Nonreciprocal cavities and the time-bandwidth limit: reply

Sander A. Mann; Dimitrios L. Sounas; Andrea Alù; Andrea Alù; Andrea Alù

doi:10.1364/OPTICA.401383

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

(1)$$\frac{\rm d}{{{\rm d}t}}a = \left({i{\omega _0} - {\gamma _r} - {\gamma _i}} \right)a + k{s_ +}.$$

Here ${\omega _0}$ is the resonance frequency and ${s_ +}$ is the incident amplitude from the port. The 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].

Fig. 1. (a) Adiabatically tapered MIM waveguide formed by silicon and gold. A pulse is launched towards the apex and progressively slows down. We record the fields inside the purple square (the area surrounding the apex). (b) The stored energy in the purple square quickly rises, as the broadband pulse approaches the termination relatively quickly but slowly decays as it is not reflected but only absorbed. (c) As the pulse approaches the termination, the pulse is compressed as the group velocity is continuously reduced.

Download Full Size | PDF

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.

Fig. 2. (a) Unidirectional delay line of length $L$. The dashed lines are input and output planes, and the section in between has low group velocity due to the narrow dielectric region. (b) A one-dimensional representation of the geometry in (a), where the effective index of each region is given by the plasmon wavenumber. In this case, ${k_s} \gg {k_0}$.

Download Full Size | PDF

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]:

(2)$$\Delta \omega \Delta t \le \frac{{L{k_c}}}{{2\sqrt 3}}\max \left[{\frac{{k_s^2\left({x,\omega} \right) - k_0^2\left(\omega \right)}}{{k_0^2\left(\omega \right)}}} \right],$$

where ${k_s}$ is the position-dependent wavenumber within the delay region with length $L$, ${k_0}$ is the wavenumber in the background region, ${k_c} = {k_0}({\omega _c})$ is the wavenumber in the background at the pulse center frequency, and the maximum value is taken over the wavenumbers within the bandwidth of the incident pulse. Hence, nonreciprocity does not help in overcoming the delay-bandwidth limit in this geometry. This is in fact expected, since reciprocity is not an assumption in the derivation of Miller’s limit, and this limit must therefore hold for both reciprocal and nonreciprocal linear systems [46].

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]:

(A1)$$\frac{\rm d}{{{\rm d}t}}a = (i{\omega _0} - {\gamma _r} - {\gamma _i})a + {k_r}{n_r}(t) + {k_i}{n_i}(t).$$

The resonance is coupled to thermal radiation incident from the exterior ${n_r}(t)$, with incoupling coefficient ${k_r}$, as well as to thermal currents inside the resonator ${n_i}(t)$, with coefficient ${k_i}$. If we Fourier transform Eq. (A1), we have for the modal amplitude

(A2)$$a(\omega) = - \frac{{{k_r}{n_r}(\omega) + {k_i}{n_i}(\omega)}}{{i({\omega _0} - \omega) - {\gamma _r} - {\gamma _i}}}.$$

Since the driving terms are stochastic white noise sources, we find [47]

(A3)$$\begin{split}&\langle {n_i}(\omega)n_i^*(\omega ^\prime)\rangle = \frac{1}{{2\pi}}{k_B}{T_i}\delta (\omega - \omega ^\prime),\\&\langle {n_r}(\omega)n_r^*(\omega ^\prime)\rangle = \frac{1}{{2\pi}}{k_B}{T_e}\delta (\omega - \omega ^\prime),\\&\langle {n_r}(\omega)n_i^*(\omega ^\prime)\rangle = 0,\end{split}$$

where the brackets indicate autocorrelations, ${k_B}$ is Boltzmann’s constant, and ${T_i}$, ${T_e}$ are the temperature of the resonator and the ambient, respectively.

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]

(A4)$$\begin{split}\iint{\langle a(}\omega ){{a}^{*}}({\omega^\prime})\rangle {\rm d}\omega {\rm d}{\omega ^\prime}& = \frac{(|{{k}_{i}}{{|}^{2}}+|{{k}_{r}}{{|}^{2}}){{k}_{B}}T}{2\pi }\\&\quad\times\iint\!{\frac{\delta (\omega -{\omega^\prime})}{{{({{\omega }_{0}}-\omega )}^{2}}+{{({{\gamma }_{r}}+{{\gamma }_{i}})}^{2}}}}{\rm d}\omega {\rm d}{\omega^\prime},\end{split}$$

where we have used Eqs. (A2) and (A3). For now, we are interested in the scenario suggested by Tsakmakidis 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

(A5)$$\iint{\langle a(}\omega ){{a}^{*}}({\omega^\prime})\rangle {\rm d}\omega {\rm d}{\omega^\prime} = \frac{(2{{\gamma }_{i}}+|{{k}_{r}}{{|}^{2}}){{k}_{B}}T}{2({{\gamma }_{i}}+{{\gamma }_{r}})}.$$

Since we posited that the cavity is in thermal equilibrium, it should satisfy the equipartition theorem $\iint{\langle a(}\omega ){{a}^{*}}({\omega^\prime})\rangle {\rm d}\omega {\rm d}{\omega^\prime} = {{k}_{B}}T$. The only way this is possible, as apparent from Eq. (A5), is if $|{k_i}{|^2} = 2{\gamma _i}$. Hence, it is impossible for absorption processes to balance radiative processes.

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.

REFERENCES

1. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012). [CrossRef]

2. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013). [CrossRef]

3. A. A. Maznev, A. G. Every, and O. B. Wright, “Reciprocity in reflection and transmission: What is a ‘phonon diode’?” Wave Motion 50, 776–784 (2013). [CrossRef]

4. C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018). [CrossRef]

5. V. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” arXiv:2001.04848 (2020).

6. K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017). [CrossRef]

7. M. Tsang, “Quantum limits on the time-bandwidth product of an optical resonator,” Opt. Lett. 43, 150–153 (2018). [CrossRef]

8. S. Buddhiraju, Y. Shi, A. Song, C. Wojcik, M. Minkov, I. A. D. Williamson, A. Dutt, and S. Fan, “Absence of unidirectionally propagating surface plasmon-polaritons at nonreciprocal metal-dielectric interfaces,” Nat. Commun. 11, 674 (2020). [CrossRef]

9. S. A. Hassani Gangaraj and F. Monticone, “Do truly unidirectional surface plasmon-polaritons exist?” Optica 6, 1158–1165 (2019). [CrossRef]

10. S. A. Mann, D. L. Sounas, and A. Alù, “Nonreciprocal cavities and the time–bandwidth limit,” Optica 6, 104–110 (2019). [CrossRef]

11. K. L. Tsakmakidis, Y. You, T. Stefanski, and L. Shen, “Nonreciprocal cavities and the time–bandwidth limit: comment,” Optica 7, 1097–1101 (2020). [CrossRef]

12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

13. S. Wonjoo, W. Zheng, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004). [CrossRef]

14. A. Ishimaru, “Unidirectional waves in anisotropic media and the resolution of the thermodynamic paradox,” Air Force Technical Rep. 69 (1962).

15. G. Barzilai and G. Gerosa, “Rectangular waveguides loaded with magnetised ferrite, and the so-called thermodynamic paradox,” Proc. Inst. Electr. Eng. 113, 285 (1966). [CrossRef]

16. U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014). [CrossRef]

17. L. Shen, X. Zheng, and X. Deng, “Stopping terahertz radiation without backscattering over a broad band,” Opt. Express 23, 11790–11798 (2015). [CrossRef]

18. M. Marvasti and B. Rejaei, “Formation of hotspots in partially filled ferrite-loaded rectangular waveguides,” J. Appl. Phys. 122, 233901 (2017). [CrossRef]

19. D. A. B. Miller, L. Zhu, and S. Fan, “Universal modal radiation laws for all thermal emitters,” Proc. Natl. Acad. Sci. USA 114, 4336–4341 (2017). [CrossRef]

20. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004). [CrossRef]

21. D. F. P. Pile and D. K. Gramotnev, “Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguides,” Appl. Phys. Lett. 89, 2004–2007 (2006). [CrossRef]

22. E. Verhagen, A. Polman, and L. K. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16, 45–57 (2008). [CrossRef]

23. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

24. P. Lalanne, S. Coudert, G. Duchateau, S. Dilhaire, and K. Vynck, “Structural slow waves: parallels between photonic crystals and plasmonic waveguides,” ACS Photon. 6, 4–17 (2019). [CrossRef]

25. D. E. Fernandes and M. G. Silveirinha, “Topological origin of electromagnetic energy sinks,” Phys. Rev. Appl. 12, 014021 (2019). [CrossRef]

26. M. A. Green, E. D. Dunlop, J. Hohl-Ebinger, M. Yoshita, N. Kopidakis, and A. W. Y. Ho-Baillie, “Solar cell efficiency tables (version 55),” Prog. Photovoltaics 28, 3–15 (2020). [CrossRef]

27. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32, 510–519 (1961). [CrossRef]

28. E. D. E. Kosten, J. H. J. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013). [CrossRef]

29. A. Polman, M. Knight, E. C. Garnett, B. Ehrler, and W. C. Sinke, “Photovoltaic materials: Present efficiencies and future challenges,” Science 352, aad4424 (2016). [CrossRef]

30. R. M. Swanson, “Photovoltaic Concentrators,” in Handbook of Photovoltaic Science and Engineering (Wiley, 2005), pp. 449–503.

31. S. A. Mann, R. R. Grote, R. M. Osgood, A. Alù, and E. C. Garnett, “Opportunities and limitations for nanophotonic structures to exceed the Shockley-Queisser limit,” ACS Nano 10, 8620–8631 (2016). [CrossRef]

32. E. D. Kosten, B. M. Kayes, and H. A. Atwater, “Experimental demonstration of enhanced photon recycling in angle-restricted GaAs solar cells,” Energy Environ. Sci. 7, 1907 (2014). [CrossRef]

33. L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys. 19, 1163–1175 (1948). [CrossRef]

34. R. M. Fano, “Theoretical limitations on the broad-band matching of arbitrary impedances,” J. Franklin Inst. 249, 57–83 (1950). [CrossRef]

35. H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019). [CrossRef]

36. A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018). [CrossRef]

37. Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007). [CrossRef]

38. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004). [CrossRef]

39. K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000). [CrossRef]

40. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Delay–bandwidth product and storage density in slow-light optical buffers,” Electron. Lett. 41, 208 (2005). [CrossRef]

41. J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. 32, 133 (2007). [CrossRef]

42. D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007). [CrossRef]

43. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004). [CrossRef]

44. M. Tymchenko, D. Sounas, A. Nagulu, H. Krishnaswamy, and A. Alù, “Quasielectrostatic wave propagation beyond the delay-bandwidth limit in switched networks,” Phys. Rev. X 9, 031015 (2019). [CrossRef]

45. S. A. Mann, D. L. Sounas, and A. Alù, “Broadband delay lines and nonreciprocal resonances in unidirectional waveguides,” Phys. Rev. B 100, 020303 (2019). [CrossRef]

46. D. A. B. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24, A1 (2007). [CrossRef]

47. H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

48. L. Zhu, S. Sandhu, C. Otey, S. Fan, M. B. Sinclair, and T. Shan Luk, “Temporal coupled mode theory for thermal emission from a single thermal emitter supporting either a single mode or an orthogonal set of modes,” Appl. Phys. Lett. 102, 103104 (2013). [CrossRef]

49. L. Zhu and S. Fan, “Near-complete violation of detailed balance in thermal radiation,” Phys. Rev. B 90, 220301 (2014). [CrossRef]

50. L. Landau, E. Lifshitz, and L. Pitaevskii, “Statistical physics part 2,” in Course of Theoretical Physics (Pergamon, 1980), Vol. 9.

Nonreciprocal cavities and the time-bandwidth limit: reply

Abstract

1. INTRODUCTION

2. NONRECIPROCAL RESONANCES

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

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

5. CONCLUSIONS

APPENDIX A: NONRECIPROCAL CAVITIES IN THERMAL EQUILIBRIUM

Funding

Disclosures

REFERENCES

Cited By

Figures (2)

Equations (7)

Optica