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White-Light Spectral Interferometry for Thin-Film Thickness and Group-Index Dispersion Metrology

This paper presents a unified theoretical and simulation framework for white-light spectral interferometry that enables high-precision, scan-free measurement of thin-film thickness (with $\pm$ 0.2 nm accuracy) and group-refractive-index dispersion using a single fiber-optic spectrometer setup.

Original authors: Shouvik Sadhukhan, Shirsendu Sarkar, Debabrata Bhadra

Published 2026-10-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shouvik Sadhukhan, Shirsendu Sarkar, Debabrata Bhadra

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light, when it travels, carries a hidden rhythm. If you take a beam of white light—which contains all the colors of the rainbow—and split it into two separate paths before bringing them back together, the waves from each path will overlap. Where the peaks of the waves line up, the light brightens; where a peak meets a trough, it dims. This creates a pattern of bright and dark bands, known as interference fringes. Scientists have long used this phenomenon to measure things with extreme precision, because the spacing of these bands changes in a predictable way if the distance the light travels changes even by a tiny fraction of a hair's width. However, when the light passes through glass or bounces off a thin coating, the glass slows the light down differently depending on its color, and the coating adds its own subtle twists to the wave. Untangling these effects to find the exact thickness of a coating or the precise way glass slows down different colors has traditionally been a difficult, multi-step process requiring complex equipment.

Two researchers from India, along with a colleague, have refined a method that turns this complexity into a single, streamlined measurement. By using a standard laboratory setup involving a light source, a beam splitter, and a spectrometer, they demonstrated that a single snapshot of the interference pattern can reveal two distinct secrets at once: the thickness of a microscopic film and the specific way a piece of glass disperses light. Their work, published in a research article, shows that by analyzing the unique "fingerprint" left on the light spectrum, they can determine the thickness of a silica film on a silicon surface with an error margin of less than one-fifth of a nanometer. Simultaneously, they can map out how the refractive index of a glass beam splitter changes across the visible spectrum, a property known as dispersion, without needing to move any parts of the machine or scan through different settings.

The team focused on two specific tasks that usually require different instruments. First, they looked at a thin film of silicon dioxide, a material commonly used in electronics and optics, sitting on top of a silicon wafer. These films are incredibly thin, ranging from 100 to 450 nanometers. When light reflects off the top of the film and the bottom where it meets the silicon, the two reflected waves interfere with each other. The researchers showed that the resulting pattern of light and dark bands contains enough information to calculate the film's thickness with high accuracy. They tested this by creating computer simulations of the light patterns for films of known thicknesses, adding realistic noise to mimic real-world imperfections, and then using a mathematical fitting process to see if the software could recover the original thickness. The results were remarkably consistent; for every film thickness tested, the calculated value was within 0.2 nanometers of the true value, and the match between the simulated data and the model was nearly perfect.

The second part of their study addressed the glass beam splitter itself, a component that divides the light beam in the interferometer. Glass does not slow down all colors of light equally; blue light travels slightly slower than red light, a property called dispersion. The researchers used an uncompensated setup, meaning the glass was present in only one arm of the interferometer, creating a slight imbalance. As they moved a mirror in the other arm to change the distance the light traveled, they observed a specific point where the interference pattern was most distinct. This point, which they called the equalization wavelength, shifted depending on the distance the mirror moved. By measuring this shift across 35 different positions, they were able to reconstruct the group refractive index of the glass across a wide range of wavelengths, from 480 nanometers to 860 nanometers. The data they collected followed the known theoretical behavior of the glass perfectly, confirming that the method could accurately track how the glass's optical properties change with color.

What makes this approach particularly powerful is that it does not require scanning the light source or moving parts to gather the data. A single spectrum, captured by a standard fiber-optic spectrometer, contains the full story. The researchers verified that the mathematical model they used to interpret the data was robust, showing that the solution for the film thickness was unique and not a result of random chance or a mathematical artifact. They also demonstrated that by combining their measurements of the group refractive index with a known mathematical relationship, they could reconstruct the phase refractive index of the glass with extreme precision, down to a difference of seven parts in a billion in their simulations. This level of detail suggests that the method is sensitive enough to detect even the smallest variations in material properties.

The study confirms that a single, relatively simple instrument can serve as a dual-purpose metrology tool. It can measure the thickness of thin films with sub-nanometer accuracy and map the dispersion of optical materials with high fidelity. The researchers noted that the precision of their results is limited primarily by the quality of the spectrometer and the stability of the light source, rather than by the method itself. In their simulations, the errors were so small that they were dominated by the random noise added to the data, not by any flaw in the underlying theory. This suggests that with high-quality equipment, the technique could be applied in industrial settings to inspect coatings on lenses, semiconductor wafers, or other optical components.

By unifying these two measurement tasks under one framework, the work offers a clearer path for characterizing optical materials. The ability to extract both thickness and dispersion information from a single, static measurement simplifies the workflow for scientists and engineers who need to ensure the quality of optical components. The researchers concluded that this method is well-suited for both laboratory research and practical industrial applications, providing a reliable way to measure the invisible dimensions of the materials that make up our modern optical world. Their findings stand as a demonstration that with the right mathematical tools, a single snapshot of light can reveal a wealth of physical detail that was previously hidden.

Technical Summary: White-Light Spectral Interferometry for Thin-Film Thickness and Group-Index Dispersion Metrology

Problem Statement
The paper addresses the need for unified, scan-free metrology techniques capable of simultaneously determining the thickness of thin films and the group-refractive-index dispersion of optical elements. Traditional methods often require separate instruments or scanning mechanisms. The authors propose utilizing the channeled spectrum generated by a slightly dispersive Michelson interferometer, recorded by a fiber-optic spectrometer, to extract both parameters from a single static measurement setup. This approach builds upon the framework established by Hlubina and co-workers, extending it to include rigorous modeling of thin-film reflection phases and group-index dispersion reconstruction.

Methodology
The study employs a unified theoretical model and experimental simulation strategy based on a dispersive Michelson interferometer using a broadband white-light source and a fiber-optic spectrometer.

Theoretical Framework:
- Channeled Spectrum: The intensity spectrum $I(\lambda)$ is modeled as a cosine modulation of a reference spectrum, modulated by a visibility function $V(\lambda)$ and an exponential term accounting for finite spectrometer resolution.
- Thin-Film Metrology: For a thin film (SiO $_2$ ) on a substrate (Si), the model incorporates the complex reflection coefficient derived from Fabry–Perot summation. This accounts for the power reflectance $R(\lambda)$ and the nonlinear phase shift $\delta_r(\lambda)$ arising from multiple internal reflections. The phase optical path difference (OPD) includes a linear geometric term and a nonlinear phase term.
- Dispersion Metrology: For an uncompensated interferometer arm containing a beam splitter (BK7 glass), the group OPD is defined. The "equalization wavelength" ( $\lambda_{eq}$ ) is identified as the wavelength where the group OPD vanishes. This condition links the group refractive index $N_g(\lambda_{eq})$ directly to the air path difference $\Delta L$ and the beam splitter thickness $t$ via the relation $N_g(\lambda_{eq}) = \Delta L/t + 1$ .
- Phase Reconstruction: The paper outlines a method to reconstruct the phase refractive index $n(\lambda)$ from the measured group index $N_g(\lambda)$ by solving the first-order linear ordinary differential equation $N_g = n - \lambda (dn/d\lambda)$ via numerical integration.
Experimental and Simulation Setup:
- Thin-Film Study: Simulations were performed for SiO $_2$ /Si films with thicknesses ranging from 100 nm to 450 nm. Synthetic spectra were generated with added Gaussian noise (1% of peak intensity).
- Dispersion Study: Simulations involved 35 interferograms with varying air path differences ( $\Delta L$ ) from 10.990 mm to 11.709 mm to trace the dispersion of a BK7 beam splitter.
- Data Analysis: The Levenberg–Marquardt (LM) algorithm was used to fit the theoretical model to the noisy spectra. The fitting parameters included film thickness, instrumental visibility, and air path difference. The quality of fit was assessed using the Pearson correlation coefficient ( $R$ ) and $\chi^2$ landscapes.

Key Contributions and Results

Thin-Film Thickness Retrieval:
- The LM fitting algorithm successfully retrieved film thicknesses for 8 samples (100–450 nm) with high accuracy.
- Precision: All retrieved thicknesses deviated from the true values by less than $\pm 0.2$ nm. The largest error was $-0.184$ nm (at 250 nm), and the smallest was $-0.009$ nm (at 100 nm).
- Fit Quality: The correlation coefficients between measured and fitted spectra were consistently high ( $R > 0.998$ ).
- Uniqueness: $\chi^2$ scans revealed a single, sharp minimum at the true thickness, confirming the robustness of the solution against local minima. The errors were found to be noise-dominated rather than model-limited.
Group-Index Dispersion of BK7:
- By measuring the shift in the envelope peak ( $\lambda_{eq}$ ) across 35 different mirror displacements, the group refractive index $N_g$ was determined over the wavelength range of 480–860 nm.
- Dispersion Curve: The results showed $N_g$ decreasing monotonically from approximately 1.558 at 480 nm to 1.525 at 860 nm, consistent with the Sellmeier model for BK7 glass.
- Specific Operating Point: At $\Delta L = 11.30$ mm, the equalization wavelength was found to be 598.5 nm, yielding a group index of 1.5401.
- Reconstruction: Using the noiseless simulated $N_g$ data, the phase refractive index $n(\lambda)$ was reconstructed via numerical integration. The difference between the reconstructed index and the Sellmeier model was less than $7 \times 10^{-9}$ , demonstrating mathematical consistency, though the authors note that real-world precision is limited by noise (typically $\sim 10^{-4}$ ).

Significance and Claims
The paper claims that this work provides a "scan-free, single-instrument approach" to two distinct metrology tasks: thin-film thickness measurement and group-index dispersion characterization.

Unified Framework: The study demonstrates that both applications can be addressed using the same channeled-spectrum model and experimental hardware (a broadband source, a Michelson interferometer, and a fiber-optic spectrometer).
Practicality: The method is described as simple and suitable for both laboratory and industrial use, requiring no mechanical scanning of the interferometer for thickness retrieval.
Modest Scope: The authors explicitly state that the high-precision phase reconstruction ( $7 \times 10^{-9}$ ) was demonstrated using noiseless data with boundary values taken from the model to show mathematical consistency. They acknowledge that in real experimental conditions, precision is limited by spectrometer resolution, source spectrum stability, and noise, with group-index precision typically attainable at the order of $10^{-4}$ .
Future Directions: The paper suggests future work could extend this framework to multilayer structures, angle-resolved configurations, and absorbing or birefringent materials, as well as linking the findings to spectral-anomaly frameworks relevant to gyroscopic and phase-singularity applications.

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