# Whispering gallery modes for 3D strain measurement

Yann Lecieux, Dominique Leduc, Corentin Guigot, Marc François, Cyril Lupi

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Yann Lecieux, Dominique Leduc, Corentin Guigot, Marc François, Cyril Lupi. Whispering gallery modes for 3D strain measurement. Optics and Laser Technology, 2022, 149, pp.107862. 10.1016/j.optlastec.2022.107862. hal-04731088

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Submitted on 10 Oct 2024

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# Whispering gallery modes for 3D strain measurement

Yann Lecieux*, Dominique Leduc, Corentin Guigot, Marc François, Cyril Lupi

GeM - Institut de Recherche en Génie Civil et Mécanique - UMR 6183 - Université de Nantes - CNRS - École Centrale de Nantes

## Abstract

This paper deals with the use of Whispering Gallery Modes to measure the six components of an homogeneous strain tensor within a sphere undergoing a uniform strain field. The device is a silica sphere in which six WGMs resonate, describing six circles positioned on the sphere's surface in planes perpendicular to the normals of a dodecahedron. The direct model relating the strain to the resonant wavelength variation is established for one circle and then linearised in accordance with the intended application, i.e. the realisation of a sensor. The expressions are then generalised to six circles and inverse system computed to obtain an expression for the six components of the strain field as a function of six wavelength shifts of WGMs. Finally, the theoretical performances of the sensor are assessed. The study shows that using a conventional 1 pm resolution detector, the uncertainty on the strain components is, after correction of systematic errors, equal to 2 με.

**Keywords:** Whispering-gallery modes, Strain, 3D sensor, Optical sensors

## 1. Introduction

Measuring strain within a structure or material is a complex task due to the nature of the strain. It is a tensor field defined by six independent components at each point of the space. To date, strain sensors are essentially unidirectional and only provide a measurement of one component of the strain tensor in projection along the axial direction of the sensor. Six unidirectional sensors are therefore needed to completely measure the strain at a point in space. Then the problem of the possible non-uniformity of the measured field arises. Since it is not possible to place six sensors at the same point, it is necessary to ensure the uniformity of the strain field in the volume where the different unidirectional sensors are positioned.

<footnote>*Corresponding author. Email: yann.lecieux@univ-nantes.fr</footnote>
10

To solve this problem, a sensor concept was recently patented [1]. Its test body is a sphere crossed by 6 fibres positioned in six directions corresponding to the normals of a dodecahedron and equipped with a Bragg grating to perform unidirectional strain measurement. Under certain conditions, the strain is uniform in the sphere [2]. The six fibres therefore measure different components of the same tensor. The validity of the measurements given by this sensor has been experimentally proven [3, 4].

15

In its current form, the body measures 4 cm in diameter. This size is well suited for civil engineering applications but is not suitable for insertion into composite materials or biological structures. However, the fact of inserting the strain gauges according to diameters in the test body limits significant miniaturisation of the device. Indeed, it is necessary to have a bonding length of several centimetres for a correct strain transfer from the test body to the fibre [5]–[9]. This requirement defines the minimum diameter of the sphere.

20

An interesting alternative would be to use a test body cut from a dielectric material as a medium for light propagation. Indeed, its spherical shape allows it to guide gallery modes [10]–[13] at its periphery. However, gallery modes are resonant modes: the phase tuning conditions after one revolution mean that there is a relationship between the wavelength of the propagating mode and the radius of the sphere, its refractive index and the refractive index of the external medium [14]. Consequently, a sphere radius variation results in a variation of the resonant wavelength. Measuring the variation of the wavelength can therefore be used to calculate the strain field within the sphere.

25

30

Several studies have already demonstrated the link between strain or mechanical loading and shift of the resonant wavelength. However, all these studies focused on one dimensional strain analysis in an a priori known direction. In some of them [15]–[19], the gallery modes propagate in microwire and they shift due to a tensile strain in the axial direction. The aim of the others[20]–[24] is to develop force sensor with whispering gallery modes in a micro-sphere. The micro-sphere is wedged between two flat plates or a similar arrangement. The force is exerted in the direction normal to the plates and the contact between the plates and the sphere induce a non homogeneous strain. The key parameter in these studies is the link between the amplitude of the force in a given direction and the shift of the wavelength of the whispering gallery mode. The aim of the study presented here is quite different. The starting point is a micro-sphere embedded in a host material and therefore strained in an unknown manner but in homogeneous strain field. The problem to solve is to determine the full strain, in all dimensions.

35

40

In order to measure strain completely, 6 different gallery modes would have to be used,
following 6 different circles. The question is whether there is a configuration that allows all components of the strain tensor to be measured using these purely tangential measurements, and if so, whether the relationship between the measured wavelengths and the components of the strain tensor is linear over a reasonable range of strains for the sensor to be truly usable in practice.

The objective of this article is to demonstrate that there are linearized relations and an architecture that satisfies these requirements, and then to propose a complete method for the inversion of measurements obtained with this type of sensor. After a brief reminder of the context, the problem statement is presented in section 2. Relationships for calculating strains from the measurement of wavelength shifts will then be established by specifying the necessary approximations to the sections 3 and 4. Then, the optimal architecture of the sensor will be defined in section 5. Finally, the relevance of the approximations with respect to the expected performances in terms of sensitivity and resolution will be discussed in connection with the practical aspects for the implementation of this type of sensor.

## 2. Problem Statement

Electromagnetic Whispering Gallery Modes propagate in structures with axial symmetry such as discs, torus or spheres [25]–[30]. These are resonant modes with quality-factor as high as $10^9$ [31] which makes them suitable for fine measurements of various parameters such as temperature [32], refractive index [14, 33] or biological parameters [29, 34, 35].

The studied structure in this paper is a silica sphere of refractive index $n$ and radius $R$ surrounded by an external medium of index $n_{ext}$. The spherical coordinate system $(\vec{e}_r, \vec{e}_\theta, \vec{e}_\phi)$ is used for the calculations. From Maxwell equations, it can be seen that two family of modes exist: transverse electric modes (TE) and transverse magnetic modes (TM). In the following, the study will be focused on TE modes. In this case, the electric field in the sphere is given by [36]:

$$ \vec{E}_{\text{elec}} = A \frac{\psi_{\ell}(knr)}{knr} \left[ j \frac{m P_m^{\ell}(\cos \theta)}{\sin \theta} \vec{e}_{\theta} - \frac{\partial P_m^{\ell}(\cos \theta)}{\partial \theta} \vec{e}_{\phi} \right] e^{jm\phi} \quad (1) $$

where $k = 2\pi/\lambda$ is the wave number, $j$ the imaginary unit, $(r, \theta, \phi)$ the spherical coordinates, $\psi_\ell$ a Ricatti-Bessel function of the first kind and $P_m^\ell$ the associated Legendre polynomials. The parameters $\ell$ and $m$ are integers such that [37]: $-\ell \le m \le \ell$ and $1 < \ell \times \lambda/2\pi R < n/n_{ext}$. $\ell$ is of the order of 50 for $R = 10$ µm and 3000 for $R = 500$ µm. For $\ell \gg 1$ the component of the electrical field along $\vec{e}_\phi$ is negligible [38]. It will then be assumed that the electrical field is purely along $\vec{e}_\theta$.
The continuity of the tangential components of the fields at the interface yields [38, 39]:

$$ \frac{n}{n_{\text{ext}}} \frac{\psi'_{\ell}(knR)}{\psi_{\ell}(knR)} = \frac{\chi'_{\ell}(kn_{\text{ext}}R)}{\chi_{\ell}(kn_{\text{ext}}R)} \quad (2) $$

where $\chi_{\ell}$ is a Ricatti-Bessel function of the second kind. The exponents prime stand for the derivation with respect to $r$. This equation gives the wavelengths of the $\ell$ order modes which can propagate around the sphere. For an unstrained sphere, modes $(\ell, m)$ are degenerated. The fundamental mode corresponds to $m = \ell$. Schematically, it can be seen as a toroidal beam centered on the equatorial plane ($\theta = \pi/2$) with radial width $\delta r \simeq \lambda$ [37, 40] and angular opening $\delta\theta$ depending on $R$.

The studied sensor architecture is made of six whispering gallery modes propagating around the sphere along six different circles. The $i^{th}$ mode propagates along a circle perpendicular to the direction $\vec{n}_i$. The previous field expressions are therefore valid for this mode, provided that the angle $\theta$ is measured from the axis carrying $\vec{n}_i$ as shown on figure 1.

<figure><img src="image_2.png" /><figcaption>Figure 1: Settings used for the study of WGM</figcaption></figure>

When the sphere is submitted to a uniform strain field $E_0$, in accordance with the intended use as sensor and the assumption of the Eshelby's inclusion theorem, it undergoes a uniform strain $E$ related to $E_0$ by Eshelby's equations [2, 41, 3]. Thus the strains leads to a shift of the wavelength of the $i^{th}$ resonant whispering gallery mode, which can be written as [20]:

$$ \frac{\Delta\lambda_i}{\lambda_i} = \frac{\Delta L_i}{L_i} + \frac{\Delta n}{n} - \frac{\ell - m}{2\ell} \Delta(e_i)^2 \quad (3) $$
It depends on three terms: a geometrical term, $\Delta L/L$, that describes the variation of the perimeter of the circle, a photo-elastic term, $\Delta n/n$, that describes the refractive index variation induced by the strain $\mathbf{E}$ and a term depending on the eccentricity of the loop. The latest is of second order in eccentricity and will be neglected (it is by the way strictly null for the fundamental mode). The remaining terms depend on the strain tensor. For each term, this dependence will be explained as well as the calculation allowing to invert the relations obtained.

Before doing so, it is worth considering the limitations of the proposed model. The first concerns the power of the incident beam. Indeed, as the cavity has a high quality factor, a low incident power can lead to a very high energy stored in the cavity and induce non-linear effects or mechanical instabilities. Such effects are observed for threshold powers of a few tens of watts with cavities of quality factor between $10^6$ and $10^9$. It is reasonable to take this order of magnitude as the maximum incident power. The second concerns the coupling between modes. In this model it is assumed that the modes are completely decoupled. One way to do this is to interrogate the different modes sequentially, leaving enough time between two interrogations for the energy of the previous mode to be fully dissipated. However, it should be noted that a coupling resulting in an energy transfer between modes with no change in resonant wavelength would be compatible with the proposed method. It would only affect the sensitivity.

### 3. Linearized relation between the variation of perimeter $\Delta L_i/L$ for a circle of normal $\vec{n}_i$ and the components of the strain tensor $\mathbf{E}$

The vectors plotted in figure 1 ($\vec{m}_i, \vec{p}_i, \vec{n}_i$) form local orthonormal bases. The components of the strain tensor $\mathbf{E}$ are defined by the equation 4, in the canonical basis ($\vec{e}_1, \vec{e}_2, \vec{e}_3$) and in the local basis ($\vec{m}_i, \vec{p}_i, \vec{n}_i$):

$$ \mathbf{E} = \begin{bmatrix} E_{pq} \end{bmatrix}_{(\vec{e}_1, \vec{e}_2, \vec{e}_3)} = \begin{bmatrix} \overline{E}_{pq} \end{bmatrix}_{(\vec{m}_i, \vec{p}_i, \vec{n}_i)} \quad (4) $$

The components of the strain tensor $E_{kl}$ projected in a plane of normal $\vec{n}_i$ are noted $E_{kl}^i$:

$$ E_{ij}^i = P_{ip}^{\perp} P_{jq}^{\perp} E_{pq} \quad (5) $$

where $P_{kl}^{\perp}$ refers to the components of the projector $\mathbf{P}^{[\vec{n}_i^{\perp}]}$ on the plane of normal $\vec{n}_i$:

$$ \mathbf{P}^{[\vec{n}_i^{\perp}]} = \mathbf{I} - \vec{n}_i \otimes \vec{n}_i \quad (6) $$
where **I** is the identity tensor. In the local basis ($\vec{m}_i, \vec{p}_i, \vec{n}_i$), the projected tensor noted **E**^i of the strain tensor **E** has the following components:

$$ \mathbf{E}^i = \begin{bmatrix} \overline{E}_{11}^i & \overline{E}_{12}^i & 0 \\ \overline{E}_{12}^i & \overline{E}_{22}^i & 0 \\ 0 & 0 & 0 \end{bmatrix}_{(\vec{m}_i, \vec{p}_i, \vec{n}_i)} . \qquad (7) $$

The directions ($\vec{m}_i, \vec{p}_i$) can be chosen such that they coincide with the **E**^i eigenvectors. The associated principal strains are noted as $\overline{E}_I^i$ and $\overline{E}_{II}^i$. The strain tensor projected onto the $\vec{n}_i$ normal plane is then written as:

$$ \mathbf{E}^i = \begin{bmatrix} \overline{E}_I^i & 0 & 0 \\ 0 & \overline{E}_{II}^i & 0 \\ 0 & 0 & 0 \end{bmatrix}_{(\vec{m}_i, \vec{p}_i, \vec{n}_i)} \qquad (8) $$

The position vectors of points describing a ring on the sphere surface in the initial and final states are denoted by $\vec{x}$ and $\vec{X}$ respectively. The possible displacement field $\vec{u}$ involves, with a possible additional rigid body motion, the final position vectors $\vec{X}$ as a function of initial position vectors $\vec{x}$ according to:

$$ \vec{u} \begin{vmatrix} \overline{E}_I^i x \\ \overline{E}_{II}^i y \\ 0 \end{vmatrix}_{(\vec{m}_i, \vec{p}_i, \vec{n}_i)} \quad \vec{X} \begin{vmatrix} X = (1 + \overline{E}_I^i)x \\ Y = (1 + \overline{E}_{II}^i)y \\ 0 \end{vmatrix}_{(\vec{m}_i, \vec{p}_i, \vec{n}_i)} \qquad (9) $$

The points, initially on the circle, satisfy:

$$ x^2 + y^2 = R^2 \qquad (10) $$

From the equations 9 and 10, it follows:

$$ \frac{X^2}{(1 + \overline{E}_I^i)^2} + \frac{Y^2}{(1 + \overline{E}_{II}^i)^2} = R^2 \qquad (11) $$

This result shows that a ring, initially circular, takes an elliptical shape under the effect of an homogeneous strain field given by **E**. As a consequence, looking for a relation between the perimeter variation of a ring in a plane of normal $\vec{n}_i$ and the components of the strain tensor projected in this same plane **E**^i implies to be interested in the calculation of an ellipse perimeter. The latter is obtained following the calculation of an elliptic integral (equation 12):

$$ \mathcal{P} = 4 \int_{0}^{\frac{\pi}{2}} \sqrt{a^2 \cos^2(t) + b^2 \sin^2(t)} dt \qquad (12) $$
where a and b are half the lengths of the minor and major axes of the ellipse defined by:

$$a = R(1 + \overline{E}_I^i), \quad b = R(1 + \overline{E}_{II}^i) \qquad (13)$$

This equation has no analytical solution. It must therefore be calculated by numerical integration or by rational approximations. Many approximations exist. The choice is made here to express the perimeter variation $\Delta L_i$ by using the Euler approximation (equation 14) since it is very precise when the ellipse is close to a circle as it is the case under the classical assumption of small perturbations:

$$L + \Delta L_i \approx \pi \sqrt{2(a^2 + b^2)} \qquad (14)$$

From this approximation, it is possible to determine the expression of the ring final length $L+\Delta L_i$ as a function of the principal strains $(\overline{E}_I^i, \overline{E}_{II}^i)$ and the radius $R$. Linearising this expression to first order in strain, it comes:

$$L + \Delta L_i = \sqrt{2\pi} \sqrt{R^2(1 + \overline{E}_I)^2 + R^2(1 + \overline{E}_{II})^2} \qquad (15)$$

$$\approx 2\pi R \sqrt{1 + \overline{E}_I + \overline{E}_{II}} \qquad (16)$$

$$\frac{\Delta L_i}{2\pi R} \approx \frac{\overline{E}_I + \overline{E}_{II}}{2} \qquad (17)$$

This expression is the trace of the $\mathbf{E}^i$ tensor in the $(\vec{m}_i, \vec{p}_i)$ basis. This quantity is an invariant, so the following expression is valid whatever the basis:

$$\frac{\Delta L_i}{2\pi R} \approx \frac{1}{2} \mathrm{trace}(\mathbf{E}^i) \qquad (18)$$

From the expression 18, it is possible to establish a relation between the perimeter variation and the components of the tensor $\mathbf{E}$ expressed in the $(\vec{e}_1, \vec{e}_2, \vec{e}_3)$ basis. Indeed, the ring length relative variation $\Delta L_i/2\pi R$ is given as a function of the strain tensor components $E_{pq}$ and $P_{pq}^{i\perp}$ the components of the projector $\mathbf{P}^{[\vec{n}_i^\perp]}$, in the canonical basis.

$$\Delta L_i / \pi R = P_{rs}^{i\perp} E_{rs} \qquad (19)$$

In expression 19 the double contraction is identified which allows to write:

$$\Delta L_i / \pi R = \mathbf{P}^{[\vec{n}_i^\perp]} : \mathbf{E} \qquad (20)$$

The equation 20 is a dot product between the two tensors resized as column vectors in Bechterew
basis [42]. Thus the following expression is obtained:

$$ \frac{\Delta L_i}{\pi R} = \langle P_{11}^{i\perp} \ P_{22}^{i\perp} \ P_{33}^{i\perp} \ \sqrt{2}P_{23}^{\perp} \ \sqrt{2}P_{31}^{i\perp} \ \sqrt{2}P_{12}^{i\perp} \rangle \left\{ \begin{array}{c} E_{11} \\ E_{22} \\ E_{33} \\ \sqrt{2}E_{23} \\ \sqrt{2}E_{31} \\ \sqrt{2}E_{12} \end{array} \right\} \quad (21) $$

<figure><img src="image_3.png" /><figcaption>Figure 2: Approximation error in the ellipse perimeter calculation, as a function of the ellipticity $\bar{E}_{II} - \bar{E}_{I}$.</figcaption></figure>

The model error when estimating the perimeter $L^{Approx}$ of a deformed sphere using equation 18 depends only on the ellipticity of the test body, i.e. the principal strains difference: $\bar{E}_{II} - \bar{E}_{I}$. To estimate this error, the elliptic integral of the equation 12 is evaluated numerically so as to obtain $L^{Num}$, the ellipse perimeter to the nearest $10^{-14}$ in relative value. Figure 2 shows that the relative error, $\Phi = \frac{L^{Approx} - L^{Num}}{L^{Num}}$, reaches at most $6.10^{-8}$ for an ellipticity of $1.10^{-3}$. This value has to be compared to the relative uncertainty on the measured wavelength which is typically of the order of 1 pm/1500 nm or about $6.10^{-7}$. The error made by estimating the perimeter length with the approximation 18 (or 21) instead of calculating the perimeter by numerical evaluation of the elliptic integral is much smaller than the measurement error. It is therefore acceptable for the expected range of strains, around $1000 \times 10^{-6}$. This approximation will be used in the following since it allows the linearisation of the perimeter variations as a function of the components of the strain tensor with a model error which remains low.
## 4. Linearized relation between the strain tensor components of E and the change in refractive index $\Delta n$

The photoelastic effects are described by the relation between the changes in the impermeability tensor components **B** and the strain tensor **E**:

$$ \Delta(B_{ij}) = p_{ijkl}E_{kl} \quad , \quad (i, j = 1, 2, 3) \qquad (22) $$

where $p_{ijkl}$ are the components of the photoelastic index tensor $\mathbb{P}^e$. The components of the impermeability tensor and those of the refractive index tensor **n** (see [43]) noted $n_{ij}$ are related by the following relation:

$$ \Delta(B_{ij}) = \Delta \left( \frac{1}{n_{ij}^2} \right) \quad , \quad (i, j = 1, 2, 3) \qquad (23) $$

For an isotropic homogeneous medium, this tensor is expressed by:

$$ \mathbf{n} = n\mathbf{I} \qquad (24) $$

where **n** is the refractive index of the medium.

As stated in section 2, the electrical field is along $\vec{e}_\theta$ for $\ell \gg 1$. In the following, it will be assumed that this direction can be equated to the normal of the gallery mode propagation plane. Simulations were performed with COMSOL in order to validate this approximation. The 2D axisymmetric system shown on figure 3 was simulated. The system is made of a sphere of refractive index $n_1$ and radius $R$ (white area on figure 3) embedded in a medium of refractive index $n_2$ (blue area on figure 3). Waves are absorbed at the external boundary by a 20 µm PML (red area on figure 3). Around the diopter, from $R-5$ µm to $R+5$µm, a structured mesh made of a 0.5 µm squares was used. This regular arrangement of the finite elements at the interface between the two media is made to avoid creating fictitious dissymmetries in the modes due to dissymmetries of the finite element mesh. The simulated equation over the whole geometry is :

$$ \vec{\nabla} \wedge (\vec{\nabla} \wedge \vec{E}_{\text{elec}}) - k^2 \epsilon_r \vec{E}_{\text{elec}} = \vec{0} \qquad (25) $$

The example of figure 3 corresponds to a silica sphere with a radius of 500 µm. This is the largest size considered in this study.

The figure 4 shows the angular distribution of the component, $E_\theta^{\text{elec}}$, of the simulated electrical field of two gallery modes propagating on two different spheres: one with a 100 µm radius and the other with a 500 µm radius. This figure illustrates the fact that the angular aperture of the
<figure><img src="image_4.png" /><figcaption>Figure 3: COMSOL modelling of a silica sphere used for WGM simulation.</figcaption></figure>
mode is weak, of the order of 2° for the 500 µm sphere. It increases when the radius of the sphere decreases, but remains smaller than 8° for a 100 µm sphere. This weak angular aperture justifies that the direction $\vec{e}_{\theta}$ can be equated to the direction $\vec{n}_{i}$. The electrical field is therefore assumed to be along $\vec{n}_{i}$. As a consequence, it is assumed that the electrical field is just influenced by the refractive index in the direction $\vec{n}_{i}$.

130

<figure><img src="image_5.png" /><figcaption>Figure 4: $E_{elec}^{\theta}$ component of a WGM propagating in a silica sphere of 500 µm diameter and in a silica sphere of 100 µm diameter.</figcaption></figure>

The component $\Delta B^i$ of the tensor $\Delta B$ along a direction $\vec{n}_i$, is given by:

$$ \Delta B^i = \mathbf{P}^{\vec{n}_i} : \Delta \mathbf{B} = \mathbf{P}^{\vec{n}_i} : \mathbb{P}^e : \mathbf{E} \quad (26) $$

where $\mathbf{P}^{\vec{n}_i}$ stands for the projector in the direction $\vec{n}_i$:

$$ \mathbf{P}^{\vec{n}_i} = \vec{n}_i \otimes \vec{n}_i \quad (27) $$

From 23:

$$ \Delta B^i = -\frac{2\Delta n^i}{(n)^3} \quad (28) $$

with $\Delta n^i$ the refractive index variation in the direction $\vec{n}_i$. This leads to:

$$ \frac{\Delta n^i}{n} = -\frac{(n)^2}{2} \mathbf{P}^{\vec{n}_i} : \mathbb{P}^e : \mathbf{E} \quad (29) $$

This equation, like expression 20, corresponds to a scalar product between the tensors resized as vectors in Bechterew's basis [42]. The system can then be written as:

$$ \frac{\Delta n^i}{n} = -\frac{(n)^2}{2} \langle P_{kl}^i \rangle [\mathbb{P}^e] \{E_{pq}\} \quad (30) $$
with the components of the projector:

$$ \langle P_{kl}^i \rangle = \langle P_{11}^i P_{22}^i P_{33}^i \sqrt{2}P_{23}^i \sqrt{2}P_{31}^i \sqrt{2}P_{12}^i \rangle \quad (31) $$

the photoelastic index tensor $\mathbb{P}^e$, in Bechterew [42] basis, is written:

$$ [\mathbb{P}^e] = \begin{bmatrix} p_{11} & p_{12} & p_{12} & 0 & 0 & 0 \\ p_{12} & p_{11} & p_{12} & 0 & 0 & 0 \\ p_{12} & p_{12} & p_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & (p_{11} - p_{12}) & 0 & 0 \\ 0 & 0 & 0 & 0 & (p_{11} - p_{12}) & 0 \\ 0 & 0 & 0 & 0 & 0 & (p_{11} - p_{12}) \end{bmatrix} \quad (32) $$

and $\{E_{pq}\}$ the components of the strain tensor $\mathbf{E}$ written as a column vector:

$$ \{E_{pq}\} = \begin{cases} E_{11} \\ E_{22} \\ E_{33} \\ \sqrt{2}E_{23} \\ \sqrt{2}E_{31} \\ \sqrt{2}E_{12} \end{cases} \quad (33) $$

In order to assess the approximation consisting in equating $\vec{e}_\theta$ to $\vec{n}_i$, an axisymmetric simulation was carried out with a mode propagating in a plane of normal $\vec{e}_3$. A strain $E_{11}$ is imposed along the axis $\vec{e}_1$. Since the strain is axisymmetric, the same strain is applied along the $\vec{e}_2$ axis. The trajectory followed by the mode is always a circle. This simulation allows to test only the hypothesis made on the variation of the optical index. The previous expressions are adapted to this geometrical configuration. Thus the resized projectors as vectors in Bechterew basis [42] are in this case:

$$ \mathbf{P}^{[\vec{e}_3^\perp]} = \mathbf{I} - \vec{e}_3 \otimes \vec{e}_3 = \begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 \end{pmatrix} \quad (34) $$

and

$$ \mathbf{P}^{[\vec{e}_3]} = \vec{e}_3 \otimes \vec{e}_3 = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} \quad (35) $$

According to the equations, 3 and 30 :

$$ \frac{\Delta L}{L} = \frac{E_{11} + E_{22}}{2} ; \quad \frac{\Delta n}{n} = -\frac{n^2}{2}(p_{12}E_{11} + p_{12}E_{22} + p_{11}E_{33}) \quad (36) $$

By adding the geometric and photoelastic effect, the wavelength shift is written:

$$ \frac{\Delta \lambda}{\lambda} = \frac{E_{11} + E_{22}}{2} - \frac{n^2}{2}(p_{12}E_{11} + p_{12}E_{22} + p_{11}E_{33}) \quad (37) $$
For the axisymmetric simulation with $E_{11} = E_{22}$, this expression becomes:

$$ \frac{\Delta\lambda}{\lambda} = E_{11} - \frac{n^2}{2}(2p_{12}E_{11} + p_{11}E_{33}) \quad (38) $$

The difference between the simulation results performed using COMSOL and considered as the reference and the simplified analytical expression is given in Figure 5 as a function of the diameter of the sphere. As expected, the larger the sphere, the lower the angular aperture and thus the lower the difference between the analytical approximation and the numerical simulation. For a 100 µm sphere, the difference is of the order of 4%, whereas for a 500 µm sphere, it is lower than 1%.

<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Relative wavelength shift calculated by simulation or using the simplified analytical expression as a function of the diameter of the sphere (for $E_{11} = 2000 \ 10^{-6}$, $n = 1.45$ and $p_{12} = 0.27$). (b) Difference between the wavelength variations calculated by simulation or using the simplified analytical expression as a function of the diameter of the sphere.</figcaption></figure>

The approximated expression 30 can therefore be used. It enables a linear relation between the refraction index variation and the strain tensor components with an error that remains low provided that the sphere is not too small.

## 5. Towards a strain measurement device using WGM propagation along 6 different circles

The choice of the measurement planes which defines the components of the normals and therefore the components of the projectors $P_{kl}^{i\perp}$ and $P_{kl}^{i}$ must be made in such a way that the
system is reversible. Among all the combinations, it is preferable to choose a regular distribution of the planes in space so that the behaviour of the sensor is not affected by its orientation with respect to the main directions of the strain field $E_0$. The best solution is to measure perimeter variations in planes perpendicular to the normals of a dodecahedron:

$$ 
\begin{aligned}
\vec{n}_1 &= (0, \varphi, 1)/\sqrt{2+\varphi} \\
\vec{n}_2 &= (0, \varphi, -1)/\sqrt{2+\varphi} \\
\vec{n}_3 &= (1, 0, \varphi)/\sqrt{2+\varphi} \\
\vec{n}_4 &= (1, 0, -\varphi)/\sqrt{2+\varphi} \\
\vec{n}_5 &= (\varphi, 1, 0)/\sqrt{2+\varphi} \\
\vec{n}_6 &= (\varphi, -1, 0)/\sqrt{2+\varphi}
\end{aligned} 
\qquad (39) $$

where $\varphi = (1 + \sqrt{5})/2$ is the golden ratio. The architecture representing this solution is shown in Figure 6. With this configuration of sensors, the system 21 becomes:

<figure><img src="image_7.png" /><figcaption>Figure 6: Distribution of the measurement perimeters on a 6-ring sensor, as also presented in [44] or in [45] for a hollow structure.</figcaption></figure>

$$ \left\{ \frac{\Delta L_i}{L_i} \right\} = \frac{1}{2} \left\{ \frac{\Delta L_i}{\pi R} \right\} = \frac{1}{2} [M_L] \{E_i\} \qquad (40) $$
where the matrix $[M_L]$, according to the property $\varphi^2 - 1 = \varphi$, is written:

$$ [M_L] = \frac{1}{2+\varphi} \begin{bmatrix} 2+\varphi & 1 & 1+\varphi & -\sqrt{2}\varphi & 0 & 0 \\ 2+\varphi & 1 & 1+\varphi & \sqrt{2}\varphi & 0 & 0 \\ 1+\varphi & 2+\varphi & 1 & 0 & -\sqrt{2}\varphi & 0 \\ 1+\varphi & 2+\varphi & 1 & 0 & \sqrt{2}\varphi & 0 \\ 1 & 1+\varphi & 2+\varphi & 0 & 0 & -\sqrt{2}\varphi \\ 1 & 1+\varphi & 2+\varphi & 0 & 0 & \sqrt{2}\varphi \end{bmatrix} \quad (41) $$

150

and

$$ \{E_i\} = \begin{Bmatrix} E_{11} \\ E_{22} \\ E_{33} \\ E_{23} \\ E_{31} \\ E_{12} \end{Bmatrix} \quad (42) $$

A similar relationship can be established between the components of the strain tensor and the wavelength variations resulting from the index variations $\frac{\Delta n_i}{n}$ (see equations 30, 31 and 32).

$$ \left\{ \begin{array}{c} \frac{\Delta n^1}{n^1} \\ \frac{\Delta n^2}{n^2} \\ \frac{\Delta n^3}{n^3} \\ \frac{\Delta n^4}{n^4} \\ \frac{\Delta n^5}{n^5} \\ \frac{\Delta n^6}{n^6} \end{array} \right\} = -\frac{(n)^2}{2} \left[ M_n \right] \left[ \mathbb{P}^e \right] \left\{ \begin{array}{c} E_{11} \\ E_{22} \\ E_{33} \\ E_{23} \\ E_{31} \\ E_{12} \end{array} \right\} \quad (43) $$

with $[M_n]$, the matrix composed by the projectors $\mathbf{P}^{\vec{n}_i}$ along the 6 normals $\vec{n}_i$, i.e.:

$$ [M_n] = \frac{1}{(2+\varphi)} \begin{bmatrix} 0 & 1+\varphi & 1 & \sqrt{2}\varphi & 0 & 0 \\ 0 & 1+\varphi & 1 & -\sqrt{2}\varphi & 0 & 0 \\ 1 & 0 & 1+\varphi & 0 & \sqrt{2}\varphi & 0 \\ 1 & 0 & 1+\varphi & 0 & -\sqrt{2}\varphi & 0 \\ 1+\varphi & 1 & 0 & 0 & 0 & \sqrt{2}\varphi \\ 1+\varphi & 1 & 0 & 0 & 0 & -\sqrt{2}\varphi \end{bmatrix} \quad (44) $$
In the equations 40 and 43, the writing in tensor basis is no longer used. The multipliers $\sqrt{2}$ of the $E_{ij}$ terms are integrated in the matrices $[M_L]$ and $[M_n]$ in order to obtain a relation between the components of the strain tensor and the WGM shifts that can be easily inverted. Finally, this relationship can be written as:

$$ \left\{ \frac{\Delta \lambda_i}{\lambda} \right\} = \frac{[M_L] - n^2[M_n][\mathbb{P}^e]}{2} \{E_i\} = \frac{[M]}{2} \{E_i\}, \quad i \in 1..6 \qquad (45) $$

where $[M]$ has the following form:

$$ [M] = \begin{bmatrix} M_1 & M_2 & M_3 & -M_4 & 0 & 0 \\ M_1 & M_2 & M_3 & M_4 & 0 & 0 \\ M_3 & M_1 & M_2 & 0 & -M_4 & 0 \\ M_3 & M_1 & M_2 & 0 & M_4 & 0 \\ M_2 & M_3 & M_1 & 0 & 0 & -M_4 \\ M_2 & M_3 & M_1 & 0 & 0 & M_4 \end{bmatrix} \qquad (46) $$

with:

$$ \begin{aligned} M_1 &= 1 - n^2 p_{12} \\ M_2 &= \frac{1 - n^2[(1 + \varphi)p_{11} + p_{12}]}{2 + \varphi} \\ M_3 &= \frac{(1 + \varphi)(1 - n^2 p_{12}) - n^2 p_{11}}{2 + \varphi} \\ M_4 &= \frac{\sqrt{2}\varphi}{2 + \varphi} [1 + n^2(p_{11} - p_{12})] \end{aligned} \qquad (47) $$

To obtain the components of the strain tensor from the resonant wavelength variations, the relationship 45 must be inverted:

$$ \{E_i\} = 2[M]^{-1} \left\{ \frac{\Delta \lambda_i}{\lambda} \right\}, \quad i \in 1..6 \qquad (48) $$

which is only possible if the determinant $D$ of the matrix $[M]$ is not zero. This determinant is as follows:

$$ D = -8M_4^3(M_1^3 + M_2^3 + M_3^3 - 3M_1M_2M_3) = \frac{32\sqrt{2}\varphi^5}{(\varphi+2)^5} P_1(p_{11}, p_{12}, n)^5 P_2(p_{11}, p_{12}, n) \qquad (49) $$

with

$$ \begin{aligned} P_1(p_{11}, p_{12}, n) &= n^2(p_{11} - p_{12}) + 1 \\ P_2(p_{11}, p_{12}, n) &= n^2(p_{11} + 2p_{12}) - 2 \end{aligned} \qquad (50) $$

The determinant therefore becomes zero when $P_1 = 0$ or $P_2 = 0$. This means that for some sets of parameters $\{p_{11}, p_{12}, n\}$, the effect of the geometry change $\Delta L/L$ on the wavelength
shift of WGM is partially compensated by the photoelastic effect $\Delta n/n$. The coloured maps in Figure 7 represent the sets of parameters for which these polynomials are zero. These are the configurations to be avoided for the realisation of a 3D strain sensor.

<figure><img src="image_8.png" /><figcaption>Figure 7: In colour are represented the points of the space of parameters {$p_{11}, p_{12}, n$} for which the determinant of the matrix $[M]$ is zero. The color-bar indicates the value of the index $n$ to be avoided for a given set of parameters {$p_{11}, p_{12}$}.</figcaption></figure>

The matrix $[M]^{-1}$, if it exists, has the following form:

$$[M]^{-1} = \begin{bmatrix} m_1 & m_1 & m_3 & m_3 & m_2 & m_2 \\ m_2 & m_2 & m_1 & m_1 & m_3 & m_3 \\ m_3 & m_3 & m_2 & m_2 & m_1 & m_1 \\ m_4 & -m_4 & 0 & 0 & 0 & 0 \\ 0 & 0 & m_4 & -m_4 & 0 & 0 \\ 0 & 0 & 0 & 0 & m_4 & -m_4 \end{bmatrix} \quad (51)$$
where:

$$
\begin{align}
m_1 &= \frac{4M_4^3}{D} (M_2 M_3 - M_1^2) \nonumber \\
m_2 &= \frac{4M_4^3}{D} (M_1 M_3 - M_2^2) \nonumber \\
m_3 &= \frac{4M_4^3}{D} (M_1 M_2 - M_3^2) \nonumber \\
m_4 &= -\frac{1}{2M_4} \tag{52}
\end{align}
$$

6. Sensitivity analysis

160

The objective of this section is to study the influence of the different parameters involved in
the measurement on the accuracy of the measurement. First of all, it should be reminded that
the radius of the sphere conditions the validity of the approximation as shown on the figure 5.
This approximation leads to a systematic error that must be corrected before analysis.

<table><caption>Table 1: Optical constants and sensitivities of different materials</caption>
  <thead>
    <tr>
      <th>Material</th>
      <th><i>n</i></th>
      <th><i>p</i><sub>11</sub></th>
      <th><i>p</i><sub>12</sub></th>
      <th><i>S</i><sub>1</sub><br>(pm/με)</th>
      <th><i>S</i><sub>2</sub><br>(pm/με)</th>
      <th><i>S</i><sub>3</sub><br>(pm/με)</th>
      <th><i>S</i><sub>4</sub><br>(pm/με)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Fused silica [46]</td>
      <td>1.458</td>
      <td>0.121</td>
      <td>0.27</td>
      <td>0.319</td>
      <td>-0.051</td>
      <td>0.178</td>
      <td>0.324</td>
    </tr>
    <tr>
      <td>Doped silica [47]</td>
      <td>1.447</td>
      <td>0.113</td>
      <td>0.252</td>
      <td>0.354</td>
      <td>-0.030</td>
      <td>0.207</td>
      <td>0.336</td>
    </tr>
    <tr>
      <td>PMMA [48]</td>
      <td>1.49</td>
      <td>0.3</td>
      <td>0.28</td>
      <td>0.28</td>
      <td>-0.28</td>
      <td>0.07</td>
      <td>0.49</td>
    </tr>
    <tr>
      <td>Diamond [49]</td>
      <td>2.4</td>
      <td>0.12</td>
      <td>-0.32</td>
      <td>2.13</td>
      <td>0.21</td>
      <td>1.4</td>
      <td>1.68</td>
    </tr>
    <tr>
      <td>Silicon Carbide [50]</td>
      <td>2.6</td>
      <td>-0.15</td>
      <td>0.04</td>
      <td>0.547</td>
      <td>0.701</td>
      <td>0.606</td>
      <td>-0.134</td>
    </tr>
  </tbody>
</table>

From an experimental point of view, the measurement consists in determining the wavelength
of a dip in the intensity of the light transmitted by the fiber which serves as a coupler. The
accuracy of the measurement therefore depends on the depth and width of the dip. With standard
detectors, a depth of -45 dBm is enough to detect the dip. The coupling has to be adjusted to
insure this minimal depth. The typical wavelength resolution is then of the order of 1 pm for
a width of a few hundred pm. This can easily be achieved since it corresponds to a quality
factor of the order of 10⁵. In this framework, the sensibility of the measurement method is
given by the components of the matrix [M] of equation 47. These components depend on the
refractive index and on the components of the photoelastic tensor. Few values of the latter are available in the literature. They depend on the material and vary slowly with wavelength in the transparency zone. For example the photoelastic constants of fused silica exhibit a 10% variation between 650 nm and 230 nm, with a more pronounced increase in the vicinity of UV [51, 52]. The table 1 synthesizes these parameters for different materials such as $p_{44} \simeq (p_{11} - p_{12})/2$ using the classical Voight notation found in the literature, together with the associated sensitivities. To better match the experimental protocol, the quantities $S_i = \lambda M_i/2$ (with $\lambda = 1500$ nm) are reported in this table since they directly give the wavelength shift induced by the strain and associated to the component $M_i$. More precisely, from equation 45, it can be seen that the wavelength shift measured for the mode 1 is given by:

$$ \Delta\lambda_1 = S_1 E_{11} + S_2 E_{22} + S_3 E_{33} - S_4 E_{23} + 0 \times E_{31} + 0 \times E_{12} \quad (53) $$

The modes 1 is then completely insensitive to shear $E_{31}$ and $E_{12}$, whatever the material. In the same way, mode 2 is insensitive to shear $E_{31}$ and $E_{12}$, mode 3 and 4 to shear $E_{23}$ and $E_{12}$, and modes 5 and 6 to shear $E_{23}$ and $E_{31}$. For fused or doped silica, modes 1 and 2 are ten times less sensitive to strain along direction $\vec{e}_2$ than to strain along the other two directions. Similar observations can be made for the other modes with direction $\vec{e}_3$ for modes 3 and 4 and direction $\vec{e}_1$ for modes 5 and 6. For the considered materials, the best sensitivities are expected for diamond. In a general manner, it can be seen from equation 47 that the sensitivities increase with $n^2$.

The last step is to determine the influence of the uncertainties on the measured wavelengths on the uncertainties on the components of the strain tensor. A statistical study can be done as a first approach. A strain state is defined by randomly chosen the strain components in the $[-1000.10^{-6}, +1000.10^{-6}]$ range. Then, the expected wavelength shifts for the 6 gallery modes are computed with the help of equation 45. In order to simulate the noise, a random value uniformly distributed in the range $[-N_{\text{oise}}, +N_{\text{oise}}]$ is added to each shift and then the strain components corresponding to this noisy set are computed with equation 48. This is repeated a thousand times. It is then observed that, for each component, the difference between the mean value of the noisy strain and the correct one is one or two order less than the standard deviation of the distribution of noisy strain. This standard deviation can then be used as uncertainty. Finally, ten thousands states of strain were considered. An example of the distribution of the uncertainties on the component $E_1$ is shown on figure 8 for a doped silica sphere and a noise equal to 1 pm. The mean value of this distribution corresponds to the uncertainty on $E_{11}$.
<figure><img src="image_9.png" /><figcaption>Figure 8: Distribution of uncertainties on E₁ for fused silica and N<sub>noise</sub> = 1 pm</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: Uncertainties on strain as a function of uncertainties on wavelength shifts</figcaption></figure>
The figure 9 shows the variation of the uncertainties on the strain with noise on wavelength for different materials. The uncertainties of modes 1, 2 and 3 behave in exactly the same way. So do the uncertainties of modes 4, 5 and 6. All vary linearly with the uncertainty on the wavelength shift, with a slope that depends on the material. This can be easily explained. Indeed, the wavelength measurements will be made with a spectrum analyser or a tunable source. Whatever the device used, it will not be necessary to change the calibration in the range of strains considered. It is therefore legitimate to consider the measured wavelength as the sum of the correct wavelength and a random noise that depends only on the detection system:

$$ \{\Delta\lambda\}_{\text{meas}} = \{\Delta\lambda\}_{\text{th}} + \{N_{\text{oise}}\} \qquad (54) $$

then:

$$ \{E\}_{\text{meas}} = 2[M]^{-1}\{\Delta\lambda/\lambda\}_{\text{meas}} = 2[M]^{-1}\{\Delta\lambda/\lambda\}_{\text{th}} + 2[M]^{-1}\{N_{\text{oise}}/\lambda\} = \{E\}_{\text{th}} + \{E\}_{\text{err}} \quad (55) $$

The error made on the strain is simply that:

$$ \{E\}_{\text{err}} = 2[M]^{-1}\{N_{\text{oise}}/\lambda\} \qquad (56) $$

Using the classical law of uncertainty composition:

$$ \sigma(E_i) = \sqrt{\sum_{j=0}^{6} \left( \frac{\partial E_i}{\partial \lambda_j} \right)^2 \frac{\sigma^2(\Delta\lambda_j)}{\lambda_j^2}} \qquad (57) $$

and assuming that the wavelength uncertainty is the same on all measurement channels: $\sigma(\Delta\lambda_j) = \sigma(\Delta\lambda) \forall j$, it comes:

$$ \sigma(E_i) = s \sigma(\Delta\lambda) \quad \text{with} \quad s = \begin{cases} s_{123} = \frac{2\sqrt{2}}{\lambda_0} \sqrt{(m_1^2 + m_2^2 + m_3^2)} & \text{for } i = 1, 2, 3 \\ s_{456} = \frac{2\sqrt{2}}{\lambda_0} m_4 & \text{for } i = 4, 5, 6 \end{cases} \qquad (58) $$

where $\lambda_0$ is the resonant wavelength of the undeformed sphere.

The coefficients $s_{123}$ and $s_{456}$ for the materials considered in this study are reported on table 2 and used to draw the straight lines in figure 9. They range from 0.4 $\mu\varepsilon$/pm for the diamond to 6 $\mu\varepsilon$/pm for the PMMA and are close to 2 $\mu\varepsilon$/pm for the silica. To put it in perspective, with a silica sphere, and with a conventional detector of resolution equal to 1 pm, the error on the strain components is, after correction of systematic errors, of the order of 2 $\mu\varepsilon$. This means that this sensor is capable to give the six components of the strain tensor with an uncertainty of the same order of that of a classical strain gauge which only gives the strain along one direction.
<table><caption>Table 2: Uncertainty coefficients</caption><thead><tr><th rowspan="2">Material</th><th>s<sub>123</sub></th><th>s<sub>456</sub></th></tr><tr><th>(με/pm)</th><th>(με/pm)</th></tr></thead><tbody><tr><td>Fused silica [46]</td><td>2.</td><td>2.2</td></tr><tr><td>Doped silica [47]</td><td>1.9</td><td>2.1</td></tr><tr><td>PMMA [48]</td><td>6.1</td><td>1.4</td></tr><tr><td>Diamond [49]</td><td>0.4</td><td>0.4</td></tr><tr><td>Silicon Carbide [50]</td><td>4.3</td><td>5.2</td></tr></tbody></table>

## 7. Conclusion

In this paper, the realization of a 3D strain sensor with a silica sphere as test body and 6 gallery modes as unidirectional strain gauges is studied. The direct model relating the strain to the resonant wavelength variation is described and linearised. The inversion method is then presented.

The direct linear model is established by considering a single whispering gallery mode. Two phenomena combine to change the resonant wavelength: a geometrical effect inducing a variation in perimeter and a photoelastic effect inducing a variation in refractive index. From a geometric point of view, when the sphere deforms, the perimeter of the circle covered by the gallery mode changes and may become an ellipse. The corresponding perimeter variation can be approximated using the trace of the strain tensor. The relative error using this approximation is less than $10^{-7}$ for a strain $(\overline{E}_{II} - \overline{E}_{I})$ less than 1000 με. Taking into account the photo-elastic effect is more delicate. To establish a linear relationship between the strain and the wavelength variation, it is necessary to use an approximation where the electric field of the wave is purely perpendicular to the plane in which the mode propagates. This approximation is more accurate as the radius of the sphere increases. The relative error on the wavelength variation is of the order of 5 % for a 100 μm sphere and falls below 1% for a 500 μm sphere. Beyond the sensor application presented here, this model can be useful to correct any measurement for the effects of strain when it is not the quantity of interest.

The direct model was then applied to 6 gallery modes regularly distributed on the sphere to obtain a matrix relationship between the wavelength variation and the strain. This system is inverted and the components of the matrix allowing to obtain the 6 components of the strain tensor from the 6 wavelength measurements were given explicitly. In practice, this matrix
contains only 4 different components and many zeros, so that the shear components depend on only two wavelength measurements along two different circles, while the other components of the strain tensor depend on all 6 wavelength measurements.

Finally, in terms of sensitivity, it has been shown that the uncertainty in [με] on the components of the strain tensor is twice the uncertainty in [pm] on the wavelength measurement. To give an order of magnitude, an uncertainty of 1 pm on the wavelength leads to an uncertainty of 2 με, which makes the performance of the proposed sensor consistent with that of conventional fibre optic sensors.

220

From an experimental point of view, it must be recognised that the development of such a sensor is likely to be very complicated. The question of mode injection is a major problem, especially as six modes have to be excited. However, there are already signs that this is possible. For example, the system described by Yan [53] where the sphere is covered by a drop of resin which also traps the tap could be used with six tapers. Similarly, the system described by Weigel [23] where a droplet of DDI solved in ethanol is inserted with help of a syringe in a silicone matrix could be used with 6 optical fibers fixed in the matrix. In both cases, the final size of the sensor, of the order of few hundreds of μm, would prohibits its use in biology at the cell level. But it could find applications in composites materials [54, 55] or bonded joints [56, 57] where the measurement of shrinkage is of major importance.

225

230

In any case, as soon as the sensor can be manufactured, the analysis method described in this article can be used as is.

235

## Acknowledgment

This work has been supported by a grant from the Région Pays de la Loire within the framework of the SMOG Project.

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