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

shouldhave 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:

- We can evaluate the maximum or minimum value of a variable in a 3D domain, but call that result globally. This is a 3D to 0D mapping and allows us to create a temperature controller. Note that this can also be used with boundaries or edges, as well as averages or spatial integrations.
- We can extrude 2D simulation results to a 3D domain. This allows you to exploit translation symmetry in one physics (2D) and use the results in a more complex 3D model.
- We can project 3D data onto a 2D boundary (or 2D to 1D, etc.) A simple example of this is creating shadow puppets on a wall, but can also be useful for analyzing averages over a cross section.

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

\overrightarrow{E}_{FFx} = emw.Efarx\times \frac{e^{-jkr}}{(r/1[m])}

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.

- Browse previous posts in the Multiscale Modeling in High-Frequency Electromagnetics blog series

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:

\overrightarrow{E}_{FFx} = emw.Efarx\times \frac{e^{-jkr}}{(r/1[m])}

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.

\begin{align}

\overrightarrow{E} & = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\

\overrightarrow{E}_{FF} & = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\

\overrightarrow{E}_{NF} & = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}

\end{align}

\overrightarrow{E} & = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\

\overrightarrow{E}_{FF} & = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\

\overrightarrow{E}_{NF} & = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}

\end{align}

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!

- Browse the other posts in the Multiscale Modeling for High-Frequency Electromagnetics blog series

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:

- An antenna converts a local signal into free space radiation.
- The radiation propagates away from the antenna over relatively long distances.
- The radiation is detected by another antenna and converted into a received signal.

*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

P_r = p(1-|\Gamma_t|^2)(1-|\Gamma_r|^2)G_t\left(\theta_t,\phi_t\right)G_r\left(\theta_r,\phi_r\right)\left(\frac{\lambda}{4\pi r}\right)^2P_t

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:

*P*refers to the power provided to a transmission line attached to an emitting antenna_{t}*P*refers to the power received from a transmission line attached to a receiving antenna_{r}

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.

\begin{align}

r& = sqrt(x^2 + y^2 + z^2)\\

\theta& = acos(z/r)\\

\phi& = atan2(y,x)

\end{align}

r& = sqrt(x^2 + y^2 + z^2)\\

\theta& = acos(z/r)\\

\phi& = atan2(y,x)

\end{align}

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 .

\begin{align}

\hat{\theta_x}& = cos(\theta)cos(\phi)\\

\hat{\theta_y}& = cos(\theta)sin(\phi)\\

\hat{\theta_z}& = -sin(\theta)

\end{align}

\hat{\theta_x}& = cos(\theta)cos(\phi)\\

\hat{\theta_y}& = cos(\theta)sin(\phi)\\

\hat{\theta_z}& = -sin(\theta)

\end{align}

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

IEEE also includes three notes about gain in their definition:

- “Gain does not include losses arising from impedance and polarization mismatches.”
- “The radiation intensity corresponding to the isotropically radiated power is equal to the power accepted by the antenna divided by 4π.”
- “If an antenna is without dissipative loss, then in any given direction, its gain is equal to its directivity.”

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

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

\overrightarrow{E} =\frac{1}{4\pi\epsilon_0}\{k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}+[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}\}

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/

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/

\begin{align}

\overrightarrow{E}& = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\

\overrightarrow{E}_{FF}& = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\

\overrightarrow{E}_{NF}& = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}

\end{align}

\overrightarrow{E}& = \overrightarrow{E}_{FF} + \overrightarrow{E}_{NF}\\

\overrightarrow{E}_{FF}& = \frac{1}{4\pi\epsilon_0}k^2(\hat{r}\times\vec{p})\times\hat{r}\frac{e^{-jkr}}{r}\\

\overrightarrow{E}_{NF}& = \frac{1}{4\pi\epsilon_0}[3\hat{r}(\hat{r}\cdot\vec{p})-\vec{p}](\frac{1}{r^3}+\frac{jk}{r^2})e^{-jkr}

\end{align}

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

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.

P_r = p(1-| \Gamma_r|^2) G_t \left(\theta_t,\phi_t\right) G_r \left(\theta_r,\phi_r\right) \left(\frac{\lambda}{4\pi r}\right)^2 P_{rad}

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 *μ*W. 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:

- In part 2, we will simulate the emission from a point electric dipole using the
*Electromagnetic Waves, Frequency Domain*interface. We will discuss the*Far-Field Domain*node, which calculates the far-field radiation from a source, and show how the*Electromagnetic Waves, Frequency Domain*interface can be coupled to the*Electromagnetic Waves, Beam Envelopes*interface to simulate fields in the intermediate zone. - In part 3, we will simulate a point dipole radiating to a half-wavelength dipole antenna an arbitrary distance away. For verification, we will calculate the power received by the half-wavelength dipole antenna and verify our results using the Friis transmission equation.
- In part 4, we will couple our emitting source, the point electric dipole, to a ray optics simulation using the
*Geometrical Optics*interface. - In part 5, we will couple the two antennas using the
*Geometrical Optics*interface. We will again verify our results and discuss how this more general method can account for inhomogeneous media and multipath transmission.

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

andthe 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:

\left (\frac{ \partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + k^2 \right )E_z = 0

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

\left ( \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}-2ik\frac{\partial}{\partial x} \right )A(x,y) = 0

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.,

\left ( \frac{\partial^2}{\partial y^2}-2ik\frac{\partial}{\partial x} \right )A(x,y) = 0

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

A(x,y)=

\sqrt{\frac{w_0}{w(x)}}

\exp(-y^2/w(x)^2)

\exp(-iky^2/(2R(x)) + i\eta(x))

\sqrt{\frac{w_0}{w(x)}}

\exp(-y^2/w(x)^2)

\exp(-iky^2/(2R(x)) + i\eta(x))

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

(\nabla^2 + k^2 )E_{\rm sc} =-(\nabla^2 + k^2 )E_{\rm bg}

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!

- P. Vaveliuk, “Limits of the paraxial approximation in laser beams”,
*Optics Letter*s, Vol. 32, No. 8 (2007) - Browse related topics here on the COMSOL Blog:

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.

- Download our Polarizing Beam Splitter app
- Browse all of the blog posts within our Applications category
- Check out our Introduction to Application Builder Videos series

*Microsoft is a registered trademark of Microsoft Corporation in the United States and/or other countries.*

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:

g(u,v) = \iint_{-\infty}^\infty {\rm rect}(x,y) \exp (-2 \pi i(xu+yv) ) dxdy.

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 the Gaussian quadrature. This function is used in the finite element integration 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.

- Check out these blog posts about simulating holographic data storage systems:
- Find more information in these introductory books on optics:
- J.W. Goodman,
*Introduction to Fourier Optics*, W. H. Freeman, 2004. - E. Hecht,
*Optics*, Pearson Education Limited, 2014.

- J.W. Goodman,

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

\phi\left(\mathbf{r}\right) = k_0\cdot\left(x-x_0\right)

\phi\left(\mathbf{r}\right) = k_0\cdot\left(\left(x_1-x_0\right) + n\cdot\left(x-x_1\right)\right)

\phi\left(\mathbf{r}\right) = k_0\cdot\left(\left(x_1-x_0\right) + n\cdot\left(x_2-x_1\right) + \left(x-x_2\right)\right)

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 SimulationDOF Used |
Beam-Envelopes SimulationDOF 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

\mathbf{E}\left(\mathbf{r}\right)e^{-j\mathbf{k_G}\cdot\mathbf{r}} = \mathbf{E_1}e^{-j\mathbf{k}\cdot\mathbf{r}} + \mathbf{E_2}e^{j\mathbf{k}\cdot\mathbf{r}}

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.

\mathbf{E}\left(\mathbf{r}\right) = \mathbf{E_1}e^{-j\left(\mathbf{k-k_G}\right)\cdot\mathbf{r}} + \mathbf{E_2}e^{j\left(\mathbf{k+k_G}\right) \cdot\mathbf{r}}

We intentionally chose the case where , which means we can simplify to

\mathbf{E}\left(\mathbf{r}\right) = \mathbf{E_1} + \mathbf{E_2}e^{j\left(\mathbf{k+k_G}\right)\cdot\mathbf{r}}.

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

\lambda_{eff} = \frac{2\pi}{\left|\mathbf{k+k_G}\right|} = \frac{2\pi}{2\left|\mathbf{k}\right|} = \frac{\lambda}{2}

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.

- Browse the COMSOL Blog for more discussions of electrical modeling
- Watch these videos:
- Take your electromagnetics modeling to the next level at a local training event
- Contact us with questions about your own model

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

E_z(x,y)=\exp \left (-\frac{y^2}{w_r^2} \right ) \exp \left (-\frac{ink_0 y^2}{2R_r(x)} \right ),

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

R_r(x)=x \left \{ 1+\left ( \frac{x_R}{x} \right )^2 \right \}.

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

w_0 = \frac{0.61 \lambda_0}{\rm NA},

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

{\rm NA} = \frac{0.61 \lambda_0}{w_0}.

On the other hand, the NA is originally defined as

{\rm NA} = \sin \theta \sim \tan \theta = \frac{Nd}{f},

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

\frac fd = \frac{N}{\rm NA}=\frac{N w_0}{0.61 \lambda_0}.

We apply the grating equation for the first order

2d\sin \alpha_1 = \lambda_0,

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

w_1=f\tan \alpha_1 \sim f\sin \alpha_1=\frac{f \lambda_0}{2d}=\frac{N w_0}{1.22}.

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

\tilde{E}(u) = \frac{1}{\sqrt{f\lambda_0}}\int_{-\infty}^{\infty}E(x)\exp(- 2 \pi i x u/(f\lambda_0))dx,

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.

- Check out the COMSOL Blog:
- Take a look at the first part of this series for an introduction to holographic data storage
- Read about the built-in integration operators for the Fourier transformation

- Learn more about Fourier optics with the book
*Introduction to Fourier Optics*by J.W. Goodman

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.

- Read the paper: “Simulating Plasmon Effect in Nanostructured OLED Cathode Using COMSOL Multiphysics“
- We’ve previously blogged about the role of simulation in optimizing light sources. Take a look at some examples:

About a decade ago, a surprising number of researchers and engineers in the U.S., Japan, and other countries worked to discover the next generation of optical storage devices to succeed the Blu-ray drive. Holography was strongly believed to be the only solution. Researchers expected that consumer demand for digital data storage would increase infinitely and in turn developed various types of holograms for a quick time-to-market. Although holographic storage was not very commercially competitive against solid-state memory, it is still a technology that any optoelectronic engineer should understand fully.

Over the last few years, as computational hardware has improved, simulation software has flourished. Software simulations let the engineer address device sensitivity, determine how much can be overwritten in one fraction of volume, and reduce the signal-to-noise ratio. Traditionally, simulation in this area has been performed by the so-called beam propagation method (BPM). The advantage of this method is that it can handle problems that involve interference, diffraction, and scattering in a domain that is 1000 times that of the wavelength. Also, the computational cost is cheap. However, the disadvantage is that it cannot correctly compute lights with a large focusing angle.

COMSOL Multiphysics has two different approaches for solving Maxwell’s equations for such holographic storage problems. One approach, the full-wave approach, can model interference and scattering, but only for modeling domain sizes that are comparable to the wavelength. The other approach, called the beam envelope method, can compute interference for a large scale, but cannot compute arbitrary scattering. In this blog series, we will look at using the full-wave approach to simulate a small-volume hologram to study how the hologram deciphers the code by the reference wave — one of the most exciting factors of holography.

As mentioned in a previous blog post, in general holography, the *object beam* is a beam scattered from an arbitrary object. In holographic data storage, the object beam is a single beam carrying one-bit data or a beam passing through a spatial light modulator (SLM) carrying multibit data. The former system is called *bit-by-bit holographic data storage*, while the latter is referred to as *holographic page data storage*.

In these processes, the object beam transmits through the aperture and comes across the reference beam to generate a complex interference fringe pattern in a holographic material. The interference fringe is the cipher that carries your information. This process is called *recording*. The light sources for the object and reference beams need to be coherent to each other and the coherence length needs to be appropriately long. To satisfy these conditions, the light source for holography is typically chosen from solid-state lasers such as a YAG laser; gas lasers such as a He-Ne laser; and nonmodulated semiconductor lasers, such as GaN and GaAs laser diodes with direct current operation.

To have a mutual coherence, the light source is originally a single laser that is split into two beams by a beam splitter. When the optical path difference between the two beams is controlled to be much less than the coherence length of the laser beam, the two beams generate an interference pattern, which is a standing wave of the laser beam at the intersectional volume in a holographic material.

Typical commercial holographic materials are made of certain photopolymers. This nonmoving stationary intensity modulation of the electric field initiates polymerization, which slightly changes the local refractive index from the original raw index. The refractive index change is , which is typically less than 1%. The value is nonlinearly proportional to the electric field intensity.

After the refractive index modulation has been set in, only the reference beam is shone on the holographic material. Then, the reference beam is scattered by the interference fringe and the scattered beam creates the objective beam as if the objective beam is present. This process is called *retrieval*. The retrieved object beam is detected by any single-pixel photodetector, such as GaP, Si, InGaAs, or Ge photodiodes for the bit-by-bit data storage, as well as by CMOS or CCD image sensors.

*A typical optical layout of page-type holographic data storage (the character code of my name is encoded in binary data in the SLM in this figure).*

Now, let’s simulate a bit-by-bit holographic data storage example. There is a single open aperture for the object beam instead of an SLM, so the object beam carries one-bit data, which can mean “1 or 0″ or “exist or not exist”. Our computational domain is a square and the layout of the beams is such that the object beam enters from the top side, while the reference beam comes from the left. Note that this 90-degree configuration is a simplified example to demonstrate the simulation setup and is not a very realistic scenario.

*A schematic of bit-by-bit holographic data storage. The objective is to compute the electromagnetic fields within a small region of the holographic material.*

Let’s go through each of the steps of the simulation process for holographic data storage, including preparation, recording, retrieval, and an overview of the appropriate settings in COMSOL Multiphysics.

Our first task is to appropriately set up the model of the laser beam. This process looks very simple, but it requires knowledge of electromagnetics and computer simulation beyond just the usage of COMSOL Multiphysics. The following points must be considered when setting up a model of a laser beam.

First of all, we want to have straight beams that uniformly propagate through the material and a wide spatial overlap between the two beams. To achieve this, the beam width has to be chosen carefully. The lower bound on the beam width is controlled by the uncertainty principle. If we try to specify a beam width that is very narrow compared to the wavelength, this means that we are trying to specify the position within a very small region. When the position is well specified, the light’s momentum becomes more uncertain, which equivalently leads to more spreading out of the beam and the beam diverges.

How much the light diverges for a given beam size is quantitatively well described by the paraxial Gaussian beam theory, which defines the beam divergence via the spread angle . This spread angle is related to the paraxial Gaussian beam waist radius as , where is the wavelength. It is obvious from this formula that the light diverges if we make the beam waist radius small compared to the wavelength. In the figure shown below and to the left, we can see a case where the waist radius equals the wavelength. You can see that the small beam waist leads to a quickly diverging beam.

If you instead specify a waist radius ten times the wavelength, then the divergence angle is , which is approximately 32 mrad. This angle is good enough for our purposes. A slightly diverging but almost collimated Gaussian beam is depicted in the figure below on the right. Super Gaussian or Lorentzian beam shapes can also be used to describe such a collimated beam.

*A beam with a narrow waist (left) diverges, while a beam with a wide waist (right) diverges negligibly. The electric field magnitude is plotted, along with arrows showing the Poynting vector.*

Our modeling domain must be large enough to capture all of the relevant phenomena that we want to capture, but not too large. This can be visualized from the image above of the two crossed beams. The modeling domain need only be large enough to enclose the region where the beams are intersecting. It doesn’t need to be too large, since we aren’t interested in the fields far away from the beam, which we know will be small. The domain also doesn’t need to be too small because we would lose information.

The boundaries of our modeling domain must achieve two purposes. First, we must launch the incoming beams, and second, the beam must be able to propagate freely out of the modeling domain. Within COMSOL Multiphysics, both of these conditions can be realized with the Second-Order Scattering boundary condition, which mimics an open boundary and also allows an incoming field representing a source from outside of the modeling domain to be specified.

It is also important that the scattering boundary conditions are placed far enough away from the beam centerline, such that the beams are only normally incident upon the boundaries. The beam should not have any significant component in parallel incidence upon the boundary, since this will lead to spurious reflections, as described in our earlier blog post on boundary conditions for electromagnetic wave problems.

We can use the information about the beam waist to choose a domain size that is sufficiently wider than the beam, such that the electric field intensity falls off by six orders of magnitude at the boundary, as shown in the figure below.

*If the domain width relative to the beam width is sufficiently large, there will be no spurious reflections (left). If the scattering boundary conditions are placed too close (right) to the beam centerline, there are observable spurious reflections.*

This problem solves for beams propagating in different directions and computes scattering and interference patterns in a material with a known refractive index. Since we know the wavelength and the refractive index, we can use this information to choose the element size. The element size must be small enough to resolve the variations in the propagating electromagnetic waves. We know from the Nyquist criterion that we need at least two sample points per wavelength, but this would give us very low accuracy. A good rule of thumb is to start with an element size of , or eight elements per wavelength in a material with peak refractive index .

Of course, you will always want to perform a mesh refinement study. For this type of problem, an element size of will typically be sufficient. Also be aware that the smaller you make the elements (the higher the accuracy), the more time and computational resources your model will take. For a detailed discussion about how to predict the size of the model, please see our blog post on the memory needed to solve a model.

Considering all of these factors, we will simulate a laser beam with a vacuum wavelength of 1 um and a beam waist profile of , a sixth-order super Gaussian beam. We will solve for the out-of-plane electric field, which means that we solve a scalar Helmholtz equation.

Now that we have appropriate settings for the beam and the domain, we are ready for the recording simulation. The figure below shows the results of the recording process. The object beam and reference beam make an interference fringe pattern at a slant angle of 45 degrees and with a periodicity of 0.524 um. This 45-degree fringe is the cipher for a single of 1 recorded in the holographic material.

*The computed electric field and intensity for the one-bit data recording.*

Next, the holographic material modulates its refractive index in the portion where the electric field intensity is above a certain threshold value. In the case of photopolymers, polymerization starts in this high-intensity region. Now, let the distribution of this high-intensity portion be denoted , as it adds up modulation on the raw index . This means that the global refractive index can be written as . is the modulation depth, which is dependent on the material’s photochemical properties.

The function shape of the modulation also depends on the material and process. The new index takes the shape of a biased and periodic rectangular function swinging around the raw index. The next figure plots the new refractive index and its cross section after recording. In this simulation example, we have used and . The modulation function can be expressed by a logical expression, `( (ewfd.normE)/maxop1(ewfd.normE) )>threshold`

, where the `maxop`

operator calculates the maximum value inside the domain, normalizing the electric field norm. ` threshold`

is a given threshold value for polymerization.

*A contour map of the electric field intensity for the binary recording that is cut off at a threshold and binarized.*

*A cross-sectional plot of the modulated index.*

Next, we simulate the retrieval process, which includes:

- Turning off the object beam
- Shining the reference beam only

After these settings change, we get the final results, as shown in the next two plots. The reference beam is diffracted/scattered by the interference fringes and creates a new beam, which restores the amplitude and the phase information overlooking a multiplicative constant. Note that the retrieved object beam is not symmetric because the reference beam slightly diverges.

*The computed electric field and intensity for the retrieval of the object beam carrying one-bit data.*

So far, we have gone through the simulation procedure in a step-by-step manner, but it is possible to perform this sequential simulation all at once. In COMSOL Multiphysics, there is a helpful feature in the Solver settings that we can use to perform this two-step sequence, the recording and retrieval processes, in one click of the *Compute* button. To do this, we select the *Modify physics tree and variables for study step* check box in each study step.

For recording, we apply the scattering boundary condition with the incident field of the super Gaussian beam (Reference SBC) on the left edge, the scattering boundary condition with the incident field of the super Gaussian beam (Object SBC) on the top edge, and the scattering boundary condition with no incidence for the rest of the boundaries (Open SBC).

*Settings for Study 1 and Step 1 of the recording process*.

*Adding the* Wave Equation, Electric 2 *node for index modulation.*

To set up a modulated refractive index, we add one more *Wave Equation, Electric* node, in which the previous result specifies a new user-defined refractive index. Here, we have used the `withsol()`

operator, which lets users apply the previous solution to evaluate an arbitrary expression. In this example, the new refractive index is given by ` n1+dn*withsol('sol2',((ewfd.normE/maxop1(ewfd.normE))^2>threshold)-0.5)`

, where `'sol2'`

is the solution for Step 1 (the recording process) and the threshold is 0.4.

*Settings for Study 1 and Step 2 for the retrieval process.*

In the retrieval process, we turn off the object beam by disabling the Object SBC. To switch to the modulated refractive index, the original *Wave Equation, Electric 1* node is disabled and the *Wave Equation, Electric 2* node is turned on. Finally, Open SBC is replaced by a new scattering boundary condition with no incidence for the top, bottom, and right boundaries (Open SBC 2).

Today, we discussed how to determine electromagnetic beam settings, which can be a very complex problem. Then, we demonstrated a simple holographic data storage simulation, called a bit-by-bit hologram. We also learned how to implement several steps in COMSOL Multiphysics to run a series of simulation steps at one time. Stay tuned for the next part of this holography series, in which we will simulate a more interesting, complicated, and realistic system of multibit holograms called holographic page data storage.

- Read the blog post Shaping Future Holography for the history, principles, applications, and implications of holograms
- Watch this archived webinar for a full demonstration on how to simulate wave optics problems in COMSOL Multiphysics
- Have any questions? Contact us for support and guidance on modeling your own holography problems in COMSOL Multiphysics

Phononic crystals are rather unique materials that can be engineered with a particular band gap. As the demand for these materials continues to grow, so does the interest in simulating them, specifically to optimize their band gaps. COMSOL Multiphysics, as we’ll show you here, can be used to perform such studies.

A *phononic crystal* is an artificially manufactured structure, or material, with periodic constitutive or geometric properties that are designed to influence the characteristics of mechanical wave propagation. When engineering these crystals, it is possible to isolate vibration within a certain frequency range. Vibration within this selected frequency range, referred to as the *band gap*, is attenuated by a mechanism of wave interferences within the periodic system. Such behavior is similar to that of a more widely known nanostructure that is used in semiconductor applications: a *photonic crystal*.

Optimizing the band gap of a phononic crystal can be challenging. We at Veryst Engineering have found COMSOL Multiphysics to be a valuable tool in helping to address such difficulties.

When it comes to creating a band gap in a periodic structure, one way to do so is to use a unit cell composed of a stiff inner core and a softer outer matrix material. This configuration is shown in the figure below.

*A schematic of a unit cell. The cell is composed of a stiff core material and a softer outer matrix material.*

Evaluating the frequency response of a phononic crystal simply requires an analysis of the periodic unit cell, with Bloch periodic boundary conditions spanning a range of wave vectors. It is sufficient to span a relatively small range of wave vectors covering the edges of the so-called *irreducible Brillouin zone* (IBZ). For rectangular 2D structures, the IBZ (shown below) spans from Γ to X to M and then back to Γ.

*The irreducible Brillouin zone for 2D square periodic structures.*

The Bloch boundary conditions (known as the Floquet boundary conditions in 1D), which constrain the boundary displacements of the periodic structure, are as follows:

u_{destination} = exp[-i\pmb{k}_{F} \cdot (r_{destination} - r_{source})] u_{source}

where **k**_{F} is the wave vector.

The source and destination are applied once to the left and right edges of the unit cell and once to the top and bottom edges. This type of boundary condition is available in COMSOL Multiphysics. Due to the nature of the boundary conditions, a complex eigensolver is needed. The system of equations, however, is Hermitian. As such, the resulting eigenvalues are real, assuming that no damping is incorporated into the model. The COMSOL software makes this step rather easy, as it automatically handles the calculation.

We set up our eigensolver analysis as a parametric sweep involving one parameter, *k*, which varies from 0 to 3. Here, 0 to 1 defines a wave number spanning the Γ-X edge, 1 to 2 defines a wave number spanning the X-M edge, and 2 to 3 defines a wave number spanning the diagonal M-Γ edge of the IBZ. For each parameter, we solve for the lowest natural frequencies. We then plot the wave propagation frequencies at each value of *k*. A band gap appears in the plot as a region in which no wave propagation branches exist. Aside from very complex unit cell models, completing the analysis takes just a few minutes. We can therefore conclude that this approach is an efficient technique for optimization if you are targeting a certain band gap location or if you want to maximize band gap width.

To illustrate such an application, we model the periodic structure shown above, with a unit cell size of 1 cm × 1 cm and a core material size of 4 mm × 4 mm. The matrix material features a modulus of 2 GPa and a density of 1000 kg/m^{3}. The core material, meanwhile, has a modulus of 200 GPa and a density of 8000 kg/m^{3}. The figure below shows no wave propagation frequencies in the range of 60 to 72 kHz.

*The frequency band diagram for selected unit cell parameters.*

To demonstrate the use of the band gap concept for vibration isolation, we simulate a structure consisting of 11 x 11 cells from the periodic structure analyzed above. These cells are subjected to an excitation frequency of 67.5 kHz (in the band gap).

*The structure used to illustrate vibration isolation for an applied frequency in the band gap.*

The animation below highlights the response of the cells. From the results, we can gather how effective the periodic structure is at isolating the rest of the structure from the applied vibrations. The vibration isolation is still practically efficient, even if fewer periodic cells are used.

*An animation of the vibration response at 67.5 kHz.*

Note that at frequencies outside of the band gap, the periodic structure does not isolate the vibrations. These responses are depicted in the figures below.

*The vibration response at frequencies outside of the band gap. Left: 27 kHz. Right: 88 kHz.*

To learn more about the 2D band gap model presented here, head over to the COMSOL Exchange, where it is available for download.

- P. Deymier (Editor),
*Acoustic Metamaterials and Phononic Crystals*, Springer, 2013. - M. Hussein, M. Leamy, and M. Ruzzene,
*Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook*, Appl. Mech. Rev 66(4), 2014.

Nagi Elabbasi, PhD, is a managing engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is the modeling and simulation of multiphysics systems. He has extensive experience in the finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, failure analysis, and material testing and modeling, and is a COMSOL Certified Consultant.

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