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Abstract
Coupled-resonator-induced transparency (CRIT) and coupled-resonator-induced absorption (CRIA) are the all-optical analogs of electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA), and enable an optical pulse to propagate subluminally and superluminally in integrated photonic systems, respectively. Here, we describe the results of a theoretical study, which reveals a reversible CRIA-CRIT transition using coupled optical ring resonators. Analogous to the dispersion reversal occurring in the EIA-EIT transition, the dispersion also reverses during the CRIA-CRIT transition. Besides the EIA-EIT crossover analogy, we describe additional analogs of the coherent atomic phenomena arising in the CRIA-CRIT transition. We also discuss the specific conditions that lead to strongly enhanced dispersion. Furthermore, in contrast to the previous studies on coupled resonators, we reveal acquiring a critically and undercoupled slow-light CRIT from an overcoupled fast-light CRIA. Our experimentally viable study provides a route for the dynamic control and reversal of dispersion, which, among other, is important for quantum information technology, the Fresnel light dragging effect, temporal cloaking, and enhanced performance of the optical devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The speed of light plays a central role in science and engineering and the control of the group velocity of optical pulses is vital for the future light-based technologies [1]. Electromagnetically induced transparency (EIT) has emerged as a preferred technique to generate subluminal light pulses. The EIT effect originates due to the atomic coherences arising in three-level atoms, where a coherent control and probe field simultaneously drive the optical transitions from the ground states to a common excited level. This dramatically alters the absorptive and dispersive properties of the medium since a strong control field can suppress the probe absorption [2]. The superluminal dispersion of the medium vanishes as a narrow transparency window emerges inside the broad probe absorption band and the resulting normal dispersion greatly intensifies near the line center, leading to distortion-free ultraslow propagation of an optical pulse [3]. In contrast, electromagnetically induced absorption (EIA) is related to the enhancement of the probe absorption in degenerate two-level systems in the presence of a control field, and in this case a narrow absorption window emerges inside the probe absorption band [4]. EIA occurs due to spontaneous transfer of coherence [5] or population [6] and results in group velocities greater than the vacuum speed of light owing to an enhancement of the superluminal dispersion of the medium [7]. The distinctive characteristics of EIT and EIA have resulted in several important developments including, atomic clocks [8], lasing without inversion [9], and sensitive magnetometry [10]. The following studies revealed the occurrence of EIT and EIA-like phenomena in the classical system of coupled optical resonators as well [11–23]. Indeed, coupled-resonator-induced transparency (CRIT) and coupled-resonator-induced absorption (CRIA), the all-optical analogs of EIT and EIA, display EIT and EIA-like spectral and dispersive features [24,25] on photonic microstructures at room temperature. The decoherence-free aspect of CRIT and CRIA and the ability to realize these resonances and the related dispersive effects over a broad wavelength regime by simply controlling the size of the optical resonators are advantageous, since the analogous effects in the atomic media can only be achieved at specific wavelengths due to the required energy level schemes. CRIT and CRIA enable applications such as optical memories [26], optical force enhancement in waveguide-resonator systems [27], optical signal routing [28], and enhanced nonlinear optical interactions owing to the long-lived EIA-like narrow absorptive resonance [29]. Experimental studies in quantum optics have demonstrated transformation of EIT into EIA using the different level schemes in Rb [30], Cs [31], and K [32]. In addition, transformation of EIA into EIT in Cs vapors has also been reported [33].
Here, we theoretically demonstrate the all-optical analog of the EIA-EIT transition based on coherently coupled optical ring resonators of high and low intrinsic quality (Q) factors. In direct analogy to the atomic media, we show that a reversal of dispersion also occurs during the reversible CRIA-CRIT transition since the superluminal (or fast) light realized initially is transformed into subluminal (or slow) light. We consider experimental schemes for the implementation of these analogous optical effects and show that tuning of the evanescent coupling between an excitation waveguide and coupled resonators mediates the reversible CRIA-CRIT transition by driving the system through critical coupling and reversing the dispersion as an overcoupled fast-light CRIA develops into a slow-light undercoupled CRIT. An alternate approach leads to the CRIA-CRIT transition as a result of tuning of coupling between the two resonators. Although both mechanisms lead to the CRIA-CRIT transition, distinct evolutions of the spectral and dispersive properties of mutually coupled resonators occur for the two cases, as detailed later in this article. In the later case, the broader resonance is almost unaffected while the amplitude and linewidth of the sharp features are modified. In the former case, the entire resonance reshapes and following the optical transitions a critically and undercoupled CRIT emerge in the transmission spectrum. In the previous investigations, critically and undercoupled CRIT were realized as a result of coupling between an undercoupled and overcoupled optical resonator, which induces a narrow peak inside the broad transmission dip of the undercoupled resonator [14,15,24]. Furthermore, we also show that under specific conditions, strongly enhanced normal and anomalous dispersion is enabled by the coupled-resonator system.
A single-resonator may be over, under, or critically coupled to an excitation waveguide, depending on the relative strengths of intrinsic and coupling losses [34]. Furthermore, the linewidth of an optical resonance depends on the intrinsic and coupling losses while intrinsic losses also attenuate the resonantly circulating optical field. Losses arising due to imperfections in the fabrication processes represent the intrinsic losses, while coupling losses stem from the decay of the confined optical field to the adjacent optical systems. The optical states of coupled resonators, however, are more complex owing to a broader category of the coupling conditions. Furthermore, similar to the single-resonator case, the dispersion of coupled resonators also depends on the coupling condition. An overcoupled CRIA produces superluminal light, while an undercoupled CRIT results in subluminal light. Transmission ceases at resonance for a critically coupled CRIA whereas zero transmission occurs at the off-resonance dips in a critically coupled slow-light CRIT [35]. The additional coupling states arising in coupled resonators include an undercoupled CRIA and an overcoupled CRIT, and both these resonances generate subluminal light.
The schematic of Fig. 1(a) shows the arrangement of the waveguide and ring resonators and also points out the possibility of variable waveguide-resonator as well as inter-resonator coupling, which modulates the coupling loss of the system. The incident optical field couples to the ring resonator RR1, which is located adjacent to the excitation waveguide and has a lower intrinsic Q-factor in comparison to the higher Q resonator RR2, which is not directly coupled to the excitation waveguide. If the two ring resonators are placed in close proximity, the evanescent coupling excites the resonant mode of RR2 as well, leading to coherent optical interactions between the two resonators. Figure 1(b) describes the phase space analysis of a single-resonator. Here, the transmitted electric field and dispersive response of an optical resonance can be examined simultaneously in a complex plane where the horizontal and vertical axis represent the real and imaginary values of the complex transmitted electric field, respectively [35,36]. As the optical field circulates inside a ring resonator, it acquires a round trip phase $\varphi$, whose value varies as the frequency of the incident optical field is swept across the optical resonance. As shown in Fig. 1(b), the plot starts at resonance ($\varphi$ = 0) and $\varphi$ increases in the anticlockwise direction. Therefore, an electric field trace emerges as $\varphi$ is varied in the complex plane. The transmission amplitude of a resonance in the phase space plot at a specific value of $\varphi$ (or frequency) is represented by the distance from the origin and the angle ${\theta _T}$ describes the corresponding effective transmission phase. The dispersive behavior of an optical resonance at any frequency can be readily observed on a phase space or phasor plot by examining how $\varphi$ and ${\theta _T}$ vary around a particular point on the loop since $d{\theta _T}/d\varphi$ is related to the dispersion of the system. It turns out that fast-light may occur at a particular location on the phasor plot where $\varphi$ and ${\theta _T}$ do not simultaneously increase or decrease along a rotational direction and this may happen if the electric field trace does not encircle the origin. This is clearly observed in the phasor plot for a single undercoupled resonator at the point of interest, which is indicated by an arrow, since in this case, as $\varphi$ increases (decreases) ${\theta _T}$ decreases (increases) [Fig. 1(b)]. The phase space description of coupled resonators is explained in section 5.
Fig. 1. (a) Schematic of the waveguide and coupled optical ring resonators (RRs) illustrates the possibility of coupling variation between the waveguide and ring resonator RR1 as well as coupling control between the two ring resonators (RR1 and RR2). (b) The phase space plot for a single-ring resonator for the undercoupled (green), critically coupled (red), and overcoupled (blue) case. The round trip resonator phase starts at resonance ($\varphi $ = 0), and increases in the anticlockwise direction.
2. Theoretical model for coupled optical resonators
The schematic of Fig. 1(a) shows the arrangement of the excitation waveguide and the two mutually coupled optical ring resonators. Here, coupling between the waveguide and RR1 is described by self-coupling coefficient ${r_1}$ while self-coupling coefficient ${r_2}$ characterizes the inter-resonator coupling. The resonantly circulating field acquires a phase ${\varphi _j} = (2\,\pi \,{n_j}\,{d_j}/c)\,(\upsilon - {\upsilon _j})$ over one round trip and the attenuation of the circulating field due to intrinsic losses is characterized by ${a_j} = \exp ({{ - {\kern 1pt} {\kern 1pt} {\alpha _j}{d_j}} / 2})$, where the attenuation coefficient ${\alpha _j}$ for the resonator j is related to the intrinsic Q-factor of the resonator ${Q_j}\,$ by ${\alpha _j} = 2{\kern 1pt} {\kern 1pt} \pi {\kern 1pt} {n_j}/({\kern 1pt} {Q_j}\,{\lambda _j}{\kern 1pt} \textrm{)}$. Here, $\upsilon$ is the frequency of the tunable probe light, ${\upsilon _j}$ the resonance frequency, ${d_j}$ the resonator circumference, nj the index of refraction, and ${\lambda _j}$ the free-space resonant wavelength of the resonator j, and j = 1, 2 specifies the ring resonators RR1 and RR2, respectively. The normalized steady-state transmitted field ${\tau _1} = {E_t}/{E_0}$ is given by [14]
The transmitted field of the RR2, arising due to coupling to RR1, is expressed as
The transmitted field-intensity of the coupled resonator system is obtained as ${|{{\tau_1}} |^2}$. For the numerical calculations, we consider co-resonant ring resonators (${\varphi _1} = {\varphi _2} = \varphi$ and ${n_1} = {n_2} = n$) having a radius of 10 μm and resonant to a free-space wavelength of 1548.36 nm. Furthermore, the rings and waveguide are assumed to be composed of silicon (n = 3.45). The group index ${n_g}$ for an optical pulse that propagates through the coupled resonator system is given by
3. CRIA-CRIT transition based on tuning of inter-resonator coupling
The results of the computational investigation of the CRIA-CRIT transition due to a variation in the inter-resonator coupling, which is described by the self-coupling coefficient ${r_2} = \sqrt {1 - t_2^2}$ and realized by moving RR2 closer to RR1, are presented in Fig. 2, where the intrinsic Q factors of RR1 and RR2 are Q1 = 2×105 and Q2 = 2×106, respectively. As ${r_2}$ decreases, the cross-coupling coefficient ${t_2}$ increases and, therefore, coupling between the two resonators increases. The results of Fig. 2 correspond to a fixed coupling between the waveguide and RR1 (r1 = 0.96326), and show the evolution of transmission, the corresponding effective phase, and group index as coupling between the resonators increases. Here, coupling between the waveguide and RR1 is analogous to the probe field Rabi frequency while inter-resonator coupling bears a similarity to the control field Rabi frequency [14]. Initially an overcoupled CRIA is realized due to coupling between an overcoupled RR1 and undercoupled RR2 (Fig. 2(a1)), which gives rise to superluminal light and increased coupling between the two resonators leads to an increase in the amplitude of the narrow dip until a critically coupled CRIA is obtained. With a further increase in the inter-resonator coupling, the dispersion reverses as an undercoupled slow-light CRIA appears in transmission (Fig. 2(a4)), which subsequently evolves into an overcoupled slow-light CRIT (Fig. 2(a6)). In the superluminal regime, the superluminal dispersion is enhanced as the amplitude of the narrow CRIA dip increases for stronger inter-resonator couplings. On the other hand, the subluminal dispersion of the system decreases as an increase in coupling shifts the resonances away from critical coupling and the linewidths of the sharp CRIA and CRIT features become broader. However, the off-resonance broader dip remains almost unaffected throughout the entire transition.
Fig. 2. Sequence of plots show transmittance T, the transmission phase $\theta _T$, and group index ng for a coupled resonator where coupling between the two resonators increases. For this case, $r_1$ = 0.96326 remains constant. Coupling between the resonators is minimum for the top row and it increases for the subsequent rows. The values of the self-coupling coefficient $r_2$ for varying couplings are stated in the figure for each case. Critical coupling occurs between (a3) and (a4).
4. CRIA-CRIT transition based on tuning of waveguide-resonator coupling
In addition to tuning of inter-resonator coupling, we show that a variation of coupling between the waveguide and RR1, accounted by the self-coupling coefficient ${r_1} = \sqrt {1 - t_1^2}$, also leads to the reversible CRIA-CRIT transition. Figure 3 shows the transmission spectrum, the corresponding effective transmission phase, and the group index for a coupled resonator where coupling between the waveguide and RR1, is gradually decreased as the waveguide is moved away from RR1. For these results, the inter-resonator coupling r2 = 0.999999 is fixed and the intrinsic Q-factors correspond to Q1 = 5×105 and Q2 = 5×106. Similar to the inter-resonator coupling case, an overcoupled fast-light CRIA resonance also appears initially in this case [Fig. 3(a1)]. As ${r_1}$ increases, coupling between the resonators decreases owing to a decrease in the corresponding cross-coupling coefficient ${t_1}$, and the resonant modes of both ring resonators are affected, as indicated by the modifications in both broader and narrower spectral features [Fig. 3(a2)-(a4)]. The corresponding dispersive response shows an enhancement of the superluminal dispersion. This trend continues until a critically coupled CRIA is realized, which later transforms into an undercoupled slow-light CRIA [Fig. 3(a5)].
Fig. 3. Sequence of plots show transmittance T, the transmission phase $\theta _T$, and group index ng for a coupled resonator where coupling between the waveguide and RR1 decreases. For this case, $r_2$ = 0.999999 remains constant. The waveguide-RR1 coupling $r_1$ is maximum for the top row and it decreases for the subsequent rows. The values of the self-coupling coefficient $r_1$ are listed in the figure for the different couplings between the waveguide and RR1. The upper part of the figure describes the evolution of coupled resonators prior to the occurrence of critical coupling while the lower part shows the behavior of the system after critical coupling.
A further decrease in coupling leads to an overcoupled slow-light CRIT [Fig. 3(a6)], which develops into a critically coupled slow-light CRIT [Fig. 3(a7)]. The critically coupled CRIT finally transforms into an undercoupled slow-light CRIT [Fig. 3(a8)]. Like the inter-resonator coupling case, both sub and superluminal dispersions maximize near critical coupling and continuously decrease as a variation in coupling moves the system away from critical coupling.
It can be readily observed that significantly greater dispersions are realized in this case in comparison to the inter-resonator coupling case. Specifically, the overcoupled fast-light CRIA [Fig. 3(a4)] demonstrates large superluminal dispersion [Fig. 3(c4)]. Similarly, enhanced subluminal dispersion [Fig. 3(c5)] occurs for the undercoupled CRIA [Fig. 3(a5)]. On the other hand, large subluminal dispersion is realized at resonance for the critically coupled CRIT [Fig. 3(a7)] while the off-resonance superluminal dispersion is also greatly enhanced [Fig. 3(c7)].
However, if coupling is increased and the Q factors are reduced, the dispersive response of the system becomes similar to the inter-resonator coupling case, which represents the actual dispersive behavior of coupled resonators. Therefore, it is a combination of weak coupling and high resonator Q factors that yield large normal and anomalous dispersion. Similarly, this approach may also be used to achieve enhanced sub and superluminal dispersions through optical transitions driven by the inter-resonator coupling.
The phase profiles of coupled resonators for both coupling control methods show that for the over and undercoupled CRIA as well as the overcoupled CRIT, the off-resonance profiles are similar to the phase of a single overcoupled resonator. A narrow feature, which possesses a negative slope, appears at resonance as an undercoupled RR2 couples to an overcoupled RR1 to yield superluminal light at resonance for the overcoupled CRIA. However, positive slope features at resonance in the case of an undercoupled CRIA and overcoupled CRIT and enhanced subluminal light is obtained at resonance for both cases since a reversal of dispersion follows critical coupling. For critically and undercoupled CRIT, the off-resonance phase profiles are similar to the phase of a single undercoupled resonator. Therefore, superluminal dispersion occurs at off-resonance frequencies while subluminal dispersion is realized at resonance for both cases owing to appearance of a positive slope feature.
Fig. 4. The phase space plots corresponding to (a) the inter-resonator and (b) wavegide-RR1 coupling control. These plots are related to Figs. 2 and 3 and describe the phase space behavior of coupled resonators for the various couplings considered in Figs. 2 and 3. (a) For the inter-resonator coupling case, initially an overcoupled fast-light CRIA is obtained (black). The amplitude of the narrow CRIA dip increases as coupling becomes stronger (red and green). An undercoupled slow-light CRIA (wine) emerges after critical coupling and this resonance moves away from critical coupling as coupling is further increased (blue). Finally, the undercoupled CRIA transforms into an overcoupled slow-light CRIT (orange). In this case, the outer loop is observed only for the overcoupled CRIA since the outer loops for the remaining cases overlap. (b) In the case of waveguide-RR1 coupling modulation, an overcoupled fast-light CRIA is realized initially, which is represented here by the black loop. Owing to a decrease in coupling, the amplitude of the narrow CRIA dip increases (red and green). The overcoupled CRIA nears critical coupling (wine). After critical coupling, an undercoupled slow-light CRIA appears (blue). The amplitude of the narrow CRIA feature decreases as coupling further decreases until an overcoupled slow-light CRIT emerges (orange). This is followed by the emergence of a critically coupled slow-light CRIT (olive), which finally converts into an undercoupled slow-light CRIT (cyan).
5. Phase space analysis of coupled resonators
The phase space diagrams for the inter-resonator and waveguide-resonator coupling case are shown in Fig. 4(a) and (b), respectively, where the real and imaginary values related to each transmission resonance appearing in Figs. 2 and 3 are plotted in the respective diagram. The phasor plots for coupled resonators also start at resonance ($\varphi$ = 0) and the round trip resonator phase $\varphi$ increases in the anticlockwise direction. However, in contrast to the single-resonator phasor plot, the resonance is located on the opposite end of the horizontal axis, where the inner loop crosses the horizontal axis but does not intersect the outer loop [35]. The phase space plots agree well with the spectral and dispersive response of coupled resonators for the two coupling cases. Furthermore, the phasor plots show that due to coupling modulation, the coupling states of coupled resonators evolve into another state, although the coupling states of individual resonators are found to be unaffected. For the inter-resonator coupling case [Fig. 4(a)], the phasor plot verifies that only the sharp feature of the resonance, depicted by the inner loop, is altered. In Fig. 4(a)], the outer loop is only observed for the overcoupled CRIA since the outer loops of the different coupling states overlap. The phasor plot for the waveguide-resonator coupling case [Fig. 4(b)] validates that a modulation of coupling alters the entire resonance. Furthermore, Fig. 4(b) also illustrates access to all coupling states of coupled resonators using this coupling control method. However, critically coupled states are not shown in Fig. 4 since the behavior of coupled resonators was not investigated at critical coupling.
6. Discussion
We show that the optical coherence effects reported in this article strongly resemble the quantum coherence phenomena occurring in driven atoms. For this purpose, we first consider the case of CRIT-CRIA transition arising due to tuning of inter-resonator coupling, which is analogous to the control field Rabi frequency [14]. In this case, essentially only the narrow feature of the optical resonance, which arises as the high-Q resonator couples to the low-Q resonator, is altered and the central CRIA dip transforms into a CRIT peak as the coupled resonator system moves through critical coupling. This behavior closely resembles the EIT-EIA transformation observed previously in degenerate three-level Λ-type Rb atoms [37]. In this case, addition of a second control field that counter propagates with respect to the probe and the other control field is found to affect only the narrow spectral features of the resonance during the transition. This happens because the co-propagating control field produces EIT in a Λ configuration while EIA emerges in the presence of both control fields due to the formation of an N-type system and Doppler averaging [37]. In the case of coupled resonators, increased inter-resonator coupling leads to an enhancement of the evanescent-wave coupling or equivalently enhanced tunneling of the resonant light from RR1 to RR2, which is verified through the calculated intracavity field intensities. As a larger fraction of nearly resonant photons tunnel to RR2, a corresponding variation in the narrow spectral features occur, as observed in the transmission spectra of Fig. 2. However, the broader off-resonance profile of the resonance is unaltered since for weak inter-resonator coupling, tunneling mainly happens for the resonant photons.
The transformation of CRIA into CRIT due to tuning of the waveguide-resonator coupling also bears a striking similarity to the coherent atomic media since the saturation effects related to a strong probe field have been found to alter the absorptive response of an atomic medium notably. A strong probe field causes enhanced absorption under the same conditions in which a weak probe would experience optical transparency. Experiments involving a ladder-based three-level EIT system in Rb have demonstrated enhanced absorption under the application a strong probe field, resulting in the conversion of EIT into an EIA-like resonance [38]. In an analogous manner, CRIT is obtained if coupling between the waveguide and RR1, which is akin to the probe field Rabi frequency [14], is weak while a stronger waveguide-resonator coupling produces a CRIA resonance. In the initial studies, an analogy was established between CRIT and EIT based on Λ-type atoms [14]. However, as we show here, the optical resonances of coupled resonators may also resemble the resonances of more complex atomic systems under specific conditions. Furthermore, the transition between EIA and EIT typically involves a modification of the energy level structure [37]. The fact that CRIA-CRIT transformation demonstrates characteristics that are analogous to the EIA-EIT crossover further strengthens the assertion that coupled resonators can mimic atoms having distinct type of level structures.
It is worth pointing out that our calculations reveal that by optimizing the waveguide-resonator coupling, the fast-light CRIA may be directly converted into a critically coupled slow-light CRIT since the occurrence of the undercoupled CRIA and overcoupled CRIT is greatly suppressed. Since the resonator transmission phase is analogous to the phase index of an EIT atomic medium [14], the transformation of an overcoupled CRIA into a critically coupled CRIT exactly matches the EIA-EIT conversion in ensemble of driven atoms.
Tunable waveguide-resonator and inter-resonator coupling strengths may be realized through integrated microlectromechanical-system-actuated deformable waveguides, which enable dynamic post fabrication control of the optical resonances due to lateral displacement of the waveguide. Variable optical coupling has been demonstrated for waveguide-coupled microdisks on silicon-on-insulator substrates [39]. This method may also be used to control coupling between two resonators by fabricating optical resonators on deformable structures. Another approach for the control of coupling is based on optical gradient forces. Free-standing slot-waveguides, realized through membrane fabrication technique, have been used to generate optical gradient forces that are strong enough for coupling control due to deformation of the waveguide [40]. Alternatively, the inter-resonator coupling may be controlled by using polymeric resonators on mechanically stretchable elastomeric substrates [41].
The transformation between fast and slow-light also occurs in a single-resonator [42] since an undercoupled resonator produces fast-light at resonance while slow-light is obtained through an overcoupled resonator and the two coupling regimes are linked via critical coupling. These optical coupling states of a single-resonator are analogous to two-level atoms where atoms with thermally governed population generate fast-light at resonance while slow-light is realized in the case of atoms with inverted population. However, for both single-resonators and two-level atoms, the sub and superluminal optical pulses are significantly attenuated and distorted. On the other hand, coupled resonators are analogous to multilevel atoms and the CRIT and CRIA effects minimize the dissipative effects. Reversal of dispersion with EIT and EIA-like dispersive behavior has not been demonstrated experimentally for coupled resonators although tuning of sub and superluminal dispersion has been achieved in indirectly coupled resonators as a result of thermal tuning of the optical resonators [25,43]. In the case of mutually coupled resonators, slow to fast-light conversion is attained due to decoupling between an under and overcoupled optical resonator [24,44], although CRIT disappears as the resonators are decoupled. A magnetic field rather than a strong control field in an atom-optomechanical cavity [45] and a cavity magnon-photon system [46] have been proposed to achieve long-lived slow-light at room temperature. However, steady-state dispersion inherently surfaces in coupled resonators in the absence of external fields. In contrast to the atomic media, the measurement or shot noise in coupled resonators can be suppressed by using a stable and low-noise laser. Furthermore, optical resonators usually do not suffer from the quantum back action noise that is fundamental to the optomechanical cavity systems [47]. The dominant noise mechanism in optical resonators is the thermal noise, which may cause a shift in the resonance frequency. However, materials displaying high transparency may be used to suppress the thermal noise significantly since the absorption of laser light that excites an optical resonance is minimized [48]. Indeed, light absorption in crystalline materials, such as calcium fluoride, is notably smaller in comparison to fused silica and silicon [49].
The conversion of CRIA into CRIT owing to manipulation the optical chiral states at an exceptional point [50] or due to tuning of coupling [51] has been demonstrated experimentally. The results presented in this article accurately describe the spectral, phase, and dispersive response of coherently interacting optical resonators and reveal CRIA-CRIT transitions with large amplitudes of the narrow spectral features. Furthermore, the various coupling states of coupled resonators have been clearly identified and the ways to move across the coupling regimes have been discussed as well (Figs. 2-4).
The present study provides an experimentally realistic strategy for controlling and reversing the dispersion owing to the EIT and EIA-like dispersion of coupled resonators. These dispersive characteristics are important for applications such as quantum information processing owing to significant and controllable group delays [52,53]. Similarly, tunable and simultaneously occurring slow and fast light is essential for specific temporal optical cloaking schemes where a time lens splits a signal field and the dispersive elements create a temporal gap owing to delay and advancement of the split fields [54,55]. The strong normal and anomalous dispersion revealed here is useful for the temporal separation and recombination of the split fields for cloaking an event occurring in time. Furthermore, it is well-known that the Fresnel light dragging effect, occurring in a non-stationary medium, is notably enhanced in the presence of sub or superluminal dispersion [56,57] and the large dispersions revealed here are certainly useful in achieving a pronounced dragging effect for light. Finally, the continuous tuning of slow and fast light is highly advantageous for enhancing the functionality of optical devices including, gyroscopes [58], interferometers [59], and buffers [60].
7. Conclusions
We have revealed the optical analog of EIA-EIT transition based on coupled optical ring resonators. Our study elucidates the spectral, phase, and dispersive characteristics of coherently interacting optical resonator in the different coupling regimes. We show that manipulation of coupling between the ring resonators and waveguide results in optical transitions as well as large and controllable sub and superluminal dispersion, which is advantageous for a broad range of optical devices and phenomena. Furthermore, by combining the two coupling techniques, a greater flexibility in tailoring the coupled-resonator effects can be accomplished since the inter-resonator coupling mainly influences the narrower features while the broader features can be controlled through the waveguide-resonator coupling. As such, optical resonances and dispersion of coupled resonators can be precisely controlled. Finally, we note that our results further foster the effectiveness of analogies in physics.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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