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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Electrical cables, also called transmission lines, are used everywhere in the modern world to transmit both power and data. If you are reading this on a cell phone or tablet computer that is “wireless”, there are still transmission lines within it connecting the various electrical components together. When you return home this evening, you will likely plug your device into a power cable to charge it.

Various transmission lines range from the small, such as coplanar waveguides on a printed circuit board (PCB), to the very large, like high voltage power lines. They also need to function in a variety of situations and conditions, from transatlantic telegraph cables to wiring in spacecraft, as shown in the image below. Transmission lines must be specially designed to ensure that they function appropriately in their environments, and may also be subject to further design goals, including required mechanical strength and weight minimization.

*Transmission wires in the payload bay of the OV-095 at the Shuttle Avionics Integration Laboratory (SAIL).*

When designing and using cables, engineers often refer to parameters per unit length for the series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G). These parameters can then be used to calculate cable performance, characteristic impedance, and propagation losses. It is important to keep in mind that these parameters come from the electromagnetic field solutions to Maxwell’s equations. We can use COMSOL Multiphysics to solve for the electromagnetic fields, as well as consider multiphysics effects to see how the cable parameters and performance change under different loads and environmental conditions. This could then be converted into an easy-to-use app, like this example that calculates the parameters for commonly used transmission lines.

Here, we examine a coaxial cable — a fundamental problem that is often covered in a standard curriculum for microwave engineering or transmission lines. The coaxial cable is so fundamental that Oliver Heaviside patented it in 1880, just a few years after Maxwell published his famous equations. For the students of scientific history, this is the same Oliver Heaviside who formulated Maxwell’s equations in the vector form that we are familiar with today; first used the term “impedance”; and helped develop transmission line theory.

Let us begin by considering a coaxial cable with dimensions as shown in the cross-sectional sketch below. The dielectric core between the inner and outer conductors has a relative permittivity () of 2.25 – j*0.01, a relative permeability () of 1, and a conductivity of zero, while the inner and outer conductors have a conductivity () of 5.98e7 S/m.

*The 2D cross section of the coaxial cable, where we have chosen a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm. *

A standard method for solving transmission lines is to assume that the electric fields will oscillate and attenuate in the direction of propagation, while the cross-sectional profile of the fields will remain unchanged. If we then find a valid solution, uniqueness theorems ensure that the solution we have found is correct. Mathematically, this is equivalent to solving Maxwell’s equations using an *ansatz* of the form , where () is the complex propagation constant and and are the attenuation and propagation constants, respectively. In cylindrical coordinates for a coaxial cable, this results in the well-known field solution of

\begin{align}

\mathbf{E}&= \frac{V_0\hat{r}}{rln(b/a)}e^{-\gamma z}\\

\mathbf{H}&= \frac{I_0\hat{\phi}}{2\pi r}e^{-\gamma z}

\end{align}

\mathbf{E}&= \frac{V_0\hat{r}}{rln(b/a)}e^{-\gamma z}\\

\mathbf{H}&= \frac{I_0\hat{\phi}}{2\pi r}e^{-\gamma z}

\end{align}

which then yields the parameters per unit length of

\begin{align}

L& = \frac{\mu_0\mu_r}{2\pi}ln\frac{b}{a} + \frac{\mu_0\mu_r\delta}{4\pi}(\frac{1}{a}+\frac{1}{b})\\

C& = \frac{2\pi\epsilon_0\epsilon'}{ln(b/a)}\\

R& = \frac{R_s}{2\pi}(\frac{1}{a}+\frac{1}{b})\\

G& = \frac{2\pi\omega\epsilon_0\epsilon''}{ln(b/a)}

\end{align}

L& = \frac{\mu_0\mu_r}{2\pi}ln\frac{b}{a} + \frac{\mu_0\mu_r\delta}{4\pi}(\frac{1}{a}+\frac{1}{b})\\

C& = \frac{2\pi\epsilon_0\epsilon'}{ln(b/a)}\\

R& = \frac{R_s}{2\pi}(\frac{1}{a}+\frac{1}{b})\\

G& = \frac{2\pi\omega\epsilon_0\epsilon''}{ln(b/a)}

\end{align}

where is the sheet resistance and is the skin depth.

While the equations for capacitance and shunt conductance are valid at any frequency, it is extremely important to point out that the equations for the resistance and inductance depend on the skin depth and are therefore only valid at frequencies where the skin depth is much smaller than the physical thickness of the conductor. This is also why the second term in the inductance equation, called the *internal inductance*, may be unfamiliar to some readers, as it can be neglected when the metal is treated as a perfect conductor. The term represents inductance due to the penetration of the magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (The term can also be expressed as .)

For further comparison, we can compute the DC resistance directly from the conductivity and cross-sectional area of the metal. The analytical equation for the DC inductance is a little more involved, and so we quote it here for reference.

L = \frac{\mu}{2\pi}\left\{ln\left(\frac{b+t}{a}\right) + \frac{2\left(\frac{b}{b+t}\right)^2}{1- \left(\frac{b}{b+t}\right)^2} ln\left(\frac{b+t}{b}\right) -\frac{3}{4} + \frac{\frac{\left(b+t\right)^4}{4} -\left(b+t\right)^2b^2+b^4\left(\frac{3}{4} + ln\frac{\left(b+t\right)}{b}\right) }{\left(\left(b+t\right)^2-b^2\right)^2}\right\}

Now that we have values for C and G at all frequencies, DC values for R and L, and asymptotic values for their high-frequency behavior, we have excellent benchmarks for our computational results.

When setting up any numerical simulation, it is important to consider whether or not symmetry can be used to reduce the model size and increase the computational speed. As we saw earlier, the exact solution will be of the form . Because the spatial variation of interest is primarily in the *xy*-plane, we just want to simulate a 2D cross section of the cable. One issue, however, is that the 2D governing equations used in the AC/DC Module assume that the fields are invariant in the out-of-plane direction. This means that we will not be able to capture the variation of the ansatz in a single 2D AC/DC simulation. We can find the variation with two simulations, though! This is because the series resistance and inductance depend on the current and energy stored in the magnetic fields, while the shunt conductance and capacitance depend on the energy in the electric field. Let’s take a closer look.

Since the shunt conductance and capacitance can be calculated from the electric fields, we begin by using the *Electric Currents* interface.

*Boundary conditions and material properties for the *Electric Currents* interface simulation.*

Once the geometry and material properties are assigned, we assume that the conductors are equipotential (a safe assumption, since the conductivity difference between the conductor and the dielectric will generally be near 20 orders of magnitude) and set up the physics by applying V_{0} to the inner conductor and grounding the outer conductor to solve for the electric potential in the dielectric. The above analytical equation for capacitance comes from the following more general equations

\begin{align}

W_e& = \frac{1}{4}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}\\

W_e& = \frac{C|V_0|^2}{4}\\

C& = \frac{1}{|V_0|^2}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}

\end{align}

W_e& = \frac{1}{4}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}\\

W_e& = \frac{C|V_0|^2}{4}\\

C& = \frac{1}{|V_0|^2}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}

\end{align}

where the first equation is from electromagnetic theory and the second from circuit theory.

The first and second equations are combined to obtain the third equation. By inserting the known fields from above, we obtain the previous analytical result for C in a coaxial cable. More generally, these equations provide us with a method for obtaining the capacitance from the fields for any cable. From the simulation, we can compute the integral of the electric energy density, which gives us a capacitance of 98.142 pF/m and matches with theory. Since G and C are related by the equation

G=\frac{\omega\epsilon'' C}{\epsilon'}

we now have two of the four parameters.

At this point, it is also worth reiterating that we have assumed that the conductivity of the dielectric region is zero. This is typically done in the textbook derivation, and we have maintained that convention here because it does not significantly impact the physics — unlike our inclusion of the internal inductance term discussed earlier. Many dielectric core materials do have a nonzero conductivity and that can be accounted for in simulation by simply updating the material properties. To ensure that proper matching with theory is maintained, the appropriate derivations would need to be updated as well.

In a similar fashion, the series resistance and inductance can be calculated through simulation using the AC/DC Module’s *Magnetic Fields* interface. The simulation setup is straightforward, as demonstrated in the figure below.

*The conductor domains are added to a *Single-Turn Coil* node with the *Coil Group* feature, and the reversed current direction option ensures that the direction of current through the inner conductor is the opposite of the outer conductor, as indicated by the dots and crosses. The single-turn coil will account for the frequency dependence of the current distribution in the conductors, as opposed to the arbitrary distribution shown in the figure.*

We refer to the following equations, which are the magnetic analog of the previous equations, to calculate the inductance.

\begin{align}

W_m& = \frac{1}{4}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}\\

W_m& = \frac{L|I_0|^2}{4}\\

L& = \frac{1}{|I_0|^2}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}

\end{align}

W_m& = \frac{1}{4}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}\\

W_m& = \frac{L|I_0|^2}{4}\\

L& = \frac{1}{|I_0|^2}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}

\end{align}

To calculate the resistance, we use a slightly different technique. First, we integrate the resistive loss to determine the power dissipation per unit length. We can then use the familiar to calculate the resistance. Since R and L vary with frequency, let’s take a look at the calculated values and the analytical solutions in the DC and high-frequency (HF) limit.

*“Analytic (DC)” and “Analytic (HF)” refer to the analytical equations in the DC and high-frequency limits, respectively, which were discussed earlier. Note that these are both on log-log plots.*

We can clearly see that the computed values transition smoothly from the DC solution at low frequencies to the high-frequency solution, which is valid when the skin depth is much smaller than the thickness of the conductor. We anticipate that the transition region will be approximately located where the skin depth and conductor thickness are within one order of magnitude. This range is 4.2e3 Hz to 4.2e7 Hz, which is exactly what we see in the results.

Now that we have completed the heavy lifting to calculate R, L, C, and G, there are two other significant parameters that can be determined. They are the characteristic impedance (Z_{c}) and complex propagation constant (), where is the attenuation constant and is the propagation constant.

\begin{align}

Z_c& = \sqrt{\frac{(R+j\omega L)}{(G+j\omega C)}}\\

\gamma& = \sqrt{(R+j\omega L)(G+j\omega C)}

\end{align}

Z_c& = \sqrt{\frac{(R+j\omega L)}{(G+j\omega C)}}\\

\gamma& = \sqrt{(R+j\omega L)(G+j\omega C)}

\end{align}

In the figure below, we see these values calculated using the analytical formulas for both the DC and high-frequency regime as well as the values determined from our simulation. We have also included a fourth line: the impedance calculated using COMSOL Multiphysics and the RF Module, which we will discuss shortly. As can be seen, our computations agree with the analytical solutions in their respective limits, as well as yielding the correct values through the transition region.

*A comparison of the characteristic impedance, determined using the analytical equations and COMSOL Multiphysics. The analytical equations plotted are from the DC and high-frequency (HF) equations discussed earlier, while the COMSOL Multiphysics results use the AC/DC and RF Modules. For clarity, the width of the “RF Module” line has been intentionally increased.*

Electromagnetic energy travels as waves, which means that the frequency of operation and wavelength are inversely proportional. As we continue to solve at higher and higher frequencies, we need to be aware of the relative size of the wavelength and electrical size of the cable. As discussed in a previous blog post, we should switch from the AC/DC to RF Module at an electrical size of approximately λ/100. If we use the cable diameter as the electrical size and the speed of light inside the dielectric core of the cable, this yields a transition frequency of approximately 690 MHz.

At these higher frequencies, the cable is more appropriately treated as a waveguide and the cable excitation as a waveguide mode. Using waveguide terminology, the mode we have been examining is a special type of mode called *TEM* that can propagate at any frequency. When the cross section and wavelength are comparable, we also need to account for the possibility of higher-order modes. Unlike a TEM mode, most waveguide modes can only propagate above a characteristic cut-off frequency. Due to the cylindrical symmetry in our example model, there is an equation for the cut-off frequency of the first higher-order mode, which is a TE11 mode. This cut-off frequency is f_{c} = 35.3 GHz, but even with the relatively simple geometry, the cut-off frequency comes from a transcendental equation that we will not examine further in this post.

So what does this cut-off frequency mean for our results? Above that frequency, the energy carried in the TEM mode that we are interested in has the potential to couple to the TE11 mode. In a perfect geometry, like we have simulated here, there will be no coupling. In the real world, however, any imperfections in the cable could cause mode coupling above the cut-off frequency. This could result from a number of sources, from fabrication tolerances to gradients in the material properties. Such a situation is often avoided by designing cables to operate below the cut-off frequency of higher-order modes so that only one mode can propagate. If that is of interest, you can also use COMSOL Multiphysics to simulate the coupling between higher-order modes, as with this Directional Coupler tutorial model (although beyond the scope of today’s post).

Simulation of higher-order modes is ideally suited for a Mode Analysis study using the RF or Wave Optics modules. This is because the governing equation is , which is exactly the form that we are interested in. As a result, Mode Analysis will directly solve for the spatial field and complex propagation constant for a predefined number of modes. We can use the same geometry as before, except that we only need to simulate the dielectric core and can use an Impedance boundary condition for the metal conductor.

*The results for the attenuation constant and effective mode index from a Mode Analysis. The analytic line in the left plot, “Attenuation Constant vs Frequency”, is computed using the same equations as the high-frequency (HF) lines used for comparison with the results of the AC/DC Module simulations. The analytic line in the right plot, “Effective Refractive Index vs Frequency”, is simply . For clarity, the size of the “COMSOL — TEM” lines has been intentionally increased in both plots.*

We can clearly see that the Mode Analysis results of the TEM mode match the analytic theory, and that the computed higher-order mode has its onset at the previously determined cut-off frequency. It is also incredibly convenient that the complex propagation constant is a direct output of this simulation and does not require calculations of R, L, C, and G. This is because is explicitly included and solved for in the Mode Analysis governing equation. These other parameters can be calculated for the TEM mode, if desired, and more information can be found in this demonstration in the Application Gallery. It is also worth pointing out that this same Mode Analysis technique can be used for dielectric waveguides, like fiber optics.

At this point, we have thoroughly analyzed a coaxial cable. We have calculated the distributed parameters from the DC to high-frequency limit and examined the first higher-order mode. Importantly, the Mode Analysis results only depend on the geometry and material properties of the cable. The AC/DC results require the additional knowledge of how the cable is excited, but hopefully you know what you’re attaching your cable to! We used analytic theory solely to compare our simulation results against a well-known benchmark model. This means that the analysis could be extended to other cables, as well as coupled to multiphysics simulations that include temperature change and structural deformation.

For those of you who are interested in the fine details, here are a few extra points in the form of hypothetical questions.

- “Why didn’t you mention and/or plot all of the characteristic impedance and distributed parameters for the TE11 mode?”
- This is because only TEM modes have a uniquely defined voltage, current, and characteristic impedance. It is still possible to assign some of these values for higher-order modes, and this is discussed further in texts on transmission line theory and microwave engineering.

- “When I solve for modes using a Mode Analysis study, they are labeled by the value of their effective index. Where did TEM and TE11 come from?”
- These names come from the analytic theory and were used for convenience when discussing the results. This name assignment may not be possible for an arbitrary geometry, but what’s in a name? Would not a mode by any other name still carry electromagnetic energy (excluding nontunneling evanescent waves, of course)?

- “Why is there an extra factor of ½ in several of your calculations?”
- This comes up when solving electromagnetics in the frequency domain, notably when multiplying two complex quantities. When taking the time average, there is an extra factor of ½ as opposed to the equation in the time domain (or at DC). For more information, you can refer to a text on classical electromagnetics.

The following texts were referred to during the writing of this post and are excellent sources of additional information:

*Microwave Engineering*, by David M. Pozar*Foundations for Microwave Engineering*, by Robert E. Collin*Inductance Calculations*, by Frederick W. Grover*Classical Electrodynamics*, by John D. Jackson

The detection and removal of landmines and IEDs is important for both humanitarian and military purposes. While the term for the process of detecting these mines — *minesweeping* — is the same in both cases, the removal process is referred to as *demining* in times of relative peace and *mine clearance* during times of war. The latter case refers to when mines are removed from active combat zones for tactical reasons as well as for the safety of soldiers.

When a war ends, landmines may still be in the ground and detonate under civilians, leading to casualties. The majority of the mines are located in developing countries that are trying to recover from recent wars. Aside from being politically unstable, these countries are unable to farm viable land that is strewn with IEDs, keeping their economies in poor positions. Unfortunately, finding and removing the dangerous devices can be rather difficult.

*A U.S. Army detection vehicle digs up an IED during a training exercise.*

In efforts to locate and remove landmines, a mechanical approach is one option. With this method, an area with known landmines is bombed or plowed using sturdy, mine-resilient tanks to detonate them safely. For a more natural approach, dogs, rats, and even honeybees are trained to detect landmines with their sense of smell, and they are usually too light to trigger detonation. Biological detection methods offer another option, utilizing plants and bacteria that change color or become fluorescent in the presence of certain explosive materials. Once the mines are detected, they are safely removed from the area.

*A trained rat searches for landmines in a field.*

One method can provide more knowledge about an area that contains IEDs: *electromagnetic detection*. An important element within electromagnetic detection is a process called *ground-penetrating radar* (GPR), which uses electromagnetic waves to create an image of a subsurface, revealing the buried objects.

GPR involves sending electromagnetic waves into a subsurface (the ground) through an antenna. The transmitter of the antenna sends the waves, and the receiver collects the energy reflected off of the different objects in the subsurface, recording the patterns as real-time data.

*Data from a traditional GPR scan of a historic cemetery.*

With recent developments in landmine cloaking technology, identifying buried objects through traditional GPR has become more challenging. Dr. Reginald Eze and George Sivulka from the City University of New York — LaGuardia Community College and Regis High School sought to improve electromagnetic IED detection by testing the method under different variables and environmental situations. By creating an intelligent subsurface sensing template with the help of COMSOL Multiphysics, the research team was able to determine better ways to safely locate and remove landmines and IEDs.

Let’s dive a bit deeper into their simulation research, which was presented at the COMSOL Conference 2015 Boston.

When setting up their model of the mine-strewn area, the researchers needed to ensure that they were accurately portraying a real-world landmine scenario. They started with a basic 2D geometry and defined the target objects and boundaries. The different layers of the model featured:

- A homogenous soil surface with varying levels of moisture
- Air
- The landmine

The physical parameters in the model included relative permittivity; relative permeability; and the conductivity of the air, dry soil, wet soil, and TNT (the explosive material used in the landmine).

Using the *Electromagnetic Waves, Frequency Domain* interface in the RF Module, the team built a model consisting of air, soil, and the landmine. Additionally, a perfectly matched layer (PML) was used to truncate the modeling domain and act as a transparent boundary to outgoing radiation, thus allowing for a small computational domain. A transverse electric (TE) plane wave was applied to the computational domain in the downward direction. The scattering results were analyzed via LiveLink™ *for* MATLAB®.

*The scattering effect of a wave on a landmine in wet soil (left) compared to dry soil (right).*

The research team studied the radar cross section (RCS), which quantifies the scattering of the waves off of various objects. Their studies were based on five key factors:

- Projected cross section
- Reflectivity
- Directivity
- Contrast between the landmine and the background materials
- Shapes of the landmine and the ground surface

With each adjustment to an environmental parameter, a parametric sweep was performed every 0.5 GHz from 0.5 GHz to 3.0 GHz. The parametric sweeps enabled an educated selection of the optimal frequency for IED detection in every possible environmental scenario.

*A parametric sweep used to identify the optimal frequency for a landmine detection system.*

The simulation results pointed out the differences in scattering patterns depending on the parameters. For example, as the depth of the target increased, the scattering effects became more negligible. The relation between how deep the mine was buried and the scattering showed a clear connection to the soil’s interference with the wave.

The results also showed that dry soil has more interference with the RF signal than wet soil. Both the size and depth of the mine were related to the amount of scattering. For instance, the more shallow the mine was buried, the more easily it was detected. The parameter sweep of the frequencies indicated that the optimal frequency to detect anomalies in the subsurface scan was 2 GHz.

*The scattering amplitude for a landmine buried in an air/wet soil/dry soil layer combination (left) compared to air/dry soil/wet soil (right).*

Studying the parameters and their effects on the scattering patterns of the waves offers insight into the objects that are being detected, including their chemical composition. Such knowledge makes it easier to identify an object, whether a TNT-based landmine, another type of IED, a rock, or a tree root.

Through simulation analyses, the researchers gained a more comprehensive understanding of the microphysical parameters and their impact on the scattering of waves off of different objects. This gave them a better idea of the remote sensing behavior, offering potential for increased accuracy in landmine detection and removal. Such advancements could lead to safer environments, particularly within developing areas of the world.

- Read the full paper: “Remote Sensing of Electromagnetically Penetrable Objects: Landmine and IED Detection“
- View the research poster, which received the Popular Choice Poster award at the COMSOL Conference 2015 Boston

*MATLAB is a registered trademark of The MathWorks, Inc.*

Take a CD in your hand. As the sun reflects off it, point the CD at a white wall. As you look at the wall, you will notice that a color reflection appears. What you are seeing is a result of small pits on one side of the CD that are arranged in a spiral. This is just one everyday example of diffraction grating.

*Image by Luis Fernández García – Own work, via Wikimedia Commons.*

Often utilized in monochromators and spectrometers, *diffraction gratings* are optical components with a periodic structure that reflects and transmits different wavelengths of light in different directions. The spacing and structure of the grating determines the directions and the relative magnitudes of the reflection and transmission. This reflection and transmission is also a function of the wavelength as well as the angle of incidence of the incoming light. As such, it is important that the grating is configured to ensure proper diffraction efficiency and thus enhance the overall performance of the optical instrument.

Testing different grating configurations can be costly and time consuming when done experimentally. Instead, simulation is a more cost-effective and efficient approach to achieving the optimal design. This virtual testing environment provides greater flexibility in analyzing different design scenarios, while eliminating costs associated with having to build prototypes to analyze each new modification.

With the Application Builder in COMSOL Multiphysics, you can now further simplify your simulation process by creating an easy-to-use app. Customized to fit your own design needs, simulation apps can be distributed throughout your organization, enabling others to run their own simulation tests. Our Plasmonic Wire Grating Analyzer demo app offers a helpful foundation for building an app of your own.

Let’s begin by discussing the model underlying the app. In the model, an electromagnetic wave is incident on a wire grating on a dielectric substrate. The example is designed for one unit cell of the grating, with Floquet boundary conditions used to describe the periodicity.

The Plasmonic Wire Grating Analyzer demo app takes the physics and functionality behind this model and makes it available in a simplified format. With this app, users can easily compute diffraction efficiencies for the transmitted and reflected waves as well as the first and second diffraction orders as functions of the angle of incidence. Additionally, this simulation app enables visualization of the electric field norm plot for various grating periods for a specific angle of incidence.

*A diffraction efficiency plot shown in the app.*

The figure above provides an overview of the app’s user interface. The left side of the interface features user-defined parameters, which are broken down into four different sections. The radius of a wire and the periodicity can be defined in the *Geometry Parameters* section, with the relative permittivity of the wire grating and the refractive index of the substrate arranged in the *Material Properties* section. The wavelength and the orientation of polarization are indicated in the *Wave Properties* section, and the current status of the app is referenced in the *Information* section.

Looking to the right side of the interface, there is a command toolbar comprised of six buttons — *Analyze, Reset Parameters, Simulation Report, Electric Field Norm Plot, Diffraction Plot,* and *Open PDF Document*. In their respective order, these buttons enable app users to run the simulation, revert input parameters back to their default values, make a simulation report, plot the electric field norm and the diffraction efficiency, and open the documentation. All results can be visualized within the graphics window in the center of the app’s interface.

When designing your own app, you can customize the look and feel of the user interface to fit your simulation needs. By including only those parameters and features that are relevant to your analysis, you can help hide the complexity of your model and create a user-friendly experience for app users.

Simulation apps offer a revolutionary approach to design that prompts greater involvement in the simulation process and thus delivers faster results. In the case of a wire grating, building an app simplifies the analysis of diffraction efficiencies, helping to identify a grating configuration that offers the optimal efficiency for its dedicated use. We encourage you to use our Plasmonic Wire Grating demo app as a resource in developing your own app.

- Download the Plasmonic Wire Grating Analyzer demo app