1. Introduction

In medical diagnostics, there is an increasing demand for biosensors that can detect specific analytes in a biological fluid. This is a challenging task, given the very low concentrations aimed, in the nM range, and the presence of many other species, sometimes in much larger concentration. Ideally, testing would be performed at the point of care, in order to save time and labour, and to be able to quickly adapt therapies to health outcomes. Several sensing platforms already provide solutions for point-of-care testing (POCT) such as electrochemical impedance spectroscopy [1] or surface plasmon resonance (SPR) [2]. On the other hand, optical circular microresonators, such as microspheres or microrings, are yet to reach the stage of POCT. Their high potential as label-free molecule detector has been demonstrated by Vollmer et al. in a set of seminal contributions [3,4] followed by numerous works. The large quality factor, $Q$, and small mode volume, $V$, of the whispering gallery modes sustained by these resonators are the key to their high sensitivity [5]. Next to resonance shift, sensing can be done through resonance broadening [6,7], transient response [8], or mode splitting (an overview is given by Vollmer and Yang [9].) Over the years, important progresses have been made to further increase the sensitivity, by cascading resonators [10,11], by actively reducing resonator losses [12,13] or, more recently, by working near exceptional points [14]. A detection limit down to single nanoparticle or molecule the size of a protein or a virus has thus been experimentally demonstrated [12,15,16]. Furthermore, with nonlinear optical processes, similar detection limit in terms of mass has been reached, but with molecules of much lower molecular weight ($<500$ Da) [17–19].

In order to bring this emerging technology closer to the ultimate goal of a low-cost, point-of-care diagnostic instrument, these microresonators have been integrated in photonic platforms, such as silicon-on-insulator [21–23], silicon nitride [24–26], silicon oxynitride [27] and aluminium oxide [28]. In these settings, multiplexed arrays have been demonstrated, where each ring microresonator is chemically functionalized for a specific response to a distinct type of molecule [29–31] and the array has been coupled to microfluidic channels [24]. Another important evolution is the move from standard samples to real, complex biological fluids; in this regard, biosensing has been demonstrated with diluted human serum [32], undiluted human plasma [33], and urine [34]. Very recently, a new landmark was made with the detection of the cancer biomarker S100A4 protein in undiluted urine at clinically-relevant concentrations and without amplification sandwhich assay [28].

Despite these advances, one important obstacle remains on the way to make microresonator arrays inexpensive and portable sensors. Their operation usually requires either a finely tunable laser source or a high-precision spectrometer, which are expensive and bulky. Heterodyne detection using a reference ring may appear to solve the problem but is in fact impractical. Indeed, minimal uncertainties in the ring fabrication immediately lead to significant changes in the reference frequency. For instance, to increase the radius of a microring from $50 \mu$m to $50.001\mu$m is sufficient to shift a given optical resonance by as much as 1GHz. Hence, calibration by accurate spectral characterization of the reference microresonator remains necessary. One interesting response to this issue is the self-heterodyne detection put forward by He et al. [12,13]. There, the adsorption of a single particle splits some resonances and leads to a low-frequency, easily detectable beating signal. However, as further particles attach to the resonator, the frequency splitting can either decrease or increase, in an unpredictable manner, which makes that detection scheme unsuitable for concentration measurements.

In this paper, we propose to structure circular microresonators with a Bragg grating in a way that resonance splitting increases monotonously with biomarker concentration. Here, mode degeneracy is considered on either side of the edge of a Brillouin zone and particle deposition is restricted to happen only on a fraction of the perimeter of the resonator, see Fig. 1. Ideally, biomarkers should bind only over a quarter, or two opposite quarters, of the resonator. Such a partial binding on the perimeter of the resonator could either be achieved with state-of-the-art functionalization technique or by using two-port microfluidic channels that let the analyte flow only over part of the resonator surface. As analytes bind to the resonator, the degeneracy between some modes of the Bragg-structured microresonator is lifted and resonance frequency splitting occurs. If the two split resonances are excited by a single broadband source, their simultaneous oscillation produces a low-frequency beating, by which the output intensity undergo a low-frequency modulation. With currently available $Q$ factors, this beating signal can be less than a few GHz and can thus be recorded with off-the-shelf electronics. In this way, the use of a spectrometer or a tunable laser can be avoided and the detector can be simplified enough to be brought to the point of care. Importantly, the magnitude of mode splitting increases monotonously with the number of binding analyte (unlike in the scheme proposed by He et al. [12]). Moreover, variations of mode-splitting, being exclusively due to selective molecular binding, make the detector self-referenced and immune to bulk changes of the environment parameters, such as temperature or concentrations of species other than the analyte. Note that due to fabrication imperfections, such as defects in the Bragg grating, splitting may pre-exist in the spectrum. However, splitting subsequently increases linearly with the number of analyte binding events [28], allowing one to separate both effects.

 

Fig. 1. Sketch of a microring cavity containing a Bragg grating and functionalized over one quarter of its perimeter. The ring is excited by injecting light in a neighbouring waveguide. If the injected wave is in resonance, energy builds up in the cavity and the intensity at the output of the waveguide is depleted, producing a sharp resonance dip in the transmission spectrum at critical coupling [20]. Molecular binding in the functionalized area splits a given resonance into two, which, under proper excitation, can produce a recordable low-frequency beating signal.

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1.1 Cavity eigenmodes

In the absence of both a Bragg grating and molecules adsorbed at its surface, the eigenmodes of a microring resonator are described by electromagnetic fields of the form $\Re \left [\phi _{\ell } e^{-i \omega _{0}(\ell )t}\right ]$, where

 

Fig. 2. Left: frequency spectrum $\omega (\ell )$ of the ring+Bragg grating, where $\ell$ is the angular number of the mode. To each resonance corresponds a dip in the transmission spectrum $I(\omega )$ of the neighbouring waveguide. Right: perturbed spectrum under proper functionalization of the ring. The degeneracy of the modes $\ell _c\pm 1$ is lifted and the corresponding transmission dip is split into two.

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As can be seen in Fig. 2 (left) or by simple inspection of Eq. (3), $\psi _{\ell _c-p}$ and $\psi _{\ell _c+p}$ have the same frequency. Hence, any mode combination $c_1 \psi _{\ell _c-p}+c_2 \psi _{\ell _c+p}$ is liable to be excited in the ring as one injects light in the neighbouring waveguide at the corresponding resonance. In the same way, $\chi _{\ell _c-p}$ and $\chi _{\ell _c+p}$ are frequency-degenerate on the upper branch. The sensing scheme proposed here consists in selectively breaking one or more of these degeneracies by appropriately functionalizing the ring, thus leading to a transmission spectrum as in Fig. 2 (right). Such a splitting, being only due to surface functionalization and particle attachment, is self-referenced and thus immune to parasitic environment changes, such as temperature or salinity. Additionally, with currently achievable $Q$ factors, splitting well within the GHz range can be resolved and thus monitored by conventional electronics.

1.2 Mode splitting due to additional polarization

In order to assess what kind of spatial functionalization is required, consider the following two modes on the low-frequency branch

If the cavity is subsequently perturbed by a material with distributed polarizability ${\boldsymbol {\alpha }}$, a polarization

It is possible to have only one peak of the transmission spectrum that split if, within the range $-\pi /4<\theta <\pi /4$, $\boldsymbol {\alpha }$ vanishes periodically with the same period as the Bragg grating instead of being uniform. If, for instance, antibodies are attached only on the high-index regions, then the high frequency modes have minimal overlap with the attached molecules and undergo negligible frequency shift. In that case, a splitting given by Eq. (11) is only expected for modes $\psi _+$ and $\psi _-$. Conversely, only modes $\chi _\pm$ are expected to split if the molecules specifically attach to the low-index part of the Bragg grating.

The above theory, based on Eqs. (4) and (5), can be seen as a minimal coupled-mode theory. Such a picture can be refined, with more general expansions, leading to more accurate but otherwise qualitatively identical results. One such expansion is given in Appendix A. In this framework, one can confirm that $\psi _\pm$ given approximately by Eqs. (6) and (7) indeed appropriately describe the cavity eigenmodes in the presence of molecules over one quarter of the perimeter.

To support the above theory, we performed a large set of finite element simulations using COMSOL. We simulated a ring cavity with refractive index corresponding to aluminium oxide (Al$_2$O$_3$). This material shows great potential as an integrated photonics platform, notably thanks to its compatibility with silicon and silicon nitride and the high doping concentration of rare-earth ions that it can accomodate to form micro-lasers [37,38]. Recently, high-$Q$ micro-ring [39,40] and photonics detection of cancer biomarkers in undiluted urine [28] have been demonstrated with Al$_2$O$_3$. Obviously, however, the validity of our analysis does not rely on this particular choice of material.

In our simulations, we assumed a 2D microring with external and internal radii given by $10.25$ and $9.75\mu$m, respectively. The ring was excited by a waveguide of width $0.5\mu$m and the waveguide-ring distance was 0.2$\mu$m. A Bragg grating containing 180 periods ($\ell _c=90$) was inscribed. Each period contained two sections of equal length, one with refractive index $n_1=1.6$ and the other with $n_2=1.65$. The difference $n_2-n_1$ was chosen to produce a band gap that is easy to discern in the spectrum given the large free spectral range of the cavity [see Eq. (3)]. To simulate the presence of antibodies and adsorbed target molecules, we increased the refractive index by an amount $\Delta n'$ over a portion of the ring perimeter. Most of the simulations were done as described in the preceding section, i.e. with $\Delta n'$ being nonzero only over one quarter of the perimeter. We also performed simulations with a perturbation $\Delta n'$ on two opposite quarters of the ring, in order to improve the sensitivity. In the sensing section of the ring we considered three possible situations:

2. Results

In all configurations studied, the calculated spectra and accompanying field distributions conformed to the above simple theory and quantitatively agreed with the more detailed theory in the Appendix. Out of all the combinations of degenerate cavity modes mode when $\Delta n'=0$, the one that maximises the intensity precisely where the refractive index has been raised is the most energetically favourable. Hence, after perturbation it acquires the lowest frequency and longest wavelenght. This mode combination and its orthogonal counterpart are, in first approximation, the eigenmodes in the presence of the perturbation $\Delta n'>0$. Figure 3 illustrates the splitting mechanism, in the case where $\Delta n'$ is only added to $n_2$, and demonstrates the monotonous increase of the magnitude of splitting with $\Delta n'$. As expected, the modes $\psi _\pm$ on the long-wavelength branch are the most affected, while $\chi _\pm$ are almost insensitive to this perturbation. Indeed, the latter have nodes of intensity in the sections of the rings with either $n=n_2$ or $n=n_2+\Delta n'$ and thus minimally feel the perturbation $\Delta n'$. The computed splitting obtained for $\Delta n'=0.012$ is $1.78$nm, giving a sensitivity $S \approx 150$ nm/RIU (bulk Refractive Index Unit). If two opposite quarters of the ring are functionalized instead of one, Eq. (9) indicates, and simulations confirm, that the sensitivity is doubled. This value of $S$ is consistent with what is computed using the extended coupled mode analysis of the Appendix.

 

Fig. 3. Transmission spectra and mode intensity distributions as a function of index perturbation $\Delta n'$ for an Al$_2$O$3$ ring in air environment (see text). Top left: unperturbed spectrum of the ring with refractive index alternating between $n_1=1.6$ and $n_2=1.65$. The number of periods along the perimeter is $2\ell _c$, with $\ell _c=90$. Each transmission dip is labelled by its angular number $\ell$. The peaks on the long wavelength side correspond to the mode $\psi _{\ell _c\pm p}$, $p=0,1,2,\ldots$. Top right and second row: spectra obtained with the changes $(n_1,n_2)\to (n_1, n_2+\Delta n')$ in one quarter of the perimeter, designated by white arrows in the bottom pictures. Splitting is observed only with the long-wavelength peak labelled $\ell _c\pm 1$. Bottom row, from left to right: unperturbed mode electric field norm at $\lambda =1.0435\mu$m, and perturbed mode at wavelengths $\lambda _-$, $\lambda _+$ for $\Delta n'=0.012$.

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In Fig. 4, we confirm the effect of microscopic functionalization on mode splitting. Starting with the same cavity as in in Fig. 3, we first perturb only the low-index section of the Bragg grating. In that case, the modes $\psi _\pm$, on the long-wavelength side of the band gap, have intensity nodes at the locus of perturbation and, hence, minimally sense the presence of the perturbation $\Delta n'$. Hence, splitting only occurs with the modes $\chi _\pm$ on the short-wavelength side of the band gap. If, on the contrary, the perturbation is uniform over a quarter of the ring, then it produces splitting on either sides of the gap, as fully confirmed by simulations.

 

Fig. 4. Simulated spectra of the same parameters as in Fig. 3 but where the perturbation over a quarter of perimeter is $(n_1,n_2)\to (n_1+\Delta n',n_2)$ (top) and $(n_1,n_2)\to (n_1+\Delta n',n_2+\Delta n')$ (bottom), corresponding to distinct microscopic functionalization of the ring.

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From the preceding theory, what is fundamentally required for the self-heterodyne detection scheme proposed here is a band gap, together with partial functionalization of the ring. By applying a periodic modulation of the refractive index along the ring perimeter, there are in fact an infinity of band gaps in the spectrum, at angular mode numbers $\ell =\ell _c, 2\ell _c, 3\ell _c,\ldots$. Similar splitting of modes can be observed near the edge of the higher-order band gaps as well. To demonstrate this point, we have carried out additional numerical simulations with the same ring as in Figs. 3 and 4 but with an angular grating period of $\pi /30$, that is $\ell _c=30$ instead of 90. We still observed splitting for modes of angular number near $\ell \approx 90$, albeit with a smaller sensitivity $S\approx 50$nm/RIU, see Appendix B. One practical advantage to work with higher-order band-gap is the possibility to reduce $\ell _c$, and hence the number of grating periods to manufacture along the ring.

Figure 5 (left) confirms the linear dependance of the splitting on the perturbation $\Delta n'$. Knowing the sensitivity $S$, given by the slope of this dependance, we can deduce the smallest detectable change $\Delta n'$, i.e. the Limit Of Detection (LOD) of the system. To this end, we require that the frequency splitting be as large as a fraction $F$ of the resonance width, which is given by $\lambda /Q$. Hence, we obtain

 

Fig. 5. Left: Mode splitting vs perturbation $\Delta n'$. The dashed line is obtained from an extended coupled theory and the dots represent COMSOL simulations. Error bars correspond to the discretization step used to scan the spectrum. The slope of the line indicates a sensitivity $S\approx 150$nm/RIU. Right: LOD vs $Q$ factor assuming $F=1$ in Eq. (12).

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3. Discussion

We have shown that stable and regular mode-splitting can be achieved with micro-ring resonators as a result of analyte binding. The strategy behind the design that leads to this property consists of two steps. Firstly, one engineers frequency-degenerate modes with distinct intensity distributions within the cavity. This is the role of the Bragg grating. Secondly, within the vectorial space spanned by two such degenerate mode one examines orthogonal pairs and their intensity distribution. What makes $\psi _+$ and $\psi _-$ given by Eq. (6) and Eq. (7) interesting from a practical point of view is that they display complementary long-wavelength interference along the ring. This suggests easy-to-manufacture functionalization patterns that can have the most contrasting overlap between these two modes –in the present case, one quarter or two opposite quarters of the perimeter.

To functionalize a ring only over a fraction of its perimeter can be done in several ways using existing technology. Using state-of-the art fabrication technique, it is possible to deposit antibodies in a controlled manner only over the desired fraction of the perimeter. Moreover, surface treatments may lead to preferable anti-body functionalization only over the high-index part of the grating period (M. de Goede, private communication), thus enforcing splitting specifically as in Fig. 3. Another possible technique is to uniformly functionalize the ring but to flow biomarkers only over a fraction of the ring surface, so that molecule attachment is achieved only over that fraction. This can be done by placing the microring under a microfluidic junction where one arm is flown with a solution containing the analyte while the other arm carries a neutral solution. By controlling the flux in the two input channels of the microfluidic device, one can control the separation streamline between the two flows [42] and let only a desired angular fraction $\Delta \theta$ of the ring exposed to the analyte. In Appendix C, we compute mode splitting for values of $\Delta \theta$ distinct from $\pi /2$ (corresponding to quarter coverage) and check that indeed the largest splitting is obtained for that value.

Thanks to continuous mode-splitting, one can operate whispering gallery mode resonators as self-referenced sensors, avoiding the use of tunable lasers or spectrometers. This may pave the way to portable sensors based on whispering gallery modes that can be used outside the laboratory and without high-level training. In particular, this family of sensors could enrich the arsenal of diagnostic tools in the hands of physicians for POCT.

Appendix

A. Outline of the coupled mode theory

In this section, we provide a couple-mode theory of the microring containing a partially functionalized (i.e. non uniform) grating. Note that the transmission characteristics of a ring covered by a uniform grating was treated in [43], while rings partially covered by a uniform grating where considered in [44,45].

A.1. Bare cavity

In the absence of both a Bragg grating (${\Delta } {n=0}$) and the perturbation due the molecular attachment ($\Delta n^{\prime }=0$), the eigenmodes of a microring resonator are the electromagnetic fields ${\mathbf {E}}_{\ell} (r,z) {e^{i \ell \theta }}$ and ${\mathbf {H}}_{\ell } (r,z) {e^{i \ell \theta }}$, where ($r,\theta ,z$) denote the cylindrical coordinates and ${\ell }$ is the angular number. For transverse electric (TE) modes, ${\mathbf {E}}_{\ell} (r,z)= (0,0, {u}_{\ell} (r,z))$ and for transverse magnetic (TM) modes, $\mathbf {H}_{\ell} (r,z)= (0,0, {u}_{\ell} (r,z))$, where ${u}_{\ell} (r,z)$ is a solution of the Helmholtz equation

(A.1)$$\begin{aligned} \nabla^{2} u_{\ell}+ \left(\frac{n^{2}}{c^{2}} \omega_{0}(\ell)^{2} -\frac{\ell^{2}}{r^{2}} \right) u_{\ell} &=0, &n&=\left\{ \begin{array}{lc} n_{2}, & R-h/2 \,< \,r < \,R+h/2,\\ n_{env}, & {\textrm{elsewhere}}. \end{array}\right. \end{aligned}$$

Above, $R$ is the mean radius of the ring and $h$ is its width. At surfaces of discontinuity of $n$, both $u_{\ell }$ and $\frac{\partial {u_{\ell }}}{\partial r}$ are continuous for TE modes, while for TM mode, continuity applies to $u_\ell$ and $n^{-2}\frac{\partial {u_{\ell }}}{\partial r}$. In the far field, as $r\to \infty$ the radiation condition implies that $u_{\ell} \propto {e}^{in_{env}\omega r/c}/\sqrt {r}$. The microring is made of material of refractive index $n_{2}$ and embedded in the environment with refractive index $n_{env}$. For simplicity, we consider air environment ($n_{env}=1$). The above Eq. is strictly valid in 2D only. In practice, the ring is made of a circular waveguide of finite vertical extend; $n_{2}$ should then be taken as the effective index $n_{eff}$ of this waveguide for the frequency considered. Finally, $\omega _{0}(\ell )$ is the pulsation of the resonance. For large $\ell$, one has, approximately, the linear dependance $\omega _{0}(\ell )=c\ell /(n_{2} R)$.

For values of $|\ell |$ in a limited neighbourhood of a reference value $\ell _{c}\gg 1$, the eigenmodes of the bare cavity can be approximated as

(A.2)$$u_{\ell}e^{i \ell \theta} \sim \left[u_{\ell_c} +O\left(\frac{\ell-\ell_c}{\ell_c}\right)\right]e^{i \ell \theta}, \quad u_{-\ell} e^{{-}i \ell \theta} \sim \left[u_{-\ell_c} +O\left(\frac{\ell-\ell_c}{\ell_c}\right)\right] e^{{-}i \ell \theta}.$$

Furthermore, $u_{-\ell _c}=u_{\ell _c}$. On the other hand, we have, approximately,

(A.3)$$\omega_0(\ell)^{2} = \left\{ \begin{array}{ll} \omega_0(\ell_c)^{2}( 1+ 2p/\ell_c), \quad\quad \textrm{for } \ell = \ell_c +p, \\ \omega_0(\ell_c)^{2}( 1- 2p/\ell_c), \quad\quad \textrm{for } \ell ={-}\ell_c +p, \end{array}\right. \quad \quad |p| \ll |\ell_c|.$$

A.2. Cavity with a Bragg grating

Let the refractive index be modulated between $n_1$ and $n_2$ with angular period $\pi /{\ell _c}$, subdividing the ring into $2\ell _c$ sections. We thus have, inside the ring:

(A.4)$$n(\theta) =n_2+\Delta nF(\theta)= n_2\left(1+ \delta F(\theta)\right), \quad \delta = \frac{\Delta n}{n_2},$$

where ${F(\theta )}$ is a unit step function with periodicity $\pi /l_{c}$. Assuming a 50% duty cycle, ${ i.e.}$ that $n_{1}$ and $n_{2}$ occupy equal proportions of a grating period, ${F(\theta )}$ has the following Fourier decomposition:

(A.5)$${F(\theta)} = \sum_{m={-}\infty}^{\infty} C_m e^{2i m \ell_c \theta}, \quad\quad C_m=\left\{ \begin{array}{ll} \frac{1}{2}, \quad\quad\quad\quad\quad \textrm{for } m=0, \\ -\frac{1}{\pi} \frac{\sin m \pi/2}{m} \quad\quad \textrm{for } m\neq0. \end{array}\right.$$

With this grating, the eigenmodes become linear combinations of the form

(A.6)$$u=\sum_{\ell={-}\infty}^{\infty} A_{\ell} u_{\ell} e^{i \ell \theta}.$$

Inside the ring, where most of the intensity is confined, Helmholtz equation thus yields

(A.7)$$\left(\nabla^{2} + \frac{n_2^{2}}{c^{2}} \omega^{2} -\frac{\ell^{2}}{r^{2}} \right) A_{\ell} u_\ell + 2 \frac{n_2^{2}}{c^{2}} \omega^{2} \delta \sum_{m={-}\infty}^{\infty} C_m A_{\ell-2m \ell_c} u_{\ell - 2m\ell_c} =0.$$

Here, $\omega$ is the eigenfrequency of the ring resonator in presence of the Bragg grating.

For a grating with a very shallow modulation of the index, $|\delta | \ll 1$, it is sufficient to consider only the dominant Fourier coefficients $C_0$ and $C_{\pm 1}$ of the expansion (A.5). Ignoring all $C_p$ with $|p|>1$ in Eq. (A.7) corresponds to coupling the modes $\ell$ and $-\ell$ of the bare ring with those of angular number with $\ell - 2 \ell _c$ and $-\ell + 2 \ell _c$, respectively. Further, in the expansion (A.6) we restrict the summation to values of $\ell$ such that $\ell = \ell _c +p$ and $\ell = \ell _c -p$ where $|p| \ll |\ell _c|$. Then, the eigenfrequencies of the ring with Bragg grating can be expressed as

(A.8)$$\omega \sim \omega_0(\ell_c) \left(1+\delta\kappa\right),$$

and it is convenient to denote

(A.9)$$a_p\equiv A_{\ell_c+p}, \quad\quad b_p\equiv A_{-\ell_c+p}.$$

Then, using Eq. (A.1), Eq. (A.7) yields

(A.10)$$\left( \kappa - \frac{ p}{ \delta \ell_c}\right) a_p u_{\ell_c} + C_0 a_p u_{\ell_c} + C_1 b_p u_{-\ell_c} \approx 0$$

(A.11)$$\left( \kappa + \frac{ p}{ \delta \ell_c}\right) b_p u_{\ell_c} + C_0 b_p u_{\ell_c} + C_{{-}1} a_p u_{-\ell_c} \approx 0$$

And since $u_{\ell _c}(r, z) =u_{-\ell _c}(r,z)$, the relative change of the eigenfrequency due to the grating only is given by $\delta \kappa (p)$, where

(A.12)$$\left( \begin{array}{cc} \kappa - \frac{ p}{ \delta \ell_c} + C_0 & C_1 \\ C_{{-}1} & \kappa + \frac{ p}{ \delta \ell_c} + C_0 \end{array}\right) \left(\begin{array}{c} a_p \\ b_p \end{array}\right)\approx 0.$$

Setting the determinant to zero yields

(A.13)$$\kappa(p) \approx{\pm} \sqrt{ C_1 C_{{-}1} + \frac{p^{2}}{\delta^{2} \ell_c^{2}} } - C_0.$$

Hence, the eigenfrequencies of the ring with the Bragg grating are given by

(A.14)$$\omega_\pm(p) \approx \omega_0(\ell_c) \left[1 - \delta C_0 \pm\delta \sqrt{ C_1 C_{{-}1} + \frac{(\ell-\ell_c)^{2}}{\delta^{2} \ell_c^{2}} } \right].$$

Neglecting $\delta C_0$ compared to $1$, one thus obtains

(A.15)$$\omega_\pm(p) \approx \omega_0(\ell_c) \pm \sqrt{ \left[v_g(\ell-\ell_c)/R\right]^{2}+\Delta\omega^{2}/4} .$$

with $v_g=c/n_2$ and $\Delta \omega =2\delta \omega _0(\ell _c) \sqrt {C_1 C_{-1}}=2\delta \omega _0(\ell _c) /\pi$.

A.3. Adding a non-uniform perturbation

We next model molecular attachment over a finite extend of the ring perimeter through an increase $\Delta n'$ of the refractive index in the range $-\Delta \theta /2 \leq \theta \leq \Delta \theta /2$. The refractive index profile in the ring becomes

(A.16)$$n(\theta) =n_2+\Delta nF(\theta)+\Delta n'G(\theta)= n_2\left[1 + \delta \left(F(\theta) +\frac{ \Delta n'}{\Delta n} G(\theta)\right) \right],$$

where $F(\theta )$ is same as before and $G(\theta )$ is a step function that is equal to 1 over the range $[-\Delta \theta /2, \Delta \theta /2]$ and zero elsewhere. Its Fourier decomposition is

(A.17)$$G(\theta) = \sum_{j={-}\infty}^{\infty} g_j e^{i j \theta}, \quad\quad g_0= \frac{\Delta\theta}{2\pi}, \quad\quad g_j = \frac{1}{j \pi}\sin (j \Delta \theta/2).$$

Using the same arguments and approximations as in the previous subsection, we get

(A.18)$$\left( \kappa - \frac{ p}{ \delta \ell_c}\right) a_p + C_0 a_p + C_1 b_p + \frac{ \Delta n'}{\Delta n} \sum_{q={-}\infty}^{\infty} g_q a_{p-q}\approx 0$$

(A.19)$$\left( \kappa + \frac{ p}{ \delta \ell_c}\right) b_p + C_0 b_p + C_{{-}1} a_p + \frac{ \Delta n'}{\Delta n} \sum_{q={-}\infty}^{\infty} g_q b_{p-q} \approx 0.$$

We truncate the system above to values of $p$ in the range $-10\leq p\leq 10$. This yields a set of 42 Eqs. which, we find, is sufficiently large to provide accurate results. The perturbed frequencies are obtained from the eigenvalues $\kappa$ of this finite linear system.

B. Sensing near the edge of a higher-order band gap

The self-heterodyne sensing scheme proposed here requires a degenerate spectra around a band gap along with a partial functionalization of the ring. A shallow period modulation of the refractive index along the ring perimeter generates an infinite number of band gap in the spectra, at angular mode numbers $\ell =\ell _c, 2\ell _c, 3\ell _c,\ldots$. In the main text and above, we focused on the modes in the proximity of the primary band gap around $\ell =\ell _c$.

The higher-order band gaps can also be probed for sensing via the proposed scheme. To demonstrate this, we consider a ring of radius $R=10 \mu$m and width $0.5 \mu m$ with a grating of angular period $\pi /30$, that is $\ell _c=30$. With $n_2=1.65$, this generates third-order band gap at $\ell =90$ with wavelength $\sim 1 \mu$m. Note that the width of the band gap is proportional to the appropriate Fourier component of the Bragg modulation. The relevant Fourier coefficients in this case are $C_{\pm 3}$, where $|C_{\pm 3}| = |C_{\pm 1}|/3$, making the band gap three times smaller than before. As a result, the band gap is barely noticeable in the spectrum. Nevertheless a close look at the intensity distribution inside the ring, Fig. 6, confirms its presence. Indeed out of the two modes labeled $\ell =3\ell _c$, the long-wavelength one has two intensity maxima in each section of high refractive index $n=n_2$ but only one intensity maximum in each section with $n=n_1<n_2$. Meanwhile, the converse is seen with its short-wavelength companion. This demonstrates that the interval between these two peaks is indeed the third-order band gap. In Fig. 6, we plot frequency spectra for increasing values of the functionalization index $\Delta n'$. Again, we see that mode splitting affects the resonance next to the edge of the band gap. On the other hand, the sensitivity in terms of wavelength splitting per RIU is smaller than in the previous simulations, being on the order of 50 nm/RIU with one functionalized quarter.

 

Fig. 6. Computed spectra in a ring of radius $R=10 \mu$m and width $0.5 \mu$m with a Bragg grating of angular period $\pi /30$. Here, $\ell _c=30$ and the third-order band gap is located at the mode angular number $\ell =3\ell _c$. Top left: unperturbed spectrum, with peaks labelled by their corresponding angular number. Top right: close-up of the intensity distribution of the two unperturbed modes with $\ell =3\ell _c$. Bottom row: mode splitting of neighbouring resonance resulting from the quarter-perimeter perturbation $\Delta n'$.

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C. Variation of angular coverage $\Delta \theta$ of the perturbation

In this section, we apply the coupled-mode theory developed above with variable coverage angle $\Delta \theta$. We compute mode splitting for a 150$\mu$m radius cavity with refractive index alternating $2\ell _c$ times between $n_1$ and $n_2$, with $\ell _c=1500$. The results, displayed in Fig. 7, confirm that a quarter coverage is optimal, in agreement with the simplified theory developed in the main text.

 

Fig. 7. Left: Mode splitting vs coverage angle $\Delta \theta$ for a fixed perturbation $\Delta n'$. Right: LOD vs $Q$ factor assuming $F=1$ for variable $\Delta \theta$. Inset: corresponding wavelength splitting vs $\Delta n'$. Calculation performed on an Al$_2$O$_3$ microring ($n_2=1.65$) with mean radius $R=150 \mu$m and width $w=500$nm.

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Funding

Fonds De La Recherche Scientifique - FNRS; Horizon 2020 Framework Programme (634928).

Disclosures

The authors declare no conflicts of interest.

References

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