Optical physicists have demonstrated an advanced common-path spectral domain low-coherence interferometry technique capable of determining the effective thermal expansion coefficient of thin transparent materials with sub-nanometer scale optical path length sensitivity. In precision optical engineering, laser development, and large-scale physics infrastructure such as ground-based gravitational wave detectors, temperature variations alter both the physical thickness and the refractive index of optical substrates. By unifying the mechanical thermal expansion coefficient and the normalized thermo-optic coefficient into a single measurable parameter known as the effective thermal expansion coefficient, the research team established an experimental framework that captures the aggregate optical path response of transparent media across thermal ramps. Demonstrating this capability at an operating wavelength of 840 nanometers over temperatures ranging from 35 to 70 degrees Celsius, the investigators verified the methodology using uncoated thin samples of fused silica and sapphire measuring merely hundreds of micrometers in thickness.
In ground-based gravitational wave detectors, thermal noise arising within mirror coatings and substrates constitutes the primary noise source within the most sensitive detection frequency band. The spectral density of this disruptive thermal noise depends directly on the mechanical thermal expansion coefficient and the thermo-optic coefficient of the substrate materials. While conventional interferometric approaches such as Michelson or Mach-Zehnder configurations have traditionally tracked fringe intensity variations to deduce path alterations, such intensity-based measurements remain highly susceptible to fluctuations in source power.
The theoretical framework underpinning this measurement relates round-trip optical path difference directly to physical sample thickness and refractive index. When broadband light reflects from the front and back surfaces of a transparent plate, the round-trip optical path difference equals twice the product of refractive index and physical thickness. As ambient temperature rises from an initial baseline, physical thickness expands through the mechanical thermal expansion coefficient while the refractive index shifts according to the normalized thermo-optic coefficient. To first order, the cumulative change in optical path difference is proportional to the product of twice the initial optical thickness, the temperature increment, and the sum of the thermal expansion and thermo-optic coefficients. Differentiating this cumulative path change with respect to temperature and normalizing by twice the initial optical thickness yields the effective thermal expansion coefficient. When the path difference displays subtle nonlinearities across broad temperature intervals, the variation can be modelled through polynomial expansions, allowing the effective parameter to be evaluated from the resulting polynomial coefficients.
To establish the phase retrieval methodology, the researchers performed comprehensive numerical simulations modelling interference spectra generated by a Gaussian light source with a central wavelength of 840 nanometers and a full width at half maximum of 35 nanometers. Simulating optical path difference values spanning 500 to 501.05 micrometers in incremental steps of 50 nanometers, the team converted the simulated wavelength spectra into uniformly spaced wavenumber domain data across 2048 points before applying a fast Fourier transform. The resulting Fourier spectrum in the depth domain produced three characteristic peaks: a central component at zero delay and two symmetric peaks at positive and negative optical depths. By isolating the peak index of the complex depth-domain array and calculating its argument via four-quadrant arctangent processing, the phase was retrieved directly from a single Fourier transformation. After phase unwrapping to eliminate cycle ambiguities, the extracted path variations matched the input values with a linear slope of 1.008 plus or minus 0.002 and a coefficient of determination of 0.9998, demonstrating sub-nanometer path retrieval without needing iterative inverse transformations.
The experimental apparatus was constructed around a common-path optical architecture using a superluminescent diode emitting at 840 nanometers with a 35-nanometer spectral bandwidth as the low-coherence source. Fiber-coupled output was split evenly using a 50:50 fused fiber coupler, with one output port terminated while the active port directed light through a collimating lens onto the uncoated test specimen. Reflections originating from the front and rear polished surfaces of the test sample were captured by the identical fiber channel, routing the recombined beams into a custom-built high-resolution spectrometer. By utilizing the sample surfaces themselves as the interferometric arms within a shared physical path, the system inherently cancelled common environmental disturbances, eliminated reference arm drift, and suppressed external mechanical vibrations. Inside the spectrometer, the beam was collimated by a 50-millimeter achromatic doublet lens, dispersed by a 1200 lines-per-millimeter transmission diffraction grating, and focused across a 16-bit line-scan camera containing 2048 sensing pixels.
Precise wavelength calibration of the detection spectrometer was achieved using a calibrated mercury-argon spectral lamp to map known atomic emission lines to specific camera pixels, achieving a maximum relative deviation of approximately 1.42 times ten to the negative fourth power. To verify phase tracking across optical delays, the collimated illumination was directed across the step boundary of a 475-micrometer-thick silicon wafer positioned upon a planar reference mirror, producing robust interference fringes that were evaluated using a Fourier filtering technique to confirm linear phase-to-wavenumber dispersion. The calibrated spectrometer demonstrated an effective spectral coverage of 120.89 nanometers and an optical resolution of approximately 0.06 nanometers, providing an operational optical path difference measurement range of nearly 5.98 millimeters. Because the corresponding depth resolution mapped to approximately 5.9 micrometers per pixel, structural peak shifting alone could not resolve minor thermal expansions in sub-millimeter plates, reinforcing the absolute necessity of depth-peak phase-sensitive extraction to resolve path adjustments at sub-nanometer dimensions.
To evaluate baseline measurement stability, the optical system recorded interference spectra every two seconds over a continuous duration of approximately 4000 seconds at ambient room temperature, acquiring 100 interferogram scans at every measurement interval with a line rate of 3.8 kilohertz and an integration time of 30 microseconds. The observed standard deviation of the optical path difference across consecutive 100-scan bursts remained steady at approximately 0.18 nanometers throughout the prolonged trial. A modest net drift of roughly 3.0 nanometers observed in the mean path difference correlated directly with ambient temperature fluctuations recorded in the laboratory environment. Thermal ramping experiments were conducted by housing each test specimen inside a customized aluminum heating stage regulated by a resistive heating element, a precision temperature controller, and a negative temperature coefficient thermistor positioned adjacent to the sample, capturing full spectral bursts across continuous heating intervals from 35 to 75 degrees Celsius.
Experimental trials were conducted on uncoated thin plates of fused silica and c-axis oriented single-crystal sapphire possessing identical 10 by 10 millimeter surface dimensions and nominal physical thicknesses of 526 and 438 micrometers, respectively. Room-temperature refractive indices were determined from Fourier depth peaks located at pixel 261, yielding 1.463 for fused silica and 1.757 for sapphire. Ten distinct thermal datasets acquired between 35 and 70 degrees Celsius were analyzed using both second-order and linear fits against temperature. Quadratic fitting yielded second-order coefficients of 0.004 for fused silica and 0.001 for sapphire, with coefficients of determination exceeding 0.9999. Because the quadratic contributions remained negligible across this temperature span, linear regression was implemented to evaluate the optical path slope with temperature. Across ten repeated experimental runs, the resulting mean effective thermal expansion coefficient was determined to be 6.8 plus or minus 0.1 times ten to the negative sixth per degree Celsius for fused silica, and 13.7 plus or minus 0.6 times ten to the negative sixth per degree Celsius for sapphire.
The experimental values demonstrated strong consistency when compared against published thermo-optic and mechanical thermal expansion data compiled from historical literature. For fused silica, prior studies by Jewell and collaborators on vitreous silica at 632 nanometers across temperatures between 25 and 150 degrees Celsius suggested effective expansion values between 6.87 and 7.14 times ten to the negative sixth per degree Celsius, while measurements by Zilio and colleagues using Heraeus fused silica at 532 nanometers between 24 and 40 degrees Celsius yielded values between 6.83 and 6.84 times ten to the negative sixth per degree Celsius. Standard material datasheets for synthetic Heraeus silica and Corning 7980 high-purity silica indicate corresponding effective coefficients ranging from 7.08 to 7.42 times ten to the negative sixth per degree Celsius near 840 to 852 nanometers.
A rigorous uncertainty analysis based on first-order differential error propagation revealed the primary factors influencing the accuracy of the effective thermal expansion measurements. The mathematical model incorporated spectral uncertainty of 0.0595 nanometers, maximum phase variations of 0.057 radians for fused silica and 0.253 radians for sapphire, temperature measurement accuracy of 1.0 degree Celsius across a 35-degree interval, screw-gauge thickness uncertainties of 1.2 micrometers for silica and 0.4 micrometers for sapphire, and derived refractive index uncertainties below 0.007. The analysis demonstrated that relative uncertainties in phase variation and temperature detection were the predominant contributors to the overall measurement error budget. Furthermore, because spectral interferograms were logged during a continuous thermal ramp featuring heating rates of approximately 0.4 degrees Celsius every two seconds, minor dynamic thermal lags and spatial temperature gradients between the heating block, sample body, and thermistor sensor introduced residual non-equilibrium offsets. Future refinements will focus on mitigating dynamic gradients and establishing simultaneous, decoupled extractions of mechanical expansion and thermo-optic coefficients in advanced optical materials.
Subject of Research: Interferometric measurement of effective thermal expansion coefficients in optical materials
Article Title: Measurement of effective thermal expansion coefficient using spectral domain low coherence interferometry
Article References: Malakar, K. K., Chakraborty, S., Pant, B. C., Thondapu, S. R., Verma, Y., & S., S. R. (2026). Measurement of effective thermal expansion coefficient using spectral domain low coherence interferometry. Applied Physics B, 132(7), Article 86. https://doi.org/10.1007/s00340-026-08687-0
Image Credits: AI Generated
DOI: 10.1007/s00340-026-08687-0
Keywords: low-coherence interferometry, effective thermal expansion, fused silica, sapphire, optical path difference, spectral domain, thermo-optic coefficient, precision metrology, Measurement, effective, thermal, expansion
News Source: Denise Maddox. (November 9, 2026). Common-Path Interferometry Accurately Measures Effective Thermal Expansion in Thin Optical Substrates. Scienmag.
