Temperature-Modulated Optical Refractometry

• Temperature-Modulated Optical Refractometry is an optical approach that probes refractive-index changes under controlled temperature variations to extract thermo-optic response functions.
• It encompasses diverse implementations, from polymer glass transition determination to on-chip liquid sensing, by decoding phase and amplitude shifts in optical signals.
• The technique leverages simultaneous measurements and calibration methods to decouple temperature-induced effects from accurate refractive-index assessments in various media.

Searching arXiv for the cited works and closely related papers on temperature-modulated optical refractometry. Temperature-Modulated Optical Refractometry (TMOR) denotes an optical measurement approach in which refractive-index changes are interrogated under controlled temperature variation, typically to extract thermo-optical response functions, compensate temperature-induced errors in refractometry, or identify thermally driven transitions. In the narrow sense used for polymers, TMOR measures the refractive-index response $n(t)$ to sinusoidal temperature fields and derives the dynamic complex thermal volume expansion coefficient $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$ from the amplitude and phase of the optical response (Klingler et al., 2023). Closely related refractometric work uses resonant wavelength shifts, optical-frequency-comb observables, Fabry–Perot beat-frequency shifts, optical path differences, or coherent Rayleigh phase evolution to decode either thermo-optic coefficients, refractive index, gas density, or temperature itself under fluctuating or deliberately modulated thermal conditions (Prasad et al., 2017, Oe et al., 2019, Zelan et al., 2017, Silander et al., 2017).

1. Thermo-optic observables and response functions

The central quantity in TMOR is the thermo-optic response of the measured medium. In the polymer formulation, the refractive index is recorded while a small, sinusoidal temperature modulation is superimposed on a linear temperature ramp or held at a constant mean temperature. The amplitude of the refractive-index modulation, $\Delta n(T)$, and the phase lag $\varphi(T,f)$ between temperature and optical response are then converted into the in-phase and out-of-phase parts of the dynamic thermal volume expansion coefficient through

$$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$

and

$$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$

Here, $\beta'$ is the in-phase component, analogous to the static coefficient of thermal volume expansion, whereas $\beta''$ is the out-of-phase component, analogous to mechanical losses due to molecular relaxations near $T_g$ (Klingler et al., 2023).

In integrated photonic implementations, the same thermo-optic logic is expressed through resonance shifts. For a silicon microring resonator, the temperature derivative of the effective index is written as a weighted sum of the thermo-optic coefficients of the silicon core, buried oxide, and fluid cladding,

$$\frac{\partial n_{eff}}{\partial T} = \Gamma_{core}\frac{\partial n_{core}}{\partial T} + \Gamma_{box}\frac{\partial n_{box}}{\partial T} + \Gamma_{fl}\frac{\partial n_{fl}}{\partial T},$$

while the experimentally measured resonant wavelength shift satisfies

$\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$0

Combining these expressions yields the liquid thermo-optic coefficient from the measured $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$1, known material coefficients, and simulated mode confinement factors (Prasad et al., 2017).

A broad methodological distinction emerges from these formulations. In some systems, temperature is the imposed modulation and refractive index is the measured response; in others, refractive index and temperature are jointly encoded in multiple optical observables and must be decoded simultaneously. This suggests that TMOR is best understood not as a single instrument class, but as a family of thermo-optic inference schemes built around temperature-sensitive optical observables.

2. Polymer glass transition determination

A specific and stringent use of TMOR is the experimental determination of the glass transition temperature $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$2 of polymers in a very narrow temperature range. The motivating problem is that, since the landmark studies of Kovacs and co-workers on the glass transition of polymers, thermally induced volume changes have been recognized as central to the nature of the glass transition; yet, due to the kinetic background of the canonical glass transition, it does not seem possible to derive a well-defined $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$3 from susceptibilities such as the thermal volume expansion coefficient $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$4, which is strongly coupled to the glass transition process. In practice, $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$5 is therefore often defined via the inflection point of the step-like $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$6 curve (Klingler et al., 2023).

TMOR introduces a different criterion based on a thermo-optical feature preceding the glass transition in the high-temperature phase. In the viscoelastic phase above $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$7, an additional loss beyond the thermal expansion loss was observed in $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$8. This additional loss grows with temperature and modulation frequency, can be linearly extrapolated from high temperatures down to the glass transition, and sharply vanishes at or near $\beta^*(f,T)=\beta'(f,T)+i\beta''(f,T)$9, indicating the cessation of temperature-modulation-induced molecular motion in the glassy state (Klingler et al., 2023).

The reported protocol is explicit. One measures $\Delta n(T)$0 as a function of temperature at low modulation frequencies of $\Delta n(T)$1–$\Delta n(T)$2, identifies the approximately linear temperature dependence of the additional loss in the viscoelastic state, extrapolates the linear segments down to zero loss, and defines

$\Delta n(T)$3

for all fitted frequencies. In the reported epoxy-resin model system, this yielded

$\Delta n(T)$4

which lies within the much broader range of approximately $\Delta n(T)$5 usually associated with step or kink anomalies in classical methods (Klingler et al., 2023).

This use of TMOR also addresses a common misconception: the glass transition temperature is not presented as a uniquely defined point that can simply be read off a susceptibility. The TMOR criterion is instead operational and dynamic. It is based on the vanishing of modulation-induced loss and therefore depends on the timescale set by the modulation frequency. The method’s sensitivity to sample–prism coupling and its present instrumental limits of $\Delta n(T)$6–$\Delta n(T)$7 and $\Delta n(T)$8 further indicate that the measured transition criterion is linked to both experimental boundary conditions and molecular dynamics rather than to a purely static thermodynamic singularity (Klingler et al., 2023).

3. On-chip liquid thermo-optic coefficient sensing

A second major branch of temperature-modulated refractometry uses integrated photonic resonators to determine the thermo-optic coefficient (TOC) of liquids. In the demonstrated silicon photonic implementation, the device consists of silicon microring resonators surrounded by platinum metal rings for temperature sensing and a PDMS liquid reservoir. As temperature changes, the refractive indices of the silicon core, liquid cladding, and buried oxide all change due to their respective TOCs, thereby shifting the resonant wavelength of the ring (Prasad et al., 2017).

A key experimental issue is accurate temperature readout at the actual sensing location. The device therefore uses a resistive metal ring surrounding each microring, with electrical resistance changing linearly with temperature at a slope of approximately $\Delta n(T)$9. The explicit purpose is to avoid errors from localized temperature variations that external sensors or even reference microrings may not track accurately because of spatial gradients or fabrication mismatch (Prasad et al., 2017).

The reported TOC measurements at $\varphi(T,f)$0 for the three standard fluids are as follows:

Liquid	Measured TOC $\varphi(T,f)$1	Comparison to literature
De-ionized Water	$\varphi(T,f)$2	$\varphi(T,f)$3, $\varphi(T,f)$4
Ethanol	$\varphi(T,f)$5	$\varphi(T,f)$6, $\varphi(T,f)$7
Isopropanol	$\varphi(T,f)$8	$\varphi(T,f)$9, $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$0

The TOC for silicon was also measured as $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$1, matching literature, and the uncertainty for water was reported as $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$2, corresponding to about $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$3 relative error. The dominant sources of measurement error were fluctuations in the measured wavelength-shift slope and variations in waveguide width (Prasad et al., 2017).

The significance of this architecture lies in the fact that the TOC of the liquid is measured directly on chip while local temperature is measured simultaneously. Potential applications stated for the demonstrated on-chip TOC sensor include improvements in accuracy of refractive-index measurements and multiparametric analysis of biochemical analytes. A plausible implication is that TOC ceases to be merely a nuisance parameter requiring compensation and becomes an additional sensing channel with chemical specificity.

4. Simultaneous refractive-index and temperature decoding

Temperature modulation also appears as a decoding problem when refractive index and temperature both perturb the same optical sensor. A clear example is the multimode-interference fiber-based optical frequency comb sensing cavity, or MMI-OFC sensing cavity, which enables simultaneous measurement of material-dependent refractive index and sample temperature by decoding the comb spacing frequency shift and the wavelength shift of the OFC (Oe et al., 2019).

The governing structure is a linearized sensitivity model,

$$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$4

with extraction of $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$5 and $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$6 through inversion of the empirically calibrated coefficient matrix. The underlying observables are the OFC comb spacing $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$7 and the MMI filter central wavelength $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$8, both of which depend on sample composition and temperature, but with different sensitivities (Oe et al., 2019).

The reported slope coefficients were $$\beta'(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \cos(\varphi(T, f))$$9 and $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$0 for $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$1, and $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$2 and $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$3 for $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$4. The system realized simultaneous and continuous measurement of RI-related concentration of a liquid sample and its temperature with precisions of $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$5 RIU and $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$6 (Oe et al., 2019).

This result addresses a recurrent limitation of conventional fiber-optic refractive-index sensors, such as MMI or SPR types, for which both refractive index and temperature affect the spectral response indistinguishably. In the MMI-OFC formulation, the ambiguity is removed by combining an optical-domain observable and an RF-domain observable. The RF readout of comb spacing is specifically emphasized as advantageous because it leverages mature electronics for frequency measurement and is not constrained by spectrometer resolution in the same way as purely wavelength-based schemes (Oe et al., 2019).

A closely related enabling technology is dual-mode optical thermometry in a silicon nitride waveguide resonator. In that work, the resonance difference between TE and TM modes served as a precision temperature channel with temperature responsivity $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$7 and sensitivity $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$8, and feedforward correction reduced laser drift from $$\beta''(f, T) = -6 \frac{N_{mean}(T)}{[N_{mean}(T)-1][N_{mean}(T)+2]} \cdot \frac{\Delta n(T)}{\Delta T} \sin(\varphi(T, f)).$$9 to $\beta'$0 while achieving a fractional frequency instability of $\beta'$1 over $\beta'$2 (Zhao et al., 2021). This suggests that integrated refractometric systems can use an internal dual-mode temperature observable to stabilize or decouple the thermo-optic background without requiring a separate external thermometer.

5. Drift-free gas refractometry and temperature dependence

In gas metrology, temperature-modulated refractometry is closely tied to the suppression of drift and the explicit analysis of residual temperature dependence. Dual Fabry–Perot-cavity-based optical refractometry measures refractivity and gas density through the change in frequency of laser light locked to a cavity mode as the refractive index inside the cavity changes, typically during evacuation. The dual-cavity configuration cancels common-mode drifts, and the fast-switching variants perform evacuation or fill–evacuation cycles rapidly enough that slow thermal or mechanical drifts are negligible on the measurement timescale (Zelan et al., 2017, Silander et al., 2017).

The reported performance is unusually strong in precision while remaining limited in absolute accuracy by correction parameters. Both refractivity and gas density can be assessed by fast-switching DFCB-OR with a precision in the $\beta'$3 range under STP conditions, whereas the relative accuracy for an uncharacterized system is typically in the $\beta'$4 to $\beta'$5 range because of uncertainty in the cavity deformation parameter, polarizability, and higher-order virial coefficients. Internal accuracy relative to an internal standard can, however, be several orders of magnitude better than the absolute accuracy (Zelan et al., 2017).

The most distinctive property of these systems is their weak temperature dependence. The temperature dependence of FS-DFCB-OR is described as exceptionally small, typically in the $\beta'$6 to $\beta'$7 range, and primarily caused by thermal expansion of the Fabry–Perot-cavity spacer material. Rapid switching times of approximately $\beta'$8 or less than $\beta'$9, use of ULE glass or Zerodur spacers, and the dual-cavity design are the main strategies identified for achieving drift-free operation (Zelan et al., 2017, Silander et al., 2017).

Methodologically, the gas case differs from polymer TMOR because temperature is not usually imposed as a sinusoidal excitation. Instead, the key problem is that cavity drifts and temperature fluctuations can corrupt refractometric readout over longer times. The fast-switching methodology moves measurements into a regime in which drifts can be disregarded, including double-measurement sequences for closed compartments and triple-evacuation protocols for leak-rate assessment (Silander et al., 2017). This establishes an important point for the broader concept of TMOR: temperature sensitivity can be exploited, decoded, or suppressed, depending on whether thermal response is the target signal or a dominant systematic.

6. Spatially resolved implementations and adjacent optical thermometry

Several recent optical methods extend the logic of temperature-modulated refractometry from scalar readouts to spatially and temporally resolved temperature fields. In phase-change devices based on GST, the strong thermo-optic effect was used to convert local reflectance changes into local temperature. The refractive indices obey

$\beta''$0

and transfer-matrix modeling showed a linear relation between reflection change and temperature change at the $\beta''$1 probe wavelength. The reported probe spot full width at half maximum was $\beta''$2, temporal resolution was sub-$\beta''$3, and the measured heating and cooling time constants were $\beta''$4 and $\beta''$5 for a Pt microheater, and $\beta''$6 and $\beta''$7 for a doped-silicon microheater. Experimental thermal maps and temporal traces showed excellent agreement with COMSOL simulations (Nobile et al., 2022).

In transparent solids, phase-sensitive optical coherence tomography provided a contactless measurement of temperature-dependent refractive-index change in a $\beta''$8-thick soda-lime glass slide immersed in a thermal bath. The measured optical-path-difference change obeyed

$\beta''$9

which combines thermal expansion and the thermo-optic effect. The OPD variation showed a strong linear correlation with temperature in the range $T_g$0–$T_g$1, with an experimentally determined sensitivity of $T_g$2, compared with a theoretical estimate of approximately $T_g$3. Simulations based on finite volume methods agreed with experiment within $T_g$4 error, and repeatability tests demonstrated sub-$T_g$5 stability with standard deviation $T_g$6 (Folgueiras et al., 18 Mar 2026).

Distributed fiber sensing extends the same principle to long baselines. In coherent $T_g$7-OTDR, the measured phase at a sensing location is modeled as

$T_g$8

so the phase evolution encodes the cumulative temperature change between the interrogator and the sensing location, while the amplitude exhibits only local sensitivity. Based on this, event-detection and temperature-profile-reconstruction algorithms were proposed, and experimentally recovered temperature evolution matched independent thermal-probe measurements within $T_g$9 uncertainty, with temperature-rate measurement uncertainties down to $$\frac{\partial n_{eff}}{\partial T} = \Gamma_{core}\frac{\partial n_{core}}{\partial T} + \Gamma_{box}\frac{\partial n_{box}}{\partial T} + \Gamma_{fl}\frac{\partial n_{fl}}{\partial T},$$0 (Ermakov et al., 30 Jun 2026).

A useful boundary case is optomechanical Raman-ratio thermometry, which is optically based but not refractometric. There the Stokes/anti-Stokes ratio

$$\frac{\partial n_{eff}}{\partial T} = \Gamma_{core}\frac{\partial n_{core}}{\partial T} + \Gamma_{box}\frac{\partial n_{box}}{\partial T} + \Gamma_{fl}\frac{\partial n_{fl}}{\partial T},$$1

provides a self-calibrating, absolute temperature measurement without requiring knowledge of mass, optical collection efficiency, or material thermo-optic coefficients. Agreement with physical thermometer readings was reported within $$\frac{\partial n_{eff}}{\partial T} = \Gamma_{core}\frac{\partial n_{core}}{\partial T} + \Gamma_{box}\frac{\partial n_{box}}{\partial T} + \Gamma_{fl}\frac{\partial n_{fl}}{\partial T},$$2 from cryogenic temperatures to near room temperature (Purdy et al., 2014). This contrast is instructive: refractometric methods usually require calibration, knowledge of material constants, or multi-observable decoding, whereas Raman-ratio thermometry derives temperature directly from quantum sideband asymmetry.

The broader implication is that TMOR and related thermo-optic methods now occupy a continuum ranging from narrowband dynamic susceptibility measurements in polymers, through temperature-compensated resonant refractometry in liquids and gases, to full spatial and temporal thermal mapping in solids, phase-change stacks, and fibers. Across these domains, the decisive technical questions are the same: which optical observable is temperature-sensitive, whether temperature is the desired signal or a confounder, and what auxiliary model or second observable makes the inverse problem well posed.