We show that a linear chain of circular macroscopic resonators coupled in parallel demonstrates the phenomena of “superabsorption” and superradiance. Both the frequency spectrum of the transmitted light through the resonator chain and the decay rate of the resonator chain being prepared in a proper initial state are proportional to the number of resonators in the chain, N, and the intensity of the emitted radiation grows as N 2. The spectral bandwidth, the growth of the decay rate, and the intensity are restricted by the dispersion of the waveguides connecting the resonators.
© 2009 Optical Society of America
Chains of linear resonators have been extensively studied for many years . Existing mathematical formalisms readily yield properties of any resonator structure, so at first glance it is hard to imagine that any novel findings may be possible in this field. Recent developments, though, contradict such an expectation. Aided with the ability to predict resonator structures and with advances fabrication technologies to obtain given properties, resonator chains have become useful in study of various phenomena, especially the processes that were previously realized only with quantum objects, like atoms and molecules [2, 4, 3, 5, 6, 7]. For instance, recently predicted [8, 9, 10, 11, 12, 13, 14] and demonstrated [15, 16, 17] coupled resonator induced transparency (CRIT) is a direct analogue of electromagnetically induced transparency (EIT) observed in coherent atomic systems [18, 19]. In this case, a pair of interacting resonators can produce spectral lines significantly narrower than the spectral width of each individual resonator. This line narrowing results from the Fano interference effect . The CRIT-related studies provide a novel way of thinking about well known linear optical phenomena observed in resonator chains , and in novel applications of the resonator systems. These studies support a more general trend towards discovery of “overlapping” areas between quantum and classical branches of physics, leading to a better understanding of the entire field [22, 24, 23].
In this work we show that the temporal decay rate of a parallel resonator chain (Fig. 1a) is proportional to the number of resonators (N), while the intensity of the decay is proportional to the square of the number of resonators (N 2), when all the resonators are prepared in the same excited state. We also show that the spectral linewidth of the resonator structure is proportional to the number of resonators (N).
The collective decay phenomenon is completely classical and is based on interaction among all the resonators (Fig. 1b). The lossless resonator chain is characterized with N-1 collective states decoupled from the environment , and one state strongly coupled to the environment. The decoupled states are CRIT-related, while the coupled state results in the superradiant decay [26, 27].
The interference of decaying radiation from the resonators is similar to the behavior of phased array of N coherent oscillators, or the far-field diffraction of plane waves from a periodic aperture of N slits, or free induction decay. In those cases the intensity is also proportional to N 2. The coherence of the decay is enforced from the beginning. This is different from the case of superfluorescence , which does not simply arise from the the coherence of the spontaneous emission, but from the interaction among atoms or molecules, which necessarily involves nonlinear couplings via the Maxwell-Bloch equations. Therefore, the resonator chain is one more entirely classical system that demonstrates superradiance.
The collective interaction manifests itself in both the frequency spectrum of the transmitted light and the temporal behavior of the resonator system. The frequency of the transmission band of the chain is proportional to the number of coupled resonators, when the distance between resonators is correctly selected. Hence, one can observe both “superabsorption” and superradiance in the resonator chain.
The width of the frequency pass-band of the structure, the decay speed, and the intensity of the emitted radiation are restricted due to the frequency dispersion of the waveguides connecting the resonators as well as variations in the resonator morphologies. On the other hand, current developments in applied science and technology makes possible creation of chains of nearly identical resonators [29, 30, 31]. Hence, to the best of our knowledge, our study provides the first practical suggestion for observing superradiance in the classical macroscopic domain. The proposed system has commonality with recently discovered superradiant emission process that through directional coupling enhances the rate and efficiency of radiation channeling from a linear array of distant atoms into a nanofiber . Unlike that study we show the feasibility of achieving superradiance behavior in an all-optical macroscopic system.
The parallel chain of resonators was formally studied in connection with high order optical filters [33, 25, 34, 35, 36]. Frequency bandgaps occurring in the infinite resonator chain have also been also discussed previously . It was noted that the spatial spacing between the resonators in the chain (“double-channel side-coupled integrated spaced sequence of resonators” ) is important because frequency resonances exist not only within the resonators but also among them. The normal modes of the chain involve all resonators and in this case the individual resonator cannot be considered as an independent entity. Band-gaps emerge in the dispersion relation of the infinite chain around each type of resonance: when the periodicity of the chain satisfies the Bragg condition and when the incoming light is resonant with the resonator modes. We show that the supperradiance takes place when the resonant conditions within resonators more strongly influences the parameters of the chain, as compared with the Bragg resonant condition. The superradiant behavior of the chain can also be envisioned from earlier works dealing with CRIT in the set of two parallel resonators. It was shown there that the two resonators have one mode uncoupled from the external environment, and one mode coupled to the environment. The last one has twice as wide a full width at half maximum (FWHM) as compared with a single resonator in the system [12, 37].
Let us consider a chain of identical lossless ring resonators shown in Fig. 1(a). The resonators are coupled via lossless single mode waveguides. For the sake of simplicity we assume that the resonators and the waveguides are made of the same material. The coupling of a resonator and a waveguide can be described with matrix equation  (see also [16, 17, 33, 34, 35, 36] for a description of resonators coupled in parallel)
where T is the coupling constant, k(ω) is the wave vector in the material, a is the radius of the resonator. The spectral properties of the jth elementary section of the resonator chain is, hence, described by
where we took into account that the resonators are separated by distance L. It is convenient to rewrite Eq. (2) in the compact form vj+1=Ujvj. The transfer matrix Uj is characterized with det(Uj)=1 because we consider a lossless system.
Spectral response of the chain of N resonators is given by the product Πj=N j=1 Uj. For a small number of resonators the spectrum of the add/drop ports is Lorentzian:
where t and r are the power transmission of the add/drop port of the chain, 2γ is the FWHM of the standing alone resonator, ω0 is the resonant frequency of the selected mode. We have assumed that k(ω)a=m 1 and k(ω)L=2πm 2, where m 1 and m 2 are integer numbers. Equations (3) and (4) show that adding a resonator to the chain broadens the spectrum of the system by 2γ. The constructive interference of the decays of the resonators into the waveguides results in the increased loading of the whole chain.
It is interesting to note that the spectral response of the chain of linear resonators Eq. (4) nearly coincides with the optical response in multi-quantum wells under Bragg conditions [38, 39, 40, 41]. This analogy clearly confirms that the phenomenon of superradiance is quite general and can be observed in both quantum and classical periodic systems.
To find the maximum number of constructively interfering resonators in the symmetric chain we consider two cases. In the first hypothetic case L=0, and Nmax formally can go to infinity. Using Bloch condition 
where κ is the effective propagation constant, we find that in the limit of infinite number of resonators the chain always reflects all the light. The spectral bandwidth of the structure grows infinitely with the increasing number of resonators in the chain (see Fig. 2(a)).
We consider next the chain with L=2πa. The maximum spectral bandwidth in this case is less than the FSR (see Fig. 2(b)). The maximum number of the constructively interfering resonators is restricted by
where F is the finesse of the individual resonator. Equation (6) was derived using numerical simulations defining Nmax as the ratio of the width of the spectral passband of the infinite resonator chain and the FWHM of a single resonator. Similar expression can be obtained from simple physical reasoning. The decay time of the set of constructively interacting resonators τd1=(2γN)-1 should exceed the phase delay resulting from light traveling through the off-resonant part of the structure τd2=nNL/c. Comparing those equations we get an expression like Eq. (6) with a slightly different numerical coefficient. Therefore, the higher is the finesse, the larger is the number of the constructively interfering resonators for a given distance between the resonators. On the other hand, the absolute spectral bandwidth of the passband decreases with increasing finesse.
It is easy to see that Nmax→∞ if L/a→0, so the decay time of the resonator structure goes to zero and the spectral width goes to infinity. This condition can be fulfilled if the resonators are located in different planes, however its practical implementation is hardly possible. Fundamentally, though, only the spectral width less than the free spectral range (FSR) of the resonators, or the spectral width of the first Brillouin zone, makes sense since the travel time of light from the input to the drop port is always larger that the round trip time within a resonator, which is necessary for the interference process to be established.
The set of the coupled resonators can be considered as a lumped system if N<Nmax. The free decay of the resonators can be described by a set of linear differential equations in this case:
where ej is the amplitude of the field in jth resonator. The set has one collective mode with a decay rate equal to γN. This is the mode hat is coupled with the external environment. The other modes are not decaying and are the “dark state” modes. If all the modes are initially prepared in the same state ej=e0, the chain of resonators decays with time τ2. The intensity of the emitted light is proportional to N 2, as in conventional superradiative systems. To observe superradiant decay experimentally one ultimately needs to excite all resonators with a single phase-coherent source, which should be uncoupled after the resonators are pumped. The coupling-uncoupling of resonators can be realized, e.g., either with MEMS technique  or electro-optically . Another approach is immersing the resonators into a lasing medium, which makes the experiment similar to the quantum well Bragg grating experiments [38, 39, 40, 41].
To conclude, we have shown that a chain of circular resonators coupled in parallel are capable of demonstrating a superradiative decay in the same way as atomic systems do. This represent the first practical system capable of exhibiting superradiance, a classical phenomenon that has otherwise been observed in microscopic systems only. The phenomenon occurs because of the constructive interference of the resonator decays. Similar interference phenomena can be observed in different configurations of coupled resonators. For instance, wide band (low-Q) compound optical and RF filters contain resonators that, being uncoupled from the chain, have much higher Qs than the filters. The Qs of those resonators are reduced because of the decay interference within the chain. The analogy is important because state of the art of fabrication of macroscopic resonator systems allows studying and predicting phenomena that are only observable in miniature quantum systems. It is interesting to generalize the treatment described in this paper to allow for nonlinear couplings and propagation among the resonators, and find if the system can evolve to a coherent decay, i.e. demonstrate the effect of superfluorescense. We plan to do it elsewhere. The dephasing processes originating from the losses in the resonator chain as well as variations in the resonator size are also under study.
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