Stability bounds on superluminal propagation in active structures

The speed of light in a vacuum, *c*₀, is a fundamental constant. Causality dictates that the speed of light in any material cannot exceed *c*₀. Faster-than-*c*₀ and negative group velocities have been observed in materials, but these are typically narrowband phenomena associated with significant absorption and pulse distortion. Passive structures exhibiting these behaviors are inherently limited by Kramers-Kronig relations. Group velocity (*dω*/ *dk*), representing the speed of a narrowband pulse's peak, can be superluminal or negative, yet the pulse's forefront cannot exceed *c*₀. Recent interest in active materials, which introduce gain, has led to explorations of broadband superluminal propagation. Optical gain and non-Foster circuit elements have enabled inverted dispersion and broadband superluminal group velocities, with potential applications in antennas, leaky-wave antennas, and electromagnetic cloaks. Microwave experiments have demonstrated superluminal velocities with minimal dispersion over broad bandwidths using active elements. However, active systems are susceptible to instabilities. Previous work often overlooked the stability of the overall structure, mistakenly assuming that causality of individual components guarantees overall causality and stability. This study addresses this gap by deriving stability-based bounds on superluminal propagation, clarifying limitations, and addressing unwarranted claims regarding broadband superluminal propagation in active systems.

Previous studies have explored superluminal wave propagation in various contexts. Withayachumnankul et al. (2010) provided a systematic overview of superluminal wave propagation, highlighting different approaches and their limitations. Early work by Chiao (1993) demonstrated gain-assisted superluminal propagation. Wang et al. (2000) and Stenner et al. (2003) further investigated gain-assisted superluminal light propagation and the speed of information in fast-light media. Hrabar et al. (2013) explored superluminal phase and group velocities in epsilon-near-zero metamaterials. Sievenpiper (2011) proposed non-Foster circuit-based superluminal waveguides for antennas. Sussman-Fort and Rudish (2009) focused on non-Foster impedance matching for electrically small antennas. Chen et al. (2013) studied broadband cloaking bandwidth enhancement using non-Foster metasurfaces. Monticone and Alù (2016) derived physical bounds on passive cloaking. Long and Sievenpiper (2017) investigated the limitations of dispersionless superluminal propagation in non-Foster loaded waveguides. Mitchell and Chiao (1997) examined negative group delay and fronts in a causal system. Tsakmakidis et al. (2019) proposed ultra-broadband 3D invisibility using fast-light cloaks. Rengarajan and White (2013) analyzed the stability of superluminal waveguides. Nistad and Skaar (2008) studied causality and electromagnetic properties of active media. Skaar (2006) investigated Fresnel equations and the refractive index of active media. Hickmann et al. (2012) explored causality-induced pulse steepening. The work builds upon Bode (1945) and Fano (1950) on network analysis and impedance matching, Rozanov (2000) on radar absorber bounds and Cameron et al. (2018) on microwave filters. Relevant experimental demonstrations and theoretical explorations of active superluminal structures are discussed, highlighting the range of settings where the reported behavior is consistent with the developed bounds. These include studies on circuits, waveguides and optical systems (Kitano et al., 2003; Song et al., 2008; Jiang et al., 2007; Stenner & Gauthier, 2003).

The study employed both theoretical analysis and experimental verification. The theoretical approach began with a simple thought experiment involving a pulse propagating through a superluminal medium of finite length, establishing a causality-based bound relating propagation velocity, length, and bandwidth. This was followed by a more rigorous analysis using filter theory. This methodology examined the complex frequency response of active structures, focusing on the stability constraints imposed by the system's poles and zeros in the complex frequency plane. A specific model of an active dielectric slab was explored numerically. This model showcased the impact of finite slab thickness and reflections at the interfaces, leading to instabilities as the slab length increased. The analysis extended to derive a general bound stemming from stability considerations, quantifying the trade-off between velocity, bandwidth, and propagation length. This involved representing the transmission response of a two-port network using filter theory, splitting the phase into free-space and structure-dependent components, and exploring the interplay between poles and zeros in determining the group delay. A figure of merit, the group advance-bandwidth product, was introduced and analyzed. A key contribution was deriving a bound on this product for a single pole-zero pair, showing that the product is inherently limited by stability. The study further investigated the applicability of this bound to more complex systems by analyzing an active dielectric slab with multiple resonances. A shunt impedance-inverted RLC circuit was also considered to demonstrate the applicability of the bound in a different circuit configuration. The experimental component involved the design and fabrication of a tunable circuit based on a transmission line loaded with an operational amplifier, effectively creating a negative capacitance. This circuit allowed independent tuning of capacitance and resistance to probe different stability regimes. The measured group delay spectra and time-domain responses were compared against theoretical predictions. Finally, the study examined the implications of the derived bounds for broadband cloaking, specifically addressing a previous proposal for active cloaking based on superluminal propagation. A theoretical model of a cloaked perfectly conducting sphere was analyzed, demonstrating that the trade-off between size and bandwidth is limited by stability, even in three-dimensional cloaking scenarios.

The study's key findings revolve around establishing fundamental limitations on superluminal propagation in active systems. A crucial finding is a causality-based bound on superluminal propagation, indicating that no portion of a pulse, particularly its peak, can surpass the forefront. This bound highlights a fundamental trade-off between velocity, bandwidth, and propagation length. The research also demonstrates the critical role of stability in active structures. It establishes that while individual components may be causal, the overall system can become unstable when aiming for extreme superluminal responses. The instability arises due to positive feedback mechanisms, such as reflections at interfaces. This instability leads to self-oscillations and an unbounded response. Employing filter theory, the paper derives a general bound on the advance-bandwidth product for any causal structure. The bound shows that this product is limited, typically to the order of unity per pole-zero pair, and improving this bound requires significant complexity. Numerical simulations and experimental results validate this bound. The numerical simulations explore an active dielectric slab and demonstrate that despite increasing complexity, only minor improvements in the advance-bandwidth product can be practically achieved. The experimental results from a designed tunable circuit verify the theoretical bound, showcasing the limitations in achieving broad bandwidth and large group advance while maintaining stability. The analysis extends to the application of cloaking devices, specifically addressing a previous proposal for broadband active cloaking. The study reveals that even in this scenario, stability considerations limit the achievable bandwidth and object size, reinforcing the general applicability of the derived bound.

The findings of this research significantly advance our understanding of superluminal propagation in active media. The derived stability-based bound, supported by numerical simulations and experimental validation, clarifies the inherent limitations in achieving broadband superluminal propagation across various device platforms. The study addresses the common misconception that causality of individual components guarantees overall system stability. It demonstrates that even with causal materials, instabilities can arise due to the complex interplay of reflections, feedback, and gain in finite structures. This work challenges previous claims of arbitrary broadband superluminal propagation in active systems. It provides a quantitative framework for assessing the trade-offs between velocity, bandwidth, and propagation distance, setting realistic expectations for the design of superluminal devices. The limitations imposed by stability suggest that extreme applications like broadband cloaking of large objects might be practically infeasible due to the onset of instabilities.

This study establishes a quantitative relationship between superluminal velocity, bandwidth, and propagation length in active materials, constrained by stability. The derived bound applies broadly, limiting the advance-bandwidth product to the order of unity. This implies significant constraints on applications like broadband cloaking of large objects. Future research could explore more complex dispersion engineering strategies, though practical limitations suggest modest improvements are likely. Investigating alternative mechanisms for achieving broadband superluminal propagation without relying on high gain is another direction for future exploration. The study's findings are critical for guiding the development of future superluminal devices and realistic assessments of their capabilities.

While the study presents a comprehensive analysis of stability bounds on superluminal propagation, some limitations exist. The theoretical analysis mainly focuses on linear systems; non-linear effects, which become important near the instability threshold, are not extensively modeled. The experimental validation uses a specific circuit design, and the generalizability to other active systems requires further investigation. The study primarily considers one-dimensional propagation, and the extension to more complex geometries warrants further exploration. Additionally, the analysis of cloaking focuses on a specific cloaking geometry and material model; the applicability of the bounds to more general cloaking schemes requires further study.