Maxwell’s equations can be used in the time domain or the frequency domain.
Describing electromagnetism in the frequency domain requires using a Fourier transform with Maxwell’s equations.
Some problems are easier to solve in the frequency domain, such as when we have sources that are superpositions of harmonic waves.
James Clerk Maxwell on a San Marino stamp.
Maxwell’s equations are fundamental for describing electromagnetism, and they are normally presented in the time domain with sources. However, many problems are often easier to solve in the frequency domain, and you can use Maxwell’s equations with a Fourier transform to move to the frequency domain. The interpretation of Maxwell’s equations in the frequency domain may be unintuitive, but the analysis benefits are clear when working with linear time-invariant (LTI) systems.
Below you’ll find the definition of Maxwell’s equations in the frequency domain and their interpretation in terms of signal behavior.
Derivation: Maxwell’s Equations Fourier Transform
Understanding how to apply a Fourier transform to Maxwell’s equations is easiest when we consider an LTI system driven with arbitrary sources (such as an external antenna or current source). In this case, the time-dependent Maxwell’s equations are written in such a way that the Fourier transform operator F can be applied to them directly. This is easiest with the differential form of Maxwell’s equations, although the Fourier transform operator can also be applied to the integral form by switching the order of integration once the F operator is applied.
Maxwell’s equations in the time domain are:
Maxwell’s equations in the time domain for macroscopic media.
In the above equations, we have time-dependent sources (J and ⍴) that may also vary in space. However, because we are in an LTI system, the dielectric properties do not vary in time, but they might vary in space. This is the most general description of the electromagnetic field in an LTI system. This is the form of Maxwell’s equations normally used to simulate the electromagnetic field in PCBs or ICs with an FDTD field solver.
By applying the Fourier transform operator to the above equations, the time-dependent terms are immediately converted into the frequency-domain. Using the derivative identity, we have Maxwell’s equations in the frequency domain:
Maxwell’s equations in the frequency domain for macroscopic media.
When to Use Maxwell’s Equations in the Frequency Domain
In the above equation, the sources (J and ⍴), electric field E, and magnetic flux density B are functions of frequency rather than time. The above equations are much easier to work with in several cases:
When the sources are harmonic functions. In this case, the frequency-domain representation of J and ⍴ will be a delta function.
When the sources are sums of harmonic functions. This is related to the previous point, but the sources will be sums of delta functions.
When the sources are continuous and broadband. Continuous signals that may have broad bandwidth are easy to work with as they will be represented by continuous functions in the frequency domain.
Modulated sources. Systems driven with modulated sources can be more easily solved in the frequency domain as their Fourier spectra are often sums of delta functions.
When broadband noise is present in the system. Random noise is very difficult to simulate in the time domain without a specialized solver, but it is quite easy to simulate in the frequency domain as it will have a continuous power spectrum.
By working in the frequency domain, we lose any nonlinear temporal information such as we might have with a piecewise source. We also lose any information on the transient response; this is one of the reasons FDTD might be used with a digital signal as the transient behavior can be calculated.
The benefit of applying a Fourier transform to Maxwell’s equations is that the resulting differential equations could be easier to solve in the frequency domain. For homogeneous media, these problems can often be solved by hand. Only spatial derivatives need to be considered when determining the eigenfunctions of the system, so there is only one step’s worth of separation of variables in the above equations. Finally, the time dependence is recovered by taking an inverse Fourier transform of the solution.
What Happens in Nonlinear Media?
In nonlinear media, the dielectric function and the magnetic permeability will be functions of E and B. In this case, the Fourier transform will be applied to products of the fields. As a result of applying the convolution theorem, we now have the Fourier transform of Maxwell’s equations being written in terms of autocorrelations. This nicely explains some phenomena like delayed spatiotemporal correlation, gain spectra, nonlinear scattering, and pulse distortion in nonlinear systems.
Spatial Fourier Transform of Maxwell’s Equations
We often talk about Fourier transforms purely in terms of the temporal Fourier transform, but what about the spatial Fourier transform? In systems with a closed boundary (such as a resonator cavity), a spatial Fourier transform can also greatly simplify a problem. However, it is often easier to work with the wave equation, rather than use Maxwell’s equations directly. If applying a spatial Fourier transform operation F to the wave equation for the electric field, we would have:
Nonlinear current vs. voltage relationship expanded as a Taylor series.
There would be a similar equation for the magnetic field. Here, we may have a similar problem as was found for the temporal Fourier transform in that we are taking a transform of a product of functions, requiring a convolution operation in the spatial coordinates.
Once the spatial Fourier transform to the above equation is found, we can formulate the solution as an initial value problem and use the electromagnetic boundary conditions to determine the allowed spatial frequencies. The solution to the homogeneous equation is then used as an orthogonal basis to expand the source term G in the inhomogeneous wave equation and determine the inhomogeneous solution.
Use Field Solvers for Complex Systems
This article provides a summary of the techniques you would find in a differential equations textbook, but they are fundamental for working with Maxwell’s equations in the spatial or temporal frequency domains. These equations can generally be solved for a range of sources as long as the system geometry is not too complex. Specialized field solvers can be used with more complex electronic systems that may not have closed-form analytical solutions. These solvers can apply finite-difference frequency-domain (FDFD) calculations to solve Maxwell’s equations in the frequency domain using the transformed equations shown above.
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