1. Introduction

Quantum coherence, one of the fundamental quantum principles that illustrates a deviation from the classical physics [1], has led to a number of extraordinary developments including, ultraslow propagation of light pulses [2–4], stopping and storing of optical pulses in coherently driven atoms [5], and slow-light-mediated all-optical switching [6]. The underlying mechanism of electromagnetically induced transparency (EIT) in multilevel atoms manifests the control of the optical response of the medium using an optical control field [7]. In particular, absorption of a resonant probe field may cease in coherently driven three-level atoms owing to destructive interference occurring between the transition probability amplitudes along two excitation pathways. The control field thus leads to a narrow transparency window within the broader probe absorption profile and transforms the anomalous dispersion of the medium into normal dispersion. The EIT concept has triggered a renewed interest in the classical optical coherence phenomenon and all-optical EIT effect has been explored extensively using diverse configurations of interacting photonic resonators including, mutually-coupled [8,9] and feedback waveguide coupled whispering-gallery [10], microdisk [11], and microtoroid [12] resonator, two-dimensional photonic crystal (PHC) cavities [13,14], double-waveguide coupled pair of wavelength-detuned [15] and co-resonant ring resonators [16], two-stage self-coupled optical waveguide resonators [17], ring-bus-ring Mach-Zehnder interferometer [18], and multimode microfiber knot based on a single-mode fiber [19]. Recently, these analogous coherence effects have been realized in silicon-based photonic structures, which is advantageous as resonances in silicon may be tuned due to the nonlinear or free carrier dispersion effects. In Ref. 20, this has been achieved owing to coherent interference between a radiant and a subradiant mode in a ring-bus-ring-bus system. Tunable EIT in an integrated photonic circuit consisting of a silicon ring resonator with embedded reflectors has been experimentally demonstrated owing to the thermo-optic tuning of reflectors [21]. In contrast to EIT, the related quantum coherence effect of electromagnetically induced absorption (EIA) results in enhanced probe absorption and yields a narrow absorption resonance at the center of the broader absorption dip [22]. Similar to the dispersion of EIA and EIT featuring atomic media, the all-optical EIA [8,15,20,23] and EIT effects in microcavities also enable super and subluminal propagation of light pulses, respectively. Interacting triple cavities were previously investigated using one and two-dimensional PHCs for parametric oscillations [24] and group delay control [25], respectively. The study of EIA and EIT phenomenon was recently extended to triple microtoroid cavities where coupling was controlled by positioning the microtorids on three precision translation stages and double anti-crossing behavior was observed [26].

A previous experimental study reported the realization of all-optical EIA and EIT effects using the much simpler platform of one-dimensional (1D) PHC [27]. The investigated 1D PHCs were based on distributed Bragg reflectors (DBRs) consisting of quarter-wave wide alternate layers of high and low-refractive-index dielectrics as well as a low and high-quality (Q) factor microcavity, which were incorporated at specific locations in the PHC. In Ref. 27, it was shown that by optimizing coherent optical coupling between such co-resonant microcavities, 1D optical analog of EIA and EIT may be realized in transmission and reflection, respectively. By varying the coupling between the two cavities, which is experimentally realized by controlling the reflectivity of the coupling DBR located between the cavities, the intracavity fields and spectra are modified owing to a change in the cavity Q factors. Therefore, tuning of coupling may be used to control the amplitudes and linewidths of the narrow EIA and EIT features, and hence the dispersion of coupled cavities. In Ref. 27, these spectral trends were experimentally observed using several dual-cavity PHCs where distinct coupling DBR reflectivities were attained as a result of variation in the number of quarter-wave bilayers of the coupling DBR. Strong coupling between the cavities leads to Autler-Townes-like splitting of EIT and EIA resonances [27]. As such, subluminality of the reflected and superluminality of transmitted pulse may be controlled by precisely controlling the intracavity fields owing to coherent interactions between the coupled cavities.

Here, we further extend the study of the microcavity analogs of EIA and EIT phenomena by considering triple-microcavity-based 1D PHCs. A key objective of our investigation is to explore the occurrence of EIA in reflection to realize superluminal pulse reflection. Furthermore, we also seek to clarify whether a variation of the field-intensity distribution in the triple-cavity also leads to spectral and dispersive effects occurring in the two-cavity system. For this purpose, we perform numerical simulations to investigate the effects of microcavity couplings on the spectral and dispersive characteristics of the triple-cavity. Furthermore, we calculate the electric-field distribution profiles, which describe the distribution of the coupled incident light at resonance across the entire multilayer structure. This allows us to correlate the triple-cavity spectra to the field-intensity distributions. Our results reveal a number of unusual spectral and dispersive characteristics of the triple-cavity system Similar to the two-cavity system, EIA and EIT also appear simultaneously at the opposite exits of the triple-cavity structure. However, in contrast to the two-cavity system, EIA and EIT in the three-cavity structure may be realized in both reflection and transmission. Furthermore, as opposed to the dual-cavity case, EIA in the triple-cavity is not restricted to the generation of superluminal light pulses. In addition, we find that, in principle, dispersion at either output may be reversed by redistributing the coupled incident light among the three cavities. This is fundamentally different from the dual-cavity dispersion where subluminality (superluminality) in reflection (transmission) sustains while altering the field distribution.

In this article, we first elucidate occurrence of EIA and EIT in triple-cavity PHC, since these analogous coherence effects may be realized in both reflection and transmission. Following this, we examine the spectral and dispersive response of a triple-cavity system where instead of tuning of coupling between the cavities, we alter the field-intensity distribution by varying the reflectivity of the terminal DBR. Finally, we consider the possibility of realizing PHC cavities with tunable distribution of the coupled light and discuss how inclusion of this important feature may enhance the functioning of PHC microcavities.

2. Theoretical background and simulation parameters

Figure 1 depicts the arrangement of DBRs and microcavities in a typical triple-cavity-based 1D PHC, where normal incidence is assumed. Following Ref. 27, we consider 1D silicon nitride (Si₃N₄)/silicon dioxide (SiO₂) PHCs, where the half-wavelength wide SiO₂ microcavities are resonant to the 600 nm wavelength. However, the Q-factor and coupling to the adjacent cavity may be different for each cavity. The entire structure is realized on a silicon substrate and for the numerical modeling, the transmitted field is assumed to propagate through a semi-infinite substrate. In the schematic of Fig. 1, ${r_j}$ describes the reflectivity of jth DBR (j=1–4) and ${r_j}$ increases if the number of quarter-wave bilayers of the corresponding DBR increase. We use the Transfer Matrix Method (TMM) [28] to calculate the reflection and transmission spectra as well as the field-intensity distributions based on Basic Linear Algebra Subprograms. For the numerical simulations, a refractive-index of 1.78 is considered for Si₃N₄ and 1.45 for SiO₂ [27]. Owing to reflection and transmission at each interface, a forward and backward propagating field arises within each dielectric layer. Depending upon the Q-factor and coupling to the adjacent cavities, a large intracavity field may surface inside a cavity as a result of multiple constructive interferences between the counter-propagating fields (Fig. 1). As these fields decay, a reflected field ${E_r} = |{{E_r}} |\,\textrm{exp} (i{\phi _r})$ and a transmitted field ${E_t} = |{{E_t}} |\,\textrm{exp} (i{\phi _t})$ emerge at the two exits of the PHC. Using the TMM determined complex reflection and transmission coefficients, the time delay or advance of an incident optical pulse upon reflection and transmission is obtained, respectively, as [29]

 

Fig. 1. Schematic of a triple-cavity-based 1D PHC shows the arrangement of distributed Bragg reflectors (DBRs) and microcavities (MCs). The incident field E₀ is sent into the system from left and a reflected (E_r) and transmitted field (E_t) emerge at the two outputs of the system, which are examined here for different compositions of MCs and DBRs in the multilayer stack. Owing to constructive interference between the counter propagating fields (arrows), a strong intracavity field buildup may occur inside a cavity. The reflected PHC field holds information about interference effects since the reflected fields of the adjacent cavities interfere destructively. This study mainly focuses on Si₃N₄ (light green) and SiO₂ (dark green) based PHCs. See the main text for the details.

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The group index ${n_q}$ is related to the time delay or advance of an incident optical pulse by

Here ${\phi _r}$ is the phase acquired on reflection, ${\phi _t}$ is the phase accumulated on transmission, ${v_q}$ is the corresponding group velocity, c is the speed of light in vacuum, d is the thickness of the PHC structure, and q denotes reflection or transmission.

As discussed in detail later in this article, in the investigated PHC structures, the cavity modes have a strong tendency to split owing to the presence of multiple cavities and the related interference effects. Accordingly, the resulting resonances are notably narrower than the two-cavity modes. We recall that as a reflected cavity field reenters an adjacent microcavity, it interferes destructively with the circulating cavity field since a π phase shift occurs for the reflected light field. As such, the reflected fields of second and third cavity as well as those of the first and second cavity interfere destructively. On the other hand, constructive interference occurs for the reflected fields of the first and third cavity. Therefore, the reflected field of multiple-cavity-based PHCs is of interest as it carries vital information about the interference effects.

It is necessary to interpret EIA and EIT in the triple-cavity structure. We retain the same definitions for EIA and EIT in the triple-cavity as those corresponding to the PHC double-cavity [27] and the closely related double whispering-gallery resonators [8], with EIT being a resonance with a sharp peak appearing at the center of the broader reflection dip, while EIA describes a resonance where a narrow dip appears at the center of the broader reflection dip. With these definitions of EIT and EIA for the reflected field, the inverted EIA (EIT) lineshape describes the transmitted EIT (EIA) resonance. This classification is justified since EIA and EIT as defined here allow comprehending the optical characteristics of the triple-cavity. However, to note a clear distinction with respect to the two-cavity case, EIA in the triple-cavity may enable both sub and superluminal light, although EIT in both cases retains the same dispersion and only generates subluminal light pulses.

3. EIA and EIT in reflectance of triple-cavity photonic crystal

We consider two specific 1D PHCs to study the EIA and EIT characteristics of the triple-cavity, since these PHC structures allow to easily comprehend the occurrence of EIA and EIT in the co-resonant triple-cavity systems. The spectra displayed in Fig. 2 are determined for the sequence ${({HL} )^{6.5}}C\,{({HL} )^{12.5}}{\kern 1pt} C\,{\kern 1pt} {({HL} )^{20.5}}$ $C\;{\kern 1pt} {({HL} )^{19.5}}S$, where H = 82 nm represents the high-index Si₃N₄ layer, L = 103 nm is the low-index SiO₂ layer, C = 2L is the SiO₂ microcavity layer, and S describes the silicon substrate. The corresponding DBR reflectivities are specified according to the relation ${r_3} > {r_4} > {r_2} > {r_1}$. For this PHC, a broad photonic stopband spanning approximately from 550 to 650 nm is realized. The corresponding field-intensity profile [Fig. 4(a)] shows localization of light mainly in the first and third cavity. Since the reflected microcavity fields in this case interfere constructively, therefore, EIA is achieved in reflection and EIT in transmission. The buildup of a weak field in the first cavity is not insignificant since for this particular case it gives rise to the broader dip and peak in the reflection and transmission spectrum, respectively. The second PHC we investigate corresponds to the arrangement ${({HL} )^{11.5}}C\,{({HL} )^{20.5}}{\kern 1pt} C\,{({HL} )^{14.5}}C\,{\kern 1pt} {({HL} )^{4.5}}S$, which results in the buildup of field only in the first and second microcavity (Fig. 3). For this structure, the DBR reflectivities satisfy the condition ${r_2} > {r_3} > {r_1} > {r_4}$. Since the reflected cavity fields interfere destructively, here EIT is attained in reflection and EIA in transmission, in perfect agreement with the field-intensity distribution for this specific PHC structure [Fig. 4(b)].

 

Fig. 2. Reflectance (a) and transmittance (c) of a triple-cavity where the field builds up mainly in MC1 and MC3. Magnified views of resonances are displayed in (b) and (d).

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Fig. 3. Reflectance (a) and transmittance (c) of a triple-cavity where the field builds up mainly in MC1 and MC2. Magnified views of resonances are displayed in (b) and (d).

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Fig. 4. Field-intensity profiles corresponding to the spectra of (a) Fig. 2 and (b) Fig. 3.

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The spectra of Figs. 2 and 3 display symmetric EIA and EIT resonances. This happens since the cavities are co-resonant owing to the identical thicknesses of the cavity layers. If this condition is not fulfilled, calculations reveal Fano-like [30] asymmetric triple-cavity resonances. The constructive and destructive interference occurring between the microcavity fields in the all-optical EIA and EIT effects is in direct analogy to the constructive and destructive interference occurring between the transition probability amplitudes in EIA and EIT featuring atomic media, respectively.

4. Evolution and transformation of EIA (EIT) into EIT (EIA)

Having clarified the critical role of interfering cavity fields, we turn to a more complex triple-cavity PHC, where EIA and EIT are obtained in spite of the buildup of the resonant field inside all three cavities. In this scenario, it is the strength of the individual cavity fields, which shape the resonant spectral features. Furthermore, unlike the two-cavity system, here a variation of electric-field distribution may transform EIA into EIT and conversely at the two outputs of the system. This is illustrated in Fig. 5 where the left panel shows the reflection spectra for decreasing number of DBR4 bilayers, while the right panel shows the corresponding transmission spectra for a PHC consisting of the multilayer sequence ${({HL} )^{6.5}}C\,{({HL} )^{13.5}}{\kern 1pt} C{\kern 1pt} {({HL} )^{16.5}}C{\kern 1pt} {({HL} )^{20.5}}S$. In fact, we find that solely by controlling the reflectivity of DBR4, EIA (EIT) may be converted into EIT (EIA) as the distribution of the coupled light among the three cavities is altered. This, in turn, modifies the nature of interference among the outgoing cavity fields and, therefore, the spectra and dispersions are altered accordingly. This is in contrast to the two-cavity case where the field distribution was modulated owing to a variation in the reflectivity of the coupling DBR [27].

 

Fig. 5. Reflectance (left) and transmittance (right) of a triple-cavity PHC where the bilayer composition of DBR4 is varied. The corresponding bilayer number for different spectra are: (a) and (g) 20.5, (b) and (h) 16.5, (c) and (i) 13.5, (d) and (j) 10.5, (e) and (k) 8.5, and (f) and (l) 5.5.

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In the investigated triple-cavity PHC, initially EIA is attained in reflection. As the reflectivity of DBR4, ${r_4}$, decreases, the amplitude of the narrow EIA dip increases until critical coupling is achieved. Beyond the critical coupling regime, the amplitude of the EIA dip decreases until EIA is converted into EIT. In the case of transmission, initially EIT is obtained and as ${r_4}$ decreases the narrow EIT peak increases until it maximizes. By further reducing ${r_4}$, this trend is reversed, and the narrow EIT peak starts to shrink and ultimately EIT is converted into EIA. The spectral profiles appearing in Figs. 5(d) and 5(j) manifest this transformation, as the diminishing narrow dip (peak) at the center of EIA (EIT) resonance in reflection (transmission) is being replaced by a peak (dip).

The field-intensity distributions for this PHC reveal that initially light is localized in the first and third cavity and a strong intracavity field emerges in the third cavity (Fig. 6). A detailed investigation, which also considers the off-resonance field distributions, shows that for the initial PHC configuration [Figs. 5(a) and 5(g)], the on-resonance field of the second cavity vanishes owing to splitting of the resonant mode into two modes. In contrast, the resonant mode of the first cavity splits into three modes, the central one being resonant to the incident light and the resonant mode of the third cavity. Owing to the high-reflectivity of the adjacent DBRs, the Q-factor of the third cavity is the largest and the coupled incident light tends to localize in the third cavity. However, as the reflectivity of DBR4 is reduced, the Q-factor of the third cavity starts to deteriorate and this trend continues as ${r_4}$ decreases. As this happens, the resonant field of the second cavity recurs and its Q-factor is enhanced since the entire PHC structure following the second cavity behaves as a high-reflectivity rear DBR for the second cavity and photons are thus driven towards the second cavity. Eventually, the resonant field is retained only in the second cavity and the EIT (EIA) resonance appears in reflection (transmission) as the reflected field at the line center is unaffected by the diminishing fields of the neighboring cavities.

 

Fig. 6. Field-intensity profiles corresponding to the spectra of Fig. 5. Here each figure part corresponds to the respective part of Fig. 5.

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The co-existence of field in the first and third cavity has been observed in a number of PHC structures investigated in this work. If the Q-factor of the first cavity is greater than the third cavity, photons predominantly localize in the first cavity while a smaller fraction of photons appears inside the third cavity. For the opposite case, a strong intracavity field surfaces in the third cavity, owing to the high Q-factor of this cavity. Under these conditions, the resonant field does not build up in the second cavity, due to the high-Q factor of the first or third cavity as well as mode splitting, which occurs due to destructive interference between the fields of the second and third cavity.

5. Dispersive characteristics of triple-cavity system

Investigation of the dispersive response of the triple-cavity system reveals a notably distinct behavior in contrast to the conventional two-cavity structure. This is illustrated in Figs. 7 and 8, where the phase and group index for the reflected and transmitted spectra shown in Fig. 5 are plotted as a function of wavelength, respectively. In Figs. 7 and 8, each figure part corresponds to the respective part of Fig. 5. For clarity, the group indices for both cases are shown in the vicinity of the resonant wavelength. These results show that prior to the occurrence of critical coupling in the refection spectrum of Fig. 5, the EIA resonance gives rise to subluminal light at resonance and the group delay increases as ${r_4}$ decreases. After critical coupling, which occurs between Figs. 5(b) and 5(c) in reflection, the dispersion switches from subliminal to superluminal and the group advance decreases with decreasing reflectivity of DBR4. As EIA converts into EIT, the dispersion again becomes subluminal. However, now the group delay decreases as ${r_4}$ is further reduced. In transmission, only subluminal light is achieved at resonance in spite of the fact that EIT obtained initially is later transformed into EIA (Fig. 8). Here, the maximum group delay is obtained for the initially realized EIT and it gradually decreases as the DBR4 reflectivity is reduced. Modest group delays and advancements are obtained for the investigated PHCs. For instance, as shown in Fig. 7, a subluminal group index of ∼50 is determined for the spectrum of Fig. 6(b) and a superluminal group index of 120 is found for the spectrum shown in Fig. 6(c). For the spectra of Figs. 2 and 3, only subluminal light is achieved at both outputs and the largest group index value of ∼380 is attained for the reflected light in Fig. 2. It is worth noting that as EIA transforms into EIT [Fig. 5(d)], the on-resonance phase for the reflected field becomes flat and the group index approaches zero [Fig. 7(d)]. Within the investigated parameter range, superluminal dispersion was not observed in transmission in the present study. Calculations show that DBRs with much higher reflectivities are required to achieve enhanced dispersion. Similarly, large DBR reflectivities are also essential for achieving superluminal transmission of an incident pulse. These unusual and complex dispersive characteristics warrant a thorough investigation of the triple-cavity dispersion.

 

Fig. 7. The phase (left) and group index (right) for the reflection spectra of Fig. 5.

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Fig. 8. The phase (left) and group index (right) for the transmission spectra of Fig. 5.

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We note that the dispersion interval over which superluminal light is obtained in reflection is directly related to the reflectivity regime where the required variation in DBR4 reflectivity for the EIA-EIT transformation occurs. Therefore, the superluminal dispersion interval may be narrowed or enlarged by designing appropriate triple-cavity PHCs. This happens because depending upon the triple-cavity structure, transformation of EIA into EIT may require only a minor decrease in the reflectivity of DBR4. As such, the EIA-EIT transformation may occur soon after critical coupling is achieved, as the required variation in ${r_4}$ is small, thus narrowing the superluminal dispersion regime. In the opposite case, a large shift in DBR4 reflectivity is necessary and, therefore, superluminal light is attained over a larger coupling regime where a variation in ${r_4}$ occurs.

A variation in DBR4 reflectivity not only affects the dispersion interval where superluminal light is realized but it also has an impact on the cavity Q factors. Specifically, this decrease in reflectivity has a major influence on the Q-factor of the second cavity. If the required variation in ${r_4}$ is small, the corresponding decrease in the Q-factor of the second cavity is minimum since the effective rear DBR for this cavity maintains high reflectivity. As a result, EIT resonance obtained after the transformation possesses narrow features and large mode amplitudes for the off-resonance dips. On the other hand, broader EIT resonance is achieved owing to a notable decrease in the Q-factor of the second cavity because a larger decrease in DBR4 reflectivity is essential for the EIA-EIT transformation, as in the present case.

6. Towards photonic crystal cavities with tunable optical response

Several approaches for fabricating wavelength-tunable 1D PHCs have been reported. In Ref. 31, a strain-tunable polymer-based 1D PHC was demonstrated experimentally. Here different concentrations of Zirconium dioxide (ZrO₂) nanoparticles in an elastomer matrix enabled the required refractive- index contrast while maintaining elasticity, which under application of strain blue shifts the photonic stopband by more than 300 nm. As strain increases, the layer thicknesses decrease due to compression, and the peak reflectance shifts to shorter wavelengths while the bandwidth of the stopband becomes narrower [31]. By incorporating an electric-field tunable defect layer of SrTiO₃ in SiO₂/CeO₂ PHC, the stopband can be tuned as the applied bias modifies the in-plane dielectric function [32]. Electric-field tuning of opals or inverse opals infiltrated with liquid crystals (LCs) provides another route to tunable PHCs as the refractive index of LCs, may be altered by applying an electric-field [33]. Furthermore, the free carrier dispersion effect (FCDE) [34] may be used as well for the wavelength tuning of PHCs composed of materials that display FCDE.

In the case of a wavelength-tunable triple-cavity PHC, the EIA and EIT effects can be realized over a broader spectral range by shifting the PHC spectra to different spectral regimes by using appropriate materials. To demonstrate this effect, we consider a triple-cavity structure based on ZrO₂ nanoparticle incorporating elastomers [31]. Here, the high-refractive-index layers of the 1D PHC are based on ZrO₂ nanoparticles incorporating elastomers while the low-refractive-index layers are realized using nanoparticle-free elastomers. The refractive index values used in calculations correspond to 1.68 and 1.48, respectively [31], and a refractive index of 1.406 [33] is considered for the SEBS substrate [35]. In Ref.31, the initial strain-free PHC spectrum was obtained for the center wavelength of ∼726 nm. By fitting the calculated spectrum to the measured PHC spectrum, the layer thicknesses for the 726 nm spectrum were found to match layer thicknesses of a PHC that is exactly resonant to the 726 nm wavelength. On the other hand, fitting to the spectrum for the center wavelength of ∼540 nm [31] shows that the layer thicknesses have been reduced by a factor of ∼1.34 in contrast to their initial values corresponding to the 726 nm spectrum. Using this information, in Fig. 9 we show the calculated EIT resonance in reflection of a triple-cavity based on materials and refractive indices described above at the resonant wavelengths of 726 and 540 nm for a PHC configuration described by ${({HL} )^{14.5}}C\,{({HL} )^{22.5}}{\kern 1pt} C\,{({HL} )^{9.5}}C\,{\kern 1pt} {({HL} )^{4.5}}S$. Clearly, the EIT resonance sustains as the spectrum shifts to the 500 nm spectral regime due to strain, although the resonance and stopband become narrower. Therefore, tunable EIT and slow light may be realized over a broad spectral range. Similarly, EIA and fast light may also be tuned over a wide spectral regime.

 

Fig. 9. Reflectance of a wavelength-tunable triple-cavity PHC in the absence (a) and under application of strain (b).

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However, if the resonant wavelength of only a segment of the PHC crystal is marginally tuned, such as the DBR4 of the PHC described in Fig. 5, the coupled-cavity spectra and dispersion may be modified owing to a redistribution of the field-intensity as a consequence of variation in DBR4 reflectivity. This change in DBR reflectivity, occurring owing to a variation in the thickness or refractive index of quarter-wave layers of DBR4, arises since the structure is no longer resonant to the initial resonant wavelength. Thus, wavelength tuning of the Bragg reflectors may be used to modulate the coupled-cavity interactions, which in turn leads to a change in the spectral and dispersive response of the system. For this purpose, among other techniques, the required change in DBR4 reflectivity may be achieved through thickness control of elastic materials [31] It is worth mentioning that instead of using several multiple-cavity-based PHCs with distinct DBR4 reflectivities, a single PHC structure, where DBR4 reflectivity is modified according to the experimental methods outlined above, is sufficient for this purpose. This makes PHC-based coupled-cavity systems attractive for a number of applications since in contrast to tunability methods considered previously [26], here the spectra, dispersion, and intracavity fields may be controlled using a single monolithic PHC with multiple cascaded microcavities.

The tunable optical response of the PHC cavities is advantageous for a wide range of applications. For instance, tunable PHC cavities are useful for on-chip dispersion engineering, and the resulting tunable all-optical slow and fast light effects are highly desirable for enhancing the performance of various optical devices such as optical buffers. By altering the intracavity field of a specific cavity via modulation of the field distribution across the PHC, temporal control of cavity quantum electrodynamics phenomenon, such as the Purcell effect, can be achieved. This concept is also applicable for controlling the on-demand generation of single photons in resonant microcavities since control of vacuum field fluctuations may be used to trigger photon emission. The narrow resonances arising in the triple-cavity system are advantageous for ultra-narrow bandwidth optical filtering in optical communication. Furthermore, by controlling the amplitude of the transparency feature of a critically coupled EIT resonance using the FCDE, all-optical switches may be realized, where free carrier lifetime, and consequently the switching operation, can be controlled using a p-i-n diode [36].

7. Discussion

It is clear from the results of the present study that the optical response of the triple-cavity is more complex in comparison to the two–cavity system, which bears a closer analogy to multilevel atomic EIT media. In the two-cavity system [27], when light couples back from the high-Q to low Q-factor cavity, it interferes destructively with the circulating field of the low-Q cavity and EIT appears in reflection, in direct analogy to the occurrence of EIT in coupled ring resonators [37]. As coupling between the two 1D PHC cavities increases, the linewidth and amplitude of the narrow features increase as well and ultimately EIT and EIA realized in reflection and transmission, respectively, are transformed into Autler-Townes-like split resonances. However, while coupling is varied, subluminality at the refection and superluminality at transmission port persists. To comprehend occurrence of EIT and EIA in the triple-cavity, it may be useful to know to what extent the similarities between the triple and conventional dual-cavity system exists. Thus, we seek to clarify whether it is possible to map the three-cavity system into the two-cavity system.

It turns out that by eliminating the resonant field inside the third cavity, the spectral response of the triple-cavity becomes quite similar to the two-cavity system, as now it is only possible to achieve EIT in reflection and EIA in transmission, as in the case of Fig. 3 where the resonant field completely extinguishes inside the third cavity. This represents the ideal case where the optical response of the triple-cavity nearly matches the dual-cavity system. However, even in the presence of the resonant field in the third cavity, the three-cavity spectra may still resemble the two-cavity case. This is shown in Fig. 6(e), where in spite of the buildup of resonant field inside all three cavities, destructive interference occurs for the reflected light owing to a larger intracavity field of the second cavity in contrast to the remaining two, and, therefore, EIT is obtained in reflection [Fig. 5(e)]. However, the triple-cavity signature is clearly apparent since unlike the two-cavity EIT, here the dominant interaction leading to EIT occurs between the second and third cavity. The response of the triple-cavity departs completely from the two-cavity system after a threshold resonant field emerges inside the third cavity and constructive interference begins to dominate the resultant reflected field. This enables the distinctive triple-cavity effects of EIA in reflection and EIT in transmission. The EIA resonance here bears a resemblance to the double-resonator EIA [8] if the fields are mainly localized in the third and first cavity [Fig. 2 and Fig. 4(a)]. With the emergence of field in all three cavities, the EIA lineshape is distorted as dips emerge at either side of the narrow central EIA dip as a consequence of the broadening of the third cavity mode [Fig. 5(c) and Fig. 6(c)]. Although the Q-factor of the third cavity remains the largest, it interacts with the narrow central split mode of the first cavity to yield EIA.

Therefore, constructive interference in the reflected light field marks the onset of characteristic triple-cavity effects, which is clearly illustrated in Figs. 5 and 6. Here, initially the large intracavity field of the third cavity interacts with the first cavity to produce EIA in reflection. In this regime, purely triple-cavity effects prevail. However, after the reflectivity of DBR4 reduces considerably, the intracavity field and the Q-factor of the third cavity are only marginally greater than the corresponding parameters of the second cavity. Accordingly, a transition from EIA to EIT is initiated and the on-resonance EIA dip in reflection is barely apparent. Beyond this coupling regime, the second cavity’s field dominates as now it has the largest Q-factor and the resonant fields diminish in the neighboring cavities, leading to appearance of EIT in reflection. Under these conditions, the central EIT peak arises owing to the second cavity while the off-resonance EIT dips appear mainly due to interaction between the fields of the second and first cavity since the off-resonance field of the third cavity is greatly suppressed. Similar trends shape the transmitted field.

Accordingly, an interplay between the resonant fields of the second and third cavity generally determines whether EIT or EIA is obtained in reflection owing to destructive or constructive interference, respectively. Destructive interference may occur if the Q-factor and intracavity field of the second cavity at resonance is the largest while the resonant fields of the remaining two cavities either diminish or the next largest field appears in the first cavity. Alternatively, EIT may be obtained if the high-Q resonant field of the third cavity interacts with the next largest second cavity field. However, for constructive interference, the Q-factor and intracavity field of the third cavity is largest and is followed by the first cavity field. Therefore, out of the second and third cavity, one that has the largest Q-factor and intracavity field defines the nature of interference and the sharp spectral features. In comparison to the dual-cavity, the presence of an additional cavity in the triple-cavity system mainly opens up a channel for constructive interference. As such, the role of the first cavity is basically to facilitate destructive or constructive interference for the reflected light field, collectively with the second or third cavity, respectively.

However, the similarities between the dual and triple cavities noted above are mainly related to spectra of the two systems. As already noted in section 5, the dispersive response of the triple-cavity is markedly different from the two cavity case, although abundant similarities are observed as far as the reflected field is concerned.

For the sake of completeness, we note a rare situation where EIA emerges in reflection if the on-resonance intracavity field of the first cavity is larger than the third cavity and field vanishes in the second cavity.

The ability to transform EIA into EIT in triple-cavity PHCs is a noteworthy phenomenon as it may provide a deeper understanding of the collective resonances of coherently interacting photonic microcavities.

8. Summary and conclusions

To summarize, we have presented results of a computational study, which reveals noteworthy spectral and dispersive features of triple–cavity PHCs. In the triple-cavity PHCs, the EIA and EIT phenomenon can be realized with greater flexibility and control. The cavity with the highest Q-factor and largest intracavity field is found to define the narrow spectral features of the optical resonances of the triple cavity. Unlike the two-cavity case, where the output cavity fields appearing in reflection interfere destructively, the inclusion of the third cavity provides a mechanism for controlling the nature of interference occurring in the reflected light field. Therefore, both EIT and EIA can be realized in the reflection spectrum owing to the possibility of both constructive and destructive interference occurring between the outgoing reflected light fields in the triple-cavity. Finally, we note that triple-cavity PHCs have the potential for tunable response in a monolithic setting, which opens up the way for more robust and versatile applications of PHC microcavities.