How to Measure Refractive Index: Prism, Refractometer, Prism Coupler, Ellipsometry and Fringes

Scope

This article describes how the refractive index of a material is measured, and how to choose a method for the sample in hand: a bulk piece of glass or crystal, a liquid, a thin film on a substrate, or a waveguide. It covers the prism minimum-deviation method, critical-angle refractometers, the prism coupler, spectroscopic ellipsometry and interferometric methods, and it separates the phase index that most of them measure from the group index that fringe spacings give. Tabulated indices for common materials are on the site's materials pages, such as fused silica.

Choosing a method

Whatever the method, the result is incomplete without the wavelength and temperature at which it was taken. Indices of common glasses change by amounts of order 10⁻⁶ per kelvin, and semiconductors by far more (see thermo-optic coefficients), which matters at the resolution of the better methods.

Minimum deviation with a prism

A prism of apex angle $A$ deviates a ray by an angle that passes through a minimum, $D_\text{min}$, when the ray crosses the prism symmetrically. At that point

$$n \;=\; \frac{\sin\!\big((A + D_\text{min})/2\big)}{\sin(A/2)}.$$

Procedure

1. Measure the apex angle $A$ on the goniometer, by reflecting the collimated beam off the two faces that meet at the apex and reading the angle between the two reflections, which is $2A$.
2. Illuminate with a collimated beam at the wavelength of interest: a spectral line lamp, or a laser.
3. Find minimum deviation. Rotate the prism table while following the refracted beam with the telescope; the beam moves toward smaller deviation, stops, and turns back. Set the telescope on the turning point.
4. Read the deviation from the undeviated beam, or, for better accuracy, measure the minimum deviation on both sides of the undeviated direction and take half the angle between them.
5. Compute $n$ and repeat at each wavelength needed; fit the results with a Sellmeier formula to interpolate.

Worked example. A 60° prism of N-BK7, $n$ = 1.5168 at 587.6 nm, has a minimum deviation of 38.65°. Near this point $dn/dD_\text{min}$ = 0.652 per radian, so an angular error of 1 arcsecond gives an index error of 3.2 × 10⁻⁶. A goniometer reading to an arcsecond therefore resolves the fifth or sixth decimal, and the practical limits become the flatness of the prism faces, the accuracy of $A$, and the temperature.

Critical-angle refractometers

An Abbe refractometer places a thin layer of liquid, or a polished solid with a contact liquid, against a measuring prism of high index. Light grazing the interface enters the prism at the critical angle, which forms a sharp boundary between light and dark in the telescope; its position gives the sample's index directly on a calibrated scale. The method needs only a drop of liquid and reads to about 10⁻⁴, usually at the sodium D line (589 nm) with the dispersion compensated by built-in prisms. It measures only samples whose index is below that of the measuring prism.

Prism coupler

A prism coupler measures a thin film on a substrate. A high-index prism is pressed against the film so that an air gap of a fraction of a wavelength remains, and a laser beam enters through the prism and strikes the base. At a discrete set of incidence angles the beam's in-plane wavevector matches a guided mode of the film, light couples into the film, and the reflected intensity dips. Each dip gives a mode's effective index. With two or more modes, the film's index and thickness are both found by fitting the mode equation; with only one, one of the two must be known. The method works for films thick enough to guide, from a few hundred nanometres upward depending on the index contrast, and measures TE and TM modes separately, which gives birefringence directly.

Spectroscopic ellipsometry

An ellipsometer measures the change in polarization state on reflection from the sample, expressed as two angles, Ψ and Δ, at several angles of incidence and across a range of wavelengths. These are not the index directly: a layer model (substrate, film thickness, and a dispersion model for each material, such as Cauchy for a transparent film or an oscillator model for an absorbing one) is fitted to the data, and the index and thickness come from the fit. Ellipsometry reaches films a few nanometres thick and gives the absorption coefficient alongside the index, but the result is only as good as the model; correlated parameters, such as the index and thickness of a very thin film, cannot be separated reliably.

Interference fringes and the group index

A plate of thickness $t$ with parallel faces transmits a spectrum of Fabry-Perot fringes whose spacing is

$$\Delta\lambda \;=\; \frac{\lambda^2}{2\,n_g\,t},$$

where $n_g$ is the group index. For a 500 μm fused silica plate at 1550 nm ($n_g$ ≈ 1.462) the fringes are 1.64 nm apart, about 205 GHz. Measuring the fringe spacing on an optical spectrum analyzer or with a swept laser gives $n_g$ if $t$ is known, or $t$ if $n_g$ is known. The same relation with a round-trip length $L$ in place of $2t$ gives the group index of a waveguide from the free spectral range of a ring resonator or of the Fabry-Perot fringes of a cleaved waveguide.

The group index is not the phase index: in fused silica at 1550 nm the phase index is 1.444 and the group index 1.462. Fringe methods give the group index; minimum deviation, refractometers, prism couplers and ellipsometry give the phase index. Converting between them needs the dispersion, $n_g = n - \lambda\,dn/d\lambda$.

Common errors

Wavelength and temperature not stated. The sixth decimal place of a glass's index changes with each kelvin, and in the visible the fifth changes with a fraction of a nanometre of wavelength (N-BK7 changes by 4 × 10⁻⁵ per nanometre near 590 nm).

Wedge in the sample. Minimum deviation assumes flat faces and an accurately known apex angle; a plate used for fringe measurements must be parallel, or the fringes wash out.

Phase and group index confused. A fringe measurement compared with a tabulated phase index disagrees by the dispersion term, a percent or more in most materials.

Ellipsometry model too flexible. A fit with more free parameters than the data can support gives a good match with physically meaningless values; check the result against an independent measurement of thickness or index.

Anisotropic samples. Birefringent crystals and stressed films have different indices for different polarizations and directions; state which was measured.

References: M. Born and E. Wolf, Principles of Optics (7th ed., Cambridge University Press, 1999); H. G. Tompkins and E. A. Irene (eds.), Handbook of Ellipsometry (William Andrew, 2005); R. Ulrich and R. Torge, "Measurement of thin film parameters with a prism coupler," Applied Optics 12, 2901 (1973).