Stability bounds on superluminal propagation in active structures

Active materials have been explored in recent years to demonstrate superluminal group velocities over relatively broad bandwidths, implying a potential path towards bold claims such as information transport beyond the speed of light, as well as antennas and metamaterial cloaks operating over very broad bandwidths. However, causality requires that no portion of an impinging pulse can pass its precursor, implying a fundamental trade-off between bandwidth, velocity and propagation distance. Here, we clarify the general nature of superluminal propagation in active structures and derive a bound on these quantities fundamentally rooted into stability considerations. By applying filter theory, we show that this bound is generally applicable to causal structures of arbitrary complexity, as it applies to each zero-pole pair describing their response. As the system complexity grows, we find that only minor improvements in superluminal bandwidth can be practically achieved. Our results provide physical insights into the limitations of superluminal structures based on active media, implying severe constraints in several recently proposed applications.


Bound derivation in shunt negative RLC element
. We assume an ideal NIC, but by including the inductor in the shunt element (which is sometimes ignored), the admittance stays causal (in time domain, no current flows before a voltage is present). We also assume the terminations to match the waveguide impedance 0 Z . Although different from the standard circuit convention, we will use the it e  − convention to be consistent with the main text. The transfer function through a length d is given by The poles of the system are given by and the zeros by where all parameters are positive. Stability is achieved when the imaginary part of all poles is 0  , requiring that 0 20 following analysis is unchanged. It is important to remember that any system must remain stable to actually operate as a waveguide, or else the behavior will more resemble an oscillator once nonlinearities and saturation comes into play.
We calculate the phase of T and derive an effective phase velocity through the accumulated phase shift as the signal is transmitted through the circuit. The group delay is given by the frequency derivative of the phase of (S.1) and the effective velocity is the distance divided by this quantity. The group velocity is explicitly ( ) The phase and group velocities are nearly flat at low frequencies, with low-frequency value In Supplementary Figure 2, we study the effect of the circuit parameters R , L and C on the response of the system. In (a), L is chosen to maximize the bandwidth for given C , and then R is brought closer to the critical stability value. It is clear that to bring the superluminal bandwidth near the limits stated by the bound, the system must be brought closer to instabilities.
These instabilities may be triggered if the actual system parameters deviate from the intended design, and there will also be enhanced noise from these elements that provide significant amplification to the system. Supplementary Figure 2 . The inductance is varied to maximize bandwidth for given C .

Limits on bandpass RLC setup
It is also possible to leverage two or more resonances to observe band-pass dispersionless superluminality, with superluminal behavior arising from more complex gain responses 3 .

Re
In the limit of Notice in this case that the poles end up being narrower than the zeros, and we end up with the same total frequency bandwidth for a given delay, even with a more complex circuit topology.
This appears to indicate that our derived bounds provide a good sense of the limits in practical situations, and more complex dispersion engineering may not be feasible in practical scenarios. In

Effects of limited gain and Q factor
So far, we have explored the bounds on group velocity defined as the velocity of a pulse peak under the assumption of narrowband excitation, corresponding to / k   . When the pulse is sufficiently broadband compared to the dispersion of the structure, the correspondence between these two quantities may break down, due to significant distortions as the pulse progresses in the medium. As discussed in 4,5 , pulse distortion adds practical limits that are in general stricter than the incurrence of instabilities on which we focus in the main paper. Kitano et. al 6 found an advancebandwidth product of roughly unity. To approach this bound and the similar bounds derived in our work, a pole must be brought very close to the real frequency axis, acting more like an oscillator rather than a transmission line. In addition, these conditions also require large gain, which is a practical challenge and may result in nonlinearities becoming important, such as gain saturation.
The obvious question is how any restrictions on the amplitude of T limit the superluminal behavior. We consider two approaches, with results provided in Supplementary Figure 4


, proportional to the system Q -factor and normalized ring-down time. Again, the bandwidth asymptotically approaches the limit as Q increases. The product reaches only half of its maximum value for 10 Q = , which is certainly expected to produce significant distortion in a signal with a spectral component at resonance.
Alternatively, a latency period of 10 oscillation cycles is needed to prevent leakage into the next pulse.
It should also be pointed out that there is no restriction stemming from stability or causality to having multiple zeros in the superluminal bandwidth, in which case , gH BW  − can be increased with each pole-zero pair. By placing n sets of identical poles and zeros, the product can grow as 1.5n . In theory, then, there is no absolute bound to the group delay-bandwidth product based on the combined contribution of a large number of pole-zero pairs. This will, however, require complex waveguide implementations and dispersion engineering. Any particular circuit topology may present additional practical restrictions not captured in our general filter theory, as evident in the case of the two RLC elements discussed previously. In other words, adding additional elements to the system is no guarantee of being able to actually increase the overall advance-bandwidth product, as there may not be enough independent degrees of freedom to arbitrarily tailor the positions of each pair. In addition, distortion effects will be enhanced with additional pole-zero pairs: in the limit of a pair made doubly-degenerate, the group advance would double, but the transmission maxima will be squared.
Supplementary Figure 4-Effect of finite Q or gain. Numerically calculated relationship between the advance-bandwidth product and the maximum allowed transmission gain, or the minimum Qfactor needed, for the low-pass circuit of Supplementary Figure 1.

 =
The instabilities that arise in active structures and fundamentally limit their response are due to feedback from mismatch at the interfaces of the active structures and the passive ports. Under certain restrictions, such as of a purely dielectric response, these mismatches are unavoidable. In theory, though, one can envision the ideal case in which the magnetic response follows precisely the same frequency response as the electric permittivity. In this case, the impedance of the slab is equal to the impedance of free space, there will be no reflections at the interfaces, and therefore no feedback to cause instabilities. Can this behavior fit within the filter theory approach outlined in the main text? The answer is yes: this ideal scenario includes a very peculiar singularity for which there are poles and zeros of infinite order that simultaneously lie at the same point in the complex  plane. In one example of such a matched system where we use the permittivity of the slab in the main text and the same relative permeability, the transmission coefficient may be written . Note that the poles are pinned to this position in the complex plane, and so this slab never becomes unstable as the length is increased (this is limited to normal incidence). The infinite order of these poles makes up for the fact that they are infinitesimally close together, so that the net effect is superluminal velocities for frequencies away from resonance when the poles are "too close" to the axis and have a smaller angular derivative than the zeros, but subluminal velocities when the frequencies are near the resonance. In essence, the zeros "feel" further from the axis and provide a broadband, superluminal response, whereas the poles are closer to the axis and provide stronger subluminal response near resonance. This is exactly the behavior we see when calculating the group velocity, shown in Supplementary Figure 5(d). Practically, achieving such a response with perfectly matched magnetic and electric properties at all frequencies is totally unrealistic, and any small deviation would imply the emergence of instabilities. The perfect matching also no longer holds when the incident angle is not at normal incidence, and so the feedback necessary for oscillations becomes present again. For example, with the same matched slab, at 5° incident angle, the transmission coefficient clearly shows poles in the upper half-plane Supplementary Figure 5(f), and so the structure may oscillate with emission at those angles. Still, this example highlights that it is not strictly forbidden with extreme dispersion engineering to achieve advance-bandwidth products significantly larger than unity.