Photonic coherence effects from dual-waveguide coupled pair of co-resonant microring resonators

Ahmer Naweed

doi:10.1364/OE.23.012573

1. Introduction

In recent years, coherent photonics has emerged as an important sub-discipline in photonics. The realization of desired functionality of coherent photonic devices is a direct consequence of the advancements in micro and nanofabrication methodologies. As a specific example, we refer to coupled microresonators (CMRs) where precision in device fabrication is critical owing to the condition of deterministic control of coherent coupling. A variety of such coherently interacting resonators have been used to achieve photonic analogs of quantum coherence effects [1] of electromagnetically induced transparency (EIT) [2] and electromagnetically induced absorption (EIA) [3], including whispering-gallery [4,5], microring [6,7], microdisk [8], and microtoroid resonators [9], one- and two-dimensional photonic crystal cavities [10,11], as well as other photonic resonator structures [12,13]. The photonic manifestations of EIT and EIA, usually referred to as coupled-resonator-induced-transparency (CRIT) and coupled-resonator-induced-absorption (CRIA), seems to be more viable than EIT and EIA mediating coherent atomic media, since material structuring alone results in occurrence of CRIT and CRIA over a wide spectral range. Furthermore, in contrast to coherence effects in atoms, the experimental realization of CRIT and CRIA does not require any special operating conditions, such as low-temperature. This represents a significant advantage as several applications based on quantum coherence may be implemented all-optically on chip-scale photonic platforms without being confronted with decoherence.

One of the CMR structures that has been experimentally explored to achieve CRIT and CRIA consists of two laterally displaced microring resonators that are side coupled to two identical and parallel waveguides [6,7]. Earlier research focusing on this structure considered microring resonators where the resonant wavelength of one resonator is slightly detuned in comparison to the other by increasing the resonator circumference by ~8 nm, which is hard to realize experimentally. However, the optical properties of the two resonators, such as the intrinsic quality factors (Q) and coupling coefficients were nearly identical. Furthermore, CRIT in this system is obtained only in transmission [7] while CRIA is achieved only in reflection [14]. Analogous to EIT and EIA, CRIT and CRIA also slow down [5] or speed up [14] the propagation of an optical pulse. Accordingly, these effects are central to applications such as optical delay lines [6], data resynchronization [15], enhancement of interferometer sensitivity [16], and realization of gyroscopes and gravitational wave detectors with enhanced sensitivity [17]. Furthermore, coupled resonators have also been explored for the storage and on-demand retrieval of optical pulses [18] as well as for directed logic circuits [19].

Here we theoretically describe photonic coherence effects using co-resonant CMRs and show that this coupled resonator scheme is not only simple to implement but is versatile as well since it not only eliminates the requirement for wavelength detuning but also allows occurrence of CRIT or CRIA under a variety of conditions, including appearance of these effects in both transmission and reflection. For this purpose, we develop a theoretical model for characterizing coherent optical interactions between the two the resonators and obtain expressions for transmitted and reflected fields. First, we discuss characteristic coupled resonator interactions that lead to appearance of CRIT and CRIA in the spectra of coupled resonators. After describing the theoretical model, calculated coupled resonator spectra are presented. This is followed by a discussion, which elucidates occurrence of CRIT and CRIA in both transmission and reflection of co-resonant coupled resonators. Additionally, we discuss the origin of CRIT and CRIA related narrow transparency and absorption features as well as their tuning using Q-modulated co-resonant coupled resonators. Finally, we discuss fabrication issues and show that the required device parameters for attaining CRIT and CRIA in co-resonant coupled resonators are compatible with the existing photonic devices.

2. Structures and resonances of coupled resonators with waveguide-mediated optical interactions

Figure 1 is a schematic of the CMR structure and shows steady-state fields related to the evanescent coupling between the resonators and waveguides. Here coupling at each resonator’s transmission (reflection) port is described by the self and cross-coupling coefficient $r_{j}$ ( $r_{j}^{'}$ ) and $t_{j}$ ( $t_{j}^{'}$ ), respectively, and owing to the assumption of lossless coupling $r_{j}^{2} + t_{j}^{2} = {(r_{j}^{'})}^{2} + {(t_{j}^{'})}^{2} = 1$ , where j = 1, 2. The resonators and waveguides are assumed to be composed of the same material, which possesses an index of refraction n. As opposed to detuned coupled resonators, where the resonator-waveguide coupling is identical throughout the entire structure for the same Q resonators ( $r_{1} = r_{2} = {r^{'}}_{1} = {r^{'}}_{2}$ ), here we consider two primary approaches to achieve CRIT and CRIA based on co-resonant coupled resonators. In the asymmetric coupling case, these coherence effects are realized by coupling two identical resonators to the waveguides with equal strength, although this coupling strength is distinct for each waveguide ( $r_{1} = r_{2}; {r^{'}}_{1} = {r^{'}}_{2}$ ). Alternatively, for the asymmetric coupling and Q-factor architecture, in addition to the outlined coupling scheme, distinct resonator Q factors are considered as well.

Fig. 1 Schematic of the dual waveguide coupled two-microring resonator (MRR) system shows coupling parameters at various ports. Coupling of the incident field E₀ to the CMR structure results in a transmitted (E_t) and a reflected (E_r) field. The separation L between the two MRRs is selected to achieve constructive interference between the fields of the two MRRs. The regions marked by grey surroundings indicate a larger resonator-like configuration, which gives rise to the supermode resonance.

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Upon excitation, resonances of the microring resonators appear as dips in transmission and peaks in reflection, and owing to constructive interference, these resonances overlap for co-resonant microresonators, while for detuned resonators they are shifted in wavelength. However, an additional resonance, referred to as the supermode resonance, also appears in this structure, and emerges as a peak in transmission and dip in reflection. The origin of the supermode may be understood by recognizing that the CMR illustrated in Fig. 1 is basically similar to a particular type of Fabry-Perot resonator and the supermode is actually the corresponding resonant mode [7]. Here the waveguides mediate the forward and backward propagation of the supermode while the microring resonators act as mirrors and channel the supermode from one waveguide to another or to the outgoing waveguide modes. The resonant wavelengths of the Fabry-Perot and ring-resonator j are determined by applying the resonance condition, which are given, respectively, as $2 Λ = 2 n k L = 2 p π$ and $ϕ_{j} = 2 π n k b_{j} = 2 q π$ . Here $b_{j}$ and $ϕ_{j}$ represent the microring radius and internal phase shift acquired by the field circulating inside the resonator j, respectively, $k$ is the propagation constant, and p and q are integers, which describe the mode order of the respective resonator. If the resonant wavelengths of the Fabry-Perot and microring resonator coincide, symmetric CRIT and CRIA are obtained, since in this case the narrow supermode peak appears at the center of the broader transmission dips of the microrings (CRIT) [7], whereas the narrow supermode dip appears at the center of the broader reflection peaks of the microrings (CRIA) [14]. If the resonant wavelengths do not match, Fano-like asymmetric resonances [20] are realized. Such asymmetric photonic resonances are also of technological significance because of their use, among others, in all-optical switching [21] and sensing applications [22].

In the case of detuned CMRs, the supermode Q-factor is larger than the Q-factor of both resonators [6,7]. The same may happen for co-resonant CMRs as well, although the narrow spectral features in this case may also arise owing to a higher Q resonator rather than supermode. This may occur if resonators with distinct Q factors are applied along with a specific coupling scheme, which is elaborated later in Section 4, and is shown to yield CRIA in transmission and CRIT in reflection. As such, the linewidth of these higher Q resonator mediated features is limited by the Q-factor of the higher Q resonator and we discuss this in greater detail later in this article.

3. Analytic model for coherently coupled resonators

According to the field distribution illustrated in Fig. 1, the normalized CMR transmitted field is determined as

\frac{E_{t}}{E_{0}} = \frac{τ_{1} τ_{2} exp(i Λ)}{1 - ρ_{1} ρ_{2} exp(i 2 Λ)},

where τ_j is given by

τ_{j} = \frac{E_{τ_{j}}}{E_{0}} = \frac{r_{j} - {r^{'}}_{j} a_{j} exp(i ϕ_{j})}{1 - r_{j} {r^{'}}_{j} a_{j} exp(i ϕ_{j})},

and describes the transmitted field of a single resonator, and the reflected single resonator field ρ_j is expressed as

ρ_{j} = \frac{E_{ρ_{j}}}{E_{0}} = \frac{- t_{j} {t^{'}}_{j} a_{j}^{1 / 2} exp(i ϕ_{j})}{1 - r_{j} {r^{'}}_{j} a_{j} exp(i ϕ_{j})} .

In the above expressions, $a_{j} = exp(- α_{j} d_{j} /2)$ accounts for the intrinsic resonator loss, α_j is as an attenuation coefficient which is related to the resonator’s intrinsic quality factor by $Q_{j} = 2 π n / (α_{j} λ_{0}),$ λ₀ is the free-space resonant wavelength, and d_j is the microresonator circumference. A steady-state analysis of the field distribution shown in Fig. 1 yields the following expression for the normalized CMR reflected field

\frac{E_{r}}{E_{0}} = \frac{ρ_{1} - (ρ_{1}^{2} - τ_{1} {τ^{'}}_{1}) ρ_{2} exp(i 2 Λ)}{1 - ρ_{1} ρ_{2} exp(i 2 Λ)},

where

{τ^{'}}_{1} = ({r^{'}}_{1} - r_{1} a_{1} exp(i ϕ_{j})) / (1 - r_{1} {r^{'}}_{1} a_{1} exp(i ϕ_{j}))

and CMR reflectance is given by

R

=

{| E_{r} / E_{0} |}^{2}

. For co-resonant coupled resonators

ϕ_{1} = ϕ_{2} = ϕ .

The derived expressions for the transmitted [Eq. (1)] and reflected [Eq. (4)] fields indeed support the Fabry-Perot resonator analogy owing to similarity to fields generated by a Fabry-Perot type resonator having resonant mirrors with cross and self-coupling coefficients

τ_{i}

and

ρ_{i}

, respectively, and a resonator round-trip phase shift of

2 Λ

. Algebraic manipulation of the transmitted field described by Eq. (1) leads to an expression for the transmittance

T

=

{| E_{t} / E_{0} |}^{2}

, which is identical to the one given in [7].

To accomplish coherent coupling, the phase acquired owing to the propagation of an electromagnetic field through the waveguide must be related to the phase imparted on the field circulating inside the microring resonator in a definitive manner. To determine this phase relation, we consider the resonance condition for the Fabry-Perot and microring resonator and find

Λ = (p / q) ​ ϕ / 2.

for the simplest case, $p = q$ and, therefore, $Λ = ϕ / 2$ . The underlying physics behind the phase relation is particularly simple to comprehend for this specific case. For instance, an electromagnetic wave coupled to the first resonator acquires a phase shift of $φ / 2$ on emerging at the reflection port, whereas an electromagnetic wave, which originates at the reflection port of the second resonator, acquires a phase shift of Λ on approaching the reflection port of the first resonator. Assuming coherence among these fields, constructive interference will occur if $Λ = ϕ / 2$ . The phase relation described here greatly simplifies numerical simulations of the spectral and dispersive response of coupled resonators since, unlike previous studies [7], no additional numerical calculations are required to account for the propagation of off-resonance fields. To determine the critical separation L between the resonators, which results in constructive interference, we use Eq. (5) to obtain

L = (p / q) π b = (p / q) d / 2.

4. Results

For the numerical simulations, we consider co-resonant microring resonators of 50 μm radius, a free-space resonant wavelength of 1550 nm and n = 3.45, which is the index of refraction for silicon. Furthermore, we assume that $Λ = ϕ / 2.$ First, we consider the case of asymmetric coupling using same Q resonators. In Fig. 2(a) we plot transmittance and reflectance of coupled resonators based on two identical microrings as a function of $φ$ , which is related to detuning. Owing to the asymmetric nature of coupling considered here, the two resonators have the same coupling to the throughput waveguide while coupling to the drop waveguide is also identical for both resonators, yet these couplings are distinct from each other. Experimentally this may be achieved by fabricating the waveguides on either side of the resonator pair with different resonator-waveguide separations. Analogous to detuned coupled resonators, here too CRIT is attained in transmittance while CRIA is achieved simultaneously in reflectance. However, if besides identical Q factors, identical coupling is also considered for all resonator-waveguide couplings of the structure, CRIT and CRIA resonances disappear, as shown in Fig. 2(b).

Fig. 2 Transmittance (red) and reflectance (blue) of CMRs with identical intrinsic quality factors. (a) Q₁ = Q₂ = 1 × 10⁵, r₁ = r₂ = 0.9878, and r₁´ = r₂´ = 0.9551. (b) The same Q as in (a) but self-coupling coefficient at each resonator’s transmission and reflection port is 0.9878.

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If the coupling detuning for the structure of Fig. 2(a) is decreased, high extinction ratios are obtained for the microring resonances as these resonances move towards the critical coupling regime. However, this coincides with a decrease in the supermode amplitude. Numerical simulations reveal that these opposing requirements may be overcome to achieve high microresonator extinction ratios and large supermode amplitudes, if high Q-factor resonators are used. However, for co-resonant coupled resonators, a simple alternative exists since in this case only one resonator needs to have a higher Q-factor to induce these spectral characteristics. This is shown in Fig. 3(a) where the second resonator possesses a higher Q-factor. Accordingly, CRIT and CRIA appearing in transmission and reflection of this system, respectively, exhibit sharper transparency and absorption features. We note that Q of both resonators here exceed the Q-factor considered in Fig. 2(a) and the coupling parameters used for the structure of Fig. 3(a) indeed minimize coupling detuning

Fig. 3 Transmittance (red) and reflectance (blue) of CMRs with distinct intrinsic quality factors. (a) Q₁ = 7 × 10⁵, Q₂ = 8 × 10⁵, r₁ = r₂ = 0.9765, and r₁´ = r₂´ = 0.9645. (b) Q₁ = 1 × 10⁴, Q₂ = 6 × 10⁵, r₁ = r₁´ = 0.9628, and r₂ = r₂´ = 0.9995. (c) Same parameters as in (a) except Q₂ = 2 × 10⁶ . (d) Same parameters as in (b) except Q₂ = 5 × 10⁶.

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By applying a significantly different coupling arrangement for resonators with distinct Q factors, CRIA is obtained in transmittance while CRIT is achieved in reflectance. This is displayed in Fig. 3(b) where coupling at both ports of each resonator is identical, although these coupling parameters are different for the two resonators. Such a coupled-resonator-geometry may be realized experimentally by fabricating waveguides which are located at the same distance on either side of the first resonator. A similar geometry is required for the second resonator as well. However, these separations are different for the two resonators. Alternatively, this type of coupling may be accomplished using multimode interference (MMI) couplers based on electrically driven position-selective p-n junctions [23]. Here a preferred fraction of the incident power may be extracted for coupling at a desired location across the MMI coupler owing to a local change in the refractive index.

The effect of Q-enhancement on coupled resonator spectra of Fig. 3(a)-3(b) is presented in Fig. 3(c)-3(d), where the Q-factor of the higher Q resonator is increased. This leads to a further narrowing and amplitude enhancement of the sharp spectral features. A continuous and controlled Q variation will provide a practical route to tunable CRIT and CRIA, resulting in dynamic and superior performance of Q-tunable co-resonant CMRs.

As already observed, a high supermode Q-factor and large amplitudes for the supermode and microring resonances are easily realized if coupling detuning is minimum but non-zero and high-Q resonators are used. However, the spectra in this case are sensitive to small variations in coupling. On the other hand, for coupled resonators with large coupling detuning, the opposite is observed in numerical simulations. This situation is similar to Fig. 2(a), where only through a substantial change in coupling, extinction of CRIT and CRIA occurs.

5. Origin of the sharp features in the spectra of coupled resonators

To comprehend occurrence of CRIT and CRIA in both transmittance and reflectance, we consider an additional coupled resonator structure as well where the drop waveguide shown in Fig. 1 is absent and compare the on-resonance transmitted field amplitudes for the dual-waveguide (DWG) and single-waveguide (SWG) based coupled resonators. We consider identical yet arbitrary waveguide structures for the two cases to accommodate various coupling schemes outlined earlier. Obviously, the supermode is not realized for the single-waveguide based resonator pair and the corresponding transmitted field is given by $τ_{1} τ_{2} exp(i Λ) .$ The difference between the on-resonance amplitudes of transmitted fields for the two cases gives the following expression for the amplitude M of the transmitted sharp resonance feature with respect to the amplitude of the resonance of the single-waveguide system

M = \sqrt{T_{DWG}} - \sqrt{T_{SWG}} = {| {\tilde{τ}}_{1} {\tilde{τ}}_{2} / (1 - {\tilde{ρ}}_{1} {\tilde{ρ}}_{2}) |}_{DWG} - {| {\tilde{τ}}_{1} {\tilde{τ}}_{2} |}_{SWG},

where tilde indicates evaluation of the single resonator transmitted and reflected fields for the on-resonance condition. For our analysis, we make the following assumptions. We assume that both microrings in the dual-wave-guide based coupled resonator system possess the same intrinsic Q-factor and the resonances of these microrings are identical as well. Therefore,

{\tilde{τ}}_{1} = {\tilde{τ}}_{2} = \tilde{τ}

and

ρ_{1} = ρ_{2} = ρ

for the dual-waveguide based coupled resonators. Additionally, we assume that the microring resonances in the single-waveguide system are also similar to those of the dual-waveguide system, which may be accomplished by adjusting the resonator parameters. These assumptions allow us to simplify Eq. (7) and we finally obtain

M = {\tilde{τ}}^{2} {\tilde{ρ}}^{2} / (1 - {\tilde{ρ}}^{2}),

which is larger than zero if neither of the two resonators are critically coupled. Accordingly, only CRIT may be attained in the transmission of co-resonant coupled resonators since M > 0 if the microring resonances are shifted away from the critical coupling, which may be accomplished through detuning of resonator-waveguide couplings. Another approach to achieve CRIT in transmission is based on detuning of the resonant wavelengths that opens up a narrow spectral window where the resonance dips of the two microrings are non-zero and, therefore, M > 0 [7]. As such, coupling or wavelength detuning is a requirement to obtain CRIT in transmission of CMRs based on nearly identical and critically coupled resonators.

We now include the possibility that Q-factor of one resonator in the coupled resonator system exceeds the Q-factor of the other. Then either a sharp transparency (CRIT) or absorption feature (CRIA) may appear within the broader transmission dip of this system owing to the supermode or narrow resonance dip of the higher Q resonator, respectively. For this case, it is sufficient to consider only a single resonator in the single-waveguide system. The transmission dip of this resonator _{${| {\tilde{τ}}_{1} |}_{SWG}$} is assumed to match the broader coupled resonator transmission dip of the dual-waveguide system. Therefore, the amplitudes of the sharp transparency or absorption features are now measured with respect to the on-resonance transmission amplitude of the single-waveguide based resonator and are given by _{$M = {| {\tilde{τ}}_{1} {\tilde{τ}}_{2} / (1 - {\tilde{ρ}}_{1} {\tilde{ρ}}_{2}) |}_{DWG} - {| {\tilde{τ}}_{1} |}_{SWG} .$} If the amplitude of the supermode is larger than the narrow resonance dip arising due to the higher Q resonator, M > 0 and CRIT is obtained in coupled resonator transmission [Fig. 3(a) and 3(c)]. However, M < 0 if the narrow resonance dip has larger amplitude in comparison to the supermode resonance and CRIA is attained in coupled resonator transmittance [Fig. 3(b) and 3(d)]. Indeed, M corresponding to the calculated transmission spectra reported here are in agreement with the conclusions drawn above. An analysis similar to the one described above shows that both CRIT and CRIA may be realized in CMR reflection as well.

6. Analysis of the simulated spectral effects and proposed implementation schemes

It is clear from the discussion of the foregoing section that appearance of sharp spectral features in the spectra of Fig. 3(b) and 3(d) is a consequence of the higher Q resonator, whereas origin of these features in the remaining figures is a result of the supermode resonance. Moreover, we also recall from our discussion of analogy between the coupled resonator and a Fabry-Perot resonator, that supermode appears only as a peak in transmission and dip in reflection. Therefore, the sharp transparency and absorption features appearing in the spectra are obtained owing to the supermode resonance and only for the case where a sharp dip (CRIA) is obtained in transmission and a sharp peak (CRIT) is realized in reflection of co-resonant CMRs composed of distinct Q resonators, these sharp features are actually the resonances of the higher Q resonator. Our calculations show that the supermode Q-factor depends on the Q of the two microring resonators. If the Q factors of microrings are increased, the supermode Q-factor increases as well, and this occurs for both detuned as well as co-resonant coupled resonators composed of same Q resonators. However, as demonstrated in Fig. 3(a), CRIT and CRIA may be obtained using co-resonant coupled resonators where Q-factor of one resonator is higher in comparison to the other. Even for such coupled resonators, the supermode Q-factor is found to be greater than Q of both resonators, even though the Q-factor of the second resonator in Fig. 3(a) is only slightly greater than Q of the other resoantor. Therefore, only for the case where CRIA (CRIT) is obtained in transmission (reflection) of co-resonant CMRs, the Q-factor of the sharp features is limited by the Q of the higher Q resonator of the system.

The integration of a higher Q-factor resonator into the coupled resonator structure certainly adds complexity to the fabrication process. However, as noted earlier, such a distinctive feature may notably enhance the functionality of coupled resonator. For instance, Q-tunable co-resonant coupled resonators may be realized experimentally using semiconductor optical amplifiers [24], and CRIT and CRIA may be tuned owing to an increase in Q, resulting in tunable slow and fast light effects. Alternatively, tunable CRIT and CRIA may be obtained using on-chip control of resonator-waveguide couplings, achieved through integrated resistive heating elements, which results in enhancement of the excited resonator Q-factor by an order of magnitude [25]. As already observed in Fig. 3, tunable CRIT and CRIA may be realized owing to Q variation of a single resonator in a co-resonant coupled resonator structure, and the resulting Q-modulation range appears to be sufficient for a host of applications. This is in contrast to detuned coupled resonators where such tunability is only possible if the Q factors of both resonators are altered exactly in the same manner. The situation is less critical for identical Q-factor based co-resonant coupled resonators. Here tunable CRIT and CRIA are possible even if the Q factors of the two resonators are not varied similarly. This is possible because for co-resonant coupled resonators, identical control of Q factors of both resonators is not essential for tuning CRIT and CRIA [Fig. 3(a) and 3(c)], although such a coupled resonator system will lose its identical Q character.

Results of a preliminary investigation that focuses mainly on the dispersive properties of co-resonant coupled resonators of the type considered in Fig. 2(a) shows that detuned resonators yield larger group delays, while co-resonant coupled resonators produce larger group velocity advancements. The resonator Q-factors assumed for the numerical simulations are experimentally achievable [26]. The published experimental coupled resonator parameters [7,14], including the inter-resonator separation L, produce CRIT and CRIA in co-resonant coupled resonators as well. Furthermore, typical gaps of a few hundred nanometers between the resonators and waveguides [27] are appropriate to realize the required couplings, as confirmed by the calculated couplings of the coupled resonator system investigated previously [7]. Our proposed scheme greatly simplifies realization of photonic coherence effects since instead of detuning resonances of two identical resonators, one only needs to ensure distinct couplings of the resonators to the two waveguides, as illustrated in the case of Fig. 2(a).

7. Summary and conclusions

In conclusion, we have proposed a simple alternative for achieving CRIT and CRIA based on co-resonant coupled resonators consisting of either identical or distinct Q resonators. Based on a theoretical model, we revealed a number of unique features of this system, including ease of attaining CRIT and CRIA via detuning of coupling, appearance of CRIT and CRIA in both transmission and reflection, and tunability of CRIT and CRIA resonances. We critically reviewed fabrication issues related to the observed effects and discussed practical roadmaps for their experimental implementation using various coupled resonator configurations. We believe the demonstrated sustaining of photonic analogs of quantum coherence effects under a variety of conditions is important for future advancements in optical quantum information processing and optical telecommunication networks.

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Photonic coherence effects from dual-waveguide coupled pair of co-resonant microring resonators

Abstract

Corrections

1. Introduction

2. Structures and resonances of coupled resonators with waveguide-mediated optical interactions

3. Analytic model for coherently coupled resonators

4. Results

5. Origin of the sharp features in the spectra of coupled resonators

6. Analysis of the simulated spectral effects and proposed implementation schemes

7. Summary and conclusions

References and links

Cited By

Figures (3)

Equations (8)

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