[Paper Review] What is Nonreciprocity? Part II

Christophe Caloz, Andrea Alù+4 more|arXiv (Cornell University)|Apr 1, 2018

Photonic and Optical Devices3 citations

TL;DR

This paper extends the theoretical framework of electromagnetic nonreciprocity by classifying systems into linear time-invariant (LTI), linear time-variant space-time (LTV-ST), and nonlinear (NL) categories, introducing generalized S-parameters for all types, and clarifying that lossy systems can be reciprocal despite breaking time-reversal symmetry. It establishes that true nonreciprocity requires field ratio asymmetry under time reversal, not just asymmetry in field levels or intensity, and warns against conflating asymmetric transmission with nonreciprocity in systems like mode converters or lenses with mirrors.

ABSTRACT

This paper is the second part of a two-part paper on \emph{Electromagnetic (EM) Nonreciprocity (NR)}. Part~I has defined NR, pointed out that linear NR is a stronger form of NR than nonlinear (NL) NR, explained EM Time-Reversal (TR) Symmetry (TRS) Breaking (TRS-B), described linear Time-Invariant (TI) NR media, generalized the Lorentz reciprocity theorem for NR, and provided a physical interpretation of the resulting Onsager-Casimir relations~\cite{Caloz_AWPL_NR_I_2018}. This part first explains the TR specificity of lossy and open systems. Next, it proposes an extended version of the S-parameters for \emph{all NR} systems. Then, it presents the fundamentals of linear-TI (LTI) NR, linear Time-Variant (LTV) Space-Time (ST) modulated NR and NL NR systems. Finally, it addresses confusions between with systems.

Motivation & Objective

• To establish a unified theoretical framework for electromagnetic nonreciprocity across linear and nonlinear systems.
• To clarify that lossy systems can break time-reversal symmetry (TRS-B) yet remain reciprocal due to preserved field ratios.
• To generalize S-parameters to LTV-ST and nonlinear systems, enabling systematic analysis of nonreciprocal behavior.
• To distinguish true nonreciprocity from deceptive asymmetries in transmission, such as those in mode converters or lens-mirror systems.
• To highlight fundamental limitations of nonlinear nonreciprocal devices, including one-way excitation and poor isolation-to-loss ratios.

Proposed method

• Extends the Onsager-Casimir relations and generalized Lorentz reciprocity theorem to include lossy and open systems.
• Introduces an extended S-parameter formalism applicable to LTI, LTV-ST, and nonlinear systems, with increasing constraints.
• Analyzes TRS-B in lossy waveguides by comparing field levels and ratios under time reversal, revealing that field ratios remain equal despite level asymmetry.
• Uses space-time modulation as a mechanism to break reciprocity in linear systems, distinct from static biasing.
• Applies the time-reversal symmetry test to various systems, including nonlinear media and asymmetric waveguides, to verify reciprocity or nonreciprocity.
• Demonstrates that systems with symmetric phase gradients or mode cancellation can exhibit apparent asymmetry while remaining reciprocal.

Experimental results

Research questions

• RQ1 How can nonreciprocity be consistently defined and classified across linear and nonlinear electromagnetic systems?
• RQ2 Why do lossy systems break time-reversal symmetry yet remain reciprocal in terms of field ratios?
• RQ3 What is the correct generalization of S-parameters for nonreciprocal systems beyond LTI media?
• RQ4 In what ways can asymmetric transmission be mistaken for nonreciprocity, and how can such confusions be avoided?
• RQ5 What are the fundamental limitations of nonlinear nonreciprocal devices in terms of isolation, insertion loss, and dynamic range?

Key findings

• Lossy systems break time-reversal symmetry due to macroscopic irreversibility but remain reciprocal because field ratios are preserved under time reversal.
• Nonlinear nonreciprocal systems exhibit one-way transmission only within a narrow intensity range and suffer from poor isolation-to-insertion loss ratios, typically below 25 dB.
• The generalized S-parameter formalism extends to LTV-ST and nonlinear systems, though with increasing restrictions on applicability.
• True nonreciprocity is defined by asymmetry in field ratios under time reversal, not by asymmetry in field levels or intensity alone.
• Systems such as lens-mirror sandwiches or mode converters can appear asymmetric but are fully reciprocal if S-parameter reciprocity holds upon source-detector exchange.
• Nonlinear nonreciprocal devices cannot achieve the high isolation (>45 dB) and low insertion loss typical of linear nonreciprocal isolators, due to fundamental physical constraints.