# How to Calculate the Impulse Response in Electromagnetic Waves

### Key Takeaways

• Propagating electromagnetic waves experience reflection and absorption when encountering wave impedance mismatches.

• Electromagnetic impulses can excite a transient response when they encounter an interface and undergo reflection and scattering.

• Network parameters are one way to formulate scattering responses for propagating electromagnetic waves and determine impulse responses.

Tropospheric scatter antennas take advantage of random scattering in the atmosphere for long-range communication

Impulse responses are fundamental measurements in high-speed transmission lines, but impulse responses are not limited to electronic circuits. Propagating waves can exhibit an impulse response that mirrors the source that excited the impulse response. One area where this has been extensively studied over the past ~60 years is in scattering problems, typically in the optical regime. Today, new techniques are being developed to describe the impulse response in electromagnetic waves for solving scattering problems with broadband signals and electromagnetic pulses.

## Scattering and Impulse Responses in Electromagnetic Waves

To calculate the scattering behavior of a propagating wave, you need to solve Maxwell’s equations for a given wave trajectory and scatterer geometry. These problems are well documented for simple geometries—such as the scattering of plane waves by spherical scatterers—and many textbooks have been written on these topics. The general form of the problem in scattering is to determine the scattering direction with respect to the incoming plane wave direction for a given scatterer size.

Scattering from more complex geometries can also be treated by solving Maxwell’s equations, but these problems typically require a numerical scheme to be solved, possibly with a 3D field solver. They also tend to only consider a single frequency, so they essentially operate only in the frequency domain. Being a frequency domain problem, scattering problems cannot be solved for fields with time-varying amplitudes, such as an electromagnetic pulse or some modulated signals. In this case, Maxwell’s equations are solved using FDTD, and the outgoing wave direction is examined over different frequencies and input angles.

The image below shows schematically what can happen when an input electromagnetic pulse is scattered from an arbitrary scatterer. The scattered wave is distributed over some solid angle, and the output pulse envelope (blue curve) can be modified compared to the input pulse envelope (green curve). This modification can occur due to interference within the scattering solid angle.

Scattering summary with input and output pulse comparisons in linear media

These problems are difficult to solve by hand and cannot be solved by simply assigning a time-varying amplitude to an electromagnetic wave in textbook scattering problems. You will only reproduce the results from textbooks for this type of input signal when the input pulse has a very slowly varying envelope, which does not encompass practical cases where impulse responses are studied. This has motivated an alternative approach where network parameters are determined for the structure being studied and used to calculate an impulse response in the system.

## Impulse Responses From Network Parameters

SI/PI engineers should already be familiar with network parameters, such as S-parameters, ABCD parameters, or other parameter sets like Z (impedance) parameters. These various parameter sets are a useful metric for treating wave propagation and transmission problems in circuits and transmission lines, but they can also be used for scattering problems. You can even determine an impulse response for an electromagnetic wave using a convolution integral because all network parameters are a certain type of transfer function.

Network parameters are normally used to determine electromagnetic responses in 0-dimensional problems (for simple linear circuits) or 1-dimensional problems (for transmission lines). The spatial distribution of the electromagnetic field is not normally considered in these problems, except for along the direction of propagation. However, by adding an additional spatial variable (scattering angle), the impulse response can be determined as a function of scattering trajectory. This allows responses to scattered electromagnetic impulse waves to be determined for a general arrangement of scatterers.

Taking a network parameter approach allows an output scattered wave in the system J(f) to be defined in terms of a network parameter matrix S(f) and an input wave g(t). In general, these quantities are matrices:

Network parameter definition relating input and output waves in a scattering problem in the frequency domain

This is basically the approach taken in circuits, but extends it to 3D. The approach is one way to solve scattering problems in photonic integrated circuits (PIC), where electromagnetic scattering is extremely important for ensuring photonic circuits will operate as designed and maintain coherence. Network parameters for certain elements and structures in PICs is a very new research topic; see Constructing the scattering matrix for optical microcavities as a nonlocal boundary value problem for an example applied to small cavities.

Just like S-parameters or ABCD parameters, a network parameter will describe a transformation between an input variable (voltage, electric field, etc.) and some other measurable output (again, voltage, field. etc.). Once a network parameter for a given scattering system is known, it can be used to determine the impulse response in the scattered wave J(t) for an arbitrary (time-varying) input function g(t):

1. Determine the network parameters for the system as a function of frequency and define these parameters as frequency domain functions S(f).
2. Calculate the impulse response function for each network parameter in #1, defined as S(t), using an inverse Fourier transform.
3. Calculate the time-domain convolution to get the output J(t) = S(t)✱g(t).

The challenge is determining the scattering parameters in the frequency domain S(f). This typically requires a field solver because, in general, structures that cause scattering can have any shape in relation to the wavelength of the incoming and scattered waves. Field solvers and some design exploration techniques can be used to determine the network parameters in the frequency domain and determine the impulse response.

Cadence’s PCB design and analysis software can be used to simulate the impulse response in electromagnetic waves while solving scattering problems in complex geometries. Designers can use a powerful field solver and circuit modeling tools to extract network parameter models from electromagnetic simulations and determine impulse responses for input pulses to simulate electromagnetic transient responses in scattering problems. When you use Cadence’s software suite, you’ll also have access to a range of simulation features you can use in signal integrity analysis, giving you everything you need to evaluate your system’s functionality.