## Abstract

In their paper in Optica **6**, 104
(2019) [CrossRef] , Mann *et al*. claim that linear, time-invariant
nonreciprocal structures cannot overcome the time-bandwidth limit and
do not exhibit an advantage over their reciprocal counterparts,
specifically with regard to their time-bandwidth performance. In this
Comment, we argue that these conclusions are unfounded. On the basis
of both rigorous full-wave simulations and insightful physical
justifications, we explain that the temporal coupled-mode theory, on
which Mann *et al*. base their main
conclusions, is not suited for the study of nonreciprocal trapped
states, and instead direct numerical solutions of Maxwell’s equations
are required. Based on such an analysis, we show that a nonreciprocal
terminated waveguide, resulting in a trapped state, clearly
outperforms its reciprocal counterpart; i.e., both the extraordinary
time-bandwidth performance and the large field enhancements observed
in such modes are a direct consequence of nonreciprocity.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The paper by Mann *et al*. [1] investigates a time-invariant, unidirectional waveguide
interacting with a cavity [Fig. 1(a)],
concluding that the behavior of the cavity remains unchanged by the presence
of the waveguide. This conclusion is then generalized to stating that
time-invariant nonreciprocal systems cannot overcome the time-bandwidth (T-B)
limit. These assertions appear to conflict with our previous work on a
terminated unidirectional waveguide [Fig. 1(b)], which we deployed to report that large (by a factor of ${\sim}{1000}$) T-B violations in linear, time-invariant
nonreciprocal systems could be achieved [2].

However, in this Comment we will show that the discrepancy in the conclusions
of these two works stems entirely from the nature of the selected tool of
analysis used in Ref. [1]: the main
conclusions reached by Mann *et al*. concerning
the T-B performance of linear time-invariant systems were on the basis of a
temporal coupled-mode theory (TCMT), thereby relying on an analytic *TCMT approximation* ansatz [1,3], whereas all the
numerical results of Ref. [2] reporting
large T-B violations in the same systems were based on full-wave
finite-difference time-domain (FDTD) simulations [2]. We will outline that the TCMT used in Ref. [1] is not suited for the study of the
structure reported in Ref. [2], which
includes a trapped state [blue shading in Fig. 1(b)], and that the extraordinary T-B performance observed in [2] is a direct consequence of the
nonreciprocal nature of the device.

To begin, we recall that TCMT [3] describes the evolution of a field inside a cavity according to the following equation:

Equation (1) is satisfied when the field inside the cavity takes the form

where,However, intrinsic to this TCMT description are several key assumptions and
approximations, making this approach inapplicable to nonresonant trapped
states (e.g., the blue shaded regions in Fig. 1). As shown below, the conclusions drawn by Mann *et al*. based on such a TCMT analysis cannot, therefore, be extended
to the trapped state. We note that these states are refered to as “wedge
mode,” “open cavities,” or “trapped states” in varying works, and for the
remainder of this Comment we shall use the latter term.

The standard form of TCMT, analyzed in detail in [1] both for reciprocal and nonreciprocal feeding, assumes that
a cavity mode must be a confined, oscillatory mode with a well-defined, single
resonance frequency ${{\omega}_0}$; that is, it describes resonances peaked at a
single frequency ${\omega _0}$ [owing to the ‘${\rm i}{\omega _0}a$’ term in Eq. (1)]. Such an assumption for a resonance, i.e., that it should
have a well-defined single peak (at an ‘${\omega _0}$’), though reasonable in ordinary cavities,
does not describe key features of the trapped state of Fig. 1(b)—a point that is now clarified and proved
below, with the aid of Fig. 2. Here (in
Fig. 2), we apply TCMT to the system
[1,3,4], with $ \omega_0 $ being the central frequency of the complete
unidirectional propagation (CUP) region [2], and compare these calculations with direct numerical solutions of
Maxwell’s equations, obtained through FDTD simulations of exactly the same
structure and conditions. Specifically, in both cases, we use the same lossy
structure (with $v = {5} \times {{10}^{-
4}}{\omega _p}$ [2],
characterizing the losses of InSb) and eliminate cavity back-reflection(s)
[3,4] in both calculation approaches. We see from Fig. 2 that the FDTD calculation predicts a broad
and *flat-top* (no single-peak/plateaued)
response, while TCMT predicts a narrowband response peaked, as always in that
theory, at a single frequency ${\omega _0}$.

Which one is the physically correct result? Clearly, the physically correct
result is that of the FDTD method because within the CUP region the trapped
state cannot radiatively escape its localization region, and therefore it can
only ‘escape’ the system nonradiatively, by eventually being 100% absorbed
*within the entire CUP region*—as shown by the red
curve in Fig. 2. As such, TCMT is,
evidently, not applicable to the trapped state, whose bandwidth is here, as
calculated from Fig. 1, in fact $ \sim 1000 $ broader than that predicted from TCMT. This
factor (${\sim}{1000}$) is in fact the degree to which the T-B limit
is overcome in the structure of Fig. 1(b), as was reported in Fig. 4 of Ref. [2], precisely because, as was also outlined above, the TCMT
always gives rise to T-B-limited resonances [2]. In other words, one cannot deploy (at least, the standard form of)
a TCMT approximation, which inherently gives rise to T-B-limited resonaces, to
investigate whether a structure might violate (or not) the T-B limit—the
result will always be negative, owing to the inherent ‘structure’ (ansatz) of
that theory. For such an analysis, *ab initio*
full Maxwell solvers are required, as Fig. 2 above shows, and as Ref. [2]
has reported. Note that, interestingly, here, TCMT fails even in the low-loss
regime where it is usually successfully applied (e.g., in silicon photonics or
in dielectric photonic crystals; cf. lossless structure studied in Ref. [1]); i.e., the failure arises not from the
second term on the right-hand side of Eq. (1), but from the first term (‘${i}{{\omega}_0}{a}$’) on the same side of Eq. (1)—a feature that, to our knowledge,
has not been identified in the past, since it does not normally arise in
ordinary (non-topological) resonant structures.

Mann *et al*. have also taken the in-/out-coupling
rates in a nonreciprocal cavity shown in Fig. 2(a) of Ref. [2] (indicated
therein, respectively, with cyan/red colors) to represent the *total* in-/out-coupled energy rate (power), whereas in
fact those rates only refer to the *radiative*
part of the power, as was explained in Ref. [2] (cf. ‘${\tau _{\text{out}}}$’ in Fig. 2(a) of Ref. [2] with ‘${\tau _{\text{out}}}$’ in Eq. (3) of Ref. [2]; i.e., the red arrow in both panels of Fig. 2(a) of
Ref. [2] is associated with the ‘${1/}{\tau
_{\text{out}}}$’ *radiative*
out-coupling power, not with the total, dissipative ‘${1/}{\tau _0}$’, plus radiative,‘${1/}{\tau
_{\text{out}}}$’, rate). In other words, for Lorentz
reciprocity to be broken in a cavity resonator, one only needs to
(radiatively) in-couple light energy to the cavity, and then the light energy
should not *radiatively* escape the cavity—but all
light energy will still, nonradiatively (that is, via heat) “escape” the
cavity, as shown in Fig. 3(b) of Ref. [2], and still further herein in Fig. 2. In fact, this is precisely the physical origin of the $ {\sim} 100\% $ absorption in the whole CUP region reported
in Fig. 2 herein. We note that this
definition of nonreciprocity in a resonator is completely analogous to the
well-known definition of nonreciprocity for a waveguide (also reported as
Eq. (2) in Ref. [2]) where the wave
*transmission* from a point A to a point B should
be different from the wave *transmission* from
point B to point A—that is, reference is made to the radiative power, to the
transmission (we do not ‘send’ Joule losses from A to B, or from B to A).
Thus, to break Lorentz reciprocity in a resonator too, one needs to make
unequal only the *radiative* parts of the
in-/out-coupled powers—as was reported and explained in Ref. [2]. The total (radiative + dissipative) in-
and out-coupled powers are always equal at steady state, as dictated from
Poynting’s theorem (which is automatically respected in FDTD simulations).

Further, in Ref. [1] Mann *et al*. observe a localized hotspot, which they refer to
as a wedge mode, i.e., the trapped state. They conclude that both their
trapped state, as well as the one observed in Ref. [2], are not due to nonreciprocity, but simply an example of
plasmonic focusing, i.e., a tapered plasmonic waveguide, with nonreciprocity
only providing impedance matching. We will now show that this is a
misconception, and that nonreciprocity is fundamental to the performance of
the device. Specifically, we will show that in the reciprocal version of the
device the electromagnetic energy is not confined to a localized region, and
while a field enhancement is observed, it does not represent the same focusing
nor enhancement factor. Crucially, we will also show that in the reciprocal
case the energy of the trapped state decays in a tiny fraction of that of the
nonreciprocal structure—that is, the T-B performance of the nonreciprocal
structure is drastically superior.

To this end, Fig. 3 reports FDTD
calculations, similar to those presented in Ref. [2], displaying (a) the energy density in the termination as a
function of time and (b),(c) the spatial distribution of the electromagnetic
energy at various times for both the (b) reciprocal and (c) nonreciprocal
cases. In all cases, the same device as in [2] is investigated [cf. Fig. 1(b)], with the difference being that for the reciprocal case no
external magnetic field is applied (${\rm B} = {0}\;{\rm
T}$), while the nonreciprocal case features an
applied external magnetic field (${\rm B} = {0.2}\;{\rm
T}$). Furthermore, the group velocity of the
incident light in both cases is almost exactly the same (${v_g} = {0.0681}c$ for ${\rm B} = {0}\;{\rm
T}$ and ${v_g} = {0.0673}c$ for ${\rm B} = {0.2\;T}$); thereby any difference(s) in behavior
cannot be attributed to conventional slow-light effects. From Fig. 3(a), we clearly see that the reciprocal
device has a much faster decay rate than its nonreciprocal counterpart. In
both cases, the pulse propagates (slowly, with the aforementioned group
velocities) towards the termination. However, once it reaches the termination,
the behavior starts to differ dramatically. For the reciprocal structure, the
pulse enters the trapping region and is then back-reflected, resulting in a
rapid decay of the energy within the open cavity region. For the nonreciprocal
case, however, the back-reflection cannot occur (as there is no backwards
propagating mode), and light is now trapped in the open cavity region,
decaying slowly *only because of dissipative
losses* (material absorption). We also note from Fig. 3(a) that if the magnetic field is reversed [${-} {0.2\;T}$ lines in Fig. 3(a)], then the pulse can be recovered at times much later than for
the reciprocal case. Therefore the nonreciprocal structure clearly outperforms
the storage capabilities of conventional plasmonic focusing, with the
increased storage time, i.e., delay, being a direct consequence of
nonreciprocity.

To further demonstrate this argument, we show the electromagnetic field within the trapping region (${\sim}{798}\;\unicode{x00B5}{\rm m} \lt x \lt {800}\;\unicode{x00B5}{\rm m})$ for both the reciprocal [Fig. 3(b)] and nonreciprocal [Fig. 3(c)] cases, at different times, normalized to the amplitude of the incident pulse. We observe that, for both cases, the maximal field enhancement occurs at a time $t = {100}\;{{\rm T}_p}$ (${{\rm T}_p} = {1/}{f_p}$, where ${f_p}$ is the plasma frequency of InSb). However, for the reciprocal case we observe only a ${\sim}{5}$ times amplitude enhancement (i.e., conventional plasmonic focusing), and we see that over the observed spatial region the field is approximately uniform. In contrast, for the nonreciprocal case we observe that at the same time instant the amplitude enhancement is by a factor of ${\sim}{3300}$—almost three orders of magnitude above the conventional result (reciprocal structure). Furthermore, the field is confined in a significantly smaller spatial region. The same pattern is observed at all times; i.e., at any point in time the reciprocal structure has a local field amplitude at the focusing tip several orders of magnitude smaller than that in the nonreciprocal structure, and spreads out uniformly in the spatial region of interest, while the nonreciprocal structure displays extraordinary amplitude enhancement and localization of the field in a smaller spatial region. As such, the argument made in Ref. [1] that the observed effect is conventional plasmonic focusing is clearly unfounded and in contradiction with the observed behavior of the device. Both the field enhancement and the T-B performance are dominated by the nonreciprocal nature of the device. These T-B-related differences between nonreciprocal (topological [7,8]) and reciprocal (ordinary) terminated structures become even more pronounced when realistic surface roughness and material imperfection effects are considered, as it is well-known that reciprocal such structures may even lose their ability to focus and localize light at their tip [9], whereas the nonreciprocal structure of Fig. 1(b), being topological [7,8], is completely immune to such effects [7].

Finally, a few points and clarifications are due with regard to the potential
role of nonlocality [10,11] on the attained, large, T-B violations,
as well as on the nature of the ‘open cavity’ considered in Ref. [2] and in (the blue spot of) Fig. 1(b) herein. First, the objective of
Ref. [2], as well as of the present
Comment, was to show that the T-B limit can be exceeded by essentially an
arbitrarily high degree in *local*
(non-spatially-dispersive), linear, time-invariant structures—that is, the
same type of structures considered in Ref. [1], as well as in similar previous works [12–14], which reasoned that no such violations may exist in such
structures for fundamental reasons [14]. The results and physical justifications presented here, as well
as in Ref. [2], rigorously show that the
T-B limit characterizing local, linear, time-invariant structures can be
overcome so long as such a violation is *topologically
enforced and protected*. Second, even when nonlocal effects are
considered, one may always redesign the terminated structure considered here
and in Ref. [2], e.g., simply by
removing the dielectric (Si) layer, such that it can robustly preserve its
unidirectional and topological character even in the presence of nonlocality,
and for arbitrarily small levels of dissipation, as has recently been shown in
Ref. [11]—thus, nonlocality cannot for
fundamental reasons, i.e., for all possible structures, destroy topological
protection (topology), since the latter is a deeper and more fundamental
property. Third, there is no need for termination and its associated large
field enhancement in a tight region [cf. Fig. 3(c) and brief discussion below], which might give rise to nonlocal
effects, as ultrabroadband light trapping [15,16] and releasing [17] can also exist in topological
(unidirectional) ‘trapped rainbow’ structures [18,19], which can stretch out
and localize (trap) a lightfield in tapered guides in a manner stable even
under fabrication disorders [15].
Fourth, for device applications of such T-B violations, other important
phenomena, such as nonlinear and thermal effects [19], will need to be considered, both of which can, however,
be addressed by, e.g., lowering the injected light power or resorting to
cryogenic conditions. It is also to be stressed that the trapped state
considered in Ref. [2] and in this
Comment is formed by a non-self-sustained [5] bulk (not surface) plasmon [6] of the terminating Ag layer: the ${E_x}$-field component, perpendicular to the
terminating Ag layer, is dramatically enhanced, inducing free charges on its
surface (bulk plasmon), and it is the near field of that bulk plasmon that the
pulse is in-coupled to, without reflection(s) across the entire CUP region.
Such plasmonic particles, and their associated bulk plasmons, are typically
referred to as ‘open cavities’ in the field of (nano)plasmonics [6]. Therefore, the lossy topological [7] ‘open cavity’ (i.e., the Ag particle) in
Ref. [2] and herein is fundamentally
different from the lossless perfect-electric-conductor ordinary cavity
terminating the unidirectional waveguide of Ref. [1], which, therefore, not surprisingly, does not reproduce the
behavior reported in Ref. [2].

In conclusion, the paper by Mann *et al*. [1] makes an interesting contribution in that
it convincingly shows that any system whose dynamics are accurately described
by (the standard, single-resonance form of) a TCMT approximation is T-B
limited, even when nonreciprocally fed. However, by not recognizing the
afore-outlined inherent limitations of such a method of analysis, and the
fundamental differences of the structure they considered compared with that in
Ref. [2], Ref. [1] reached the generalized conclusion that all (local) linear,
time-invariant structures are T-B limited, including the one shown in
Fig. 1(b) herein, studied previously in
Ref. [2]—a conclusion that is
unwarranted, as explained in some detail above. Moreover, the assertion of
Ref. [1] that nonreciprocity does not
beget any specific advantage(s) in terms of the T-B performance of a device is
unjustified too, as was clearly shown in Fig. 3 above. Thus, overall, this Comment helps to clarify that the
time-bandwidth limit *can* be exceeded, in fact to
an arbitrarily high degree as Ref. [2]
has previously reported, even in (local) linear, time-invariant structures,
that topology and nonreciprocity play a crucial role in achieving this feat,
and that the standard form of, otherwise powerful, quasi-analytic techniques,
such as TCMT that was deployed in Ref. [1], fails to accurately describe the dynamics and physics of (open)
nonreciprocal cavities, even in the low-loss regime where they are normally
successfully applied.

## Funding

General Secretariat for Research and Technology; Hellenic Foundation for Research and Innovation (1819).

## Acknowledgment

The authors thank Sebastian A. Schulz, Jeremy Upham, and Robert W. Boyd for
many stimulating discussions and useful comments made on this work. The
first of these colleagues, in particular, drew our attention to the key
‘iω_{0}a’ assumption of standard TCMT, explained in the main text,
and has provided Fig. 1.

## Disclosures

The authors declare no competing financial interests or conflicts of interest.

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