The time–bandwidth limit inherently relates the lifetime of a resonance and its spectral bandwidth, with direct implications for the maximum storage time of a pulse versus its frequency content. Recently, it has been argued that nonreciprocal cavities may overcome this constraint by breaking the strict equality of their incoupling and outcoupling coefficients. Here, we study the implications of nonreciprocity on resonant linear, time-invariant cavities and derive general relations, stemming from microscopic reversibility, that govern their dynamics. We show that nonreciprocal cavities do not provide specific advantages in terms of the time–bandwidth limit, but enable other attractive properties for nanophotonic systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The demands of integrated photonic circuits have motivated a large body of research with the objective of scaling down and integrating crucial optical components for information processing [1–13]. A common approach to enable small footprints is to use resonant cavities [6–13], which can dramatically enhance light–matter interactions by storing light over the resonance lifetime. This reduced footprint, however, comes at the cost of operational bandwidth: a single resonant cavity adheres to a strict correspondence between its resonance bandwidth and its lifetime (see Supplement 1). As a relevant example in nanophotonics, resonators are frequently used to impart delays on optical pulses, given their ability to temporarily store light. The strict relation between lifetime and bandwidth of a cavity implies a trade-off between the spectral bandwidth of the pulse that can be stored and the temporal delay that can be imparted on it. It is straightforward to delay pulses for longer times, for example, by combining multiple cavities , or even by using non-resonant structures such as slow-light waveguides  or dispersive multilayer stacks . However, these structures still follow an analogous trade-off between the delay–bandwidth product and the overall footprint of the device . Time–bandwidth trade-offs can be overcome by using time-varying or nonlinear systems [18–21], but efficiently implementing such schemes is currently technologically challenging.
A controversial proposal to alleviate the strict correspondence between the bandwidth and lifetime of linear, passive, time-invariant structures has been recently put forward, based on breaking reciprocity with a static magnetic bias . The authors argue that the supported bandwidth scales with the rate at which energy enters the system, whereas the lifetime is inversely proportional to the rate at which energy exits it. As depicted schematically in Fig. 1(a), these rates are known to be equal in reciprocal cavities , but Ref.  suggests that nonreciprocal cavities may support different input and output rates and, as a result, can surpass the time–bandwidth limit. While this idea would have a large impact on many photonic applications, given its appealing realization in a passive, time-invariant system, it also raises concerns regarding its thermodynamic validity : for example, a system with unequal input and output rates violates the second law of thermodynamics.
Inspired by this proposal, in what follows, we develop a general temporal coupled-mode theory describing the dynamics of nonreciprocal cavities (previously studied in Refs. [6,25,26]). We focus on passive nonreciprocity in linear, time-invariant, magnetically biased cavities, in order to isolate the specific effect of nonreciprocity on the time–bandwidth limit and on cavity dynamics. Recent work on magnet-free nonreciprocity based on temporal modulation [2,4,13] and nonlinear phenomena [27–29] has shown that there are other viable ways to realize nonreciprocity, and certainly these approaches open new doors to overcome the time–bandwidth limit as well [18–21]. However, in what follows, we demonstrate that nonreciprocity itself has no particular benefit in the context of the cavity time–bandwidth limit. In particular, we show that the bandwidth is exclusively determined by the total decay rate, and therefore the time–bandwidth product is independent of any asymmetry between the coupling rates in nonreciprocal systems. Furthermore, our theory proves that the incoming and outgoing rates are strictly related in any linear, time-invariant cavity, independent of whether reciprocity holds, through a number of identities that generalize the known relations for reciprocal systems. For example, we prove that in nonreciprocal cavities the total input and output rates of energy transfer must always be equal. Finally, we derive bounds for the coupling coefficients of nonreciprocal cavities and design cavities operating at these bounds.
2. DYNAMICS OF NONRECIPROCAL CAVITIES
We start our discussion by studying a general resonant cavity with a complex mode amplitude connected to ports, with complex input amplitudes . Such a system is described by the equation of motion [23,30]1(b), where the aforementioned input amplitudes and coupling coefficients are depicted schematically. The output amplitudes of the system are given by 1(b). Power conservation ensures that  1) for the cavity amplitude in frequency domain 1)–(4) therefore apply to both reciprocal and nonreciprocal cavities. Equation (4) already tells us that the bandwidth of nonreciprocal resonances is solely governed by the total decay rate of the cavity, and therefore the time–bandwidth product stems directly from the linearity of the system, without a specific relation to whether it is reciprocal.
When assuming reciprocity, we can additionally show from time-reversal symmetry that and (asterisk denotes complex conjugate) . However, if the cavity is biased with a magnetic field, these two identities do not necessarily apply. This is because the magnetic field bias reverses its direction under a time-reversal operation, resulting in a system different from the original one. This also applies to systems subject to other static biases that are odd symmetric (i.e., reverse sign) under time reversal, such as electrical currents, angular momentum, or linear motion. To distinguish between the original and time-reversed scenarios, we will therefore denote the coefficients in the time-reversed scenario with a tilde, such as , , and . As a first important result of this work, we show that, if the resonant frequency is not perturbed by a time-reversal operation, the following relationships necessarily hold for both reciprocal and nonreciprocal linear, time-invariant systems (proofs are provided in Supplement 1):30,31], particularly because they involve the original system and its time-reversed counterpart (which, as we will show, can differ strongly in their response). This correspondence is a consequence of microscopic reversibility : while the system itself is not time-reversal symmetric, global time-reversal invariance still applies, and the original and time-reversed systems are therefore necessarily related.
Equation (5f) is the fluctuation–dissipation relation, relating dissipation () to how the cavity responds to external fluctuations () [24,33]. Equations (5e) and (5f) are arguably the most important results for the following discussions, as they put stringent restrictions on the achievable asymmetry in the coupling coefficients: e.g., while it is possible to achieve at any individual port, the positive-definite norm of and have to be equal [Eq. (5f)]. Finally, we note that Eqs. (5a)–(5f) readily reduce to the known identities for reciprocal cavities , if we assume that the system is time-reversal symmetric, i.e., , , and .
3. TIME–BANDWIDTH LIMIT
Equations (5a)–(5f) have important general consequences in the context of nonreciprocal cavities. As a relevant example, consider a nonreciprocal waveguide supporting a unidirectional mode, i.e., no backward channel into which energy can be reflected, and terminated into a cavity, similar to the geometries studied in Refs. [34,35] and shown schematically in Fig. 2(a). For the nonreciprocal waveguide, we use the interface between a magnetized magneto-optic semiconductor and a dielectric material [25,36–38] that for the combination of InSb and Si is known to support a unidirectional surface wave around 1.5 THz [22,39,40] (see geometry details and mode dispersion in Supplement 1). The waveguide is terminated into a resonant lossless rectangular cavity (20 μm by 30 μm in size), enclosed by perfect electric conducting (PEC) walls, connected through a small aperture (0.5 μm). This geometry is almost identical to the geometry studied in Refs. [22,40], except for the resonant cavity at the termination; in Refs. [22,40], the geometry under consideration consists of only the terminated unidirectional waveguide.
We use a home-built finite-difference time domain code (see Supplement 1) to excite the unidirectional waveguide with a broadband pulse (, ) and record the fields inside the cavity as functions of time. Figure 2(b) shows the Fourier transform of the electric field intensity induced at the center of the cavity (red line), showing a sharp Lorentzian resonance at 1.52 THz, with a linewidth of . When exciting the cavity from the interior, we find exactly the same decay rate of by examining the field decay [inset in Fig. 2(b)]. This confirms numerically that Eq. (4) applies independent of whether the system is reciprocal or not. The bandwidth of any linear, time-invariant cavity is thus solely determined by the total decay rate , i.e., by the internal loss and the outcoupling coefficients . This is also supported by the observation that the fields at the termination outside the cavity have much larger bandwidths [Fig. 2(b), blue line], equal to the bandwidth of the incident pulse, but these fields cannot be forced into the cavity. In other words, as an important consequence of Eq. (4), the time–bandwidth product of any linear, time-invariant cavity satisfies , and in lossless cavities, it is controlled only by the outcoupling coefficients . The incoupling coefficients in only control the amount of stored energy in the resonator.
More surprisingly, the cavity has a finite decay rate, despite the fact that it is both lossless () and connected to a waveguide that does not support backward modes (). While the reason behind the decay process is not immediately apparent, the decay is consistent with Eq. (5f): if the cavity can be excited, it must also be able to decay to maintain equilibrium. This paradox can be resolved by inspecting the electric field distribution during decay (without external excitation), shown in Fig. 2(c). We notice a strong hotspot at the corner formed by the Si–InSb interface and the PEC termination, which, given the finite material loss in InSb, dissipates all incoming energy and thus sustains the cavity decay. This form of hotspot is commonly referred to as a “wedge mode” [41,42], as it arises at the sharp corner of, e.g., a metallic wedge in a dielectric environment (or more complex geometries consisting of dielectrics/metals and sharp corners). For a wide range of angles, these wedges sustain a quasi-static resonance at their apex, supporting a mode with an field dependence at short distances, where is the distance to the corner and the coefficient depends on the geometry and permittivities [41,43]. In the present case, the metallic/dielectric wedge has a 90 degree angle.
The wedge mode is excited also when power is incident from the waveguide port, both on- and off-resonance [Figs. 2(d) and 2(e)] [22,40,44–46]. Since the cavity and the input port can both excite this localized mode, they interfere in the process of dissipation [as evidenced by the sharp spectral feature near the resonance in the blue curve in Fig. 2(b)]. As a result, the wedge mode cannot be treated as a regular internal loss process (), but needs to be treated as an additional output port (see Supplement 1). Then, writing and , respectively, for the input and output coefficients for the radiation from/into the waveguide (subscript ) and wedge (subscript ), we find that is permitted while is simultaneously satisfied. In other words, the input and output coefficients may differ at each individual port, but the total input and output rates must always be equal. Hence, it follows that in the special case of a truly one-port system like the one generally described in Ref.  and shown in Fig. 1(a), the input and output coefficients must necessarily be equal in magnitude but may still differ in phase. The nonreciprocal scenario in Fig. 1(a), with input and output rates of energy transfer , is thus impossible.
One may wonder what happens in the limit in which material loss in the unidirectional waveguide is zero, for which the wedge mode is expected to become non-dissipative. This problem has been extensively discussed in the literature (see [44,47–50] for a selection) and has been referred to as the “thermodynamic paradox”, since it was believed to produce an inconsistency between Maxwell’s equations and thermodynamics. This general paradox was resolved by Ishimaru  who pointed out that a lossless terminated one-way waveguide is actually an ill-posed boundary-value problem, and that the field distribution at the termination of a unidirectional waveguide is non-integrable if it is assumed lossless a priori. In contrast, if a small loss term is included in the analysis, then the termination sustains a finite absorption even in the limit of vanishing material loss. As such there is no paradox because the rates at which energy enters a volume near the termination and is dissipated in that volume remain equal. This is analogous to Eq. (5f) in this work, which requires that in steady state the rates at which energy enters and exits a cavity are equal. Aside from the wedge mode in this terminated unidirectional waveguide [22,40,44–46], similar singularities that support finite absorption in the limit of vanishing material loss can be found in other extreme, yet reciprocal, electromagnetic systems [43,51], and are a reminder that ideal lossless scenarios should be generally considered an artifact in electromagnetics and may sometimes lead to singularities and non-unique solutions , especially with negative permittivities.
Equations (1)–(5) strictly prove that the time–bandwidth product of a linear, time-invariant cavity is always equal to 2, and that therefore it is impossible to force broadband fields into a long-lived resonance, independent of reciprocity. References [22,40,44–46] do however demonstrate broadband focusing of photons in an ultrasmall volume near the termination, whose decay rate is unrelated to, and can be much slower than, the excitation. Consistently, our simulations confirm focusing of incident photons at the InSb/Si/PEC corner [Figs. 2(d) and 2(e)] over a wide range of frequencies [Fig. 2(b), blue line], irrespective of the properties of the cavity resonance. We stress, however, that this broadband focusing is not directly a consequence of nonreciprocity: adiabatically tapered terminated plasmonic waveguides [53–55], which slowly focus the incoming fields toward an apex, perform the same function. The benefit of applying nonreciprocity is that the termination is automatically matched, due to the absence of a backward mode in the waveguide [34,35], relaxing the need for a carefully designed adiabatic transition that minimizes reflections.
4. BOUNDS ON COUPLING COEFFICIENTS
After having demonstrated that nonreciprocity is not beneficial to break the trade-off between lifetime and bandwidth in linear, time-invariant cavities, in what follows we explore to what degree the incoupling and outcoupling coefficients may be made different in nonreciprocal cavities, and what functionalities can be enabled by such asymmetry. Achieving asymmetry in the coupling coefficients is important for, e.g., circulators  and unidirectional heat transfer [57,58]. In the previous example, the absence of a backward mode ensured . However, we now show that the input and output coefficients at a given port can differ significantly in nonreciprocal cavities even if there is a backward mode. To do so, we examine the same system as in Fig. 2, but now excited in the bidirectional regime at frequencies just below 1.25 THz. We therefore tune the cavity to 1.24 THz by increasing the cavity size to 35.4 μm × 20 μm (see Supplement 1 for geometry details). With full-wave simulations, we retrieve the complex cavity and reflection amplitudes, and through a fitting procedure, we obtain and (see Supplement 1 for spectra and fits). As expected, now because of the presence of a propagating backward mode, but the two magnitudes are shown to be significantly different due to the presence of the wedge mode. When operating just below the unidirectional gap, the system is still strongly asymmetric (in this case, the forward and backward effective indices are 4.47 and 9.92, respectively), and it is to be expected that as the asymmetry reduces, and will also approach each other.
While the input and output coefficients at the input port may be made different, both in magnitude and phase, Eq. (5f) requires that the total rates of the cavity must be the same; in the system under consideration, this is guaranteed by the wedge mode, which balances out any asymmetry in and . Interestingly, this is, however, not the only requirement: the direct pathway places an additional bound on these coefficients through Eq. (5e). For a general two-port system with and , we find using the Cauchy–Schwarz inequality3(a) as a function of for ( quantifies the fraction of radiated power flowing toward port as the cavity decays). This specific example corresponds to the previously discussed bidirectional system, where the subscript indicates radiation into/from the port (e.g., ). The cross in the figure highlights the magnitude of and for this geometry at 1.24 THz, falling within the bounds delimited by the solid line. The shape of the bound on strongly depends on : Fig. 3(b) displays similar bounds for and . In the system in Fig. 2, the absence of a backward mode dictates that both and , in which case Eq. (6) becomes : as expected, energy can only enter the cavity at the rate in which it is dissipated by the wedge mode, i.e., the other output port. Note that this holds generally, also when . Similarly, if , which implies that the two ports are not directly connected, Eq. (6) requires that , and thus the input and output coefficients for each port can be nonreciprocal only in phase, as in a gyrator. This is a consequence of the fact that the ports only exchange power with the cavity itself, and, hence, cannot compensate for any asymmetry in its response.
Having discussed the general bounds on the incoupling and outcoupling coefficients of nonreciprocal cavities, we now investigate an extreme condition allowed by the bounds in Eq. (6), which may provide interesting functionalities for nanophotonic systems. We consider the scenario of a cavity connected to a bidirectional waveguide ( that cannot be excited from its input port (), but that does decay into it (). To design such a cavity, it is sufficient to place its connecting aperture at a position with complete destructive interference between the forward and backward modes, as shown in Fig. 4(a). Since nonreciprocal waveguides have different forward and backward mode profiles , we need to ensure that the backward mode has higher fields at the top PEC plate, and that the reflection coefficient compensates for differences in magnetic field amplitude. As shown in Fig. 4(b), we achieve this condition by reversing the magnetic field bias with respect to the previous examples and optimizing the reflectivity of the dissipative waveguide termination to achieve . The necessity of a reduced reflection coefficient is consistent with our previous finding in Eq. (6) that a nonreciprocal response can only be achieved if . Note that here we consider the dissipative termination as the second output port of the cavity directly (see Supplement 1 for additional discussion).
We now find a locally vanishing magnetic field at the top PEC wall [Fig. 4(a)], implying that a cavity coupled to the waveguide at that location cannot be excited. In Fig. 4(c), we place a cavity above the waveguide with its opening at the location of destructive interference, and a forward wave impinging from the channel indeed does not excite the cavity. If the waveguide were reciprocal, a vanishing magnetic field at the cavity opening when excited from the port would imply that the cavity also cannot decay into the port. However, in the nonreciprocal case, this is not so: if we excite the cavity from inside, we see it decay freely toward the output port [Fig. 4(d)]. This is a result of the fact that, due to nonreciprocity and the different profiles of the forward and backward modes of the waveguide, for excitation from inside the cavity, these modes are excited at the aperture with amplitudes that do not lead to destructive interference.
Interestingly, according to Eqs. (5c) and (5d), we can swap the values of and by reversing the direction of the magnetic field bias (which is equivalent to a time-reversal operation). This is shown in Figs. 4(e) and 4(f), which present the same structure but with opposite magnetic bias. We now see that the cavity can be efficiently excited from the port [Fig. 4(e)], but when the cavity is excited from inside, it decays only toward the termination and the wave gets fully dissipated there. It is interesting to point out that, in contrast to the example in Fig. 2, here whereas : even though there is an available backward mode, the cavity cannot couple to it.
5. DISCUSSION AND CONCLUSIONS
To conclude, in this article, we have presented a general theoretical framework describing the dynamics of nonreciprocal cavities. Our results show that, for single port systems, and can only differ in phase. The requirements for and at any individual port to be different in magnitude are twofold: (1) at least one additional channel is necessary, and (2) the direct pathway and cavity output must be able to interfere in that additional channel (). This is consistent with the general principle that to realize a linear isolator, it is necessary to have a loss channel [60,61] or an additional radiative port, as in a circulator. For systems with multiple ports, we have shown that the sums of all input and output rates are necessarily equal: . This implies that nonreciprocal cavities still follow strict bounds on their input and output coefficients, but nonetheless can be employed to realize highly non-trivial phenomena, such as cavities that can be pumped only one-way and release the energy into a totally different channel. We have shown that, due to time-reversal invariance, under a reversal of magnetic field bias, the functionalities of such a system strictly reverse. Finally, we have shown that the bandwidth of a linear, time-invariant cavity is always inversely proportional to its decay rate, both in reciprocal and nonreciprocal systems. The decay rate of a cavity is solely determined by the internal loss and the outcoupling coefficients , not by the incoupling coefficients. Thus, one cannot use nonreciprocity alone to force broadband fields into a long-lived resonance, which we have exemplified with a numerical demonstration. Instead, to overcome the time–bandwidth limit, time-varying systems or nonlinearities are required. Our results clarify claims that nonreciprocity may alleviate the strict limitations imposed by the trade-off between delay and bandwidth in photonic systems, and may help envisioning new efficient nonreciprocal components for, e.g., information processing or unidirectional transport.
Defense Advanced Research Projects Agency (DARPA) Nascent program; Air Force Office of Scientific Research (AFOSR) (FA9550-18-1-0379); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
See Supplement 1 for supporting content.
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