1. Introduction

Optical microresonators are at the heart of a wide range of applications and studies [1]. Particularly, WGM microcavities such as microspheres [2,3], microdisks [4], microbottles [5] and microtoroids [6] resonantly confine light to small volumes and can serve as highly sensitive label-free, fast detectors [7–11]. WGM microresonators were harnessed for single-particle detection and sizing of nano-objects [12–18], as well as of monolayers [19,20]. The characterization of non-spherical particles was performed by measuring the shifts of a single Lorentzian dip resonance of the microresonator [2,21]. This method was applied later to characterize a layer of uniaxial particles by measuring the spectral changes of the TE and the TM modes [19]. A number of works proposed to use mode splitting to acquire information on a single arbitrarily shaped nanoparticle [22,23]. Mode splitting arises when two counterpropagating modes, ideally degenerate, spectrally split to two non-degenerat modes due to perturbations in the surface or the surrounding of the microcavity [13,24–26]. The eigenmodes of open-systems are also known as quasinormal-modes (QNMs) [27]. The perturbation may be caused by a nanoparticle entering the optical mode of the microcavity; the spectral properties of the mode splitting can then be correlated to the optical properties of a nanoparticle. Xu et al. proposed a theoretical method to extract the orientation and polarizability tensor of a nanoparticle with a negligible imaginary part and no prior mode splitting from the splitting and broadening signals [23].

In this work, we suggest a complementary approach for 3D characterization of arbitrarily shaped nanoparticles with complex polarizability based on measuring the induced shift, splitting and broadening of a single or split Lorentzian resonance, which is often the case in high-Q WGM sensing. We experimentally characterize a single WS$_2$ nanotube using a high-Q toroidal microresonator by independently probing the whispering gallery TE and TM modes along the equator, and then correlate the induced spectral changes to the optical properties of the nanotube.

The analytical model we develop is an extension of the formalism introduced by Mazzei et al. [28], which was later generalized to treat multiple scatterers [29], and non-spherical Rayleigh scaterrers [23]. The model describes how the spectral changes (of both TE and TM) depend on the coupling and loss rates, and how these rates depend on the linear (TE) or elliptical (TM) [30–32] polarization of the optical mode (where linear polarization is always parallel to the local WGM surface, whereas elliptical polarization includes also a longitudinal component) and the polarizability properties of arbitrarily shaped nanoparticle [23]. Our model takes into account the often pre-existing inherent coupling between the counter-propagating modes (exhibited in a split lorentzian spectrum), and for the asymmetric backscattering between the counterpropagating modes that arise when the losses of the system are significant [16,33–35]. It also accounts the effect of the imaginary part of the polarizability tensor [36] and the finite size of the nanoparticle on the induced spectral changes. Finally, to overcome the strong influence of the exact position of the nanoparticle on the measurement, we propose a self-referenced method of measurement that allows real-time monitoring of the particle position in the standing wave pattern.

2. Analytical model

2.1 Hamiltonian of the cavity modes

WGM resonators support two counterpropagating modes coupled to the tapered fiber modes (see Fig. 1(a). Ideally, these two modes are degenerate; however surface roughness and other defects couple between the counterpropagating modes, lifting this degeneracy. Such a system is described by the QNMs, which appear as the split-mode spectrum [13,24–26].

The eigenfrequencies of the QNMs, either of the TE or the TM mode, of a bare cavity are [28]: $\displaystyle \omega _{\pm }=\omega _c \pm h - i \kappa _t$, where $\omega _c$ is the resonance central frequency, $\kappa _t$ is the total loss rate of a bare cavity defined as $\kappa _t = \kappa _i + \kappa _{ex}$, and $\textit {h}$ is the intrinsic coupling rate between the clockwise (cw) and the counterclockwise (ccw) counter-propagating modes. The origin of this coupling is Rayleigh scattering caused by defects, imperfections, and surface roughness of the resonator [24,37].

 

Fig. 1. Bare cavity model sketch. Illustration of the fiber-coupled WGM resonator where light is sent from one direction. $a_{cw}$ and $a_{ccw}$ denote the clockwise and counterclockwise modal amplitudes, respectively. $a_{in}$ denote the input amplitude from the fiber that couples to the counterclockwise mode and $a_{out}$ denote the fiber modal amplitude after the interaction with the cavity. $h$ is the intrinsic coupling rate and $\kappa _i, \kappa _{ex}$ are the intrinsic and extrinsic (fiber coupling) rates.

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When a nanoparticle is positioned within the optical mode, it absorbs or scatters light. The scattered light can couple coherently back into the same mode or backscatter into the counterpropagating mode, or incoherently scatter to the free-space modes. The backscattering causes an additional coupling between the counterpropagating modes, hence modifies the splitting of the spectrum. Scattering into the same mode causes a total shift of both QNMs. Loss due to free-space scattering or absorption causes broadening of the QNMs spectrum.

To find the new complex eigenfrequencies of the cavity we need to solve a new set of equations of motion. We write the non-Hermitian Hamiltonian [29] for the counter-propagating modes, which includes coupling and loss due to the nanoparticle, in the following way:

From the given Hamiltonian one can derive the following set of steady-state coupled linear equations:

The shifts and broadening changes of the resonances are obtained by taking the difference between the new eigenmodes and the bare-cavity QNMs. We denote the shifts (broadening changes) of the symmetric and antisymmetric modes by $\delta \omega _+$ ($\delta \Gamma _+$) and $\delta \omega _-$ ($\delta \Gamma _-$), respectively:

2.2 Cavity-particle interaction

The induced coupling rate and losses depend on the material properties and shape of the nanoparitcle, given by the polarizability tensor, $\bar { \bar \alpha }$, and on the orientation of the (non spherical) particle in the optical field. We use the dipole approximation [28], that dictates a linear relation between the applied external electric field $\vec {\mathcal {E}}$ and the induced polarization $\vec P$ in the particle: $\vec {P} =\mathcal {R}e [\bar { \bar \alpha }] \vec {\mathcal {E}}$. Accordingly, the coupling energy between the counterpropagating modes, given by the perturbation energy $\vec {P} \cdot \vec {\mathcal {E}}^*$ caused by the particle is:

 

Fig. 2. Particle-cavity interaction. a, Rates. The grey arrows illustrate the backscattering of light from the particle back to the same mode, $g_{cw,cw}$ and $g_{ccw,ccw}$, or to the counterpropageting one, $g_{cw,ccw}$ and $g_{ccw,cw}$. The absorption, $\gamma _{a,k,k'}$, and scattering, $\gamma _{s,k,k'}$, are illustrated by a yellow and maroon arrows. b, Ellipticity of the TM mode. The overlap with circular polarization (at $735nm$) of a spherical WGM resonator as a function of the refractive index difference between the resonator (SiO$_2$) and its surrounding, for different resonator diameters. As evident, in aqueous environment (dashed blue line) the TM mode is moderately elliptical (overlap of 0.68-0.82, compared to 0.5 overlap in the case of linear polarization), whereas in air or vacuum (dashed green line) the TM polarization is nearly circular (>0.95 overlap)

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The loss rate is the sum of the absorption loss rate $\gamma _a$, which is proportional to the imaginary part of the polarizability tensor $\mathcal {I}m[\bar { \bar \alpha }]$ and the scattering loss rate $\gamma _s$, which is proportional to $|\alpha |^2$ [39]. The loss rate is also dependent on the polarization of the field and the orientation of the particle relative to it.

As presented here, all the information on the particle polarizability and orientation is embedded in the shift, splitting and broadening signals. For example, the real part of the effective polarizability tensor (shape and orientation) can be deduced from the induced shift and splitting signals between two TE QNMs (probing one axis) and two TM QNMs (probing two axes). A method for separating between the particle’s shape and orientation is presented by Xu et al. [23]. In the case of birefringent material, the effective polarizability tensor does not correspond directly to the particle’s shape, and prior knowledge is required to extract one from the other. Here, to extract the particle’s polarizability and material properties, we experimentally monitor and control the particle’s position and orientation under a SEM. Specifically, we position a nanotube directly on the equator of a toroidal resonator, which simplifies the extraction of the polarizability ratios (see section 4.3 and Appendix A). Note, however, that the polarization vectors of the TE and TM modes are defined locally, as parallel and perpendicular to the surface of the WGM, respectively. Under this definition, our model is valid to a particle positioned anywhere on the WGM’s surface, not necessarily on the equator.

2.3 Open-system and particle size effects

When the cavity losses are considerably smaller than the coupling rate between the counter-propagating modes, these coupled modes can be considered as a closed system. In the spectral picture, this means that the splitting between the modes (determined by the coupling rate) is larger than the width the modes (determined by the losses), and accordingly is well-resolved. In the time domain, this means that the energy exchange between the modes is much faster than the cavity decay. As a result, both modes have similar steady-state field amplitudes, even though only one of them was driven by the input probe field. For example, when the coupling between the modes is four times larger than their loss, the ratio between the steady-state modes amplitudes is, $\left | a_{ccw}\right |/\left |a_{cw} \right | \approx 97\%$ (see Appendix B).

On the other hand, when the cavity losses are comparable to the coupling rate, the system cannot be considered anymore as a closed one. In the time domain this means that light coupled to one of the modes will exhibit significant loss before coupling to the counter-propagating mode. In the spectral domain, this is exhibited not only by splitting that is not well-resolved, but also by asymmetry of the spectrum - unequal steady-state amplitudes of the normal modes [16,33,34]. In order to find the steady-state solution in this case, we need to take into account the fact that the loss effectively decreases the coupling between the modes. To do this, we use the asymmetry, quantified by the steady-state amplitude ratio $f$, (see [16,33,34] and Appendix B), to correct the corresponding coupling-rates. For example, in the case of driving the counterclockwise mode, the corrections will be:

Another parameter that affects the measured splitting is the size of the particle. When the size of the nanoparticle is not negligible compared to $\lambda /4$ - the scale of the standing-wave pattern - it cannot be approximated as point-sized. This effect can be taken into account by assuming the coupling rate of the nanoparticle is the integral of the coupling rate per unit length over its width, which effectively can be seen as the convolution of the nanoparticle with the standing wave. For example, if we assume to have a square overlap region, as for the nanotube case, the correction will have the form of (see Appendix B):

3. Methods

3.1 Calculation: sensing of a spherical particle with different WGM polarizations

The induced spectral shifts and broadening strongly depend on the polarization of the mode. In Fig. 3, we illustrate the sensitivity of linearly polarized TE modes as well as circular and elliptical polarization of the TM mode to the presence of a spherical particle. For example, Fig. 3(e) shows that the splitting signal of a perfect circularly polarized mode is insensitive to spherical particles at all (the red and the black curves that represent the shift of the symmetric and antisymmetric QNMs, respectively, coincide). This is because coherently scattered light from a polarization-preserving spherical particle cannot couple light into the orthogonally polarized mode and induce splitting, therefore both QNMs are red-shifted together (and have the same broadening). Detection of spherical particles using a perfectly circularly polarized TM mode is still possible by measuring the average shift (or average broadening) of the QNMs.

 

Fig. 3. Sensing of a spherical dielectric particle using the TM mode. a-c, The spectral changes upon introduction of a spherical, $120nm$ in diameter, polystyrene nanoparticle inside the optical mode (for details see methods). The bare cavity spectrum (blue curve) is changed due to the presence of the particle (red curve) inside a linearly polarized mode (a). The induced spectral changes of a perfectly circularly polarized TM mode (b) and of typical TM mode with an elliptical polarization (c). d-f, show the shifts ($\delta \omega$) of the symmetric (red curves) and antisymmetric (black curves) QNMs as a function of the position of the particle on the equator (Eq. (8)) of the linearly (d), circularly (e) and elliptically (f) polarized TM vs. the position of the nanoparticle along the equator. The induced splitting signal is given by subtraction between the red and the black curves in (d)-(f) (not shown). The average shift is shown with the dash blue lines and marked by S$\omega$.

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In Fig. 3(c) we show the case of elliptically polarized TM mode, such as in silica WGM microresonators in air or water. These fields give rise to a slightly intensity-modulated pattern due to the particle presence (Fig. 3(f)). This means that a splitting signal of a typical TM mode could indeed be used even for the sensing of spherical nanoparticles, although showing lower signal compared to the linearly polarized TE mode (Fig. 3(d)). For example, for a silica microresonator in air, the maximum induced change in the splitting of the TM mode is only $\approx 34\%$ of the induced change in splitting of the TE mode (for a silica microresonator in water it is $\approx 70 \%$).

3.2 Experiment: Optical characterization of a single WS$_2$ nanotube and model verification

A WS$_2$ nanotube with an outer diameter of $60nm$ is brought close to the surface of a silica microtoroid and placed at the equatorial level, perpendicular to the surface with nanometer accuracy. This was achieved by conducting the experiment in a customized SEM vacuum chamber (Fig. 4(a)). While monitoring the exact position of the nanotube using the SEM, we measure the induced spectral shifts and broadening of the TE and TM polarized QNMs. These are then correlated to the polarizability tensor of the particle or its orientation.

 

Fig. 4. Measurement of the induced spectral changes. a, Image of the experimental setup inside the SEM (Zeiss Supra $^{TM}$). b, Image of the nano-fiber coupled microtoroid with a nanotip brought close to the microresonator surface. Inset: Close-up of the nanotube mounted on the nanotip near the surface of the resonator. c, A recorded spectra of the microtoroid with the nanotube outside (blue curve) and inside (red curve) the optical mode. The black curves are fits from which the center frequencies and widths were extracted. The bare cavity fitted parameters are: $h=29$MHz and $\kappa _t=28$MHz. d, The recorded center frequencies of the two QNMs and the splitting between them change in time. Top: The asymmetric QNM center frequency. Center: The symmetric QNM center frequency. Bottom: The resulting splitting. The blue circles mark the values when the nanotube is out of the mode (far) and the red circles mark the values when the nanotube is positioned inside the evanescent field, as close as possible to the surface without making contact. The induced splitting is twice of $\Delta \omega = 3.4$MHz. The average shift of the center frequency, $\textit S \omega$, is $10$MHz. This gives us a ’normalized induced splitting’ of $0.34$, in this example. The presented signals of center frequency were corrected for a linear drift due to thermal effects. The splitting signal is the difference between the raw data of the central frequencies.

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An example of a single-shot spectrum with the nanotube brought into the evanescent field of a microtoroidal WGM resonator is shown the red and blue curves in Fig. 4(c), respectively. Figure 4(d) shows the shift of the symmetric and asymmetric modes and the induced splitting (the difference between the shifts) of the nanotube brought periodically in-and-out of the evanescent wave. The modulation of the nanotube position near the surface allows a repeatable measurement with and without the perturbation and an accurate estimation of the induced shifts and broadening of the QNMs. Note that in this specific position the counter-propagating coupling induced by the nanotube is opposite in sign to the intrinsic coupling $h$, and therefore decreases the overall measured splitting.

4. Results

4.1 Induced spectral changes of the TE mode vs. the position along the equator

In our theoretical model, we derive the new eigenfrequencies of the cavity with a non-spherical nanoparticle and arrive at the normalized induced splitting vs. the position along the equator (For full expression, see Appendix A). For the TE mode in the limit of: $h\gg g, \gamma$, we can use the following approximated expression:

In the experiment, we scan the nanotube positioning along the equator while it is being held horizontally, perpendicular to the surface of the microtoroid. The spectra are collected with the nanotube inside and outside the optical mode at intervals of 45nm for a total range of at least one period ($\approx \lambda /2n$). Then we measure the normalized induced splitting and difference in broadenings vs. the position of the nanotube for the TE and TM modes independently. The measurement with linearly polarized TE mode allows the extraction of the polarizability of the nanotube along a single axis, assuming that the mode properties and the position inside the optical mode are known.

The induced splitting and the induced average shift of the TE QNMs vs. the position along the equator are plotted in Fig. 5(a),(b). The blue circle markers represent the measured data: half the splitting ($\Delta \omega$) and the average shift ($\textit {S} \omega$) and the blue curves are guide to the eye (using the non-approximated expression: $\Delta \omega = \frac {1}{2}Re\{\delta \omega _- -\delta \omega _+\}-h$.).

 

Fig. 5. Scan results of a TE mode and demonstration of the self-referenced method and open system nature. (a) The measured half induced splitting between the QNMs of the TE mode vs. the position of the nanotube along the equator. The blue curve is the theoretical prediction of $\Delta \omega$ using the interpolated line (dashed blue line in (b)) from $\textit {S}\omega$ measured data. (b) The measured average shift $\textit {S}\omega$ of the two QNMs in the same measurement. (c) The normalized induced splitting data (the red circles) along with three options for a fit. The light red curve represents the fit to Eq. (15) with the backscattering between the counterpropagating modes being symmetric. This represents the case where the coupling rate between the counterpropagating modes and the coupling rate into the same mode are equal, i.e., $g_{k,k'} = g_{k,k}$. The light dashed red curve represents an improved fit, where the reference to the size of the particle is included in the model. The dark red solid curve is the best fit to the expression in Eq. (15), where both the particle size and open system nature is taken into account.

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From these results, we can extract the real part of the polarizability along a single axis. In the current configuration, the TE field probes the short axis of the nanotube, giving $\mathcal {R}e[\alpha _{\perp }]$. To estimate the polarizability, we can use the average induced shift ($S\omega = (2.7 \pm 0.85)$MHz in the current experiment) or the coupling rate between the counterpropagating modes from fit to the ($\Delta \omega$) data. However, these measurements also depend on the parameters of the optical mode, such as the mode volume and the exact position of the nanotube inside the evanescent field in the radial direction, which are sometimes hard to evaluate. For example, we see (Fig. 5(b)) that although the average shift signal $S\omega$ is not supposed to be modulated with the position on the equator, it still varies. The same dependence on the mode properties exists in the induced splitting signal.

To cancel out this position-dependent variation and to overcome the randomness of the nanotube positioning within the standing wave pattern, we propose here a method for a self-referenced measurement that involves normalizing the induced splitting, $2\Delta \omega$, with the induced total shift, $2S\omega$. This ratio, taken at each measurement point, is plotted in Fig. 5(c) along with three types of fits to the data.

The strength of the model is demonstrated by first plotting the fit according to the expression that assumes a closed system approximation and does not take into account the size of the particle relative to the standing wave pattern (light red). This fit has a modulation depth of one, as the coupling rate between the counterpropagating modes and the coupling rate into the same mode are equal ($g_{k,k'}=g_{k,k}$ in Eq. (15)). We see that this curve deviates from the measured data. Secondly, we plot a fit that takes into account that the diameter of the nanotube ($60nm$) and the periodicity of the standing wave ($\sim 250nm$) for the wavelength of $740nm$ (dash red). This reduces the modulation depth to $\sim 0.9$. Lastly, we plot the fit that takes into account both the size of the particle and that the approximation to a closed system does not describe the system accurately and the standing wave pattern has a traveling component (solid dark red). In this case, the modulation depth is only $\sim 0.6$. This demonstrates that these aspects, that we introduced into our model, influence drastically (by up to $40\%$) the accuracy of the particle characterization and sizing.

4.2 Characterization of a uniaxial particle using the TM mode

Using prior information about the nanoparticle’s structure or orientation enables extracting more specific properties out of the measured spectra. As an example, here we use the knowledge that the nanoparticle is uniaxial (single WS$_2$ nanotube) and positioned on the equator of the WGM, to extract the ratios between the different components of the polarizability tensor using solely the TM mode.

The polarizability ratios are often used to characterize non-spherical structures such as carbon nanotubes [41,42] or various inorganic nanotubes, such as WS$_2$ nanotubes [43,44] and are defined here as,

 

Fig. 6. a, The normalized induced splitting (top) and the normalized induced difference in broadening (bottom) between the symmetric and the antisymmetric QNMs of the TM mode vs. the position of the nanotube along the equator. Top: The normalized induced splitting is obtained according to the method described for the TE measurement, (Fig. 5). The data ($\Delta \omega / \textit {S} \omega$) is marked by the blue circles. The blue curve is the best fit to the data using Eq. (17) and the shaded range indicates the $95\%$ confidence intervals of the fit. The dashed blue light curve is the best fit for the normalized induced splitting without the correction for an open system. Bottom: The data ($\Delta \Gamma / \textit {S} \Gamma$) marked with the red circles. The red curve is the best fit to the normalized and the shaded range indicates the $95\%$ confidence intervals of the fit. b, Summary of the measured $R_{re}=0.38$, $R_{im}=0.12$, $R_{im/re}^{||}=0.5$ and $R_{im/re}^{\perp }=0.16$ polarizability ratios on top of a TEM (Transmission electron microscope) image of a typical WS$_2$ nanotube as used in our work. The scale-bar of the TEM image of the nanotube is $10nm$.

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The induced normalized splitting of the TM mode according to our theoretical model is:

Table 1. Summery of measured and calculated polarizability ratios

View Table

For the calculation of the ratio values, we approximate the end of the nanotube that enters the optical mode by a ellipsoid [45–47] and calculate the polarizability of an ellipsoid [48]. We use the measured diameter of the nanotube $58nm \pm 2nm$. The exact length of the nanotube inside the mode is not well-defined so we make our calculations assuming an ellipsoid with a varying length. The ellipsoid length corresponds to the overlap of the nanotube with the optical mode (overlap length). The permittivities that are used for the calculations are $\epsilon _{||}=14.26-i0.55$ and $\epsilon _{\perp }=10.3-i0.2$ [49]. The calculated polarizability ratios for the nanotube are presented in Table 1.

Figure 7(a) presents the comparison of the measured value for the ratio between the real parts of the polarizability $R_{re}$, (blue dot) to the calculated value of $R_{re}$ ratio vs. the overlap length(blue curve). The estimated overlap length in our measurement is $130nm$, which is given by our FEM simulation in COMSOL. For comparison the dashed black curve in Fig.  7(b) represents the polarizability ratio for bulk WS$_2$.

 

Fig. 7. The calculated and the measured polarizability ratios of the WS$_2$ nanotube. a, The calculated ratio between the real parts of the polarizabilities $R_{re}$ of a WS$_2$ nanotube vs. the overlap length between the nanotube and the optical mode (blue curve). The measured polarizability ratios are shown: The first (blue circle) $0.38 \pm 0.05$ is measured in our experiment and the second (green circle) $0.16.\pm 0.05$ was measured previously for a similar type of nanotubes at a wavelength of 633nm [43]. The green dash-dot curve is the calculated polarizability ratio for a $25nm$ outer diameter nanotube illuminated with $633nm$ light. The dashed black curve represents the polarizability ratio for bulk-WS$_2$. b, The calculated ratio between the imaginary parts of the polarizabilities $R_{im}$ of a WS$_2$ nanotube vs. the overlap length between the nanotube and the optical mode assuming effective permittivity (blue curve) and when taking the normal one (dotted curve). The measured imaginary part of the polarizability ratio is shown to be in good agreement with the curve plotted using the corrected effective permittivity (blue circle).

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The ratio $R_{re}=0.16\pm 0.05$ of a single suspended nanotube was measured previously in the far-field regime, using Raman spectroscopy [43] (green dot in Fig. 7(b)). In that work, an individual multiwalled WS$_2$ nanotube with an outer diameter of $15-25nm$ was placed on an Atomic force microscope (AFM) tip and illuminated with linearly polarized light ($633nm$) at different angles to the axis of the nanotube. The overlap length corresponds to the spot size of $\sim 1um$ that was used. The green dash-dot curve in Fig. 7(a) represents the calculated $R_{re}$ for a nanotube with an outer diameter of $25nm$ at a wavelength of $633$nm.

In Fig. 7(b), the measured ratio between the imaginary parts of the polarizability $R_{im}$ is plotted (blue dot) along with the calculated ratio vs. the overlap length (blue solid curve). Note that in all of our calculations of the polarizability along the short axis of the nanotube, we use an effective value of permittivity: $\sqrt {\epsilon _{\perp }\epsilon _{||}}$ [49]. The reason is the fact that in the multiwall nanotube of such small radius ($\sim 30nm$) the principal crystallographic axis (defined perpendicular to the WS$_2$ sheets) changes rapidly on the scale of the nanostructure, thus both the in and out-of-plane components of the dielectric tensor contribute to the out-of-plane polarizability calculation. The dash-dotted blue curve in Fig. 7(b) represents the calculated polarizability ratio without this adaptation. We see that indeed using the adapted value leads to a much better agreement. The dashed black curve represents the calculated value for the bulk-WS$_2$.

We see that the in-plane ratio ($R_{im/re}^{||}$) for the WS$_2$ nanotube is $\sim 12$ times larger than the ratio for bulk-WS$_2$. The out-of-plane ratio ($R_{im/re}^{\perp }$) is $\sim 106$ times larger than the bulk ratio. We believe this is attributed to the fact that the imaginary part of the permittivity is higher in the nanotube than in bulk, whereas there is a decrease in the real part. The optical absorption of nanotubes was believed to have the same features [50] as the bulk. Specifically, the direct band-gap of bulk 2H-WS$_2$ is at $2.04$eV ($\lambda =608nm$) [51], the indirect band-gap at $1.3$eV ($\lambda =954nm$) [52] and exciton A is at $1.95$eV ($\lambda =636nm$) [50]. However, recent work [53] proves the presence of both excitonic and plasmonic features. Namely, the nanotubes preserve their WS$_2$ bulk properties but also maintain a plasmonic resonance at $\sim 600nm$. Our illumination, at $735nm$, is detuned from all these transitions and the plasmonic resonance. Hence, we can assume that we measure the plasmonic off-resonance response, being red-detuned by $\sim 135nm$ from the plasmon resonance center. Different loss mechanisms contribute to the imaginary part of the dielectric function, and one of them might be oscillations of free charges, namely a plasmon [53]. This could explain the increase in the imaginary part of the permittivity by an order of magnitude.

In another supporting recent work by Levi at al. [54], which measured the conductivity of a single WS$_2$ nanotube using a nanotube-based transistor and find it to be six orders higher than in similar dichalcogenide multilayer-devices. The authors attribute the measured conductivity to unexpectedly high charge density of the WS$_2$ nanotube relatively to 2H-WS$_2$ bulk.

5. Discussion

In this work, we presented a model for light-matter interactions of arbitrarily shaped nano-object with linear (TE) or elliptical (TM) polarization optical modes. We then implemented a measurement scheme that allows 3D characterization and measurement of the polarizability ratios of arbitrarily shaped nano-object using a WGM microresonator. In particular, the spectral changes of the two QNMs of the TM mode induced by a presence of a WS$_2$ nanotube are correlated to its polarizability tensor and the polarizability ratios between the real parts, the imaginary parts and the imaginary to real parts of a single WS$_2$ nanotube are measured. The measured polarizability ratios of the WS$_2$ nanotube are in accordance with the reported value [43] and our estimation form the measured bulk properties.

The present model introduces important aspects that were often not taken into account. The implementation for sensing with an open-system configuration, which is typical in WGM sensing, could potentially influence the accuracy of particle characterization by tens of percent. Also, the influence of the nanoparticle size compared to the standing wave pattern was also taken into consideration. For $60nm$ radius particle in a $740 nm$ wavelength, as for this work, we estimate the effect to be $\approx 9\%$ on the modulation depth. In our work, we did not consider the influence of the evanescent field gradient over the nanoparticle as was presented in the recent work [55]. We estimate this effect could add $\sim 6 \%$ decrease in the modulation depth that is still in correspondence with our experimental results.

To support the proposed method we used the measurement of the nanotube with the TE mode. We showed there could be tens of percents of deviation of the estimated coupling rate (and hence the polarizability) when the values are derived under the closed system approximation, and that the full model we proposed corrects these deviations.

Additionally, we demonstrated a complementary approach of a self-referenced measurement, which involves normalization of the splitting or difference in broadening of the two QNMs by the average shift or average broadening of the two QNMs, respectively. In previous works [13,18,23], where a non-absorbing particle was used, the particle position and mode volume dependence, which are parameters that are hard to accurately predict, were eliminated by dividing the shift with the broadening signal $\delta \omega / \delta \Gamma$. Similarly to the $\delta \omega / \delta \Gamma$ method, the current $\delta \omega / \textit {S} \omega$ normalization technique also eliminates these dependencies. The importance of our method is in a system where the detected particle perturbs an already existing standing wave pattern. This is because the $\delta \omega / \textit {S} \omega$ measurement tells us were along in the standing wave the particle landed. For example, in a measurement of a small nanoparticle that perturbs a well resolved TE mode, when $\delta \omega / \textit {S} \omega$ equals one, it tells us that the particle is positioned at a node of the symmetric standing wave pattern. This enables more accurate and less position-dependent, real-time sizing of the particle.

Finally, note that although we used our ability to vary the position of the nanoparticle to scan its coupling parameters to the WGM, one could alternatively, in cases of low coupling between the counter-propagating modes ($h \lesssim \kappa _t$), excite the QNMs directly by probing from both directions of the coupling fiber. In this case, varying the relative phase between the counter-propagating probes scans the position of QNMs (and the corresponding coupling parameters) over the stationary nanoparticle, thereby creating an equivalent tomographic measurement. For the cases of ($h > \kappa _t$), one can use the method presented by Xu et al. [23], which uses measurements from different free spectral ranges, thereby providing a number of data points, each with a different effective relative position of the particle and the QNMs.

Beyond enabling more accurate and sensitive label-free detection schemes, we believe this comprehensive three-dimensional model opens the path to tomographic characterizations of single, arbitrarily shaped, nano-objects and particles, and the detailed measurement of various properties, such as chirality and conformational changes.

Appendix A: Full expressions

The full expression for $\mathbf {D_1D_2}$:

(20)$$D_1D_2 = g_{k,k}g_{k,k}^{\dagger} + h \left( g_{k,k'} + g_{k,k'}^{\dagger} \right) + h^2$$

(21)$$-\gamma_{t,k,k}\gamma_{t,k,k}^{\dagger} - i\left( g_{k,k'}\gamma_{t,k,k'}^{\dagger} + g_{k,k'}^{\dagger}\gamma_{t,k,k'}\right) - ih \left( \gamma_{t,k,k'} + \gamma_{t,k,k'}^{\dagger}\right)$$

where: $g_{k,k'}^{\dagger } = g_{k',k}$.

The approximated broadening change for the TE (linear) polarization:

(22)$$\Delta \Gamma = -\gamma \frac{g + h \cos{\left( 2 k x \right)}}{\sqrt{g^2 + 2gh\cos{\left( 2 k x \right)} + h^2}}$$

where we defined: $g_{k,k'} = g e^{-i \Delta \beta _{k,k'} x}$ and $\gamma = \left | \gamma _{k,k'} \right |$.

The approximated broadening change for the TM (elliptical) polarization:

(23)$$\Delta \Gamma = -\gamma \frac{g + h \cos{\left( 2 k x \right)}}{\sqrt{g^2 + g^2_{cp} + 2gh\cos{\left( 2 k x \right)} - 2g_{cp}h\sin{\left( 2 k x \right)} + h^2}}$$

(24)$$-\gamma_{cp} \frac{g_{cp} - h \sin{\left( 2 k x \right)}}{\sqrt{g^2 + g^2_{cp} + 2gh\cos{\left( 2 k x \right)} - 2g_{cp}h\sin{\left( 2 k x \right)} + h^2}}$$

where we defined: $g_{k,k'} = \left (g \pm g_{cp} \right ) e^{-i \Delta \beta _{k,k'} x}$ and $\gamma _{k,k'} = \left (\gamma \pm \gamma _{cp} \right ) e^{-i \Delta \beta _{k,k'} x}$.

$\mathbf {g_{k,k'}}$ for a nanotube along the equator:

(25)$$g_{k,k'} \propto \left[ \underbrace{C_{\parallel,re}^{k,k'} \operatorname{Re} \left[ \alpha_{\parallel} \right] + C_{\perp,re}^{k,k'} \operatorname{Re} \left[ \alpha_{\perp} \right]}_{g} \pm i \underbrace{C_{im} \left( \operatorname{Re} \left[ \alpha_{\parallel} \right] - \operatorname{Re} \left[ \alpha_{\perp} \right] \right)}_{g_{cp}} \right] e^{\pm 2 i k x}$$

where,

(26)$$C_{\parallel,re}^{k,k'} \,= d^2 \sin^2{\theta} \left[(c+1)^2 \sin^2{\phi} - \cos^2{\phi}\right]$$

(27)$$C_{\perp,re}^{k,k'} = d^2 \left\{ \cos^2{\phi} \left[ \right( 1 + c \left)^2 - \cos^2{\theta} \right] + \sin^2{\phi} \left[ \right( 1 + c \left)^2 \cos^2{\theta} - 1 \right] \right\}$$

(28)$$C_{im} \ \: = d^2(c+1) \sin^2{\theta} \sin{2\phi}$$

(29)$$C_{\parallel}^{k,k} \;\:\! = d^2 \sin^2{\theta} \left[(c+1)^2 \sin^2{\phi} + \cos^2{\phi}\right]$$

(30)$$C_{\perp}^{k,k} \;\:\! = d^2 \left\{ \cos^2{\phi} \left[ \right( 1 + c \left)^2 + \cos^2{\theta} \right] + \sin^2{\phi} \left[ \right( 1 + c \left)^2 \cos^2{\theta} + 1 \right] \right\}$$

and $d$ and $c$ define the polarization of the optical mode: $\varepsilon _{cw} = d \left (\varepsilon _y + i \varepsilon _x \right ) +dc\varepsilon _y$.

Appendix B: Open-system and size corrections

Open-system factor. The asymmetry between the field amplitudes of the clockwise and counter-clockwise modes can be estimated by taking the steady-state solution of the equations of motion.

(31)$$f = \frac{\left| a_{cw} \right|}{\left| a_{ccw} \right|} = \frac{\left| i\left( g_{cw,cw} + h\right) - \gamma_{cw,ccw}\right|}{\left| i \sqrt{D_1 D_2} - \left( \gamma_t + \kappa_t \right) \right|}$$

For a bare-cavity it takes the form of:

(32)$$f = \frac{\left| a_{cw} \right|}{\left| a_{ccw} \right|} = \frac{h}{\left| i h - \kappa_t \right|}$$

Note that Eqs. (31) and 32 depend on the field amplitudes ratio, and therefore are independent of the light intensity coupled to the resonator.

Finite size effect. In order to account for the finite size of the nanoparticle inside the optical mode we assume that the coupling rate $g$ of a large particle are being a summation over the rates per unit length, $\hat {g}$.

$$g = w\hat{g}$$

where $w$ is the size of the particle-cavity mode overlap.

(33)$$\delta \omega_{\pm} \left(x_{pos}, w \right) = \frac{1}{w} \int_{x_{pos}-w/2}^{x_{pos}+w/2} \delta \omega_{\pm} \left(x \right)$$

which is effectively the convolution between the particle and the QNMs

(34)$$g_{k,k'} = w \hat{g}_{k,k'} \frac{\sin{wk}}{wk} = g_{k,k'} \frac{\sin{wk}}{wk}$$

Appendix C: Polarizability

Ellipsoid polarizability. In Fig. 7 the calculations of the polarizability ratios of the nanotube and bulk-WS$_2$ assumes the following permittivities along its two axes: $\epsilon _{||}=14.26-i0.55$ and $\epsilon _{\perp }=10.3-i0.2$ [49,56] for the wavelength of $735nm$ and an ellipsoidal particle [48].

Comsol simulations: The decay length of the TM field in the microtoroid that was used (major diameter $52um$, minor diameter $8um$) is $120nm$ (see inset in Fig. 7(a)) and together with the orientation of the nanotube we get approximately $130nm$ of overlap.

Extraction of the polarizability ratios. The polarizability ratios where extracted using the following expressions, which for the loss components assume small scattering cross section.

(35)$$\tilde{g} = \frac{C_{\parallel,re}^{k,k'} + R_{Re}C_{\perp,re}^{k,k'}}{C_{\parallel,re}^{k,k} + R_{Re}C_{\perp,re}^{k,k}}$$

(36)$$\tilde{\gamma} = \frac{C_{\parallel,re}^{k,k'} + R_{Im}C_{\perp,re}^{k,k'}}{C_{\parallel,re}^{k,k} + R_{Im}C_{\perp,re}^{k,k}}$$

(37)$$\frac{g_{k,k}}{\gamma_{a,k,k}} = R^{\parallel}_{im/re} \frac{C_{\parallel,re}^{k,k} + R_{Re}C_{\perp,re}^{k,k}}{C_{\parallel,re}^{k,k} + R_{Im}C_{\perp,re}^{k,k}}$$

Fig. 8 represents the scattering and absorption contributions. The two data points represent measurements done in our setup. The left is the data point for the nanotube in the TM data set, from which the polarizability ratios were extracted. The right point was measured for a parallel nanotube (overlap of $\sim 1 \mu m$). This points represents the most extreme case of measured loss-to-coupling ratio with the biggest particle-cavity overlap, which corresponds to strongest scattering contribution. The lines represent the different contributions of the loss mechanisms. The calculation assumed an ellipsoid shape and used the measured polarizability ratios of the TM data set.

 

Fig. 8. Scattering and absorption partial contributions for different nanotube-to-optical-mode overlap.

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Funding

Crown Photonics Center; Minerva Foundation (2014AA014402); Israel Science Foundation (1798/17).

Acknowledgments

This research was made possible in part by the historic generosity of the Harold Perlman family. B.D. acknowledges support from the Israeli Science Foundation, the Minerva Foundation, and the Crown Photonics Center. B.D. is also supported by a research grant from Charlene A. Haroche and Mr. and Mrs. Bruce Winston, and by a research grant from Mr. and Mrs. Howard Laks.

Disclosures

The authors declare no conflicts of interest.

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