the \bf{a_1} vector. This is done by picking a Periodic Port Reference Point, which must be one of the corner points of the incident port. The software uses the in-plane edges coming out of this point to define two vectors, \bf{a_1} and \bf{a_2}, such that \bf{a_1 \times a_2 = n}. In the figure below, we can see the four cases of \bf{a_1} and \bf{a_2} that satisfy this condition. Thus, the Periodic Port Reference Point on the incoming side port should be the point at the bottom left of the x-y plane, when looking down the z-axis and the surface. By choosing this point, the \bf{a_1} vector becomes aligned with the global x-axis.
Now that \bf{a_1} and \bf{a_2} have been defined on the incident side due to the choice of the Periodic Port Reference Point, the port on the outgoing side of the modeling domain must also be defined. The normal vector, \bf{n}, points in the opposite direction, hence the choice of the Periodic Port Reference Point must be adjusted. None of the four corner points will give a set of \bf{a_1} and \bf{a_2} that align with the vectors on the incident side, so we must choose one of the four points and adjust our definitions of \alpha_1 and \alpha_2. By choosing a periodic port reference point on the output side that is diametrically opposite the point chosen on the input side and applying a \pi/2 rotation to \alpha_2, the direction of \bf{a_1} is rotated to \bf{a_1'}, which points in the opposite direction of \bf{a_1} on the incident side. As a consequence of this rotation, \alpha_1 and \alpha_2 are switched in sign on the output side of the modeling domain.
Next, consider a modeling domain representing a dielectric half-space with a refractive index contrast between the input and output port sides that causes the wave to change direction, as shown below. From Snell’s law, we know that the angle of refraction is \beta=\arcsin \left( n_A\sin(\alpha_1)/n_B \right). This lets us compute the direction of the wavevector at the output port. Also, note that this relationship holds even if there are additional layers of dielectric sandwiched between the two half-spaces.
In summary, to define the direction of a plane wave traveling through a unit cell, we first need to choose two points, the Periodic Port Reference Points, which are diametrically opposite on the input and output sides. These points define the vectors \bf{a_1} and \bf{a_2}. As a consequence, \alpha_1 and \alpha_2 on the input side can be defined with respect to the global coordinate system. On the output side, the direction angles become: \alpha_{1,out} = -\arcsin \left( n_A\sin(\alpha_1)/n_B \right) and \alpha_{2,out}=-\alpha_2 + \pi/2.
The incoming plane wave can be in one of two polarizations, with either the electric or the magnetic field parallel to the x-y plane. All other polarizations, such as circular or elliptical, can be constructed from a linear combination of these two. The figure below shows the case of \alpha_2 = 0, with the magnetic field parallel to the x-y plane. For the case of \alpha_2 = 0, the magnetic field amplitude at the input and output ports is (0,1,0) in the global coordinate system. As the beam is rotated such that \alpha_2 \ne 0, the magnetic field amplitude becomes (\sin(\alpha_2), \cos(\alpha_2),0). For the orthogonal polarization, the electric field magnitude at the input can be defined similarly. At the output port, the field components in the x-y plane can be defined in the same way.
So far, we’ve seen how to define the direction and polarization of a plane wave that is propagating through a unit cell around a dielectric interface. You can see an example model of this in the Model Gallery that demonstrates an agreement with the analytically derived Fresnel Equations.
Next, let’s examine what happens when we introduce a structure with periodicity into the modeling domain. Consider a plane wave with \alpha_1, \alpha_2 \ne 0 incident upon a periodic structure as shown below. If the wavelength is sufficiently short compared to the grating spacing, one or several diffraction orders can be present. To understand these diffraction orders, we must look at the plane defined by the \bf{n} and \bf{k} vectors as well as in the plane defined by the \bf{n} and \bf{k \times n} vectors.
First, looking normal to the plane defined by \bf{n} and \bf{k}, we see that there can be a transmitted 0^{th} order mode with direction defined by Snell’s law as described above. There is also a 0^{th} order reflected component. There also may be some absorption in the structure, but that is not pictured here. The figure below shows only the 0^{th} order transmitted mode. The spacing, d, is the periodicity in the plane defined by the \bf{n} and \bf{k} vectors.
For short enough wavelengths, there can also be higher-order diffracted modes. These are shown in the figure below, for the m=\pm1 cases.
The condition for the existence of these modes is that:
for: m=0,\pm 1, \pm 2,…
For m=0 , this reduces to Snell’s law, as above. For \beta_{m\ne0}, if the difference in path lengths equals an integer number of wavelengths in vacuum, then there is constructive interference and a beam of order m is diffracted by angle \beta_{m}. Note that there need not be equal numbers of positive and negative m-orders.
Next, we look along the plane defined by the \bf{n} and \bf{k} vectors. That is, we rotate our viewpoint around the z-axis such that the incident wavevector appears to be coming in normally to the surface. The diffraction into this plane are indexed as the n-order beams. Note that the periodic spacing, w, will be different in this plane and that there will always be equal numbers of positive and negative n-orders.
COMSOL will automatically compute these m,n \ne 0 order modes during the set-up of a Periodic Port and define listener ports so that it is possible to evaluate how much energy gets diffracted into each mode.
Last, we must consider that the wave may experience a rotation of its polarization as it gets diffracted. Thus, each diffracted order consists of two orthogonal polarizations, the In-plane vector and Out-of-plane vector components. Looking at the plane defined by \bf{n} and the diffracted wavevector \bf{k_D}, the diffracted field can have two components. The Out-of-plane vector component is the diffracted beam that is polarized out of the plane of diffraction (the plane defined by \bf{n} and \bf{k}), while the In-plane vector component has the orthogonal polarization. Thus, if the In-plane vector component is non-zero for a particular diffraction order, this means that the incoming wave experiences a rotation of polarization as it is diffracted. Similar definitions hold for the n \ne 0 order modes.
Consider a periodic structure on a dielectric substrate. As the incident beam comes in at \alpha_1, \alpha_2 \ne 0 and there are higher diffracted orders, the visualization of all of the diffracted orders can become quite involved. In the figure below, the incoming plane wave direction is shown as a yellow vector. The n=0 diffracted orders are shown as blue arrows for diffraction in the positive z-direction and cyan arrows for diffraction into the negative z-direction. Diffraction into the n \ne 0 order modes are shown as red and magenta for the positive and negative directions. There can be diffraction into each of these directions and the diffracted wave can be polarized either in or out of the plane of diffraction. The plane of diffraction itself is visualized as a circular arc. Note that the plane of diffraction for the n \ne 0 modes is different in the positive and negative z-direction.
All of the ports are automatically set up when defining a periodic structure in 3D. They capture these various diffracted orders and can compute the fields and relative phase in each order. Understanding the meaning and interpretation of these ports is helpful when modeling periodic structures.
]]>Before digging further into this example, let’s make sure we understand what a Gaussian beam is. In short, it is an electromagnetic beam where a Gaussian function can be used to describe the electric field profile in a plane that is perpendicular to the axis of the beam. Most lasers emit a beam that very nearly approximates a Gaussian beam, thus it occurs often when analyzing photonic devices. If you want to learn more about modeling Gaussian beams, please check out the following models in the Model Gallery:
Now, let’s get to our optical scattering example. Suppose we want to investigate whether or not there is electromagnetic coupling between the nanorods in an array. In theory, we know that under the right conditions there will be coupling — by modeling the phenomenon, we can verify that it really is the case.
Assume we have a Gaussian beam with a 750 nm wavelength and a spot radius of equal length. We also have an array of 40 metallic nanorods, each with a radius of 20 nm. I mentioned in the beginning that the rods are very close to each other; to be precise, the separation between the rods is only 150 nm. The tightly focused beam is striking the nanorods, and propagates in the negative y-direction.
Gaussian beam incident on an array of nanorods.
In order to get an idea of the coupling, we can set up and solve for this model using COMSOL Multiphysics and the Wave Optics Module. As you can see in the image above, only a few of the nanorods are illuminated by the electromagnetic beam. By setting up a surface plot, we can visualize the beam’s electric field norm when it’s polarized in the grating vector direction (the x-direction). Below to the left, you can see that half of the beam is illuminated by the nanorod array, and there is no noticeable reflection or diffraction. If you want a closer look at the rods, you can create a second, zoomed-in version of the surface plot (below, right). This demonstrates that there is dipolar coupling between the rods.
Electric field norm surface plot when the beam is polarized in the x-direction. |
Close-up of the nanorods, showing dipolar coupling. |
As you can see in the line graph below, the coupling extends much farther than the light beam’s intensity distribution.
Graph of the electric field’s x component (blue) and the background field of the beam (green). The
beam is polarized in the x-direction.
We can also plot a close-up of the beam as polarized in the z-direction instead of in the direction of the vector grating. Doing so will illustrate a scenario where the nanorod array behaves like an opaque metal sheet to the beam, reflecting the beam and causing strong edge diffraction. This effect is clearly demonstrated in the bottom-right plot.
Close-up of the nanorods when the beam is polarized in the z-direction, causing the array to act like an opaque metal sheet rather than individual rods. | Reflection and edge diffraction effects. |
The main intended application of the model outlined here is short-range optical communication, where light follows the path laid out by the array of nanorods. As the nanorods are smaller than the wavelength of light, and light is tightly coupled to the nanorods, the width of the light path is narrower than the wavelength. Therefore, this technology could open up for spatially dense optical communication channels, for instance between electrical circuits on chips and printed circuit boards.
You may have also heard about nanorods in a different context. In addition to what is demonstrated in this modeling example, nanorods can be used in cancer treatment.
The book starts, as all good textbooks do, with an introduction to the governing Maxwell’s equations as well as the various frequency-domain forms that are commonly used: the full-vectorial and scalar form, the Beam Propagation Method (BPM), slowly varying envelope approximation, and paraxial approximations.
The second chapter is the lengthiest, and introduces the finite element method in detail for both frequency domain and eigenvalue analysis. A lot of these details are at a very low level, perhaps more than is needed by the typical COMSOL user. However, the information here is sufficient for a student to write their own finite element code from scratch, which I always advocate for people who really want to understand the inner workings of the software. It is also nice to see that practical issues like mesh refinement and the practicalities of limited computational resources are addressed, since we cannot always solve all problems on a supercomputer!
The third chapter addresses the application of the finite element method to the BPM. This chapter will be less useful to the COMSOL user, unless they want to implement the BPM themselves using the software’s equation-based modeling capabilities. We at COMSOL have instead chosen to implement the beam envelope method, recently released with the Wave Optics Module. We look forward to seeing the authors extend this book to discuss this formulation — a natural extension of what is already presented.
The fourth chapter addresses time-domain finite element modeling. This chapter is quite short, as the second chapter has already laid most of the foundations. Here, it would be nice to see more of a discussion of the various time-marching algorithms, but the extensive references provide good starting points for further investigations.
The fifth and sixth chapters introduce a few practical applications of what we would call multiphysics analysis; incorporating thermal and stress effects into a model. Nonlinear examples are also presented, and directions for further work are suggested.
In summary, Finite Element Modeling Methods for Photonics is useful for those who would like a very focused introduction to the mathematical and numerical concepts behind the RF and Wave Optics modules, and it is nice to see that the authors have written the book with that purpose in mind.
Learn more about the textbook directly from the author in this interview:
You can also download sample chapters of the book from the website of the publisher, Artech House.
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When attempting to simulate electromagnetic wave propagation in optical waveguide structures, such as optical fibers, engineers are faced with a seemingly unsolvable problem: the electromagnetic waves are oscillating rapidly in the direction of propagation. For standard computational methods (such as the finite element method) to handle such cases efficiently, they require a very fine mesh, much smaller than the wavelength. Enter the Beam Envelope Method. This method provides a clever way of capitalizing on the fact that in optical components you often have a well-defined single direction of propagation. Instead of solving for the incredibly computationally-intensive electric field, you solve for the slowly varying electric field envelope.
Solving directly for the electric field requires the oscillations to be resolved with sample points that are dense enough to avoid aliasing. The Nyquist–Shannon sampling theorem tells us that you need at least three sample points per wavelength to resolve the wave. With numerical methods for electromagnetic wave propagation, you usually need even more than three sample points, or discretization points, per wavelength. This is due to technical reasons surrounding the numerical scheme used. The finite element method is one method offered by COMSOL to solve for electromagnetic field propagation. Using finite element terminology, it is recommended that you have a least five quadratic finite elements per wavelength. By instead solving for the slowly varying field envelope, we get by with a lot fewer sample points; that is, a much coarser mesh can be used.
Built into this method is a factorization trick — you separate out the fast-varying portion of the wave as illustrated in the picture above: . The user provides the wave-vector k_{1} as an input to the method. To recover the “real” field you just multiply the solution E_{1}(x) with the fast-varying phase factor. The picture below illustrates the difference between the sample density for the electric field and the electric field envelope.
The simulations you can perform with this new method are quite impressive, especially considering that there are no particular approximations involved other than that of the applied finite element method using vector (also known as edge, or Nedelec) elements.
The picture above (top) shows the true aspect ratio representation of a self-focusing laser beam simulation and (bottom) a compressed view with the variation in refractive index as an isosurface.
The Beam Envelope Method is one of the main features of the new Wave Optics Module that was released May 3^{rd} with COMSOL Version 4.3b. The method, as implemented in the module, comes in two flavors: unidirectional and bidirectional. The bidirectional method actually solves for two waves: one incident and one reflected. In optical devices you may very well get reflections back the same direction the wave originated in and the bidirectional method is designed for that case. The Beam Envelope Method complements the conventional electromagnetic full-wave propagation methods that are also featured in the Wave Optics Module. These user interfaces are very similar to those of the RF Module but are tailored for optics simulations, having the refractive index as the default option rather than the permittivity default of the RF Module.
The Wave Optics Module gives you several analysis options including 2D, 3D, frequency-domain, eigenfrequency, and time-domain simulations. We anticipate that it will be used for wave propagation through optical media including engineered metamaterials and gyromagnetic materials. It has built-in support for anisotropic permeability, permittivity, and refractive index tensors, with or without losses.
As you may know, COMSOL Multiphysics has been used for quite some time by the optics community. The Wave Optics Module expands on previous capabilities of COMSOL Multiphysics and the RF Module, and it is the first dedicated optics product we have released. Tune into our free Wave Optics Simulations webinar on June 13^{th} to learn more.