When an electromagnetic field is applied to a dielectric material, the field displaces the material’s electron from its original orbit, making the electron oscillate at a particular frequency. In other words, the field polarizes the material. The displacement field in this case is represented as a function of the applied electric field as follows:
where E is the applied electric field vector, P is the induced polarization vector, is the vacuum permittivity, and is the isotropic susceptibility.
In the case of an anisotropic dielectric material, the induced polarization vector is a function of the susceptibility tensor as follows:
Finally, in the case of a nonlinear dielectric material, the induced polarization can be expressed as a function of the electric field within the medium through the susceptibility () of the medium and, using a power series expansion, as follows:
where E is the applied electric field, ε_{0} is the vacuum permittivity, and is the ith-order susceptibility.
It is assumed that there is no polarization that is independent of E. To have a thorough derivation of the polarization with nonlinear terms, refer to the book by Y. R. Shen (Ref. 5).
First-order susceptibility () deals with the change in refractive index due to the dipole oscillation of the bound and free carriers, such as electrons. Hendrik Lorentz originally came up with the idea of creating a mathematical oscillator model that can relate the dipole oscillation of bound electrons to the susceptibility of the material. Paul Drude originated the idea of oscillation within semiconductors, which deals with free carriers within the material. After combining the effects of bound and free carriers, the new model was called the Drude-Lorentz model.
In the COMSOL Multiphysics® software, the Drude-Lorentz model can be used to define the relative permittivity of a material. To define the Drude-Lorentz model, the relative permittivity at high frequency, plasma frequency, resonance frequency, and damping coefficient need to be given as inputs, as shown below. Multiple oscillators can also be added while assigning each of the oscillators’ contributions.
where ε_{r} is the complex relative permittivity of the material, ε_{∞} is the interband transition contribution to the permittivity, ω_{p} is the plasma frequency, and Γ is the damping coefficient.
To showcase the capability of COMSOL Multiphysics in modeling the Drude-Lorentz material model, a waveguide with a metal-insulator-metal (MIM) interface is modeled. Here, the metal and insulator are modeled as silver and air, respectively. In this configuration, the width of the insulator is made to periodically vary along the waveguide (see the figure below). This particular arrangement of the insulator causes the waveguide to work like a filter known as a plasmonic waveguide filter.
This example shows that the waveguide rejects the electromagnetic radiation of the wavelength between 1.4 um and 1.6 um, but allows the rest of the wavelength (see the figure below). The silver material can be modeled using the Drude-Lorentz approximation, with ε_{∞} = 3.7, ω_{p} = 13.8 rad/s, and Γ = 2.736 rad/s, while the insulator is modeled using air. As an alternative to the Drude-Lorentz material model approximation, the material property determined by Johnson and Christy’s experimental data (Ref. 4), which is available in the Material Library as Ag (Johnson), can be used.
Note that the output characteristic of this plasmonic waveguide filter is similar to that of the fiber Bragg grating (FBG) configuration.
Schematic of the plasmonic waveguide filter. Blue and gray are the insulator and metal domain, respectively. The dashed line depicts a unit cell that is periodically repeated.
The transmittance and reflectance through the plasmonic grating filter (with 10 unit cells) using both the Drude-Lorentz model and Ag (Johnson) from the Material Library.
There are nonlinear crystals that display relatively high second-order susceptibility (). When a monochromatic beam of light is introduced and passes through such a nonlinear crystal, not only does the output frequency spectrum show the original frequency (ω), but it also indicates the second-order harmonic frequency (2ω). Hence, this phenomenon is called second-harmonic generation (SHG).
The application of SHG lies in the field of laser design and engineering, where it’s challenging to find a material to eject light with shorter wavelengths. For example, when an infrared light source (1064 nm) is pumped through a potassium-dihydrogen-phosphate (KDP) crystal, the crystal ejects a green (532 nm) source of laser light.
In COMSOL Multiphysics, this approach can be modeled either with a transient or frequency-domain analysis in which the polarization is defined using the nonlinear coefficient (d), as shown below. In the transient analysis of the beam, the electric-field-dependent nonlinearity term needs to be introduced to the electric displacement field (D). The way it is introduced in this model is by clever use of the remnant electric displacement (D_{r}). As a matter of fact, the remnant electric displacement can also accept a nonlinear field quantity, which involves the square of one of the electric field components here. This approach displays the sum frequency generation as well as the difference frequency generation.
where , is the nonlinear coefficient, and E_{z} is the z-component of the electric field.
With the frequency-domain analysis of the beam, only one particular frequency can be analyzed at one instance. (In other words, only one frequency can be analyzed with the Helmholtz equation.) Hence, the model sets up two interfaces and couples the two physics. The first interface represents the fundamental wave, while the second interface represents the second-harmonic frequency. The polarization of the first interface, , and that of the second interface, , can be defined as the following:
where d is the nonlinear coefficient, is the y-component of the electric field at the fundamental frequency, and is the y-component of the electric field at the second-harmonic frequency.
Left: The output frequency spectrum. The small peak on the left side of the large peak shows the difference frequency generation, while the small peak on the right shows the SHG. Right: The y-components of the electric fields for the fundamental and second-harmonic wave.
The materials that have dominant third-order susceptibility () display phenomena such as the optical Kerr effect, self-phase modulation, cross-phase modulation, third-harmonic generation, and four-wave mixing. To illustrate the optical Kerr effect in COMSOL Multiphysics, a high-intensity (in the order of GW/cm^{2}) monochromatic beam of light (such as a Nd:YAG laser source) is propagated through a nonlinear crystal made of BK-7. Due to the dominant third-order material nonlinearity in BK-7, the refractive index changes as a function of the beam intensity (I) of the monochromatic input light as follows:
where n_{0} is the constant (linear) part of the refractive index, γ is the nonlinear refractive index coefficient, and I is the beam intensity.
The spatially Gaussian-launched beam creates a spatial Gaussian profile of the refractive index, with the peak at the center and decreasing radially outward. This profile of the refractive index causes the beam of light to be more focused at the center during its propagation through the crystal. This phenomenon is called the self-focusing of a laser beam, specifically because the source beam is responsible for its own focusing. This effect is particularly useful in laser engineering, where the self-focusing of a high-power light source in such a narrow central regime can permanently damage the crystal, hence the need to model and compensate for these effects in the design process.
Left: The induced refractive index change, γI, for a high peak intensity, I_{0} = 14 GW/cm^{2}. Right: The spot radius at the end of the propagation domain versus the peak intensity.
There are materials where the refractive index of the medium can be a function of the applied external electric field, as described in the introduction of this blog post. This applied electric field can be from the DC potential or a slowly varying harmonic potential applied through the coils or contact pads attached to the material. The refractive index optical material property is considered from here on instead of the susceptibility χ.
Mathematically, the refractive index can be represented as a Taylor series expansion of the applied E-field.
For electro-optic materials, the refractive index translates to the following:
where n is the refractive index of the material with no applied E-field, while d_{1} and d_{2} are the electro-optic coefficients.
Crystals such as KDP, lithium niobate (LiNbO_{3}), and cadmium telluride (CdTe) have a refractive index with the dominant first and second terms above. Such media are known as Pockels media, where d_{1} is known as the linear optic coefficient because the refractive index is a linear function of the electric field.
In COMSOL Multiphysics, the Pockels effect has been demonstrated using an optical modulator. In this model, light propagates through a single silicon waveguide that branches out into two waveguides. Contact pads are applied to the upper branch and excited with a DC voltage, as shown in the image below. This branch of the waveguide can also be defined as a Pockels medium with d_{1} = 30e-12 m/V.
When no voltage is applied to the contact pads, the light flows unimpeded through both the upper and lower branches and interferes constructively at the point where the branches merge together. However, when a certain voltage is applied to the contact pads, a local electric field is created in the region within the contact pads. The material property in the region under the influence of an external electric field modifies the refractive index of this medium, effectively changing the speed of the light propagating through the upper waveguide. When the light propagating in these upper and lower branches interferes at the location where the branches merge, it leads to destructive interference, with no light propagating forward.
The potential application of the Pockels effect lies in designing optical switches; for example, in the field of photonic integrated circuits. In a tutorial model, we demonstrate a particular kind of optical switching element known as a Mach-Zehnder optical modulator.
Schematic of the Mach-Zehnder optical modulator.
Transmission of the upper output branch 1 and lower output branch 2 versus the applied DC voltage.
There are certain gases, liquids, and crystals that exhibit a centrosymmetric property, where the first and third term of the Taylor expansion are dominant. In such cases, the refractive index can be defined as a quadratic function of the applied E-field as follows:
Here, we have discussed different optical materials such as KDP, BK-7, LiNbO_{3}, CdTe, and silicon under an external electric field. These materials show different linear and nonlinear phenomena, such as SHG, self-focusing, and linear and quadratic electric field effects. We have also dealt with the applications of these materials in the laser engineering field, filter design, and optical switches.
In electromagnetic simulations, the wavelength always needs be resolved by the mesh in order to find an accurate solution of Maxwell’s equations. This requirement makes it difficult to simulate models that are large compared to the wavelength. There are several methods for stationary wave optics problems that can handle large models. These methods include the so-called diffraction formulas, such as the Fraunhofer, Fresnel-Kirchhoff, and Rayleigh-Sommerfeld diffraction formula and the beam propagation method (BPM), such as paraxial BPM and the angular spectrum method (Ref. 1).
Most of these methods use certain approximations to the Helmholtz equation. These methods can handle large models because they are based on the propagation method that solves for the field in a plane from a known field in another plane. So you don’t have to mesh the entire domain, you just need a 2D mesh for the desired plane.
Compared to these methods, the Electromagnetic Waves, Beam Envelopes interface in COMSOL Multiphysics (which we will refer to as the Beam Envelopes interface for the rest of the blog post) solves for the exact solution of the Helmholtz equation in a domain. It can handle large models; i.e., the meshing requirement can be significantly relaxed if a certain restriction is satisfied.
A beam envelopes simulation for a lens with a millimeter-range focal length for a 1-um wavelength beam.
We discuss the Beam Envelopes interface in more detail below.
Let’s take a look at the math that the Beam Envelopes interface computes “under the hood”. If you add this interface to a model and click the Physics Interface node and change Type of phase specification to User defined, you’ll see the following in the Equation section:
Here, is the dependent variable that the interface solves for, called the envelope function.
In the phasor representation of a field, corresponds to the amplitude and to the phase, i.e.,
The first equation, the governing equation for the Beam Envelopes interface, can be derived by substituting the second definition of the electric field into the Helmholtz equation. If we know , the only unknown is and we can solve for it. The phase, , needs to be given a priori in order to solve the problem.
With the second equation, we assume a form such that the fast oscillation part, the phase, can be factored out from the field. If that’s true, the envelope is “slowly varying”, so we don’t need to resolve the wavelength. Instead, we only need to resolve the slow wave of the envelope. Because of this process, simulating large-scale wave optics problems is possible on personal computers.
A common question is: “When do you want the envelope rather than the field itself?” Lens simulation is one example. Sometimes you may need the intensity rather than the complex electric field. Actually, the square of the norm of the envelope gives the intensity. In such cases, it suffices to get the envelope function.
The math behind the beam envelope method introduces more questions:
To answer these questions, we need to do a little more math.
Let’s take the simplest test case: a plane wave, , where for wavelength = 1 um, it propagates in a rectangular domain of 20 um length. (We intentionally use a short domain for illustrative purposes.)
The out-of-plane wave enters from the left boundary and transmits the right boundary without reflection. This can be simulated in the Beam Envelopes interface by adding a Matched boundary condition with excitation on the left and without excitation on the right, while adding a Perfect Magnetic Conductor boundary condition on the top and bottom (meaning we don’t care about the y direction).
The correct setting for the phase specification is shown in the figure below.
We have the answer , knowing that the correct phase function is or the wave vector is a priori. Substituting the phase function in the second equation, we inversely get , the constant function.
How many mesh elements do we need to resolve a constant function? Only one! (See this previous blog post on high-frequency modeling.)
The following results show the envelope function and the norm of , ewbe.normE
, which is equal to . Here, we can see that we get the correct envelope function if we give the exact phase function, constant one, for any number of meshes, as expected. For confirmation purposes, the phase of , arg(E1z)
, is also plotted. It is zero, also as expected.
The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the correct phase function k_{0}x, computed for different mesh sizes.
Now, let’s see what happens if our guess for the phase function is a little bit off — say, instead of the exact . What kind of solutions do we get? Let’s take a look:
The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the wrong phase function, 0.95 k_{0}x, computed for different mesh sizes.
What we see here for the envelope function is the so-called beating. It’s obvious that everything depends on the mesh size. To understand what’s going on, we need a pencil, paper, and patience.
We knew the answer was , but we had “intentionally” given an incorrect estimate in the COMSOL® software. Substituting the wrong phase function in the second equation, we get . This results in , which is no longer constant one. This is a wave with a wavelength of = 20 um, which is called the beat wavelength.
Let’s take a look at the plot above for six mesh elements. We get exactly what is expected (red line), i.e., . The plot automatically takes the real part, showing . The plots for the lower resolutions still show an approximate solution of the envelope function. This is as expected for finite element simulations: coarser mesh gives more approximate results.
This shows that if we make a wrong guess for the phase function, we get a wrong (beat-convoluted) envelope function. Because of the wrong guess, the envelope function is added a phase of the beating (green line), which is .
What about the norm of ? Look at the blue line in the plots above. It looks like the COMSOL Multiphysics software generated a correct solution for ewbe.normE
, which is constant one. Let’s calculate: Substituting both the wrong (analytical) phase function and the wrong (beat-convoluted) envelope function in the second equation, we get , which is the correct fast field!
If we take a norm of , we get a correct solution, constant one. This is what we wanted. Note that we can’t display itself because the domain can be too large, but we can find analytically and display the norm of with a coarse mesh.
This is not a trick. Instead, we see that if the phase function is off, the envelope function will also be off, since it becomes beat-convoluted. However, the norm of the electric field can still be correct. Therefore, it is important that the beat-convoluted envelope function be correctly computed in order to get the correct electric field. The above plots clearly show that. The six-element mesh case gives the completely correct electric field norm because it fully resolves the beat-convoluted envelope function. The other meshes give an approximate solution to the beat-convoluted envelope function depending on the mesh size. They also do so for the field norm. This is a general consequence that holds true for arbitrary cases.
No matter what phase function we use in COMSOL Multiphysics, we are okay as long as we correctly solve the first equation for and as long as the phase function is continuous over the domain. When there are multiple materials in a domain, the continuity of the phase function is also critical to the solution accuracy. We may discuss this in a future blog post, but it is also mentioned in this previous blog post on high-frequency modeling.
So far, we have discussed a scalar wave number. More generally, the phase function is specified by the wave vector. When the wave vector is not guessed correctly, it will have vector-valued consequences. Suppose we have the same plane wave from the first example, but we make a wrong guess for the phase, i.e., instead of . In this case, the wave number is correct but the wave vector is off. This time, the beating takes place in 2D.
Let’s start by performing the same calculations as the 1D example. We have and the envelope function is now calculated to be , which is a tilted wave propagating to direction , with the beat wave number and the beat wavelength .
The following plots are the results for θ = 15° for a domain of 3.8637 um x 29.348 um for different max mesh sizes. The same boundary conditions are given as the previous 1D example case. The only difference is that the incident wave on the left boundary is . (Note that we have to give the corresponding wrong boundary condition because our phase guess is wrong.)
In the result for the finest mesh (rightmost), we can confirm that is computed just like we analyzed in the above calculation and the norm of is computed to be constant one. These results are consistent with the 1D example case.
The electric field norm (top) and the envelope function (bottom) for the wrong phase function , computed for different mesh sizes. The color range represents the values from -1 to 1.
The ultimate goal here is to simulate an electromagnetic beam through optical lenses in a millimeter-scale domain with the Beam Envelopes interface. How can we achieve this? We already discussed how to compute the right solution. The following example is a simulation for a hard-apertured flat top incident beam on a plano-convex lens with a radius of curvature of 500 um and a refractive index of 1.5 (approximately 1 mm focal length).
Here, we use , which is not accurate at all. In the region before the lens, there is a reflection, which creates an interference. In the lens, there are multiple reflections. After the lens, the phase is spherical so that the beam focuses into a spot. So this phase function is far different from what is happening around the lens. Still, we have a clue. If we plot , we see the beating.
Plot of . The inset shows the finest beat wavelength inside the lens.
As can be seen in the plot, a prominent beating occurs in the lens (see the inset). Actually, the finest beat wavelength is in front of the lens. To prove this, we can perform the same calculations as in the previous examples. The finest beat wavelength is due to the interference between the incident beam and reflected beam, but we can ignore this because it doesn’t contribute to the forward propagation. We can see that the mesh doesn’t resolve the beating before the lens, but let’s ignore this for now.
The beat wavelength in the lens is for the backward beam and for the forward beam for n = 1.5, which we can also prove in the same way as the previous examples. Again, we ignore the backward beam. In the plot, what’s visible is the beating for the forward beam. The backward beam is only a fraction (approximately 4% for n = 1.5 of the incident beam, so it’s not visible). The following figure shows the mesh resolving the beat inside the lens with 10 mesh elements.
The beat wavelength inside the lens. The mesh resolves the beat with 10 mesh elements.
Other than the beating for the propagating beam in the lens, the beating in the subsequent air domain is pretty large, so we can use a coarse mesh here. This may not hold for faster lenses, which have a more rapid quadratic phase and can have a very short beat wavelength. In this example, we must use a finer mesh only in the lens domain to resolve the fastest beating.
The computed field norm is shown at the top of this blog post. To verify the result, we can compute the field at the lens exit surface by using the Frequency Domain interface, and then using the Fresnel diffraction formula to calculate the field at the focus. The result for the field norm agrees very well.
Comparison between the Beam Envelopes interface and Fresnel diffraction formula. The mesh resolves the beat inside the lens with 10 mesh elements.
The following comparison shows the mesh size dependence. We get a pretty good result with our standard recommendation, , which is equal to . This makes it easier to mesh the lens domain.
Mesh size dependence on the field norm at the focus.
As of version 5.3a of the COMSOL® software, the Fresnel Lens tutorial model includes a computation with the Beam Envelopes interface. Fresnel lenses are typically extremely thin (wavelength order). Even if there is diffraction in and around the lens surface discontinuities, the fine mesh around the lens part does not significantly impact the total number of mesh elements.
In this blog post, we discuss what the Beam Envelopes interface does “under the hood” and how we can get accurate solutions for wave optics problems. Even if we get beating, the beat wavelength can be much longer than the wavelength, which makes it possible to simulate large optical systems.
Although it seems tedious to check the mesh size to resolve beating, this is not extra work that is only required for the Beam Envelopes interface. When you use the finite element method, you always need to check the mesh size dependence for accurately computed solutions.
Try it yourself: Download the file for the millimeter-range focal length lens by clicking the button below.
The beam of light that Erasmus Bartholinus observed traveling straight through the crystal is called an ordinary ray. The other light beam, which bends while traveling through the crystal, is an extraordinary ray. Anisotropic materials, such as the crystal from the stone and bench experiment described above, are found in applications ranging from detecting harmful gases to beam splitting for photonic integrated circuits.
Ordinary and extraordinary rays traveling through an anisotropic crystal.
In a physical context, when an unpolarized electromagnetic beam of light propagates through an anisotropic dielectric material, it polarizes the dielectric domain, leading to a distribution of charges known as electric dipoles. This phenomenon leads to induced fields within the anisotropic dielectric material, wherein two kinds of waves experience two different refractive indices (ordinary and extraordinary).
The ordinary wave is polarized perpendicular to the principal plane and the extraordinary wave is polarized parallel to the principle plane, where the principal plane is spanned by the optic axis and the two propagation directions in the crystal. Because of this behavior, the waves propagate with different velocities and trajectories.
In a previous blog post, we discussed silicon and how its derivative, silicon dioxide, is used extensively in photonic integrated chips due to its compatibility with the CMOS fabrication technique. Bulk silicon, which has an isotropic property, is used to develop prototypes for photonic integrated chips. However, due to unique optical properties such as splitting beams and polarization-based optical effects, anisotropy comes into play at a later stage.
Anisotropy in silicon photonics occurs unintentionally due to the annealing process while fabricating the waveguide. The difference in thermal expansion between the core and cladding causes geometry mismatch due to stress optical effects, which results in effects such as mode splitting and pulse broadening. Anisotropy could also be intentionally introduced by varying the porosity of silicon dioxide. This enables researchers to work with a range of effective refractive indices from silicon dioxide (n ~1.44) to air (n ~1), giving them the edge to perform very sensitive optical sensor applications.
To perform qualitative analyses of anisotropic media, researchers investigate how optical energy propagates within planar waveguides (also known as modes of propagation). In planar waveguides, we define modes using and terminology (Ref. 2), where x and y depict the direction of polarization and p and q depict the number of maxima in the x- and y-coordinates.
Picture it this way: You are walking on an “landscape” (as shown below). The “winds” (polarization) are along ±x direction, and you encounter two distinct peaks when traveling from the -x to +x direction. When you move from the -y to +y direction, you observe both of the peaks simultaneously.
Mode analysis of the planar waveguide. Top row, left to right: and . Middle row, left to right: and . Bottom row, left to right: and . The arrow plot represents the electric field; contour and surface plot represent out-of-plane power flow (red is high and blue is low magnitude).
Before launching a beam of light through a waveguide using a laser source, it is important to know which optical modes could persist within a specified core/cladding dimension of the waveguide. Performing a mode analysis using a full vectorial finite element tool, such as the COMSOL Multiphysics® software, could be very helpful to qualitatively and quantitatively analyze the optical modes and dispersion curve respectively.
Performing a modal analysis on any isotropic material requires the definition of a single complex value, while in the case of an anisotropic material, a full tensor relative permittivity approach is required. The electric permittivity essentially relates the electric field with the material property. Here, tensor refers to a 3-by-3 matrix that has both diagonal (_{xx}, _{yy}, _{zz}) and off-diagonal (_{xy}, _{xz}, _{yx}, _{yz}, _{zx}, _{zy}) terms as shown below.
However, for all materials, you can find a coordinate system in which you only have nonzero diagonal elements in the permittivity tensor, whereas the off-diagonal elements are all zero. The three coordinate axes in this rotated coordinate system are the principal axes of the material and, correspondingly, the three values for the diagonal elements in the permittivity tensor are called the principal permittivities of the material.
There are basically two kinds of anisotropic crystal: uniaxial and biaxial crystal. With a suitable choice of coordinate system, where only the diagonal elements of the permittivity tensor are nonzero, in terms of optical properties, uniaxial crystal considers only the diagonal terms, that is _{xx} = _{yy} = (n_{o})^{2}, _{zz} = (n_{e})^{2}, where n_{o} and n_{e} are the ordinary and extraordinary refractive indices. However, when , it is known as a biaxial crystal.
To put this argument into a modeling perspective, we can extend the buried rib waveguide example from this blog post on silicon photonics design. We can perform a modal analysis on the 2D cross section of the waveguide with the square core and cladding length of 4 um and 20 um, respectively (shown below). The operating wavelength for all the cases is considered as 1.55[um].
Schematic of 3D buried rib optical waveguide where the mode analysis was performed at the inlet 2D cross section. The intensity plot and arrow plot representing the mode and polarization of E-fields respectively.
Core of the rib waveguide depicting the optic axis (red) along the x-axis and the principal axis (blue).
In the classic case of a uniaxial material, we assume the optic axis (i.e., c-axis) is along the principal x-axis (as shown above) and consider the diagonal relative permittivity _{xx} and _{zz} terms (which are orthogonal to the c-axis) as the square of ordinary refractive index (~1.5199^{2} ~ 2.31). The _{yy} component element that is along the c-axis is considered as the square of extraordinary refractive index (~1.4799^{2} ~ 2.19) (as per Ref. 3). In addition, the off-diagonal terms are considered zero (as shown below) and the cladding has an isotropic relative permittivity (~1.4318^{2}). The optical modes derived are the 6 modes shown above. Note the difference in the refractive indices: “n_{xx} – n_{yy}” is known as birefringence, where n_{xx} = and n_{yy} = .
Relative permittivity tensor with diagonal elements.
By evaluating the optical modes, we can visually comprehend the behavior of the optical waveguide. However, the dispersion curves could also be handy for performing quantitative analyses. A dispersion curve represents the variation of the effective refractive index with respect to the length of the waveguide or the operating frequency.
A modal analysis is performed while parametrically sweeping the length of the waveguide from 0.5 um to 4 um to derive the dispersion curve for the anisotropic core, as shown in the figure below. We assume the earlier case stated, with diagonal anisotropy terms of the core (i.e., _{xx} = 2.19, _{yy} = _{zz} = 2.31 and all of the diagonal elements are zero). The results are compared with Koshiba et al. (Ref. 3).
Dispersion curve with transverse anisotropic core.
When the optic axis (i.e., c-axis) lies in XY plane and makes an angle of with the x-axis, the diagonal components _{xx}, _{yy}, _{zz} and off-diagonal components _{xy} and _{yz} are nonzero, while the rest of the components are zero. The full relative permittivity tensor could be evaluated by using the rotation matrix [R] as shown below, where the rotation matrix [R] is specifically for rotating the c-axis in the XY plane. _{xx} is the square of the extraordinary refractive index (~2.19), because the c-axis lies along the principal x-axis, while _{yy} and _{zz} are the square of the ordinary refractive index (~2.31). The off-diagonal elements _{xy} and _{yz} are derived from the multiplication of the matrices as stated below.
The c-axis lying in the XY plane and making an angle of with the x-axis.
The relative permittivity tensor ε is treated along with a rotation matrix, rotating the c-axis in the XY plane with angle .
Finally, the modal analysis of the waveguide with off-diagonal anisotropic core and isotropic cladding, where the optic axis makes angles of 0, 15, 30, and 45 degrees with respect to the principal x-axis, as shown below. Here, it could be observed that the direction of the in-plane magnetic field changes according to the change in the angle of the optic axis. The dispersion curve could also be plotted by parametrically sweeping the length of the core and cladding from 0.5 um to 4 um, while considering the angle as 45°. The dispersion curve tends to be similar to the dispersion curve of the diagonal anisotropy, as discussed above.
Mode analysis, including off-diagonal terms, for θ = 0° (top-left), θ = 15° (top-right), θ = 30° (bottom-left), and θ = 45° (bottom-right). The figure represents the magnetic field lines within the core for different rotation angles.
Finally, when considering the longitudinal anisotropy where the optic axis (i.e., c-axis) lies in the YZ plane and makes an angle of with the y-axis, the diagonal components _{xx}, _{yy}, _{zz} and the off-diagonal components _{yz} and _{zy} are nonzero, while the rest of the components are zero. The relative permittivity tensor could be evaluated by using the rotation matrix [R] as shown below, where the rotation matrix [R] is specifically for rotating the c-axis in the YZ plane. _{yy} is the square of the extraordinary refractive index (~2.19), because the c-axis lies along the principal y-axis, while _{xx}, _{zz} is the square of the ordinary refractive index (~2.31). The off-diagonal elements _{yz} and _{zy} are derived from the multiplication of the matrices as stated below.
The c-axis lying in the YZ plane and making an angle of with the x-axis.
The relative permittivity tensor ε is treated along with a rotation matrix, rotating in the YZ plane with angle .
A modal analysis is then performed where the length of the waveguide is parametrically swept from 0.5 um to 4 um to derive the dispersion curve for the longitudinal anisotropic core, as shown in the figure below. In this case, = 45° (i.e., the c-axis lies in the YZ plane and makes 45° with the y-axis) (Ref. 3).
Dispersion curve with longitudinal anisotropic core.
In this blog post, we performed qualitative analyses (modes of propagation) and quantitative analyses (dispersion curves) of the anisotropic optical waveguide using modal analysis in COMSOL Multiphysics. Diagonal anisotropy as well as off-diagonal transverse and longitudinal anisotropy were considered to derive their dispersion relationships. These types of analyses give us more flexibility when carrying out optimization of material and geometric parameters to help us gain an in-depth and intuitive understanding of the physics of anisotropic materials.
A simple tutorial model to help you started would be the Step-Index Fiber, which involves mode analysis over a 2D cross section of the 3D optical fiber.
For decades, researchers aimed to find a way to control light and use it for the transmission and processing of information, an area of study known as photonics. In the meantime, electrons took this responsibility on their shoulders. More recently, scientists were able to viably manufacture nanostructure devices and control the flow of light, due to the extensive development of technologies such as photolithography, molecular beam epitaxy, and chemical vapor deposition. The packets of light (photons) were projected as a prospective candidate to assume the responsibilities of sustaining Moore’s law.
The goal of researchers studying photonics was to deliver an analogue of an electronic integrated chip that could perform all of the required computational processes using photons while being space and time efficient. Scientists termed this technology photonic integrated circuit (PIC), devices that could integrate different optical components on a single substrate. This chip should, in principle, be able to perform various optical operations, such as focusing, splitting, isolation, polarization, coupling, modulation, and (eventually) detecting light.
Schematic of the photonic integrated circuit (not to scale), showcasing different optical components. For more information, see Ref. 1.
In this blog post, which is the first in a new series on silicon photonics, we discuss optical waveguides. Later in the blog series, we will contemplate how these optical components came to be an inevitable part of PICs.
The different optical components constituting a fully functioning PIC were subject to research. Scientists determined that the way to create the light source was through lasers, which could deliver a narrow-band light source to the integrated chip component. As for optical fibers, they could transport light from one end to the other for thousands of kilometers. Then there was the most common component in a PIC: the optical waveguide. This waveguide could link different components on the substrate.
Input couplers were developed to efficiently couple the light from lasers or optical fibers to the optical waveguide placed on the substrate, while directional couplers were created to control the coupling of light between two parallel optical waveguides. Then came the ring resonator, which served the same purpose as an optical filter (that is, allowing only a narrow band of frequency) and could also couple two optical waveguides in opposite directions.
An example of an optical ring resonator notch filter.
Some scientists probed the much underappreciated nonlinear optical effects to devise second-harmonic and third-harmonic waves. With these waves, it would be possible to perform operations between two optical beams, such as frequency doubling, differencing, and mixing.
Another invention was optical modulators. These components could modify the light intensity based on the applied DC bias potential using the nonlinear electro-optic effects.
From nature, it was observed that with the periodic arrangement of high- and low-refractive-index materials in 1D, 2D, and 3D, it was possible to reflect a certain band of frequency while allowing another band of frequency to pass. Hence, these materials could act as both a filter and resonator in a certain periodic arrangement. The periodic arrangement of different dielectric materials was termed a photonic crystal.
With the idea of creating optical waveguides to propagate light on chip-scale packages, scientists were left to wonder which materials to use. One of the materials was high-refractive-index GaAs. This was used as the core and was surrounded by low-refractive-index AlGaAs. More advanced techniques were developed to dope titanium in the lithium niobate substrate to increase its refractive index and form a core.
The focus was narrowed down to silica, which is more easily available than any other material. The technology came to be known as silica on silicon (Si-SiO2) or silicon on insulator (SOI), where the silicon (high refractive index of ~3.5) was embedded within silica (lower refractive index of ~1.4). The fabrication techniques for silicon were well established (courtesy of electronic chips) and at the same time, silicon was compatible with other CMOS techniques, which helped boost research into silicon photonics technology.
The crux of the silicon waveguide lies with the high contrast of the refractive index, around a 50% difference. Prior work relied on total internal reflection to confine the energy. In this case, energy was confined in a higher-refractive-index core that was surrounded by a low-refractive-index cladding. However, recent work confined the energy in the lower-refractive-index slot neighbored by the high-refractive-index slabs, inherently helping to lower losses.
The first technique involved confining the energy in a higher-refractive-index medium, where the inner core (in the order of hundreds of nanometers) is devised with a high-refractive-index material (silicon) surrounded by a low-refractive-index cladding (silica). The difference in the refractive index must be as much as 50%.
The fundamental mode is confined in the core, as shown in the image below on the left, and the confined normalized power, as shown in the image below on the right.
Left: The fundamental mode for an operating wavelength of 1.55 um. The white and black arrows depict the magnetic and electric field. Right: The normalized power density through the center of the waveguide.
Although counterintuitive, the energy could also be trapped in a low refractive index. Moreover, it was found that more energy stays put in an even and narrow region (20 to 80 nm), which makes a low refractive index more compatible for integration with photonic circuits.
Such a design involves two slabs of high refractive index neighboring a nanoslot of low refractive index. Further, considerable energy is bounded in the slot.
Left: The transverse (Ex) field for a slot width of 50 nm. Right: The normalized transverse electric field (Ex) through the center of the waveguide.
To analyze the required width of the nanoslot for delivering maximum power through the waveguide, it was imperative to perform a sweep of the width, as shown below.
The normalized power and intensity in the slot versus slot width.
Fabricating such an optical waveguide prototype and then analyzing it is resource intensive. An alternate, preferred approach is to use numerical tools such as the COMSOL Multiphysics® software. With this simulation tool, one could quickly set up prototypes and investigate further before finalizing the prototype to be fabricated.
We can use COMSOL Multiphysics to perform a mode analysis on the 2D cross section of the silicon waveguide (both for high- and low-refractive-index cases). This enables us to evaluate the effective refractive index of the waveguide and the fundamental mode, which helps us understand the normalized power distribution.
We implement the full 3D propagation for both types of waveguides by first having a 3D geometry of the optical waveguide and assigning Numeric Port boundary conditions on both ends of the waveguide. The Boundary Mode Analysis study (similar to a mode analysis in 2D) could be applied on these numeric ports to figure out their fundamental mode. The fundamental mode could be used to propagate within the waveguide using the Frequency Domain study, as shown in the animations below.
The y-component of the H-field propagating in the high-refractive-index confinement case for a silicon waveguide with a length of 10 um.
The y-component of the E-field propagating in the low-refractive-index confinement case for a silicon waveguide with a length of 10 um.
This is the first blog post in the Silicon Photonics blog series, where we will discuss different optical components in detail and how a finite element analysis tool such as COMSOL Multiphysics can help design these components. On our journey from laser cavities to photodetectors, we will meet some fascinating scientists and discuss how they attempted to control light.
Stay tuned!
Before we get to the rough surface, let’s start with something simple: a thin uniform layer of gold coating on top of optically flat glass, as shown in the image below. Such a model exhibits negligible structural variation in the plane of glass. In addition, it can be modeled quite simply in the COMSOL Multiphysics® software by considering a small two-dimensional unit cell that has a width much smaller than the wavelength.
This computational model is based on the Fresnel equation example, one of the verification models in the Application Gallery, but is modified to include a layer of gold with a wavelength-dependent refractive index. This type of index requires that we manually adjust the mesh size based on the minimum wavelength in each material as well as the skin depth, as described in a previous blog post.
Light incident on a metal coating on top of a glass substrate is reflected, transmitted, and absorbed.
The model includes Floquet periodic boundary conditions on the left and right sides of the modeling domain and a Port boundary condition at the top and bottom. The Port boundary condition at the top launches a plane wave at a specified angle of incidence and computes the reflected light, while the one at the bottom calculates the transmitted light. We can integrate the losses within the metal layer to compute the absorbance within the gold layer.
The computational model that calculates the optical properties of a metal film on glass.
If we are interested in computing incident light at off-normal incident angles, then we also have to concern ourselves with the height of the modeling domain — the distance between the material interfaces and the Port boundary conditions. This distance must be large enough such that any evanescent field drops off to approximately zero within the modeling domain.
The reason for this has to do with the Port boundary conditions, which can only consider the propagating component of the electromagnetic field. Any evanescent component of the field that reaches the Port boundary condition is artificially reflected, so we must place the port boundary far enough away from the material interfaces. In the most general cases, it is difficult to determine how far the evanescent field extends. A simple rule of thumb is to place the Port boundary conditions at least half a wavelength away from the material interfaces and to check if making the domain larger alters the results.
The sample results below show the transmitted, reflected, and absorbed light as well as their total — which should always add up to one. If these do not add up to one, then we must carefully check our model setup.
The transmittance, reflectance, and absorbance of light normally incident on a flat glass surface with a metal coating as a function of wavelength.
The transmittance, reflectance, and absorbance of 550-nm light at various angles of incidence.
Let’s now make things a little bit more complicated and introduce a periodic structural variation: a sinusoidal ripple. Clearly, we now need to consider a larger unit cell that considers a single ripple.
A surface with periodic variations reflects and transmits light into several different diffraction orders.
We can still apply the same domain properties and all of the same boundary conditions. However, if spacing is large enough, then we can have higher-order diffraction. In other words, light can be reflected and transmitted into several different directions. To properly compute the reflection and transmission, we need to add several diffraction order ports. The software computes the appropriate number of ports based on the domain width, material properties, and specified incident angle. If we are studying a range of incident angles, we must make sure to compute all of the diffraction orders present at the limits of the angular sweep.
There can be multiple diffraction orders present, depending on the ratio of wavelength to domain width, refractive index, and incident angles.
The conditions under which higher-order diffraction appears and the appropriate modeling procedure is presented in depth in the example of a plasmonic wire grating, so let’s not go into it at length here. In short, the wider the computational domain relative to the wavelengths in the materials above and below, the more diffraction orders can be present (the number of diffraction orders varies with the incident angle). The results shown below plot the total transmittance and reflectance; i.e., all of the light reflected into the different diffraction orders is added up, as is all of the transmitted light.
The transmittance, reflectance, and absorbance of light normally incident on a rippled glass surface with a metal coating.
The transmittance, reflectance, and absorbance of 550-nm light at various angles of incidence.
Let’s now move on to the most computationally difficult case: a surface with many random variations in the surface height. To model the randomness, we must model several different domains of increasing widths and different subsets of the rough profile. As the domain width increases — and as different subsets of the surface are sampled — the average behavior computed from these different models converges. That is, we generate a set of statistics by sampling the rough surface. Rather than going into detail on how to calculate these statistics, let’s focus on how to model one domain that approximates a rough surface by defining the height variation as the sum of different sinusoids with random height and phase, as described here.
A rough surface with random variations reflects and transmits light in random directions. The computational model must sample a statistically significant subset of the roughness profile.
Our computational domain must now be very wide, many times longer than the wavelength. As we still want to model a plane wave incident at various angles on the structure, we use the Floquet periodic boundary conditions, which require that we have an identical mesh on the periodic boundaries. Practically speaking, this means that we may need to slightly alter the geometry of our domain to ensure that the boundaries on the left and right side are identical. If we do use a sum of sine functions, as described here, then the profile will automatically be periodic.
We still want to launch the wave with a Port boundary condition. However, it is no longer practical to use diffraction order ports to monitor the reflected and transmitted light, as this can result in hundreds (or thousands) of diffraction orders. Furthermore, since this model represents a statistical sampling, the relative fraction of light scattered into these different orders is not of interest; we’re only interested in the sum of the total reflected and transmitted light. That is, this modeling approach computes the total integrated scatter plus the specular reflection and transmission of the surface.
The computational domain for a model of a rough surface. Light is launched from the interior port toward the material interface. Light reflected back toward this port passes though it and is absorbed in the PML, as is the transmitted light. Two additional boundaries are introduced to monitor the total reflectance and transmittance.
Thus, we introduce an alternative modeling strategy that does not use ports to compute reflection and transmission. Instead, we use a perfectly matched layer (PML) above and below to absorb all reflected and transmitted light as well as probes to compute reflection and transmission. PMLs absorb any fields incident upon them, as described in this blog post on using PMLs for wave electromagnetics problems.
The PML absorbs both propagating and evanescent components of the field, but we only want it to absorb the propagating component. Thus, we again need to ensure that we place the PMLs far enough away from the material interfaces. We use the same rule of thumb as before, placing the PML at least half a wavelength away from the material interfaces.
As we approach grazing angles of incidence, even the PML domain does not, by default, absorb all of the light. At nearly grazing angles, the effective wavelength in the absorbing direction is very long, and we need to modify the default wavelength in the PML settings (shown below). This change to the settings is only necessary if we are interested in angles of incidence greater than ~75°.
The PML settings modified to account for grazing angles of incidence.
Since our domain is now bounded by PMLs above and below, the port that launches the wave must now be placed within the modeling domain. To do this, we use the Slit Condition option to define an interior port that is backed by a domain. This means that the port now launches a wave in one direction, emanating from this interior boundary. Any light reflected back toward the boundary passes through unimpeded and then gets absorbed by the PML.
Although this is a good way to launch the wave, we will no longer use the Port boundary condition to compute how much light is reflected, since we would have to add hundreds of diffraction ports, and similarly, we’d need hundreds of ports to compute the total transmittance.
To monitor the total transmitted and reflected light, we instead introduce two additional interior boundaries to the model, placed just in front of the PML domains (shown in the schematic above). At these two boundaries, we integrate the power flux in the upward and downward directions, normalized by the incident power, which gives us the total reflectance and transmittance. To more accurately determine the integral of the power flux at these boundaries, we also introduce a boundary layer mesh composed of a single layer of elements much smaller than the wavelength.
On the incident side, we place this monitoring boundary above the interior port. The launching port introduces a plane wave propagating toward the material interface. The light reflected at the interface passes through this interior port, then moves through the boundary at which we monitor reflectance, and is absorbed in the PML.
The plots below show sample results of the transmittance, reflectance, and absorbance. They are notably different from the smooth surface and periodically varying surface results. Note that the sweep over the angle of incidence terminates at 85° off normal. Of course these plots will look slightly different for each different random geometry case that we run.
The transmittance, reflectance, and absorbance of light normally incident on a rough glass surface.
The transmittance, reflectance, and absorbance of 550-nm light at angles of incidence up to 85° off normal.
Here, we have introduced a modeling approach that is appropriate for computing the optical transmission and reflection from a rough surface. This method contrasts with the approach for modeling a uniform optically flat surface as well as the one for modeling surfaces with periodic variations. The modeling method for rough surfaces can also be used for the modeling of periodic structures that have a very long period, such as when the scattering into different diffraction orders is not of interest.
Modeling truly random surfaces does require some care, as the geometry needs to be altered to ensure that it is periodic. Furthermore, the domain size and number of different random geometries studied must be large enough to give statistically meaningful results. Since this requires solving many different variations of the same model and postprocessing the results, it is helpful to use the Application Builder, LiveLink™ for MATLAB®, or LiveLink™ for Excel® in our modeling workflow.
MATLAB is a registered trademark of The MathWorks, Inc. Microsoft and Excel are either registered trademarks or trademarks of Microsoft Corporation in the United States and/or other countries.
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Sensors, chemical imaging, and optoelectronics: These are just some of the applications in which midinfrared wavelengths are of significance. With their growing importance across various industries and technologies, the need for identifying optical fibers that can produce midinfrared light in a large wavelength range is increasing as well.
One approach is to fuse silica — a typical material for optical fibers — with infrared transparent semiconductors like germanium, zinc selenide, and silicon. To be more specific, silica makes up the cladding, or outer layer, of the optical fiber design and the selected semiconductor makes up its core. This combination provides opportunities for developing new types of midinfrared multimaterial optical fibers.
A multimaterial optical fiber design. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
In order for these optical fiber designs to be effective, an important step is to better understand their optical losses. While this can be time consuming and costly via experiments, simulation tools like the COMSOL Multiphysics® software provide a more efficient route for modeling this behavior and identifying means of optimization. Let’s look at an example from researchers at Pennsylvania State University and Pacific Lutheran University that involves the analysis and design of a germanium-based optical fiber.
For their analysis, the researchers used the RF Module, an add-on product to COMSOL Multiphysics. After defining their respective geometries, they applied refractive indices to both the core and cladding of the optical fiber at a particular wavelength. Along the outside of the cladding, the electric field is said to be zero. The plot below shows the electric field distribution for a characteristic HE_{11} mode that is confined and guided through a 6-μm core diameter made of germanium.
The electric field distribution of a characteristic HE_{11} mode within the core diameter. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
To identify the optimal fiber geometry and operative wavelength range, the research team simulated the mode’s propagation attenuation as a function of wavelength and core diameter. Between 2 and 4 μm, a window of low optical loss occurs. As the wavelength becomes longer, the loss begins to increase. This is due to the evanescent wave extending further into the cladding region and the increase in silica’s extinction coefficient at a longer wavelength. Note that the loss related to this effect is larger when the core diameter is smaller.
Left: The optical losses for the fibers. Right: The electric field distributions of the HE_{11} mode in the core diameter for varying wavelengths. Images by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
For these longer wavelengths, one strategy for reducing optical loss and achieving a wider window of high transmission is to add another layer of material between the germanium and silica. This material needs to have smaller refractive indices than germanium and a smaller extinction coefficient than silica over a large wavelength range. Two good candidates for this are silicon and zinc selenide. In the plot below, we can see the characteristic HE_{11} mode confined and guided through each of these new fiber structures.
The electric field distribution of a characteristic HE_{11} mode within the core diameter when an interfacial layer is added. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
Once again, the researchers simulated the optical loss as a function of the wavelength for a 6-μm core diameter. As the results indicate, introducing the additional layer significantly reduces the optical propagation losses, particularly at longer wavelengths. What’s more: It does so without having to sacrifice the size of the core diameter.
Left: The optical losses for the fibers that include an added layer. Right: The electric field distributions of the HE_{11} mode in the core diameter at a wavelength of 10 μm. Images by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
From the above results, it is clear that the reduction in optical loss is more pronounced in the zinc selenide case. This is because the refractive index between germanium and zinc selenide is greater than that between germanium and silicon, allowing light to be better confined. However, smaller refractive index differences often reduce the constraint of a small core diameter for single-mode guidance.
To account for this, the researchers calculated the single-mode guidance for each germanium-based core diameter configuration. The results show that the zinc selenide structure requires a smaller germanium core diameter to achieve single-mode guidance.
The germanium-based core diameter requirements for the zinc selenide (a) and silicon (b) structures. The gray areas represent conditions for single-mode guidance. Image by X. Ji, R. Page, and V. Gopalan and taken from their COMSOL Conference 2016 Boston paper.
With the flexibility of COMSOL Multiphysics, the research team was able to easily modify different parameters in their optical fiber design and analyze the impact on fiber performance. From there, they implemented further strategies to minimize optical losses and enable greater transmission. Their optimized fiber design has potential use in the medical field, specifically in endoscopes for spectroscopic imaging.
Implementing the Fourier transformation in a simulation can be useful in Fourier optics, signal processing (for use in frequency pattern extraction), and noise reduction and filtering via image processing. In Fourier optics, the Fresnel approximation is one of the approximation methods used for calculating the field near the diffracting aperture. Suppose a diffracting aperture is located in the plane at . The diffracted electric field in the plane at the distance from the diffracting aperture is calculated as
where, is the wavelength and account for the electric field at the plane and the plane, respectively. (See Ref. 1 for more details.)
In this approximation formula, the diffracted field is calculated by Fourier transforming the incident field multiplied by the quadratic phase function .
The sign convention of the phase function must follow the sign convention of the time dependence of the fields. In COMSOL Multiphysics, the time dependence of the electromagnetic fields is of the form . So, the sign of the quadratic phase function is negative.
Now, let’s take a look at an example of a Fresnel lens. A Fresnel lens is a regular plano-convex lens except for its curved surface, which is folded toward the flat side at every multiple of along the lens height, where m is an integer and n is the refractive index of the lens material. This is called an m^{th}-order Fresnel lens.
The shift of the surface by this particular height along the light propagation direction only changes the phase of the light by (roughly speaking and under the paraxial approximation). Because of this, the folded lens fundamentally reproduces the same wavefront in the far field and behaves like the original unfolded lens. The main difference is the diffraction effect. Regular lenses basically don’t show any diffraction (if there is no vignetting by a hard aperture), while Fresnel lenses always show small diffraction patterns around the main spot due to the surface discontinuities and internal reflections.
When a Fresnel lens is designed digitally, the lens surface is made up of discrete layers, giving it a staircase-like appearance. This is called a multilevel Fresnel lens. Due to the flat part of the steps, the diffraction pattern of a multilevel Fresnel lens typically includes a zeroth-order background in addition to the higher-order diffraction.
A Fresnel lens in a lighthouse in Boston. Image by Manfred Schmidt — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.
Why are we using a Fresnel lens as our example? The reason is similar to why lighthouses use Fresnel lenses in their operations. A Fresnel lens is folded into in height. It can be extremely thin and therefore of less weight and volume, which is beneficial for the optics of lighthouses compared to a large, heavy, and thick lens of the conventional refractive type. Likewise, for our purposes, Fresnel lenses can be easier to simulate in COMSOL Multiphysics and the add-on Wave Optics Module because the number of elements are manageable.
The figure below depicts the optics layout that we are trying to simulate to demonstrate how we can implement the Fourier transformation, applied to a computed solution solved for by the Wave Optics, Frequency Domain interface.
Focusing 16-level Fresnel lens model.
This is a first-order Fresnel lens with surfaces that are digitized in 16 levels. A plane wave is incident on the incidence plane. At the exit plane at , the field is diffracted by the Fresnel lens to be . This process can be easily modeled and simulated by the Wave Optics, Frequency Domain interface. Then, we calculate the field at the focal plane at by applying the Fourier transformation in the Fresnel approximation, as described above.
The figures below are the result of our computation, with the electric field component in the domains (top) and on the boundary corresponding to the exit plane (bottom). Note that the geometry is not drawn to scale in the vertical axis. We can clearly see the positively curved wavefront from the center and from every air gap between the saw teeth. Note that the reflection from the lens surfaces leads to some small interferences in the domain field result and ripples in the boundary field result. This is because there is no antireflective coating modeled here.
The computed electric field component in the Fresnel lens and surrounding air domains (vertical axis is not to scale).
The computed electric field component at the exit plane.
Let’s move on to the Fourier transformation. In the previous example of an analytical function, we prepared two data sets: one for the source space and one for the Fourier space. The parameter names that were defined in the Settings window of the data set were the spatial coordinates in the source plane and the spatial coordinates in the image plane.
In today’s example, the source space is already created in the computed data set, Study 1/Solution 1 (sol1){dset1}, with the computed solutions. All we need to do is create a one-dimensional data set, Grid1D {grid1}, with parameters for the Fourier space; i.e., the spatial coordinate in the focal plane. We then relate it to the source data set, as seen in the figure below. Then, we define an integration operator intop1
on the exit plane.
Settings for the data set for the transformation.
The intop1
operator defined on the exit plane (vertical axis is not to scale).
Finally, we define the Fourier transformation in a 1D plot, shown below. It’s important to specify the data set we previously created for the transformation and to let COMSOL Multiphysics know that is the destination independent variable by using the dest
operator.
Settings for the Fourier transformation in a 1D plot.
The end result is shown in the following plot. This is a typical image of the focused beam through a multilevel Fresnel lens in the focal plane (see Ref. 2). There is the main spot by the first-order diffraction in the center and a weaker background caused by the zeroth-order (nondiffracted) and higher-order diffractions.
Electric field norm plot of the focused beam through a 16-level Fresnel lens.
In this blog post, we learned how to implement the Fourier transformation for computed solutions. This functionality is useful for long-distance propagation calculation in COMSOL Multiphysics and extends electromagnetic simulation to Fourier optics.
Modern optical communication systems commonly use EO routers and macroscale MO devices. Each device, however, has its own drawbacks. EO routers require an electric field and often have operation voltages in the kilovolt range, while macroscale MO devices don’t allow for scalable solutions. The quest for small, low-power optic routing alternatives is thus an important focus for specialized researchers within this field. This search is complicated, as various specializations within physics and engineering are required to address the study and interaction of magnetism, magnetic materials, and light.
Scanning electron micrograph of an unpolished silicon-on-insulator rib waveguide used for optical routers on a chip. Image credit: J. Tioh, “Interferometric switches for transparent networks: development and integration,” 2012, Graduate Theses and Dissertations. Paper 12487.
One potential solution involves integrating optical components onto a silicon substrate to create an MO routing solution on a chip. This option reduces both the size and operation power of the device and can potentially enable new technologies like light processors. But before monolithically integrated MO routers become commonplace, there are still a few hurdles that this technology needs to overcome.
Standard industry practices, for instance, present a challenge when it comes to manufacturing monolithically integrated MO routers. To introduce new technology with minimal industry disruptions, standard practices must be used. In this case, silicon should be used as a base substrate and the combining materials must be compatible with silicon for successful monolithic integration. But bonding silicon and the suitable magneto-optic materials can be quite difficult using standard industry practices due to their crystal structures. As a result, the materials tend to become brittle and crack, significantly increasing optical losses.
Multiphysics simulation offers potential solutions to such challenges. These tools can help the research community identify optimal designs and manufacturing techniques for monolithically integrated MO routers. For his doctoral dissertation at Iowa State University, John Pritchard, an engineer who works within this field, turned to the COMSOL Multiphysics® software to provide new insight into the design and future of on-chip MO routers.
When analyzing an on-chip MO system, Pritchard chose to focus on a few key design elements. One point of focus was analyzing a codirectional coupler, a device that is commonly found in interferometer designs. The power coupling coefficients of a codirectional coupler vary based on the distance between the coupling section length and coupled waveguides. Through his simulation work, Pritchard determined how to generate an ideal power coupler coefficient by choosing a specific coupler length.
3D simulation results for the codirectional coupler. Copyright © John Pritchard.
Another point of analysis was an on-chip optical waveguide. The goal here was to design a rib waveguide that minimizes energy loss and maintains a sufficient beam profile throughout the device. To achieve this, Pritchard used silicon as the rib waveguide’s transmission medium, since it is suitably transparent to infrared light and useful for integration with electronic devices. Further, a low-index cladding model was placed between the substrate and waveguide to stop the optical mode from leaking out.
The optical mode of an SOI rib waveguide. Copyright © John Pritchard.
As for the waveguide’s silicon-on-insulator (SOI) platform, Pritchard used a buried oxide insulator on a silicon substrate. This waveguide configuration enabled him to confine relatively large optical modes in the waveguide and avoid harming the single-mode operation. Subsequent simulations of the configuration revealed that the optical mode is well confined within the waveguide and that this geometry can be used to design an interferometer. Pritchard also performed a frequency analysis of the top view of the design, as seen in the animation below and to the left. This waveguide design was deemed a success and is a significant step toward fully realizing MO routers on a chip.
Left: Wave propagation at the top of a dual waveguide and coupler at 1550 nm. Right: Mode profile of a coupler and dual waveguide. Copyright © John Pritchard.
The coupler and waveguide designs we’ve discussed thus far are ideal for the single-mode confinement of light at 1550 nm. Now, let’s see how adding a top layer of MO material to the SOI rib waveguide affects the device. Specifically, the goal is to find out the amount of light that is exposed to the Faraday rotation. This indicates when light with a rotated state of polarization interferes with nonrotated light.
Mode analysis of an SOI waveguide with a top layer of MO material. Copyright © John Pritchard.
The results, highlighted above, show that the MO material contains 3.9% of the light. Despite being a small percentage, previous research suggests that this creates sufficient Faraday rotation to observe interference at the electrical output. But for this to happen, the material needs to be magnetized with a permanent magnet or controlled magnetic field generator. Finding appropriate monolithically integrated magnetic field generators was therefore a final point of consideration.
While magnetic field generation techniques are important for creating on-chip MO modulators, the process itself is complex. The small size of MO systems makes it difficult to develop monolithically integrated magnetic field generators. To address this, Pritchard used simulation to validate the design of a novel dynamic magnetic field generator: a four-turn integrated coil.
Left: Geometry of an integrated magnetic field generator. Right: Geometry of the integrated magnetic field generator with the MO material highlighted in pink and the silicon waveguide shown in purple. Copyright © John Pritchard.
The results show that the coil generated 260 G near the center of the optical waveguide when energized with a 35 mA current. Within the tested waveguide dimensions, this magnetic field strength can magnetize Ce:YIG film on silicon.
It is possible to expand such research by investigating the field at the center of the MO material, which is part of the core of the four-turn integrated coil. Here, the simulation studies indicate that with a current of 35 mA, the magnetic field at the center of the MO material layer has a maximum field of about 210 G, a reduction possibly explained by the difference in properties of the material. Such findings speak to the potential of on-chip MO routing solutions and can be used as a resource in improving their design and manufacturing processes.
Simulation results for the magnetic field generator at the center of the coil. Copyright © John Pritchard.
Future on-chip optical network architects will have a variety of active switching and routing options, allowing them to make their networks more robust by using both EO and MO devices. While it’s important to note that the results mentioned here are preliminary and more research is needed, the simulations and design methodologies act as a proof of concept for on-chip magnetic field generators and silicon rib waveguides. They can serve as a useful foundation for continued studies on such devices, creating a path for furthering their optimization.
As John Pritchard notes: “Some of the most beautiful connections between light, magnetism, and quantum theory have led to breathtaking technologies ranging from superconductor imaging to gigawatt laser pulses. These have enabled revolutionary inventions in transportation; measurement instruments used to understand the birth of the universe; and, in the near future, optical integrated circuits.” Looking to the future, we are eager to see the continued role of multiphysics simulation in advancing optical research, a field with wide-reaching applications.
In Part 2 of the blog series, we used the Electromagnetic Waves, Frequency Domain interface, which we call a Full-Wave simulation, and a Far-Field Domain node to determine the electric field in the far field. We then coupled a Full-Wave simulation to the Electromagnetic Waves, Beam Envelopes interface (or a Beam-Envelopes simulation) in order to precisely calculate fields in any region, regardless of the distance from the source.
The Far-Field Domain and Beam-Envelopes solutions that we looked at in the previous blog post are effective, but they share one noteworthy restriction. In each case, we assumed that a homogeneous domain surrounded the antenna in all directions. For many situations, this information is sufficient. In other simulations, you may not have a homogeneous domain surrounding your antenna and you need to account for issues like atmospheric refraction or reflection off of nearby buildings. These simulations require a different approach.
A model of several hotels in Las Vegas. A directional antenna emits rays toward the ARIA® Resort & Casino.
The Geometrical Optics interface in the Ray Optics Module, an add-on product to the COMSOL Multiphysics® software, regards EM waves as rays. This interface can account for spatially varying refractive indices, reflection and refraction from complicated geometries, and long propagation distances. However, these features come with a tradeoff. Since waves are treated as rays, this approach neglects diffraction. In other words, we are assuming that the wavelength of light is much smaller than any geometric features in our environment. You can read a more thorough description of ray optics in a previous blog post.
As you may recall, we introduced an approach to coupling a radiating and receiving antenna in Part 3 of this series. When incorporating ray optics into our multiscale modeling, we are required to use a similar but more generalized approach. Before we show you how to set up a geometrical optics simulation in COMSOL Multiphysics, let’s first review this alternate method.
As a quick refresher, we are interested in calculating the fields at the location of the receiving antenna using the following equation:
We previously used an integration operator on a single point to calculate this along the line directly between the two antennas. We now wish to retain the angular dependence, so we need to recalculate this equation for each point in the receiving antenna’s domain. Since it is impractical to add numerous points and integration operators, we need to establish a more general technique.
To do so, we replace the integration operator with a General Extrusion operator. As before, we create a variable for the magnitude of r. We then use the General Extrusion operator to evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point in which we are actually interested.
To demonstrate this concept, we use a figure that is slightly more involved than that from the previous post. Note that the subscripts 1, 2, and r in represent a vector in component 1, a vector in component 2, and the offset between the antennas, respectively.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
As we previously outlined, the primary complication is determining where to calculate the scattering amplitude. We want the fields at the point , which requires calculating the scattering amplitude at . The complication, of course, is that each point in the domain around the receiving antenna (each vector ) will have its own evaluation location . We evaluate this by again rescaling the Cartesian coordinates, but instead of doing it for a single point, we define it inside of the general operator so that it can be called from any location. From the above figure, we know that this point is , with corresponding equations for y and z. The operator is defined in component 1, so the source will be defined in that component. It will be called from component 2, so the x, y, z in the following expressions refer to x_{2}, y_{2}, z_{2} in the above figure.
The General Extrusion operator used for the scattering amplitude calculation. Note that this is defined in component 1.
As a bookkeeping step, we store the calculated fields in a “dummy” variable. By a dummy variable, we mean that we add in an extra dependent variable that takes the value of a calculation determined elsewhere. We do this for two reasons.
The first reason is that most variables in COMSOL Multiphysics are calculated on demand from the dependent variables. In an RF simulation, for example, the dependent variables are the three Cartesian components of the electric field: Ex, Ey, and Ez. These are determined when computing the solution. In postprocessing, every other value (electric current, magnetic field, etc.) is calculated from the electric field when required. In most cases, this is a fast and seamless process. In our case, each field evaluation point requires a general extrusion of a scattering amplitude, and each scattering amplitude point requires a surface integration as defined in the Far-Field Domain node. This can take a while and we want to ensure that we perform this calculation only once.
The second reason why we do this has to do with the element order. The Scattered Field formulation requires a background electric field. COMSOL Multiphysics then calculates the magnetic field using the differential form of Faraday’s law (also known as the Maxwell-Faraday equation). This requires taking spatial derivatives of the electric field. There are no issues when taking the spatial derivatives of an analytical function like a plane wave or Gaussian beam, but it can cause a discretization issue when applied to a solved-for variable. This is a rather advanced topic, which you can find out more about in an archived webinar on equation-based modeling.
By using a cubic dummy variable to store the electric field, we can take a spatial derivative of the electric field and still obtain a well-resolved magnetic field for use in the Scattered Field formulation. Without the increased order of the dummy variable, the magnetic field used would be underresolved. Below, you can see what it looks like to put the General Extrusion operator together with the dummy variable setup. The variable r is identical to the one used in Part 3 of this blog series and is defined in component 2.
The dummy variable implementation. Notice that the dummy variable components are called Ebx, Eby, and Ebz.
The only remaining step is to use the dummy variables — Ebx, Eby, and Ebz — in a background field simulation of the half-wavelength dipole discussed in Part 1 and Part 3.
This technique isn’t actually very good for this particular problem. There may be situations where it is useful, but the technique from Part 3 is preferred in the vast majority of cases. The received power from the two simulations is extremely close, but this method takes much longer to calculate and the file size increases drastically. In the demo examples for this post, this method took several times longer than the previous simulation method. While you may conclude that this is not a terribly useful step overall, it is useful when we incorporate ray optics into our multiscale modeling, as discussed in the next section.
A geometrical optics simulation implicitly assumes that every ray is already in the far field. Earlier in the blog series, we saw that the Far-Field Domain feature correctly calculates the electric field at arbitrary points in the far field. Here, we use that information as the input for rays in a geometrical optics simulation. The simulation geometry, symmetry, and electric dipole point source used are the same as in Part 2.
The domain assignments for the simulation. The Full-Wave simulation is performed over the entire domain, with the outer region set as a perfectly matched layer (PML). The geometrical optics simulation is only performed in this outer region. Note that this image is not to scale.
With the domains assigned, we select the Geometrical Optics interface, change the Intensity computation to Compute intensity, and select the Compute phase check box. These steps are required to properly compute the amplitude and phase of the electric field along the ray trajectory.
Settings for the Geometrical Optics interface. The Intensity computation is set to Compute intensity and the Compute phase check box is selected.
We also apply an Inlet boundary condition to the boundary between the Full-Wave simulation domain and Geometrical Optics domain. The inlet settings can be seen in the image below, but let’s walk through them one at a time. First, the Ray Direction Vector section is configured. This will launch the rays normal to the curved surface we’ve selected for the inlet — in other words, radially outwards. The variables Etheta and Ephi are calculated from the scattering amplitude according to
with a similar assignment for Ephi.
This equation comes from our previous blog post about using the Far-Field Domain node to calculate the fields at an arbitrary location. These variables are used to specify the initial phase and polarization of the rays. The variable specifies the correct spatial intensity distribution for the rays (as antennas generally do not emit uniformly) and is calculated according to , where Z is the impedance of the medium.
The initial radius of curvature has two factors. The parameter is the radius of the spherical boundary that we are launching the rays from and will correctly initialize the curvature of the ray wavefront.
Finally, we use the Cartesian components of our spherical unit vector to specify the initial principal curvature direction. This ensures that the correct polarization orientation is imparted to the rays. The wavefront shape here must be set to Ellipsoid — even though the surface is technically a sphere — because we need to be able to specify a preferred direction for polarization. If we choose Spherical, then each orientation is degenerate and we cannot make that specification.
The settings for the Inlet boundary condition in the Geometrical Optics interface. Note that you can click the image to expand it.
Beyond setting the correct frequency, the only other setting here is the placement of a Freeze Wall condition on the exterior boundary to stop the rays. Let’s take a look at the results vs. theory. As before, we express the full solution for a point dipole as a sum of two contributions, which we have labeled near field (NF) and far field (FF).
The electric fields from a geometrical optics simulation compared against theory. Geometrical optics is always in the far field, so we see excellent agreement as the distance from the source increases. For reference, the far-field domain results from the previous post would overlap exactly with the ray optics and FF theory lines.
As mentioned before, the Geometrical Optics interface is necessarily in the far field, so we do not expect to be able to correctly capture the near-field information as we did in the Beam-Envelopes solution in Part 2. This can also be seen because we seeded the ray tracing simulation with data from the Far-Field Domain node calculation. It is therefore unsurprising that there is disagreement near the source, but we can clearly see that the results match with theory as the distance from the source increases.
From looking solely at the above plot, we have to ask ourselves: “What have we actually gained here?”
This is a fair question, because the plot shown above could have been constructed directly from any of the techniques covered in the series so far. To make this clear, let’s review each of them.
Multiscale Technique | Regime of Validity | Modules Used | Notes |
---|---|---|---|
Far-Field Domain node | Far field | RF or Wave Optics | Requires the antenna to be completely surrounded by a homogeneous domain. |
Beam-Envelopes | Any field | Wave Optics | Requires specification of the phase function or wave vector. |
Geometrical Optics | Far field | Ray Optics | Can account for a spatially varying index as well as reflection and refraction from complex geometries. Diffraction is neglected. |
A summary of the multiscale modeling techniques we have covered in this blog series.
Note that any of these techniques will require a Full-Wave simulation of the radiation source. This generally requires the RF Module, although there is a subset of radiation sources that can be modeled using the Wave Optics Module instead. The Far-Field Domain node is available in both the RF and Wave Optics modules.
We originally motivated this discussion by talking about signal transmission from one antenna to another, and solved that simulation using the Far-Field Domain node in the last post. In the next blog post in this series, we’ll redo that simulation using the Geometrical Optics interface introduced here.
Access the model discussed in this blog post and any of the model examples highlighted throughout this blog series by clicking on the button above.
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In the simulation of our receiving antenna, we will use the Scattered Field formulation. This formulation is extremely useful when you have an object in the presence of a known field, such as in radar cross section (RCS) simulations. Since there are a number of scattered field simulations in the Application Gallery, and it has been discussed in a previous blog post, we will assume a familiarity with this technique and encourage you to review those resources if the Scattered Field formulation is new to you.
The Scattered Field formulation is useful for computing a radar cross section.
When comparing the implementation we will use here with the scattering examples in the Application Gallery, there are two differences that need to be referenced explicitly. The first is that, unlike the scattering examples, we will use a receiving antenna with a Lumped Port. With the Lumped Port excitation set to Off, it will receive power from the background field. This is automatically calculated in a predefined variable, and since the power is going into the lumped power, the value will be negative. The second difference, which we will spend more time discussing, is that the receiving antenna will be in a separate component than the emitting antenna and we will have to reference the results of one component in the other to link them.
What does it mean when we have two or more components in a model? The defining feature of a component is that it has its own geometry and spatial dimension. If you would like to have a 2D axisymmetric geometry and a 3D geometry in the same simulation, then they would each require their own component. If you would like to do two 3D simulations in the same model, you only need one component, although in some situations it can be beneficial to separate them anyways.
Let’s say, for example, that you have two devices with relatively complicated geometries. If they are in the same component, then anytime you make a geometric change to one, they both need to be rebuilt (and remeshed). In separate components this would not be the case. Another common use of multiple components is submodeling, where the macroscopic structure is analyzed first and then a more detailed analysis is performed on a smaller region of the model. When we split into components, however, we then need to link the results between the simulations.
In our case, we have two antennas at a distance of 1000 λ. Separating them into distinct components is not strictly required, but we are going to do it anyways to keep things general. We will add in ray tracing later in this series and some users may find this multiple component method useful with an arbitrarily complex ray tracing geometry.
While we go through the details, it’s important that we have a clear image of the big picture. The main idea that we are pursuing in this post is that we first simulate an emitting antenna and calculate the radiated fields in a specific direction. Specifically, this is the direction of the receiving antenna. We then account for the distance between the antennas and use the calculated fields as the background field in a Scattered Field formulation for the receiving antenna. The emitting antenna is centered at the origin in component 1 and the receiving antenna is centered at the origin in component 2. Everything we will discuss here is simply the technical details of determining the emitted fields from the first simulation and using them as a background field in a second simulation.
Note: The overwhelming majority of the COMSOL Multiphysics® software models only have one component and only should have one component. Ensure that you have a sufficient need for multiple components in your model before implementing them, as there is a very real possibility of causing yourself extra work without benefit.
There are a number of coupling operators, also known as component couplings, available in COMSOL Multiphysics. Generally speaking, these operators map the results from one spatial location to another. Said in another way, you can call for results in one location (the destination), but have the results evaluated at a separate location (the source). While this may seem trivial at first glance, it is an incredibly powerful and general technique. Let’s look at a few specific examples:
As mentioned above, we want to simulate the emitting antenna (just like we did in Part 2 of the series) and calculate the radiated fields at a distance of 1000 λ. We then use a component coupling to map the fields to being centered about the origin in component 2.
If we look at the far-field evaluation discussed in Part 2, we know that the x-component of the far field at a specific location is
The only complication is determining where to calculate the scattering amplitude. This is because component couplings need the source and destination to be locations that exist in the geometry. We don’t want to define a sphere in component 1 at the actual location of the receiving antenna, since that defeats the entire purpose of splitting the two antennas into two components. What we will do instead is create a variable for the magnitude of r, and then evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point we are actually interested in. In the image below, we show the point where we would like to evaluate the scattering amplitude.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
We add a point to the geometry using the rescaling of the Cartesian coordinates shown in the above figure. Only x is shown in the figure, but the same scaling is also applied to y and z. For the COMSOL Multiphysics implementation, shown below, we have assumed that the receiving antenna is centered at a location of (1000 λ, 0, 0), and the two parameters used are ant_dist = and sim_r = .
The required point for the correct scattering amplitude evaluation.
Note that we create a selection group from this point. This is so that it can be referenced without ambiguity. We then use this selection for an integration operator. Since we are integrating only over a single point, we simply return the value of the integrand at that point similar to using a Dirac delta function.
The integration operator is defined using the selection group for the evaluation point.
The above discussion was all about how to evaluate the scattering amplitude at the correct location. The only remaining step is to use this in a background field simulation of the half-wavelength dipole discussed in Part 1. When we add in the known distance between the antennas, we get the following:
The variable definition for r. Note that this is defined in component 2.
The background field settings.
In the settings, we see that the expression used for the background field in x is comp1.intop1(emw.Efarx)*exp(-j*k*r)/(r/1[m]), which matches the equation cited above. Also note that r is defined in component 2, while intop1() is defined in component 1. Since we are calling this from within component 2, we need to include the correct scope for the coupling operator, comp1.intop1(). The remainder of the receiving antenna simulation is functionally equivalent to other Scattered Field simulations in the Application Gallery, so we will not delve into the specifics here.
It is interesting to note that running either the emission or background field simulations by themselves is quite straightforward. All of the complication in this procedure is in correctly calculating the fields from component 1 and using them in component 2. All of this heavy lifting has paid off in that we can now fully simulate the received power in an antenna-to-antenna simulation, and the agreement between the simulated power and the Friis transmission equation is excellent. We can also obtain much more information from our simulation than we can purely from the Friis equation, since we have full knowledge of the electromagnetic fields at every point in space.
It is worth mentioning one final point before we conclude. We have only evaluated the far field at an individual point, so there is no angular dependence in the field at the receiving antenna. Because we are interested in antennas that are generally far apart, this is a valid approximation, although we will discuss a more general implementation in Part 4.
We have now reached a major benchmark in this blog series. After discussing terminology in Part 1 and emission in Part 2, we can now link a radiating antenna to a receiving antenna and verify our results against a known reference. The method we have implemented here can also be more useful than the Friis equation, as we have fully solved for the electromagnetic fields and any polarization mismatch is automatically accounted for.
There is one remaining issue, however, that we have not discussed. The method used here is only applicable to line-of-sight transmission through a homogeneous medium. If we had an inhomogeneous medium between the antennas or multipath transmission, that would not be appropriately accounted for either by this technique or the Friis equation. To solve that issue, we will need to use ray tracing to link the emitting and receiving antennas. In Part 4 of this blog series, we will show you how we can link a radiating source to a ray optics simulation.
Let’s begin by discussing a traditional antenna simulation using COMSOL Multiphysics and the RF Module. When we simulate a radiating antenna, we have a local source and are interested in the subsequent electromagnetic fields, both nearby and outgoing from the antenna. This is fundamentally what an antenna does. It converts local information (e.g., voltage or current) into propagating information (e.g., outgoing radiation). A receiving antenna inverts this operation and changes incident radiation into local information. Many devices, such as a cellphone, act as both receiving and emitting antennas, which is what enables you to make a phone call or browse the web.
Antennas of the Atacama Large Millimeter Array (ALMA) in Chile. ALMA detects signals from space to help scientists study the formation of stars, planets, and galaxies. Needless to say, the distance these signals travel is much greater than the size of an antenna. Image licensed under CC BY 4.0, via ESO/C. Malin.
In order to keep the required computational resources reasonable, we model only a small region of space around the antenna. We then truncate this small simulation domain with an absorbing boundary, such as a perfectly matched layer (PML), which absorbs the outgoing radiation. Since this will solve for the complex electric field everywhere in our simulation domain, we will refer to this as a Full-Wave simulation.
We then extract information about the antenna’s emission pattern using a Far-Field Domain node, which performs a near-to-far-field transformation. This approach gives us information about the electromagnetic field in two regions: the fields in the immediate vicinity of the antenna, which are computed directly, and the fields far away, which are calculated using the Far-Field Domain node. This is demonstrated in a number of RF models in the Application Gallery, such as the Dipole Antenna tutorial model, so we will not comment further on the practical implementation here.
One question that occasionally comes up in technical support is: “How do I use the Far-Field Domain node to calculate the radiated field at a specific location?” This is an excellent question. As stated in the RF Module User’s Guide, the Far-Field Domain node calculates the scattering amplitude, and so determining the complex field at a specific location requires a modification for distance and phase. The expression for the x-component of the electric field in the far field is:
and similar expressions apply to the y- and z-component, where r is the radial distance in spherical coordinates, k is the wave vector for the medium, and emw.Efarx is the scattering amplitude. It is worth pointing out that emw.Efarx is the scattering amplitude in a particular direction, and so it depends on angular position , but not radial position. The decrease in field strength is solely governed by the 1/r term. There are also variables emw.Efarphi and emw.Efartheta, which are for the scattering amplitude in spherical coordinates.
To verify this result, we simulate a perfect electric dipole and compare the simulation results with the analytical solution, which we covered in the previous blog post. As we stated in that post, we split the full results into two terms, which we call the near- and far-field terms. We briefly restate those results here.
where is the dipole moment of the radiation source and is the unit vector in spherical coordinates.
Below, we can see the electric fields vs. distance calculated using the Far-Field Domain node for a dipole at the origin with . For comparison, we have included the Far-Field Domain node, the full theory, as well as the near- and far-field terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the Far-Field Domain node and the far-field theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the Far-Field Domain node will only account for radiation that goes like 1/r, and so the agreement improves with increasing distance as the contribution of the 1/r^{2} and 1/r^{3} terms go to zero. In other words, the Far-Field Domain node is correct in the far field, which you probably would have guessed from the name.
A comparison of the Far-Field Domain node vs. theory for a point dipole source.
For most simulations, the near-field and far-field information is sufficient and no further work is necessary. In some cases, however, we also want to know the fields in the intermediate region, also known as the induction or transition zone. One option is to simply increase the simulation size until you explicitly calculate this information as part of the simulation. The drawback of this technique is that the increased simulation size requires more computational resources. We recommend a maximum mesh element size of for 3D electromagnetic simulations. As the simulation size increases, the number of mesh elements increases, and so do the computational requirements.
Another option is to use the Electromagnetic Waves, Beam Envelopes interface, which here we will simply refer to as Beam-Envelopes. As discussed in a previous blog post, Beam-Envelopes is an excellent choice when the simulation solution will have either one or two directions of propagation, and will allow us to use a much coarser mesh. Since the phase of the emission from an antenna will look like an outgoing spherical wave, this is a perfect solution for determining these fields. We perform a Full-Wave simulation of the fields near the source, as before, and then use Beam-Envelopes to simulate the fields out to an arbitrary distance, as required.
The simulation domain assignments. If the outer region is assigned to PML, then a Full-Wave simulation is performed everywhere. It is also possible to solve the inner region using a Full-Wave simulation and the outer region using Beam-Envelopes, as we will discuss below. Note that this image is not to scale, and we have only modeled 1/8 of the spherical domain due to symmetry.
How do we couple the Beam-Envelopes simulation to our Full-Wave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the Full-Wave and Beam-Envelopes domains. First, we set the exterior boundary of the Full-Wave simulation to PMC, which is the natural boundary condition for that simulation. The second step is to set that same boundary to an Electric Field boundary condition for Beam-Envelopes. We then specify the field values in the Beam-Envelopes Electric Field boundary condition according to the fields computed from the Full-Wave simulation, as shown here.
The Electric Field boundary condition in Beam-Envelopes. Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the Beam-Envelopes domain to absorb the outgoing spherical wave. The remaining boundaries are set to PEC and PMC according to symmetry. We must also set the solver to Fully Coupled, which is described in more detail in two blog posts on solving multiphysics models and improving convergence from a previous blog series on solvers.
If we again examine the comparison between simulation and theory, we see excellent agreement over the entire simulation range. This shows that the PMC and Electric Field boundary conditions have enforced continuity between the two interfaces and they have fully reproduced the analytical solution. You can download the model file in the Application Gallery.
A comparison of the electric field of the Full-Wave and Beam-Envelopes simulations vs. the full theory.
In today’s blog post, we examined two ways of computing the electric field at points far away from the source antenna and verified the results using the analytical solution for an electric point dipole. These two techniques are using the Far-Field Domain node from a Full-Wave simulation and linking a Full-Wave simulation to a Beam-Envelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using Beam-Envelopes has the additional advantage in that it also computes fields in the intermediate region. In the next post in the series, we will combine the calculated far-field radiation with a simulation of a receiving antenna and determine the received power. Stay tuned!