1. Introduction

Over the past few years, biological and chemical sensors based on microresonators have attracted much attention because they provide the significant advantages of high sensitivity, real-time response, minimal sample requirement, compatibility with microfluidics, and eliminating the requirement for fluorescent labeling [1–13]. In the resonator-based sensors, the light photons interact with the target molecules multiple times, which builds up the changes of the signal induced by the interactions. Specifically, the mode shift or transmission change at a fixed wavelength in response to variation of the effective optical path length is utilized as the sensing signal. Using this high-sensitivity mechanism, a whispering-gallery-type cavity with a quality factor in excess of 100 million has demonstrated single molecule resolution [14].

Resonance of a single cavity usually has a Lorenzian lineshape, which is symmetrical with respect to the resonant wavelength. Recently, a coupled-cavity configuration, consisting of a Fabry-Perot and ring cavities, has been proposed to give an asymmetric lineshape. This structure provides a larger transmission slope as compared to a single resonator, and enables improved sensitivity in sensing [15–17]. Here we propose an alternatively coupled microcavity geometry, which can generate a sharp asymmetrical line shape and offer one order of magnitude increase in transmission slope and, thereby, increase the sensitivity of the device.

2. Theoretical model

We consider a system in which two microresonators are coupled through a waveguide. One resonant mode is present in each of the cavities, as shown in Fig. 1a. In the case of slowly varying field amplitudes, the modes of this system can be described by the coupled harmonic oscillator model (which is well-known coupled-mode theory in the semi-classical limit), and the motion of the j ^th cavity mode (j=1, 2) is described by [18]

where c_j(t) is the bosonic annihilation operator of the j ^th cavity mode with resonant frequenc y ω_j; Δ_j=ω-ω_j denotes the detuning of the input carrier from the resonant frequency; κ_T,j represents the total cavity decay rate with κ_T,j=κ_I,j+2κ_E,j, such that κ_I,j is the intrinsic ca vity decay rate and κ_E,j is the external cavity decay rate; a ^(j) _in(b ^(j) _in) and a ^(j) _out(b ^(j) _out) describe the input and output fields in the left and right ports, respectively.

 

Fig. 1. (a) Schematic diagram of two coupled cavities. (b) The typical reflection (R), transmission (T), and loss (cavity absorption) spectra. (c) The enlarged reflection and loss spectra of the coupled-cavity system. Used parameters: κ_E=20κ_I and θ=-π/8.

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We are interested in the steady-state regime of the system. The internal cavity mode operator can be expressed as

Considering the coupling between the resonators and the waveguide that gives rise to standard cavity input-output formalism X ^(j) _out(ω)=Y ^(j) _in(ω)+κ ^1/2 _E,jc_j(ω), we obtain the output fields

where X,Y=a,b and X≠Y. Eq. (3) can be written in a matrix form

with the transfer matrix

This is a general description for propagation of light through a single-cavity, wave-guide system. In the case of two coupled cavities, the whole transfer matrix can be written as [19]

with T=T ₂ T ₀ T ₁. T ₀ denotes the transfer matrix for propagation through the waveguide,

where θ is the phase shift acquired by the wave-guide mode as it travels for a distance L, with θ=kL where k is the propagation constant. Assuming that there is no wave entering the right port, i.e., b ⁽²⁾ _in(ω)=0, we can derive the field transmission and reflection rates, defined by T(ω)≡|b ⁽²⁾ _out/(ω)/a ⁽¹⁾ _in(ω)|² and R(ω)≡|a ⁽¹⁾ _out/a ⁽¹⁾ _in|², respectively, with

For simplicity but without loss of the underlying physics, we assume the microcavities are identical, i.e., ω ₁=ω ₂, κ _E,1(2)=κ _E, κ _I,1(2)=κ_I, κ _T,1(2)=2Γ; and, κ_I≪κ_E (over-coupling regime). The over coupling is especially used to reduce the coupling instability (from perturbation of surrounding air and attractive gradient force of cavity mode field) because we can place the waveguide in contact with the microresonator in this case. With simple algebraic calculations, the maximum and minimum reflection rates are found at 0 and κ_E tan θ, respectively. It should be noted that these positions will be slightly modified if there are significant intrinsic losses and cavity-cavity detuning. In general, the point of the maximum reflection tends to drift from the minimum point, and it reduces the slope of the reflection.

To demonstrate the unique property of this coupled system, Fig. 1(b) shows the typical reflection spectra calculated from Eqs. (8) and (9) when the two cavities are resonant (ω ₁=ω ₂). Compared with the response of a single cavity, which behaves as a symmetrical Lorenzian lineshape, the coupled cavities possess a unique spectrum appearing as an asymmetric lineshape. This asymmetry originates from the conventional Fano effect [20–22], in which coherent, destructive interference occurs between two coupled cavities. For instance, the whole reflection is a sum of two optical paths: direct reflection from the first cavity and reflection from the second cavity. This configuration of coupled cavities has been proposed for storing photons [23–24] and implementing quantum information [25].

 

Fig. 2. (a) Slopes of the reflection spectra (dR/dω) in Fig. 1(b). (b) The slope of the reflection spectrum for various working conditions θ and ω-ω ₀. Here κ_E=20κ_I.

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3. Refractometric sensing based on asymmetric transmission lineshape

We propose that this unique asymmetric line shape with enhanced resonance slope can be used for ultra-sensitive, label-free, microcavity-based biosensing. On one hand, both the reflection and transmission exhibit asymmetric line shapes. However, since the reflection signal possesses lower noise, it was chosen as the sensing signal in our example. On the other hand, as shown in Fig. 1a, our theoretical mode is built on the standing-mode assumption. This is valid for microcavity in photonic crystal, but here we focus on whispering gallery modes (WGMs) in realistic microcavities since high-Q standing WGMs can form in the presence of strong modal coupling [26]. The resonant wavelength λ for a WGM in ring-type microcavity (with angular momentum number l) is

where a=30 µm is the radius of the microcavities; η denotes the fraction of the light energy traveling in the surrounding medium; n_s and n_c represent the refractive indexes (RI) of the surrounding medium and the cavity, respectively. Therefore, we have

For a given detection wavelength, the derivation of the reflection rate R is thus obtained as

where c is the light velocity in vacuum.

The reflection derivation, dR/dn_s defines the detection sensitivity of the coupled-cavity structure. As shown in Fig. 2, it provides more than an order of magnitude enhancement in sensitivity (directly proportional to the reflection slope) compared with a single-cavity configuration. With λ~640 nm, l~400, η=0.05,Q_I,j=ω_j/2κ_I,j~10⁸ [26], and from Fig. 2b, the detection sensitivity approaches 10⁷ when the probe wavelength and θ are optimized, which means that the reflection rate will deviate by 0.1 even when n_s changes by only 10^-8. It is noted that the reflection slope is sensitive to the phase shift θ. The destructive interference of Fano resonance is strongly dependent on the distance L. In Fig. 2b, θ of the red area with the highest slope covers about π/30, corresponding to the precision in distance of ~20 nm. This precision in distance is not difficult to reach by designing the initial mask for microcavity preparation, since L is dominantly dependent on the center positions of the microcavities.

In addition, such a measurement system could be constructed such that only one of the cavities is used as the sensing head in the test solution, while the other is utilized under normal conditions as a feedback cavity. In this case, both the cavities will almost respond synchronously to the environmental variations (such as thermal fluctuations) on the device because they are located in a micro-scale space. As a result, the reflection spectrum of the coupled system has a shift, but no distortion of the lineshape occurs. When target biomolecules interact with the first cavity, the reflection spectrum will undergo distortion in addition to the shift. Therefore, we can remove the effect of the environmental variations if we can define this kind of spectrum aberrance as the sensing signal, which is our future work.

 

Fig. 3. Typical reflection spectra (a) and their slopes (b) when the two cavities have a detuning ω ₂₁. Inset in (b) shows the maximum slope as a function of ω ₂₁. Here κ_E=20κ_I, θ=-π/8.

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4. Discussion

Note that the analysis described above is based on ideal conditions in which two identical microcavities are coupled at a constant temperature with negligible, non-linear effect. It is important, however, to address these following issues. First, when the two resonators have a detuning of ω ₂₁=ω ₂-ω ₁, we find that the detection sensitivity does not degrade dramatically, as shown in Fig. 3. On the other hand, a single cavity mode is in principle tunable with wet chemical etching [27–28], thermally [29] or electrically [30] tuning. Therefore, various techniques could be used to tune the resonance frequency of one of the cavities so that the initial detuning between the two microcavities could be optimized.

Second, the thermal effects (thermo-optic effect and thermal expansion) may play a significant role in the detection. The absorbed power (other than scattering loss) of the coupled structure will heat the cavities and lead to a shift of the coupled mode. We will briefly discuss this effect since it may affect the detection sensitivity and restrict the detection limit. When biomolecules interact with the cavity modes, the reflection spectrum of the coupled cavities undergoes a shift (usually a red shift), δλ=(dλ/dn_s)δn_s. If the device works in the right side of the reflection spectrum, there will be a decrease of the reflection rate (point A in Fig. 1(c)) at λ=λ ₀ with δR=(dR/dn_s)δn_s. However, the rate of absorption of the probe power (point B) by the coupled system increases due to the red shift of the spectrum, as shown in Fig. 1(c). As long as the device material has a positive thermo-optic coefficient (dn/dT), another red shift of δλ′ is added to the reflection peak due to the heating effect. Therefore, the total reflection rate changes δR+δR′ where δR′=(dR/dλ)δλ′, which is larger than the change expected without thermal effect. It shows that the detection sensitivity can be improved by introducing the probe-induced thermal effect if the device is chosen to work in the right side of the spectrum. It must be remembered that there are always some instabilities in the probe power and surrounding temperature. Those instabilities will significantly degrade the advantages of sharp asymmetric line shapes, thus increasing the detection limit [31]. The detailed analysis should consider the dynamical thermal behavior and thermal self-stability. This will be addressed in the future work.

Third, the optical Kerr effect cannot be ignored in a high-Q microcavity due to large intra-cavity field intensity [32]. For example, when Q factors are as high as 10⁸ and the diameter of the cavity is around 60 µm, only 1 mW of coupled power can produce an enormous circulating intra-cavity field intensity (I) on the order of 1 GW/cm², The RI of the cavity material can be described by n_c=n ₀+n ₂ I, where n ₀ and n ₂ are linear and non-linear RI, respectively. Similar to the probe-induced thermal effect, as long as the device is working in the right side of the spectrum, the probe-induced Kerr effect can also be used to improve the sensitivity. When the coupled mode has a red shift induced by biomolecules, the intracavity field intensity is increased, which will introduce another red shift due to the Kerr effect. Thus, the optical Kerr effect helps increase the cavity mode shift caused by a small change in the RI of the surrounding medium under appropriate conditions. In other words, the detection sensitivity is enhanced due to the probe-induced Kerr effect.

5. Summary

In conclusion, we theoretically analyzed the applicability of a system, which comprises of two coupled microcavities, as a RI sensor with enhanced sensitivity. Due to the sharp asymmetrical Fano resonance, the output transmission variation (at a fixed wavelength) from the coupled system is much larger than that from a single cavity resonance, thus resulting in an order of magnitude enhancement in detection sensitivity. The effects of thermal and optical Kerr effects on the performance of the system of coupled cavities are also discussed. It is found that both these effects can be used to improve the sensitivity by choosing the appropriate working points.