# Electromagnetic Nonreciprocity: It's a One-Way Street

Key Takeaways

• Linear time-invariant (LTI) systems with symmetric topology are reciprocal.

• Electromagnetic nonreciprocity arises when a system’s response depends on the direction in which electromagnetic waves flow through the system.

• Some instances nonreciprocity arises in are hysteretic circuits, nonlinear coupled circuits with feedback, parametric amplification, and wave propagation through nonlinear media.

Electromagnetic nonreciprocity is like a one-way street.

When most designers plan out signal paths in a new PCB or IC, they probably don’t worry about the reciprocity of the system. In other words, you plan to have the signal move through a chain of components in one direction only, so why worry about what happens in the opposite direction? For typical components like passives, amplifiers, and other integrated circuits, it’s perfectly fine to ignore reciprocity.

For advanced components, upcoming photonic technologies, reciprocity becomes critical during the design phase. Electromagnetic nonreciprocity defines a sort of symmetry that occurs in electromagnetic wave propagation, both in photonic and electronic systems. This is sometimes confused with time-reversal symmetry (invariance), but these are not the same concepts. If you want to design in these new areas, then electromagnetic nonreciprocity is a fundamental phenomenon that you can use to your advantage.

## Electromagnetic Nonreciprocity Explained

When we say that a propagating wave is traveling through a reciprocal medium, it means that the interaction between the field and the system is independent of the direction traveled through the system. All lossless LTI circuits with symmetric topology between the input and output are reciprocal. If you send a signal into the circuit, you will get a certain signal out; you would gather the same measurements if the locations of the source and measurement point were reversed. This conceptual example should illustrate what might cause nonreciprocity, and it was even used in some of the first descriptions of optical reciprocity in the 19th century.

Think about the way light moves from a source to an observer in a typical room where the environment doesn’t change. Suppose the observer has an intensity meter and they measure the intensity of light being emitted from the source. If you switch the locations of the source and observer, you’d naturally expect the observer to measure the same intensity. Even though the propagation direction was reversed, you still measure the same light intensity.

This does not happen in a nonreciprocal system. When a system is nonreciprocal, the signal measured at the output of a circuit or network will be different depending on the direction the signal travels through the network. This is often equated with time-reversal symmetry (reversing the direction of time in the forward direction is equivalent to reversing direction with time moving forward only in time-invariant LIH media), but the two phenomena are not equivalent mathematically or physically. This ability to control how a system responds to light based on the direction it travels through a system is exploited in advanced photonics applications, such as:

• Components for photonic integrated circuits

• Microwave circuits used in microwave photonics or quantum computing

• Dynamically modulated optical or electro-optical devices (e.g., parametric amplifiers)

• Magneto-optical devices

Faraday, who discovered that the polarization of a light beam can be rotated when the light travels through glass in the presence of a magnetic field (parallel to the light propagation direction), provides the first report of electromagnetic nonreciprocity (this term was not used by Faraday). When light is sent in the opposite direction, we do not see a reciprocal effect; the polarization direction is reversed.

Electromagnetic nonreciprocity and the Faraday effect.

This behavior is quantified mathematically in terms of the Poynting vector for electromagnetic waves traveling through the system. This gives the Lorentz reciprocity theorem, which can be extended to the case of electromagnetic fields in real media with some current density (given by the J1 and J2 terms). If the following integral equation is satisfied when states 1 and 2 are swapped, then the system is reciprocal:

Electromagnetic nonreciprocity is defined with the Lorentz reciprocity theorem.

Note that there are many reciprocity theorems, but Lorentz’s is widely viewed as the most general. This equation can be easily reformulated in terms of electrical networks by breaking the LHS volume integral into terms parallel to the field components (giving the voltage) and a surface integral (giving a total current), and the RHS becomes the power transmitted through the network. A reciprocity theorem can also be derived from Poisson’s equation (in terms of arbitrary charge density) using Green’s theorem.

## What Makes a System Nonreciprocal?

To exploit nonreciprocity in these applications, it helps to know what causes it and how to use nonreciprocity for practical applications. There are a number of effects that can cause a system to exhibit nonreciprocal behavior:

• Bi-isotropic or fully anisotropic material properties (with either linear or nonlinear response)

• Rectification

• Ferrimagnetism/ferromagnetism and magneto-optic effects

• Static or dynamic electro-optical effects, such as:

• Pockels effect (change in birefringence with applied field)

• Kerr effect (change in dielectric constant with applied field)

• Franz–Keldysh effect (change in a semiconductor’s optical absorption with an applied electric field)

• Hysteresis

• Biased nonlinear dielectric/magnetic media

• Time-varying material dielectric properties induced by dynamic versions of any of the above effects

• Successive frequency conversion with delay during signal propagation

As we can see from this list, there is a rich set of effects that can be exploited to produce electromagnetic nonreciprocity. Note that nonlinearity is not a prerequisite in any of the above conditions for nonreciprocity; the above effects can be linear or nonlinear yet still produce electromagnetic nonreciprocity. Electrical and photonic systems can exploit these effects to produce nonreciprocity in practical devices and networks of circuit elements.

## Nonreciprocal Electrical and Photonic Networks

Once you’ve identified a material and/or effect that produces electromagnetic nonreciprocity, you can start to design circuit elements for a practical application. The upcoming use of electromagnetic nonreciprocity is in optical and microwave photonic systems for EPICs/PICs and quantum computing. Some structures that are being investigated for use in nonreciprocal electrical and photonic networks are:

• Nonlinear versions of isolators, circulators, and other conventional microwave circuits

• Topological arrays as circuit elements

• Linear resonators as ion traps for qubits

• Spin wave resonators

As nonreciprocity creates the optical analog of rectification, these unique structures can be used as optical analogs of transistors when tuned with an external signal. Think of electro-optic or magneto-optic effects with an applied field which changes the state of an electromagnetic wave; this is effectively the same type of switching seen in a transistor. Engineers who want to dive into optical computing should take time to understand these electro-optical and magneto-optical effects to generate nonreciprocity.

Exploiting nonreciprocity, combined with current semiconductor manufacturing processes, allows this type of optical system to be scaled down to the wafer level.

When you need to design electrical or photonic systems that exploit electromagnetic nonreciprocity, you need the best PCB design and analysis software to create your layouts and integrated circuit network modeling utilities. The PSpice Simulator from Cadence lets you build circuit networks from COTS and custom components, and you can use a modeling application to create simulation models for use in your circuits.

If you’d like to keep up-to-date with our System Analysis content, sign-up for our newsletter curating resources on current trends and innovations. If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.