# Measurement of small linear birefringences in chiral photonic devices

Hoshang Sahib, Amina Bensalah-Ledoux, Alban Gassenq, Bruno Baguenard, Laure Guy, Stephan Guy

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Hoshang Sahib, Amina Bensalah-Ledoux, Alban Gassenq, Bruno Baguenard, Laure Guy, et al. Measurement of small linear birefringences in chiral photonic devices. Optical Materials, 2026, 171, pp.117749. 10.1016/j.optmat.2025.117749. hal-05392696

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Measurement of small linear birefringences in chiral photonic devices

Hoshang Sahib<sup>a,1,*</sup>, Amina Bensalah-Ledoux<sup>a</sup>, Alban Gassenq<sup>a</sup>, Bruno Baguenard<sup>a</sup>, Laure Guy<sup>b</sup>, Stephan Guy<sup>a,*</sup>

<sup>a</sup>Univ. Lyon Institut Lumiere matiere UMR 5306 CNRS-Universite Claude Bernard Lyon 1 10 rue Ada Byron 69622 Villeurbanne Cedex France.

<sup>b</sup>Univ. Lyon ENS de Lyon CNRS Laboratoire de Chimie UMR 5182 Universite Claude Bernard Lyon 1 F-69342 Lyon France

**Abstract**

We present a straightforward and accurate method for measuring the birefringence of chiral waveguides. Due to the chiral nature of these devices, the supported modes are elliptical, enabling direct access to their linear birefringence characteristics. Consequently, determining the linear birefringence reduces to measuring the device's Mueller matrix. This method is particularly effective when the linear birefringence is comparable in magnitude to the circular birefringence (on the order of $10^{-5}$). The technique is applied to study the impact of humidity on sol-gel-based chirowaveguides. Four distinct linear birefringence contributions are identified and measured: (1) Opto-geometrical birefringence, resulting from the waveguide geometry and modal TE/TM decomposition; (2) tensile stress-induced birefringence (~ $10^{-4}$), linked to the waveguide fabrication process; (3) compressive stress-induced birefringence, which increases with humidity due to material swelling; and (4) residual diagonal birefringence (~ $10^{-6}$), attributed to slight asymmetries in the rib waveguide cross-section.

**1. Introduction**

Waveguide modal birefringence (MB), the effective refractive index difference between the two fundamental transverse magnetic (TM) and transverse electric (TE) modes, is a critical parameter for photonic integrated circuits. For instance, polarization-independent devices require the same propagation properties regardless of the polarization and, therefore, zero birefringence (MB=0) is targeted [1, 2, 3]. Measuring modal birefringence has therefore been extensively studied. The optical waveguides investigated in this work are fabricated from chiral materials, forming so-called chirowaveguides. These structures represent promising platforms for chiral evanescent-wave sensing due to their ability to sustain circularly polarized eigenmodes [4]. However, accurate characterization and control of their birefringence properties are crucial, motivating the development of a dedicated experimental apparatus for their precise measurement.

Traditional characterization techniques such as ellipsometry, M-line spectroscopy, and grating coupling are generally limited to an accuracy on the order of $10^{-4}$. To enhance this precision to the $10^{-5}$–$10^{-6}$ range, several advanced methodologies have been developed, including electro- and magneto-optic techniques [5, 6], Sénarmont ellipsometry [7], interferometric approaches [8], ring-cavity resonance methods [9], and polarimetric configurations [10]. However, these methods often involve complex experimental architectures, such as ring resonators, or necessitate specialized perturbation devices, multi-wavelength sources, or destructive procedures requiring the sample to be sectioned. In contrast, polarimetric approaches are based on the mode beating between the two eigenmodes as they propagate. They require only the measurement of the change in the state of polarization (SOP) of a beam passing through the device, without the need for a specifically designed structure. However, the mode beating phenomenon is a periodic function of the phase shift between the two eigenmodes and, therefore, the zero birefringence value, obtained for a zero phase shift, is indistinguishable from all other birefringence values corresponding to a $2\pi$-multiple phase shift.

<footnote>*</footnote>

<footnote>Email addresses: hoshang.sahib@uoh.edu.iq (Hoshang Sahib), stephan.guy@univ-lyon1.fr (Stephan Guy)</footnote>

<footnote n="1">¹Present address: Department of Physics, College of Science, University of Halabja, Halabja, Iraq</footnote>
We demonstrate that the introduction of chirality into optical waveguides enables birefringence characterization through straightforward polarization measurements—using a single-wavelength, non-destructive configuration, achieving an accuracy on the order of $10^{-6}$ and allowing unambiguous determination of the zero-birefringence condition. Indeed, chirowaveguides are governed by two birefringences: an “achiral” linear birefringence ($LB$) arising from the geometrical structure, and a “chiral” circular birefringence (CB) originating from the chiral material. These two birefringences combine not only to produce the modal birefringence, MB, but also to modify the polarization nature of the eigenmodes [11]. Indeed, eigenmodes are no longer the TE and TM modes, but left- and right-handed elliptical polarizations characterized by their ellipticity angle $\chi$ and azimuth $\alpha$ (with respect to the TE axis). While MB remains a periodic function of $LB$, the nature of the eigenmodes, measured via $\chi$ and $\alpha$, varies continuously from TE–TM polarized to left- and right-circular polarization as a function of the CB/$LB$ ratio. Because CB, as an intrinsic property of the chiral molecule, does not vary, any change in the polarization of the eigenmodes can be related to a change in $LB$.

In this paper, we use this approach to measure the $LB$ dependence on humidity in rib chirowaveguides made of sol-gel materials, which are especially sensitive to humidity, as previously reported [12, 13, 14, 15, 16]. We first recall the main properties of elliptical birefringent devices and how the measurement of modal ellipticities enables the direct determination of $LB$. This approach is subsequently applied to assess the humidity dependence of samples held in a chamber with controlled relative humidity (RH). The measurement principle, described in the second part, allows us to fully characterize the eigenmode ellipticity and MB, from which $LB$ can be accurately extracted. Our optical setup allows us to track the evolution of $LB$ as a function of RH with an accuracy of about $10^{-6}$, thereby shedding new light on the complex relationship between humidity and optical properties of waveguides made of sol-gel materials. Furthermore, this study reveals that the linear birefringence of these sol-gel waveguides has two main origins: (i) opto-geometrical, resulting from the geometrical symmetry and the guiding effect, and (ii) mechanical constraints that induce anisotropy of the guiding material. By measuring the Mueller matrix of the guiding systems, and, thanks to their chirality, we were able to identify and accurately measure the contributions related to the opto-geometry of the waveguides and two contributions related to mechanical stresses, one of which is humidity dependent.

## 2. Measurement principle

### 2.1. Chirowaveguides as elliptically birefringent devices

Waveguides made from chiral materials were first studied by Yamamoto [17] in 1970. The name “chirowaveguide” was introduced by Engheta and Pelet in 1989 [18]. The eigenmodes are generally calculated to be elliptically polarized [19, 18, 20], with unique dispersion relations and field structures [20, 21, 22, 23]. The birefringent properties can be simply modeled by stating that the device mixes linear and circular birefringences, which originate from the geometric structure and the refractive index contrast of the waveguide and the chiral material, respectively [11, 19, 24].

<figure><img src="image_2.png" /><figcaption>Figure 1: a) Device cross-section of the rib optical chirowaveguides and b) definition of birefringence axes.</figcaption></figure>

A schematic representation of the rib chirowaveguides studied is shown in Fig. 1-a). The device exhibits three distinct types of birefringence: TE–TM linear birefringence (LB), diagonal linear birefringence at 45° (LB'), and circular birefringence (CB) (Fig. 1-b). LB results from the waveguide geometry, which induces an effective refractive index difference between the TE and TM modes. The LB' component captures asymmetries in the structure that disrupt TE–TM symmetry, typically due to geometric distortions. Thus, the components LB and LB' determine the norm and orientation of the device’s overall linear birefringence $LB$. Circular birefringence (CB), on the other hand, arises from the incorporation of chiral molecules into the waveguide material, introducing circular anisotropy.
<figure><img src="image_3.png" /><figcaption>Figure 2: a) Visualization of the transformation of the SOP by an elliptically birefringent medium on the Poincaré sphere. Any input SOP (blue point) traces a circular path around the red axis, which connects the two polarization eigenstates. The length of this trajectory (blue arrows) is determined by the phase retardation $\Delta\Phi$ resulting in the output SOP (orange point). b) The latitude $2\chi$ and longitude $2\alpha$ define the polarization ellipses of the two eigenstates. The blue ellipse is orthogonal to the red one and exhibits opposite handedness.</figcaption></figure>

## 2.2. Poincaré sphere representation

A refined approach for studying the evolution of the SOP in an elliptically birefringent system involves analyzing the rotation of the Stokes vector on a sphere [25, 26]. The Stokes vector consists of four parameters, denoted as $S_{i=0,1,2,3}$: the total intensity ($S_0$), the intensity difference between horizontally and vertically polarized light ($S_1$), the intensity difference for linear polarizations at $\pm 45^\circ$ ($S_2$), and the intensity difference between left- and right-handed circular polarizations ($S_3$). In non-depolarizing media, the total intensity $S_0$ remains constant, while the remaining components $S_1$, $S_2$, and $S_3$ trace a trajectory on the surface of a sphere with radius $S_0$ as the light propagates. This sphere of polarizations, known as the Poincaré sphere, offers a geometric representation of the polarization state. Each point on its surface corresponds to a specific SOP, with its latitude and longitude indicating the ellipticity and azimuth of the associated polarization ellipse, respectively.

Fig.2-a) shows how the Stokes vector evolves on the Poincaré sphere during propagation through an elliptically birefringent medium. The vector rotates around the axis connecting the two orthogonal polarization eigenstates, with the rotation angle $\Delta\Phi$ corresponding to the phase retardation between the fast and slow axes of the medium [26]. Therefore, to fully characterize an elliptically birefringent system from a polarization standpoint, three angles are required: $\Delta\Phi$; the latitude $(2\chi)$, which defines the ellipticity of the eigenstates; and the longitude $(2\alpha)$, which specifies the ellipse orientation (azimuth) angle. These parameters are directly related to the material's intrinsic optical properties, as described below:

$$ \tan(2\chi) = \frac{CB}{LB} \qquad LB = \sqrt{LB^2 + LB'^2} \qquad (1) $$

$$ \tan(2\alpha) = \frac{LB'}{LB} \qquad (2) $$

$$ \Delta\Phi = \frac{2\pi}{l} MB \cdot l, \qquad MB = \sqrt{CB^2 + LB^2} \qquad (3) $$

$LB$ is the device linear birefringence independent of the chosen referential [27]. MB denotes the modal birefringence and $l$ is the device length.

The latitude and longitude of the rotation axis define the two polarization eigenstates, specifically, their azimuth and ellipticity, as illustrated in Fig. 2. These parameters are intrinsic to the material and independent of its physical length, as evidenced by Eqs. (1)–(2). Notably, in a typical achiral waveguide, the rotation axis lies along the equator of the Poincaré sphere ($2\chi = 0$). The rotation around this axis gives rise to a periodic trajectory on the Poincaré sphere, commonly referred to as “polarization beating”.

## 2.3. Polarimetric measurements

In general, polarimetric analysis involves measuring the Mueller matrix of a sample, which represents the geometric transformation on the Poincaré sphere. This matrix is derived by analyzing the relationship between the input and
output states of polarization (SOP) of light as it propagates through the sample, using various input SOPs [28]. Once the Mueller matrix is known, various optical parameters can be extracted. In the case of an elliptically birefringent device, the polarization transformation, following propagation, is described by the three angles ($\Delta\Phi$, $2\alpha$ and $2\chi$) as illustrated in Fig. 2.

<figure><img src="image_4.png" /><figcaption>Figure 3: a) Calculated $\Delta\Phi$ vs LB for CB=10⁻⁵. b) Calculated $2\chi$ vs LB/CB.</figcaption></figure>

On the Poincaré sphere, the angle $\Delta\Phi$, which represents the rotation of the injected modes around the eigenmodes axis, is measured within the interval [0, $2\pi$]. As such, it is defined modulo $2\pi$ and does not reflect the total number of full rotations. The dependence of the measured $\Delta\Phi$ on linear birefringence (LB) is illustrated in Fig.3 (a). It is evident that a single value of $\Delta\Phi$ can correspond to multiple values of LB. At low birefringence levels, the phase shift is primarily influenced by circular birefringence (CB), whereas at higher LB values, $\Delta\Phi$ exhibits the expected linear dependence on LB. In this high-birefringence regime, Eq.3 indicates that LB is only defined modulo $\lambda/l$. Therefore, to resolve this ambiguity and accurately determine the absolute value of LB from $\Delta\Phi$, it is necessary to perform measurements using samples of varying lengths, thereby circumventing the $2\pi$ periodicity.

The angles $2\alpha$ and $2\chi$, representing the longitude and latitude of the eigenmodes axis on the Poincaré sphere, are independent of the sample length and are defined modulo $\pi$. This $\pi$-periodicity introduces an inherent ambiguity, preventing distinction between the fast and slow eigenmodes. The angle $2\alpha$ characterizes the orientation of the sample's birefringence axis relative to the laboratory frame—i.e., the rotation of the polarization ellipses with respect to the measurement axes. The latitude $2\chi$ relationship to the linear birefringence LB, as given in Eq. 1, establishes a one-to-one correspondence between $2\chi$ and the ratio LB/CB. This relationship, illustrated in Fig.3 (b), enables unambiguous determination of LB for a given CB.

## 2.4. Sensitivity Analysis of Small Linear Birefringence

The detection of extremely weak linear birefringence effects requires distinguishing small deviations from the ideal circular birefringence. In this regime, the smallest measurable linear birefringence corresponds to an elliptically birefringent element that slightly departs from a purely circular one. The rotation axis of such a medium lies near the North Pole of the Poincaré sphere, and its latitude can be expressed as:

$$2\chi = \frac{\pi}{2} - \Delta\chi. \qquad (4)$$

where $\Delta\chi \ll \pi$. In the experimental configuration, the sample is probed with input states of polarization located along the meridian $2\alpha = 0$. The difference between the trajectories corresponding to a purely chiral medium ($2\chi = \pi/2$) and a slightly elliptical birefringent medium ($2\chi = \pi/2 - \Delta\chi$) is given by (see appendix for the calculation details):

$$ \begin{bmatrix} -(1 - \cos \Delta\Phi)S_3 \\ \sin \Delta\Phi S_3 \\ -(1 - \cos \Delta\Phi)S_1 \end{bmatrix}. \qquad (5) $$

where $\Delta\Phi$ denotes the optical rotation induced by the purely chiral medium. This relation defines a helical trajectory on the Poincaré sphere, characterized by a radius $\Delta\chi(1 - \cos \Delta\Phi)$ in the $S_1-S_3$ plane and a length $\Delta\chi|\sin \Delta\Phi|$ along the $S_2$ axis. Fig. A1 in the appendix shows the functions $1 - \cos \Delta\Phi$ and $|\sin \Delta\Phi|$. For $\Delta\Phi < \pi/2$, the variation of the
output SOP is dominated by the $S_2$ component, while for $\Delta\Phi > \pi/2$, it is primarily governed by rotation within the $S_1-S_3$ plane.

The trajectory deviation becomes experimentally resolvable only when it exceeds the uncertainty of the Stokes parameters, denoted $\Delta S_i$. From the relationship between $\mathcal{L}\mathcal{B}$ and CB (Eq. 1), the minimum detectable birefringence is therefore:

$$ \mathcal{L}\mathcal{B}_{\min} = CB\Delta\chi_{\min} = CB\Delta S_i \begin{cases} \frac{1}{|\sin\Delta\Phi|}, & \text{for } \Delta\Phi < \pi/2[\pi], \\ \frac{1}{1 - \cos\Delta\phi}, & \text{otherwise.} \end{cases} \quad (6) $$

This expression defines the sensitivity limit of the measurement system. As expected, the limit scales proportionally with the accuracy of the Stokes parameter measurements. Furthermore, since the analysis is normalized with respect to the circular birefringence, the sensitivity is also proportional to the CB of the chiral medium. The dependence on the waveguide length arises through the optical rotation, which governs the variation of the SOP trajectory induced by linear birefringence via the $\sin\Delta\Phi$ and $(1 - \cos\Delta\Phi)$ terms. As shown in Fig. A1 (Appendix), the sensitivity increases with the optical rotation up to a half a turn, where $\mathcal{L}\mathcal{B}_{\min} = CB/2\Delta\chi$. These results therefore establish a quantitative criterion for evaluating the minimum detectable linear birefringence in chiral media.

For small optical rotations, where $\sin\Delta\Phi \approx \Delta\Phi$, the sensitivity reduces to

$$ \mathcal{L}\mathcal{B}_{\min} = \frac{\lambda}{2\pi l} \Delta S_i \quad (7) $$

This expression coincides with the sensitivity obtained in a direct LB measurement. Physically, this equivalence arises because the method involves measuring a difference in SOP relative to that of the chiral medium, and is therefore limited by the same polarimetric accuracy. However, as previously discussed, this approach offers the advantage of an unambiguous zero reference, free from the modulo-$2\pi$ phase uncertainty.

In summary, polarimetric measurements can be used to characterize the linear birefringence ($\mathcal{L}\mathcal{B}$) of a homogeneous device. The azimuth of the optical axes can be determined from the parameter $2\alpha$, while the strength of the linear birefringence can be evaluated using either the rotation angle $\Delta\Phi$ or, in the presence of chirality (i.e. $CB \neq 0$), the latitude of the eigenstates $2\chi$. In typical achiral systems, the measurement of linear birefringence relies solely on mode beating, with $\Delta\Phi$ known only modulo $2\pi$. As a result, ($\mathcal{L}\mathcal{B}$) can only be determined modulo $\lambda/l$, and not as an absolute value (Fig. 3 a). However, devices exhibiting both circular and linear birefringences (giving rise to elliptically polarized eigenstates) offer the unique advantage of allowing access to unwrapped phase values of $\Delta\Phi$. Consequently, an unambiguous, absolute determination of the device linear birefringence is possible using the polarization latitude change for the low LB values (compare to CB) and using the phase shift $\Delta\Phi$ for high LB values.

### 3. Samples and optical measurements setup

The chirowaveguides are made of a chiral organically modified sol-gel glasses (ormosils) composed of an enantiopure binaphthyl unit covalently connected to a trialkoxysilane. This material presents a high $CB=10^{-5}$, with PEG200 as an anti-cracking agent, it can be coated as a thin film [29, 30] and even micro-structured to produce the rib optical chirowaveguides displayed in Fig. 4 [11]. Optical chips under investigation are composed of rib waveguides with different widths located on the same substrate. In order to assess the role of the chirality, we have compared materials made from the (+)-enantiomer (positive $CB=+1 \times 10^{-5}$), (-)-enantiomer (negative $CB=-1 \times 10^{-5}$) and the (+/-)- racemic mixture ($CB=0$). In doing so, all the characteristics are the same (RI, height, length,...) except the CB (+, - or +/-) or the width (w).

Polarimetric measurements were carried out using an experimental setup previously described in detail by Sahib et al. [11]. The system includes a SOP generator (polarization preparation), consisting of a Glan laser Calcite linear polarizer followed by a Babinet-Soleil compensator. The extinction ratio of the polarizer ($> 10^6$) combined with the accuracy of the Babinet-Soleil compensator ($\lambda/1000$) guarantees an input polarization accuracy of approximately $10^{-3}$ in term of Stokes components, which is ten times better than the polarimeter accuracy (see below)

The samples are deposited onto silicon wafer substrates, enabling them to be cleaved to yield smooth, well-defined planar facets. A red light source ($\lambda = 630$ nm) from a fiber-coupled LED is coupled into and out of the waveguide
<figure><img src="image_5.png" /><figcaption>Figure 4: Scanning electron microscope image and Schematic (inset) of the rib channel waveguides with cover ($n_c = 1$), guiding ($n_g = 1.625$), and buffer ($n_b = 1.61$) refractive indices. The dimensions of the rib channels are height $h = 2.1\mu$m, rib slab thickness $d = 0.2\mu$m, and different width $w$.</figcaption></figure>

under investigation using microscope objectives with a numerical aperture of 0.25. In principle, the coupling of light could induce polarization-dependent effects, such as linear dichroism, arising from Fresnel reflection coefficients at the input and output facets, if the sample is not perfectly orthogonal to the incident beams. To mitigate this, the sample alignment with respect to the optical axes is carefully optimized. No evidence of linear dichroism was observed in any of the measurements, indicating that the coupling and decoupling processes do not affect the polarization state of the transmitted light.

The light emerging from the waveguides is analyzed using a SOP analyzer comprising a rotatable zero-order quarter-wave plate, a polarizer, and a camera, as schematically depicted in Fig. 5. Intensities are recorded for 36 angular positions of the quarter-wave plate, from which the Stokes parameters are retrieved following the procedure described in [31]. To calibrate the setup, the input and output polarizations were first characterized in the absence of the waveguide. For this purpose, the two microscope objectives were positioned at a separation equal to the sum of their respective working distances. Input SOPs corresponding to the four meridians of longitude 0°,±40° and 90° were generated using a Babinet Soleil and measured with the polarimeter. The results, shown in the Appendix on Fig. A2, indicate that the maximum input–output deviation of the Stokes parameters remains below 0.05 over the entire polarization range. This confirms that neither significant birefringence nor dichroism originates from the chamber or the microscope objectives. Finally, for a 5 mm-long sample with a circular birefringence of CB = $10^{-5}$ and a wavelength of $\lambda= 630$ nm, Eq. 6 yields a minimum detectable linear birefringence on the order of $\sim 10^{-6}$.

Finally, eight distinct input SOPs, defined by $\alpha = 0$ and $2\chi = p\pi/4$ with $p = 0, 1, 2, ..., 7$, were launched into the waveguides, and the corresponding output SOPs were subsequently recorded. From these measurements, by fitting the input and output SOPs, the characteristics of the rotation axis (i.e., the angles $2\alpha$ and $2\chi$), and the phase retardation angle (ΔΦ angle) are determined as shown in Fig. 2-a). As the circular birefringence (CB) arises from the organic component of the structure, it is assumed to remain unaffected by humidity variations. Consequently, as discussed earlier, the orientation of the eigenmodes in polarization space directly enables the extraction of the linear birefringence $LB$ via Eq. 1.

<figure><img src="image_6.png" /><figcaption>Figure 5: Schematic representation of Stokes polarimetry optical setup for the measurement of humidity effect on a waveguide. P: polarizers. O1-O2: objectives. BS: Babinet-Soleil compensator. RQWP: rotating quarter-waveplate.</figcaption></figure>
## 4. Eigenmode polarization vs humidity

<figure><img src="image_7.png" /><figcaption>Figure 6: Evolution of slow eigenmodes in relation to RH for three rib waveguides with $w = 2.4\mu m \pm 0.1\mu m$, $h = 2.1\mu m \pm 0.1\mu m$, $n_g = 1.618$, $n_b = 1.605$. **a)** Representation on the Poincaré sphere. **b)** Latitude and **c)** longitude evolution of the slow eigenmodes as a function of the humidity. Devices made with the chiral (+) and (-) enantiomers are represented in blue–green and red–gold, respectively. The black–gray points indicate devices made of racemic (+/-) achiral material. On the Poincaré sphere representation, the colors variation code the RH variation.</figcaption></figure>

Our chirowaveguides made from sol-gel materials being sensitive to humidity, we investigate its effect on the optical properties of our devices. Therefore, in this section, we focus on the eigenmode SOP variation with the relative humidity. The eigenmode evolution with RH presents the same characteristics regardless of the type of sample. We describe here in detail the results for one chirowaveguide. Fig. 6 displays the measured eigenmode SOP (blue points) for the (+)-enantiomer of a rib chirowaveguide (2.4µm wide) versus RH. The eigenmode SOP variation following an increase in RH, is characterized by a change in the latitude and reaches a maximum at RH~45%, for which the SOP is the nearest to the pole (Fig. 6-b). At the same time, the longitude ($2\alpha$) evolves as represented in Fig. 6-c), going from 0° to 180° for RH changing from 11% to 75%. In other words, between these two RH values, the LB sign is inverted, and the effective RI for the TM polarized light becomes higher than for the TE polarized light. Overall, as the RH increases, the SOPs leave the initial TE location at the $S_1=1$ position on the Poincaré sphere (in Fig. 6-a), where the latitude and the longitude are equal to 0 and describe a northward trajectory. After a maximum at RH=45%, the direction changes towards the TM location at $S_1=-1$, which is reached at RH=75%.

The eigenmode SOP of the (-)-enantiomer waveguide (red points in Fig. 6) moves southward with a similar trajectory to that of the (+)-enantiomer. This mirror image behavior, relative to the plane of the equator, is expected from Eq. 1, while the only difference between the two samples is the sign of CB.

To elucidate the role of chirality, a comparative analysis was conducted on rib waveguides constructed from a racemic material. Both the RI and dimensions were identical; however, the organic composition of the guiding layer was a racemic mixture, which rendered the chirality parameter (CB) equal to zero. The SOPs evolution is plotted in black points in Fig. 6. As expected, the eigenmodes' SOPs for racemic waveguides present $2\chi = 0$ contribution. However, the longitude $2\alpha$ gradually increases from 0° to 180° with humidity, and consequently the eigenmodes' SOPs are all located on the equatorial circle. Again, the value of RH=45% corresponds to a peculiar SOP where $2\alpha = 90°$ and it has Stokes parameter $S_2 = 1$.

## 5. Evolution of the linear birefringences vs humidity

Quantitatively, the localization of the eigenmodes in the polarization space gives a direct access to CB, LB and LB' using Eq. 1-3. Only the measured $2\alpha$ and $2\chi$ for RH between 40 and 50% have been selected as their angles are greater than 20°. LB and LB' extracted in this RH range are plotted in Fig. 7. LB shows a linear evolution versus RH and it crosses the zero axis at RH~45% which is consistent with the highest latitude angle in the Poincaré sphere representation described before.

Since LB' remains constant across the selected range of relative humidity, it can be regarded as an intrinsic parameter of the waveguide, reflecting geometric distortions arising from imperfections in the fabrication process.
<figure><img src="image_8.png" /><figcaption>Figure 7: Linear birefringences deduced from the eigenmodes SOP orientations displayed in Fig. 6. LB and LB' are the TE–TM and ±45° oriented linear birefringences respectively.</figcaption></figure>

However, as LB' is as small as $(7 \pm 3) \times 10^{-6}$, its contribution can be neglected except when $LB \le CB$ where it makes the elliptically polarized eigenmodes azimuth not aligned any more with the TE–TM axes but rotated by $\alpha$ (Eq. 2).

Knowing CB and LB', determination of LB outside the 40-50% RH range has been deduced by combining the SOP measurements and Eq. 3, i.e. the phase retardation between the two eigenmodes. In this way, LB was obtained for a planar waveguide and for rib waveguides of different widths. As Eq. 3 gives the phase retardation modulo $2\pi$, to access the absolute values of the modal birefringences over the full humidity range, we made the values superimpose with the most accurate data obtained from the latitude measurement in the RH=40-50% range that are displayed in Fig. 7.

<figure><img src="image_9.png" /><figcaption>Figure 8: LB as a function of humidity deduced from the eigenmodes SOP orientation and modal phase mismatch.</figcaption></figure>

The results of LB as a function of RH in the 11% to 75% RH range are plotted in Fig. 8. For all the studied waveguides, LB is positive at low RH and decreases linearly with humidity. For the rib structures, the plots are nearly parallel to each other with a negative slope.

## 6. Discussion

To elucidate the TE–TM linear birefringences (LB) evolutions with humidity, we first, computed the opto-geometric linear birefringence ($LB_{opt-geometry}$) corresponding to the non-deformed cross section shapes of the waveguides under study. We used EMpy mode solver (a Python-based eigenmode expansion solver–version 2.1.5) to complete our work [32]. The mesh grid parameters were selected such that modifying the grid size by a factor of two (i.e., doubling the size or the mesh density) resulted in a change in birefringence of less than 5%. Taking into account the refractive index contrast of our waveguiding systems and a perfect shape as schematized in Fig. 1(a), the $LB_{opt-geometry}$ values for
waveguides with different (w) are computed and provided in the second column of Table 1. These calculated values are much smaller than the measured LB over the entire RH range. Furthermore, to test for potential variation in the refractive index of the material due to humidity, we simulated the effect of variation in the refractive index up to (10⁻²) on LBopt-geometry of the waveguides, and the calculated variation on LBopt-geometry was less than (10⁻⁶) RI units. Given that this value is substantially smaller than the calculated LBopt-geometry and than the measured LB variation over the considered humidity range, we can assume that the LBopt-geometry remains constant regardless of RH.

These, together with the linear dependency of LB versus RH, shows that in addition to the calculated LBopt-geometry, extra LB sources have to be taken into account. We attribute these LB sources to a stress-optic effect in the sample, as has been observed many times in optical waveguides [33]. This stress-optic effect can induce a positive LB when the waveguide layers contract in-plane relative to the substrate (shrinkage), making a tensile stress occur in the waveguide. It can also induce negative LB when the waveguide layers expand in-plane relative to the substrate, making a compressive stress in the waveguide [34]. Whatever the source of the stresses in the waveguide, they favour the appearance of cracks in the constrained material.

Fig. 8 shows that regardless of whether the guide is structured or planar, at RH = 0, the overall linear birefringence is positive even for waveguides with intrinsic LBopt-geometry ≤ 0. This indicates that the stress-optic induces necessarily a positive LB contribution at RH = 0. It can have mainly two origins: the thermal expansion mismatch between the ormosil and substrate layer, as well as the layer shrinking during the deposition and baking of the sol-gel solution [34]. This positive linear birefringence (LBtensile) is constant for each considered waveguide and can be deduced from the intersection of the linear fits of Fig. 8, with the RH = 0% axis, corresponding to (LBopt-geometry + LBtensile). Knowing the theoretical values of LBopt-geometry, we can get LBtensile for each waveguide as reported in table 1.

As humidity increases, the overall LB decreases regardless of the nature of the guide, until it passes through LB = 0 which clearly shows that the initial LB<0 at RH = 0, has been balanced by another stress-optic effect inducing LB of opposite sign, i.e. negative. This negative linear birefringence is compressive and is humidity dependent (LBx × RH%). It is related to the nature of the material. Indeed, we can assume that water interaction with the PEG200 causes a swelling of the structures. Microscopic observations support this assumption. Layers that were not of optimal quality and exhibited micro-cracks in their structure showed evolution of these cracks when exposed to high humidity, as depicted in Fig. 9. This highlights the progression of the surface condition and the growth of cracks under humidity-induced stress. As the relative humidity increases, water diffuses into the films. This causes the films to swell, leading to compressive stress in the waveguide layer, which is known to account for negative LB values [35]. There are therefore two contributions to the overall linear birefringence due to stresses: One is tensile (LBtensile>0)

<figure><img src="image_10.png" /><figcaption>Figure 9: Optical microscope images of planar ORMOSIL waveguides a) before and b) after the humidity effect.</figcaption></figure>

and the other is compressive and humidity dependent (LBx × RH% <0).

Finally, the measured LB can be written as the sum of the opto-geometrical linear birefringence, calculated for a symmetric shape of the considered waveguide (LBopt-geometry, constant), and the two components of the stress-optic effect; a contribution induced by tensile stress (LBtensile, constant), and the one induced by humidity-dependent compressive stress (LBx × RH%, variable):

$$ LB = LB_{opt-geometry} + LB_{tensile} + LB_{x} \times RH\% \quad (8) $$

The parameters LBopt-geometry, LBtensile and LBx used for fitting the experimental LB for the studied waveguides with various geometry are listed in Tab. 1. It is seen that LBtensile and LBx are, respectively, approximately two and four times lower in ribs than in planar waveguides. This trend is consistent with previous reports that mechanical stresses are reduced in channels compared to unstructured flat layers [36].
<table><caption>Table 1: Humidity LB parameters for planar and Channel waveguides of different widths w. Uncertainty on LB<sub>tensile</sub> and LB<sub>x</sub> are ±4 × 10<sup>-6</sup> and ±10<sup>-7</sup> respectively. w = ∞ denotes a planar waveguide (film).</caption><thead><tr><th>w<br>(μm)</th><th>LB<sub>opt-geometry</sub><br>(×10<sup>-5</sup>)</th><th>LB<sub>tensile</sub><br>(×10<sup>-5</sup>)</th><th>LB<sub>x</sub>[RH<sup>-1</sup>]<br>(×10<sup>-5</sup>)</th></tr></thead><tbody><tr><td>∞</td><td>20</td><td>57,5</td><td>-1.2</td></tr><tr><td>3.3</td><td>0.8</td><td>23.9</td><td>-0.31</td></tr><tr><td>3</td><td>-3.1</td><td>24.8</td><td>-0.32</td></tr><tr><td>2.4</td><td>-17</td><td>28</td><td>-0.24</td></tr></tbody></table>

## Conclusion

Polarimetric measurements on elliptical birefringent devices provide a precise means of characterizing linear birefringence within the magnitude range comparable to circular birefringence. Utilizing this method, we investigated the influence of humidity on sol-gel derived chirowaveguides and quantitatively determined both the magnitude and orientation of the contributing birefringence components. Our analysis identified five distinct sources of birefringence, along with their respective relationships to humidity:

* Circular birefringence: Originating from organic chiral molecules; unaffected by humidity.

* Opto-geometrical linear birefringence: Resulting from waveguide geometry and corresponding to non-deformed cross section shapes of the waveguides, independent of humidity.

* Tensile stress-induced birefringence: Caused by sol-gel film shrinkage and thermal expansion mismatch between the substrate and ormosil; also humidity-independent. Notably, it is lower in rib waveguides (~2 × 10<sup>-4</sup>) compared to planar ones (~6 × 10<sup>-4</sup>).

* Compressive stress-induced birefringence: Increases with humidity due to PEG-induced swelling; its eigen-mode axes align with the TE/TM directions, and its magnitude opposes the tensile component. This birefringence is significantly decreased in rib structures compared to planar ones.

* Residual birefringence: Resulting from waveguide geometry and corresponding to a very small component ((7 ± 3) × 10<sup>-6</sup>), with eigenmodes at ±45° to the TE/TM axes (i.e. LB'). It appears only in rib waveguides when other birefringences counterbalance each other. This component is humidity-independent and is attributed to deformation of the rib cross-section.

This comprehensive study underscores the humidity sensitivity of sol-gel derived materials and enhances our understanding of their behavior in integrated optical devices.

## Acknowledgement

Authors acknowledge support from Agence Nationale de la Recherche (project CAPTOR ANR-18-CE06-0010).

## 7. Experimental Section

*Material.* Thick layers (> 5 µm) of organically modified silica (ORMOSIL) are prepared by dip coating ORMOSIL solutions on silicon wafers, and then they are baked. The experimental detail of the synthesis and deposition of the films is explained elsewhere [29]. These thick ORMOSIL layers are then used as optical buffer layers upon which guiding layers with a higher refractive index than the buffer one are deposited. The refractive index of the deposited layers is adjusted by adding a given amount of pre-hydrolyzed MTEOS into the ORMOSIL solution. The thickness of the planar waveguide layer used in this study is measured by M-line spectroscopy and found to be 2 µm ± 0.1. The refractive indices of the buffer ($n_b$) and guide ($n_g$) layers are 1.605 ± 0.001 and 1.618 ± 0.001 respectively.
*Waveguide Preparation.* Micro-structured films (rib waveguides) are prepared by the same procedure as the planar ones, with the exception that the guiding layers are patterned using imprint lithography. The experimental detail of the channel waveguide fabrication is explained elsewhere[11]. All the channel waveguides analyzed in this study are found to be rib channels with channel height *h* = 2.1 ± 0.1 µm, rib slab thickness *d* = 0.2 ± 0.1 µm, but different width *w* as schematically shown in Fig. 4. The cover layer of the waveguides is air and humidity is modified in the vicinity of this layer. The refractive indices of the buffer and guiding layers of the channel waveguide samples are measured from planar waveguides made with the same materials composition and found to be *n*<sub>b</sub> = 1.61 ± 0.001 and *n*<sub>g</sub> = 1.625 ± 0.001, respectively. The refractive indices of the planar and channel waveguides are measured at room temperature (from 20 to 27 °C) and ambient relative humidity (35 ± 5 %).

*RH chamber.* The waveguides are analyzed in a chamber where the relative humidity (RH) is controlled by flowing RH-controlled air. A mechanical air pump and valves are used to control the flow of air bubbling through a saturated LiCL salt (RH= 11 ± 5%) and water solution (RH= 75 ± 5%) into the sample analysis chamber. The air mixture controlled by the mechanical valves is air with RH∈ [11 ± 5 : 75 ± 5%]. A commercial electronic humidity sensor (DHT11), placed next to the analyzed optical waveguides, is then used to measure the RH values. We calibrated the humidity sensor from the RH of air in equilibrium with different saturated salt solutions[37]. All measurements are performed at room temperature (from 20 to 27 °C), but it only changed within ±2 °C throughout any experimental run. The optical analyzes are made with a red laser diode (λ = 633nm).
# Appendix

## Mueller matrix of an elliptical birefringent medium

The Mueller matrix is a rotation in the $S_1, S_2, S_3$ polarization space writing [26]:

$$ M(2\alpha, 2\chi, \Delta\Phi), = \begin{bmatrix} u_x^2(1-c) + c & u_x u_y(1-c) - u_z s & u_x u_z(1-c) + u_y s \\ u_x u_y(1-c) + u_z s & u_y^2(1-c) + c & u_y u_z(1-c) - u_x s \\ u_x u_z(1-c) - u_y s & u_y u_z(1-c) + u_x s & u_z^2(1-c) + c \end{bmatrix} \quad (9) $$

where $\pm u_{x,y,z}$ are the coordinates of the Stokes eigenvectors representing the two elliptically polarized eigenstates. In term of ellipticity and azimuth of the eigenstates:

$$ u_x = \cos(2\alpha) \cos(2\chi) $$

$$ u_y = \sin(2\alpha) \cos(2\chi) $$

$$ u_z = \sin(2\chi) $$

c and s denotes the rotation of the states of polarization about the rotation axis:

$$ c = \cos \Delta\Phi, \quad s = \sin \Delta\Phi \quad (10) $$

## Mueller matrix for a weak linear birefringent medium

This medium is characterized by a rotation axis defined by $2\chi = \pi/2 - \Delta\chi$ and $2\alpha = 0$. To first order, the corresponding unit vector components are

$$ u_x = \Delta\chi, \quad u_y = 0, \quad u_z = 1, \quad (11) $$

where $\Delta\chi \ll \pi$. The Mueller matrix describing such a medium is given by

$$ M(0, \frac{\pi}{2} - \Delta\chi, \Delta\Phi) = \begin{bmatrix} c & -s & \Delta\chi(1-c) \\ s & c & -\Delta\chi s \\ \Delta\chi(1-c) & \Delta\chi s & 1 \end{bmatrix}, \quad (12) $$

where only first-order terms in $\Delta\chi$ are retained.

## SOP trajectory variation from a chiral to a slighty linear birefringent medium

In the experimental configuration, the sample is probed with input states of polarization (SOPs) located along the meridian $2\alpha = 0$. The difference between the trajectories corresponding to a purely chiral medium ($2\chi = \pi/2$) and a slightly elliptical birefringent medium ($2\chi = \pi/2 - \Delta\chi$) is given by

$$ [M(0, \frac{\pi}{2} - \Delta\chi, \Delta\Phi) - M(0, \frac{\pi}{2}, \Delta\Phi)] \begin{bmatrix} S_1 \\ 0 \\ S_3 \end{bmatrix} = \Delta\chi \begin{bmatrix} -(1-c)S_3 \\ sS_3 \\ -(1-c)S_1 \end{bmatrix}. \quad (13) $$
<figure><img src="image_11.png" /><figcaption>Figure A1: 1 - cos $\Delta\Phi$ and |sin $\Delta\Phi$|</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure A2: Comparison of input and output SOP by the optical setup. a) Input/output SOP are drawn on the Poincaré sphere b,c,d) Correlation between the input and output Stokes components $S_{1,2,3}$.</figcaption></figure>
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