Waveguides enable a range of unique RF designs that take advantage of the superposition of electromagnetic waves.
Possible parallel plate waveguide-based designs include novel emitters and couplers, both of which rely on determining or exciting a dominant mode in a parallel plate waveguide.
Unique emitters can be designed to excite a specific mode in a parallel plate waveguide or the waveguide can be operated at a single mode.
This RF amplifier board uses waveguide routing
If you think about it, every signal that exhibits wave propagation along a trace is traveling through a waveguide. Any trace that carries a signal is actually guiding an electromagnetic wave through the substrate between the driver and receiver. We sometimes visualize this as a nice, clean sine wave that travels through the board. However, when working with a broadband excitation source that excites a general waveguide structure, you won’t always know which mode is dominant.
To determine the dominant mode in a parallel plate waveguide, or in any other type of waveguide, you’ll need to do some calculations and simulations using the structure of the exciting emitter in your design. While the theory can be worked out by hand, it’s not so simple to do this in a practical example in a PCB. However, designers can use a 3D field solver to help them optimize their emitters to excite a dominant mode and engineer the mode structure of their waveguides.
The Modes in a Waveguide
In theory, the number of possible modes (eigenmodes) in any waveguide is infinite. How you excite specific eigenmodes, and which mode is dominant for a given excitation, is another matter altogether. The modes in a waveguide are determined by the structure of the waveguide, the material constants in the system, and the boundary conditions in the system. If you know the eigenmodes of a waveguide structure, you can engineer a system that excites a specific mode or determines the dominant mode for an arbitrary excitation.
To see how the mode structure in a waveguide can be used to engineer electromagnetic wave propagation in a PCB or other device, we can look at a parallel plate waveguide as a simple example. The image below shows a parallel plate waveguide structure with defined dielectric constant and magnetic permeability constant.
Parallel plate waveguide structure
In this structure, we have propagation along one defined direction between two parallel plates of infinite extent. In general, the dielectric constant and permeability can be complex, which will account for losses along the propagation direction in the waveguide.
Mathematically, electromagnetic wave propagation in a waveguide (assuming an LIH system) is a Sturm-Liouville problem. The general solution to the problem is a linear combination of the structure’s eigenmodes, where the eigenmodes form an orthogonal set. Each eigenmode has a corresponding eigenvalue:
Next, the eigenmodes have a particular solution involving sines and complex exponentials:
If you excite the structure with a broadband source or you excite with a single frequency above the n = 2 mode, there will be multiple modes propagating into the waveguide. Now we would want to know, what is the dominant mode in a parallel plate waveguide?
The Dominant Mode in a Parallel Plate Waveguide
The modes excited in a parallel plate waveguide depend on the spatial and temporal distribution of the excitation source in your system as well as the eigenmodes of the waveguide structure. Determining the dominant mode in a parallel plate waveguide requires determining the mode with the largest coefficient Cn. This involves the inverse Fourier transform of the orthogonality relation from Sturm-Liouville theory:
Determining the dominant mode in a parallel plate waveguide
In this equation, f(x, y, ω) is the excitation function in the frequency. In other words, this is the distribution of the electromagnetic field in the x-y plane and in the frequency domain, which might excite multiple modes. Using orthogonality, we can possibly engineer the electromagnetic field to excite a specific mode in the parallel plate waveguide structure.
Exciting a Specific Mode
The typical way to use a waveguide in a PCB is to operate in a single mode. In this case, we simply excite the n = 1 mode by sourcing a frequency between the n = 1 and n = 2 modes:
Single-mode excitation conditions
This is the easiest way to excite a single mode in a parallel plate waveguide. Exciting a higher order mode than n = 1 is more difficult and cannot be achieved by simply sourcing a harmonic sinusoid. For example, if you source in the frequency range for n = 2, you might excite both the n = 1 and n = 2 modes, unless you engineer the spatial part of the sourcing function f(x, y, ω).
To excite a specific mode with an arbitrary frequency, we need to take advantage of orthogonality (or orthonormality if you’ve normalized your eigenfunctions). This requires a unique emitter that can match the spatial distribution of the electromagnetic field in the x-y plane for the desired mode.
Why Worry About Dominant Modes?
The dominant mode in a waveguide is important for many RF applications, specifically for high frequency communication or sensing at wireless frequencies. Unique RF applications could benefit from selective mode excitation on a PCB, similar to what is currently done in a multimode optical fiber using a spatial light modulator. Other applications include radar sensing and imaging, wherein a waveguide could be a useful alternative to diverse antenna arrays in MIMO radar.
Because of the difficulty in designing emitters to excite a specific higher order mode or a mix of modes, designers will typically just source a harmonic wave into the waveguide at the lowest order frequency. In this case, the excitation spectrum is a delta function, and the inverse Fourier transform gives a propagating sinusoidal wave:
The value of the eigenmode coefficient Cn involves taking the inverse Fourier transform of a delta function in the frequency domain, giving a sinusoidal wave
In the above case, it may be that the spatial integral evaluates to zero because the forcing function is orthogonal to the eigenmode. This further underscores how the spatial distribution of the electric field in the forcing function can be engineered to excite a specific mode in the waveguide.
If you can engineer the current density in the emitter (in this case, the feedline into the waveguide), you can select the dominant modes you want to excite. This will determine the spatial distribution of the electromagnetic field you excite in the structure and how this couples into a receiver or aperture. Because of the difficulty in deriving source terms and the geometry for emitter structures in a PCB, it’s best to use a 3D field solver to determine the current distribution needed to excite your desired mode structure.