## Abstract

We reply to the comment on our recent article [Optica **6**,
104 (2019) [CrossRef] ], in which we
demonstrated that nonreciprocal cavities comply with the resonance
time-bandwidth limit. We fully stand by our original claims and
further elucidate how breaking of reciprocity is not required to
achieve the large field enhancements and time-bandwidth products
observed in the comment and our article. We also further clarify that
these hotspots do not overcome the conventional time-bandwidth
limit.

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

## 1. INTRODUCTION

Lorentz reciprocity fundamentally governs the symmetry in electromagnetic
wave propagation for opposite directions. Even though this property was
discovered over a century ago, it remains a tricky and often poorly
understood subject. Unwarranted claims on nonreciprocity that regularly
appear in the scientific literature are testimony to the confusion the
topic generates when improperly handled and digested, as discussed in a
number of recent reviews and comments [1–5]. In 2017, Tsakmakidis *et al.*
[6] theoretically proposed that
breaking Lorentz reciprocity may enable broadband yet long-lived
resonances. Their claim raised significant attention, and soon after their
publication a number of papers [7–9], including ours [10],
disputed most of their arguments. Continuing the discussion in [10], we take this reply to the comment
[11] as an opportunity to further
clarify the relationship between the bandwidth of a pulse and its temporal
interactions with a resonant cavity as well as the role that breaking
reciprocity can or cannot have in delaying pulses. We stand by all
conclusions in Ref. [10], and hope
that these clarifications may be helpful to the authors of Refs. [6,11] and to the readers interested in nonreciprocity,
time-bandwidth limits and, more broadly, nanophotonic systems with exotic
electromagnetic responses.

## 2. NONRECIPROCAL RESONANCES

Resonances follow an inverse relationship between their bandwidth $\Delta \omega$ and the time over which they can store
energy, $\Delta t$. Specifically, linearity and time
invariance require that $\Delta \omega \Delta t =
2$, where the bandwidth is defined as the
full width half-maximum of the stored energy, and the stored energy decays
by a factor ${e^2}$ every $\Delta t$ [8]. The *time-bandwidth product* $\Delta \omega \Delta t =
2$ is a consequence of the mathematical
properties of the Fourier transform applied to the resonance dynamics, and
it implies that, e.g., a broader resonance bandwidth corresponds to a
faster decay rate for any resonant element. This trade-off is particularly
relevant in the context of delay lines: resonators are commonly used to
impart a delay $\tau$ on an incoming pulse with bandwidth $\Delta {\omega
_p}$, and the inverse relation between
bandwidth and decay results in the delay-bandwidth limit for a resonator, $\Delta {\omega _p}\tau \le
2$. The upper limit is approached when the
pulse bandwidth is smaller than the cavity bandwidth ($\Delta {\omega _p} \ll
\Delta \omega$).

The general dynamics of a resonator can be conveniently modeled using coupled-mode theory (CMT) [12,13], which we briefly review here to properly define the terminology. When excited by a single port, the equation of motion for the complex amplitude $a$ of a resonator, normalized so that ${| a |^2}$ is the stored energy, is

*loss rates*${\gamma _r}$ and ${\gamma _i}$ determine the rate at which the mode amplitude decays through radiation and absorption, respectively. The

*incoupling coefficient*$k$, with units of $\sqrt {{\rm Hz}}$, relates the incident amplitude to the mode amplitude, and the reflected amplitude ${s_ -}$ is given by ${s_ -} = C{s_ +} + da$, where $C$ is the direct reflection coefficient at the port and $d$ is the

*outcoupling coefficient*. From power conservation, it is easy to show that the radiative loss rate is related to the outcoupling coefficient as ${| d |^2} = 2{\gamma _r}$, and for this reason we can refer to ${| d |^2}$ as the radiative

*output rate*. In analogy to this definition, and consistent with [6,11], we will refer to ${| k |^2}$ as the radiative

*input rate*of the cavity. We should remark, however, that the rate equation for the stored energy is $d{| a |^2}/dt = 2\Re ({a{k^*}s_ + ^*}) - 2({{\gamma _r} + {\gamma _i}}){| a |^2}$ [12], indicating that the actual input rate of a cavity is not simply described by ${| k |^2}$ for coherent excitation (but it is for incoherent excitation).

In their 2017 paper, Tsakmakidis *et al.*
propose that “if Lorentz reciprocity is by some means broken in [a]
passive, linear, and time-invariant *resonant
system* [emphasis added], … the product … can be engineered at
will and take on arbitrarily large values—i.e., in such a case we can
exceed the conventional time-bandwidth limit by an arbitrarily large
factor” [6]. In an attempt to prove
this bold claim, the authors theoretically study the time evolution of a
resonant cavity using CMT. They conjecture, incorrectly, that its input
rate determines the bandwidth of a signal that can feed the cavity ($\Delta \omega \propto 2{| k
|^2}$), while the output rate determines the
decay time $\Delta t$. Given that Lorentz reciprocity governs
the symmetry between forward and backward propagation of signals, they
then argue that breaking reciprocity may make input and output rates
different {see Fig. 2(A) in [6]} and
hence enable an arbitrarily large time-bandwidth product $\Delta \omega \Delta
t$. Their hand-waving argument may sound
agreeable at first: if we can feed a resonant cavity through a
nonreciprocal waveguide, with different forward and backward propagation
properties, the cavity may be able to accept energy with a fast rate (and
supposedly thus engage a pulse with a large bandwidth), but decay through
the same channel with a much slower rate, thereby overcoming the
time-bandwidth limit. These claims are reiterated,
sometimes with slightly modified arguments [11].

Next, in [6] the authors study a
nonreciprocal plasmonic waveguide, supporting over a given bandwidth a
single forward mode but no backward mode, terminated on a fully reflective
plane. This configuration is well known to sustain a broadband
electromagnetic hotspot at the termination [14–18], which Tsakmakidis *et al.* call a “zero-dimensional cavity resonator” [6] or a “trapped state” [11], and we referred to as a “wedge mode”
[10]. As originally shown in [14], this hotspot has interesting
electromagnetic properties: it necessarily fully absorbs all incident
radiation for any level of (nonzero) loss over the entire unidirectional
bandwidth. Tsakmakidis *et al.* present this
well-established feature as practical evidence for their claim that a
nonreciprocal resonance can overcome the time-bandwidth product.
Specifically, they notice that, since the hotspot bandwidth is determined
by the unidirectional frequency window and its decay time is governed by
material losses, their product can be made arbitrarily large if low-loss
materials are employed [6,11].

The potential of surpassing the time-bandwidth limit in a passive, linear,
and time-invariant system by breaking reciprocity (e.g., by applying a
static magnetic bias) would indeed be groundbreaking for many photonic
applications. Unfortunately, for various reasons this claim is too good to
be true, as we and others have pointed out previously [7–10] and
will further clarify in the following. Our article [10] was indeed motivated by the claims in the first part
of Ref. [6], with the main goal of
exploring the actual dynamics of nonreciprocal resonant systems. In [10], after rigorously extending CMT to
nonreciprocal systems, we generally demonstrated several results of direct
relevance to the claims of [6,11], including that: (i) the bandwidth $\Delta \omega$ of the signal that any resonant system,
reciprocal or not, can accept is *always
proportional* to its decay rate, and it is not controlled at all
by the input rate as claimed in [6];
(ii) the time-bandwidth product of any resonator is $\Delta \omega \Delta t =
2$, independent of whether it is reciprocal
or nonreciprocal; and (iii) the radiative input and output rates, as
defined above, of any linear cavity are not independent and *must be equal*, independent of reciprocity or of the
presence of absorption loss, i.e., ${| d |^2} = 2{\gamma _r} =
{| k |^2}$.

In [10], the aforementioned results are more generally derived for nonreciprocal cavities with multiple ports, and they are verified numerically. Because the hotspot at the termination of the unidirectional waveguide is not a resonant system, and hence cannot be described using CMT, we purposefully added a resonant cavity to the geometry considered in [6]. By doing so, we realized a nonreciprocal cavity that could be studied to validate our theory [10]. Indeed, our numerical results fully agree with our conclusions and demonstrate that nonreciprocal cavities do not adhere to the claims in [6].

Quite surprisingly, Tsakmakidis *et al.* now
argue in [11] that we should not
have used CMT to study the unidirectional waveguide termination. Quoting
from their comment: “…the temporal coupled-mode theory, on which Mann
*et al.* base their main conclusions, is not
suited for the study of nonreciprocal trapped states ….” Of course, we
fully agree with this statement: precisely because the hotspot is not a
conventional resonator and should not be studied with CMT, we added a
resonant cavity to the unidirectional waveguide termination to properly
examine the electromagnetic response of nonreciprocal cavities. The
authors of [11] appear to forget
that CMT and the resonance framework were introduced by Tsakmakidis
*et al.* themselves in [6] to motivate their concept and explain how
nonreciprocity could be used to “break the time-bandwidth limit” in
resonant cavities. In this pursuit they made a series of erroneous claims,
inconsistent with basic thermodynamic [7] and foundational electromagnetic principles, which motivated us
to properly study the dynamics of nonreciprocal resonators in [10]. We are relieved that Tsakmakidis
*et al.* now appear to agree that the CMT
framework used in [6], together with
the associated claims used to describe the terminated unidirectional
waveguide, are incorrect.

Unfortunately, despite this initial agreement, later in [11] the authors challenge one of the
fundamental results in [10]: our
proof that the total radiative input and output rates must be equal in any
linear, time-invariant resonator. They claim that “… 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 …” and “…to break Lorentz reciprocity in a resonator,
one needs to make unequal only the *radiative*
parts of the in-/out-coupled powers…” [11]. Due to the imprecise phrasing, it is unclear what the authors
are claiming here. If we interpret their statement to discuss *steady-state power flows*, they are describing the
scenario in which light is coupled into a cavity radiatively, but instead
of being reflected back it is fully absorbed. This trivially corresponds
to a critically coupled cavity, which is a well-known (and completely
reciprocal) phenomenon that results in perfect absorption in resonators
when radiation and absorption loss rates are equal [12]. It is obvious that incoming and outgoing power flows
can be different in a cavity with absorption loss, and this phenomenon has
nothing to do with breaking reciprocity. Given the context, however, the
authors may be referring to the input and output *rates* ${| k |^2}$ and ${| d |^2}$ as defined earlier in this reply. We have
already rigorously proven in [10]
that these rates *must* be equal for any
resonance, but to make this point even clearer we provide an additional,
independent proof in Appendix A based
on thermodynamic arguments. In particular, we show that the *only* way a linear time-invariant cavity can satisfy
the equipartition theorem and the second law of thermodynamics is if the
radiative input and output rates are indeed equal, independent of
reciprocity or the presence of absorption. Similar arguments may be
extended to more general electromagnetic systems, not restricted to
resonators [19].

## 3. IS NONRECIPROCITY NECESSARY TO REALIZE A SLOWLY DECAYING BROADBAND HOTSPOT?

In the previous section we have clarified once more that the premise of
Ref. [6], i.e., that nonreciprocity
decouples the bandwidth and lifetime of a resonator, is incorrect. We will
now discuss the broadband response of the hotspot arising at the
termination of a unidirectional waveguide. The product of the hotspot
bandwidth (determined by the unidirectional frequency window) and its
decay time (governed by material losses) can indeed become large relative
to the time-bandwidth product of a resonator, a feature taken to be of
particular significance by Tsakmakidis *et al.* However, is the large difference between the hotspot
bandwidth and its decay rate through absorption a feature that inherently
requires nonreciprocity? The authors of [11] are adamant: “…the extraordinary time-bandwidth performance
observed in [6] is a direct
consequence of the nonreciprocal nature of the device.”

To the contrary, in [10] we
emphasized that “this broadband focusing is not directly a consequence of
nonreciprocity: adiabatically tapered terminated plasmonic waveguides
[20–22], which slowly focus the
incoming fields toward an apex, perform the same function.” In other
words, nonreciprocity not only does not play any role in determining the
time-bandwidth product for resonant cavities, it also is not at all
necessary to achieve broadband focusing of energy in a hotspot with a long
decay time due to absorption. The authors of [11] attempt to disprove our claim by trivially observing
that, if they switch off the magnetic bias in their geometry, no
discernable energy is stored at the termination. This “proof” in itself is
quite absurd: it is obvious that if the unidirectional feature of the
waveguide is removed *and nothing else is done to
avoid reflections*, no hotspot or concentration of energy can be
expected at the termination. To further clarify our claims, however, we
now explicitly show how a completely reciprocal waveguide, with carefully
designed tapering to avoid reflections, supports an electromagnetic
response in every way analogous to the nonreciprocal hotspot
discussed in [6,11].

Figure 1(a) shows a tapered metal-insulator-metal (MIM) waveguide [23] made of a silicon slab interfaced on both sides by gold. MIM waveguides are well known to support a group velocity dependent on the width of the dielectric layer, approaching zero for vanishing widths [24]. By adiabatically tapering the width as in Fig. 1(a), the group velocity can thus slowly be brought to zero without causing reflections [20–22]. As a result, a broadband signal can be focused at the apex of the tapered structure, decaying only through absorption. The bandwidth of this phenomenon is mostly controlled by the taper length, whereas the decay time of the energy focused at its apex is controlled largely independently by the absorption coefficient of the involved materials. In close analogy to the nonreciprocal hotspot, the product of the decay rate and bandwidth can thus be made as large as desired. By integrating the energy density over a small area surrounding the apex [shown by the purple square in Fig. 1(a)], Fig. 1(b) shows a fast rise in stored energy as the pulse approaches the apex, but a much slower decay as the pulse is slowly absorbed by the metal. In addition, as the group velocity is reduced near the apex, the pulse is compressed [Fig. 1(c)]. Notice the staggering similarity between the response in Figs. 1(b) and 1(c) and the results in Refs. [6,11]. These results unequivocally demonstrate that nonreciprocity is not required to realize broadband field concentration with slow decay, in contrast to the bold claims in Refs. [6,11] and fully consistent with the large body of work on tapered plasmonic waveguides (see e.g. [20–22]).

The clear advantage of a nonreciprocal termination compared with the geometry in Fig. 1(a) is that the unidirectional waveguide forbids reflections and therefore forces the emergence of the hotspot independent of how abrupt the termination is. This peculiar feature has motivated the study of nonreciprocal hotspots for several decades [14–18,25], including in Ref. [19] by some of the authors of Refs. [6,11]. Hence, as we stated in [10], nonreciprocity may be practically beneficial in that the termination can be arbitrarily abrupt, but it does not play a fundamental role in achieving a slowly decaying broadband hotspot.

## 4. DOES THE HOTSPOT OVERCOME “THE TIME-BANDWIDTH LIMIT”?

Since the combination of a wide bandwidth and slow decay rate observed by
Tsakmakidis *et al.* is clearly not unique to
nonreciprocal systems, the remaining claim we want to address is whether
these (reciprocal or nonreciprocal) systems that focus broadband
illumination into a subwavelength hotspot actually “break the
time-bandwidth limit.”

Before addressing this question, we would like to briefly revisit the utility of fundamental limits. In general, limits are useful because (under a given set of constraints) they can tell us whether a proposed device functionality is impossible, whether a device performs well or if there is room for improvement, and what trade-offs to expect in optimizing a certain design. Consider single-junction solar cells: the maximum recorded conversion efficiency is currently 29.1% [26], which may appear low, until one considers that the fundamental limit (the Shockley–Queisser limit [27]) for that particular material is ${\sim}{33.5}\%$ [28,29]. Understanding this limit has also led to multiple demonstrated ways to surpass it: by combining different semiconductors or using solar concentrators [30] and directive absorbers [31,32], for example. Other well-known examples of fundamental limits are the Chu limit [33], which provides a lower bound on the Q factor of electrically small antennas and has been a metric of performance since its inception, and the Bode–Fano bound [34], which limits the product of the bandwidth and minimum reflectance that can be achieved in impedance matching a given passive load. Both of these limits can be surpassed through, e.g., temporal modulation [35,36] and gain, because time invariance and passivity are assumptions in their derivation. The same holds true for the resonance time-bandwidth product: by modulating cavities in time, for example, this product can be made arbitrarily large [37,38].

In Ref. [11], Tsakmakidis *et al.* maintain that their structure breaks the
resonance time-bandwidth limit, as, e.g., stated in the conclusion: “Thus,
overall, this comment helps clarify that the time-bandwidth limit *can* be exceeded, in fact to an arbitrarily high
degree as Ref. [6] has previously
reported…”. This statement, however, appears to follow from a conceptual
misunderstanding of what the time-bandwidth limit means, when it applies,
and when it is a useful reference. Recall that the same authors admit in
[11] that the nonreciprocal hotspot
is not a resonant phenomenon and should therefore not be framed or
analyzed in the context of resonances. While the product of the bandwidth
and decay rate of the hotspot indeed exceeds the resonance time-bandwidth
product, this is precisely because $\Delta \omega \Delta t =
2$ specifically applies to single
resonances, and the hotspot is not a resonator. As such, the comparison
between the resonance time-bandwidth product and the broadband hotspot is
a logical fallacy with no practical implications.

To put the unidirectional waveguide studied by Tsakmakidis *et al.* in the context of an appropriate fundamental
limit, its exact functionality must first be determined. The termination
takes a broadband incident pulse and fully absorbs it, which implies that
perhaps it may be better compared to limits describing broadband
absorbers, such as the Rozanov limit [39]. However, more work is required to make a fair comparison (for
example, the unidirectional waveguide is now infinitely long, for which it
is not surprising to expect a very large bandwidth of complete
absorption).

When it comes to delaying pulses over a finite distance, fundamental limits have been derived for numerous platforms, such as slow-light waveguides [40], coupled resonator waveguides [41], and even a fundamental limit for 1D (or single-mode) systems by Miller [42]. In order to determine whether nonreciprocity can provide a fundamental benefit for pulse delays, the system should have a well-defined input and output port between which a pulse is delayed (and not fully absorbed). As a solution to this problem, Refs. [6,11] suggest to instantaneously reverse the magnetic bias after the broadband pulse is slowed towards the hotspot, before it is fully absorbed, thereby releasing it towards the port from which it originally came. Not considering the difficulties of quickly reversing a strong magnetic field, this operation could implement a delay device with mildly interesting properties. More importantly, since the magnetic field bias changes at a certain instant in time, this system is not time invariant, and it is well established that time-bandwidth products can become as large as infinity in time-varying structures [37,43,44]. A trivial example is a cavity that is closed after a pulse enters it, to be released as late as desired.

To actually be able to fairly compare a time-invariant nonreciprocal system with the broad class of reciprocal devices used to delay pulses, we adjusted the geometry of the terminated unidirectional waveguide studied in [6,11]. The waveguide is now almost fully terminated, except for a narrow slow-light channel [Fig. 2(a)], so that an output port is now present and a meaningful delay time can be attributed to the structure. We analyze this structure in detail in Ref. [45] and demonstrate that unidirectionality indeed provides the benefit of impedance matching through abrupt geometrical transitions, enabling robust and smaller footprints compared to other reciprocal geometries using slow plasmonic waves. However, considering the wavenumber in each region of the device [Fig. 2(b)], these structures fully comply with the delay-bandwidth limit derived by Miller (modified to apply to single-mode waveguides) [42]:

## 5. CONCLUSIONS

To conclude, in this reply we have reiterated the fundamental principles of
nonreciprocal cavities, in particular with respect to the time-bandwidth
product and the equality of radiative input and output rates, correcting
the claims in [6,11]. In Appendix A, we have provided an independent proof of the equality
of input and output rates rooted in basic thermodynamic principles, which
complements the proof in [10] based
on CMT. We subsequently discussed two bold claims made by Tsakmakidis
*et al.* in [11]: (i) Is nonreciprocity required to achieve large
time-bandwidth products based on broadband field concentration that decays
slowly through absorption? and (ii) Does this phenomenon “break the
time-bandwidth limit”? We have shown that the answer to both questions is
a resounding “no.” We hope that this reply and our article [10] may further help the fundamental
understanding of nonreciprocal resonant and nonresonant systems, and
provide a reality check for some of the unrealistic or unwarranted claims
related to reciprocity that occasionally come to surface.

## APPENDIX A: NONRECIPROCAL CAVITIES IN THERMAL EQUILIBRIUM

Consider the equation of motion for a cavity without a coherent input signal but excited thermally [47,48]:

We assume that the cavity is in thermal equilibrium with the environment so that $T = {T_i} = {T_e}$. We evaluate how much energy is stored in the resonance in equilibrium, calculating the double integral over the autocorrelation [47]

*et al.*, where the radiative input and output rates of this one port cavity may become different due to nonreciprocity. Linear, time-invariant nonreciprocal systems still obey the relationship between the coefficient ${k_i}$ and the absorption loss rate ${\gamma _i}$: the relationship $|{k_i}{|^2} = 2{\gamma _i}$ still holds [48–50]. Evaluating the double integral over the Lorentzian, we then find

One might boldly suggest that the fluctuation-dissipation relation $|{k_i}{|^2} = 2{\gamma
_i}$ *could*
be broken through time-invariant nonreciprocity. In this case
equipartition could be maintained in the scenario where $|{k_r}{|^2} = 2{\gamma
_i}$ and $|{k_i}{|^2} = {\gamma
_r} = 0$, which is the scenario that
Tsakmakidis *et al.* seem to imply in
[11]. However, such a
configuration clearly violates the second law of thermodynamics:
starting in equilibrium ${T_i} =
{T_e}$, the system would evolve, without the
expenditure of work, to ever-increasing ${T_i}$ through the one-way flow of
power.

## Funding

Air Force Office of Scientific Research; The Simons Foundation;Dutch Research Council (NWO).

## Disclosures

The authors declare no conflicts of interest.

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