Fizeau interferometer

A Fizeau interferometer measures the shape of an optical surface by placing it facing a reference surface of known shape and observing the interference between the light reflected from each. A collimated beam passes through a transmission flat, whose last surface is the reference and reflects a few percent of the light; the rest crosses a small air gap, reflects from the test surface and returns. The two reflections interfere, and each fringe is a contour of constant gap. Because the gap changes by half a wavelength from one fringe to the next, the fringe map is a height map of the test surface relative to the reference, with a contour interval of λ/2, 316.4 nm for a helium-neon laser at 632.8 nm. Commercial phase-measuring instruments report surface error with sub-nanometer repeatability and an accuracy of a few nanometers rms, which is how the λ/10 and λ/20 flatness grades on optics catalogs are verified.

Geometry and fringes

The reference and test beams share almost the whole path through the instrument, separating only in the gap between the two surfaces. Errors in the collimator, beam splitter and imaging optics are therefore common to both beams and largely cancel; the instrument measures the difference between the reference and test surfaces. A deliberate small tilt between the surfaces produces straight fringes, and the tilt angle follows from the fringe count $N$ across an aperture $D$:

$$\theta = \frac{N\lambda}{2D}.$$

Ten fringes across a 100 mm aperture correspond to a tilt of 31.6 µrad. On a flat test surface the fringes are straight and evenly spaced; curvature makes them bow, and a departure of one quarter of the fringe spacing from a straight line means a height error of λ/8, 79.1 nm. Testing a spherical surface uses a transmission sphere, a lens whose last surface is a concentric reference sphere, placed so that its focus coincides with the center of curvature of the test surface. The transmission sphere must be fast enough to fill the test aperture: its f-number must not exceed the ratio $R/D$ of the test surface's radius to its diameter.

Phase-shifting measurement

Modern Fizeau instruments move the reference surface along the axis with a piezoelectric actuator and record several frames. Because the light crosses the gap twice, a displacement of λ/8, 79.1 nm, changes the phase by π/2. Four or more frames at known phase steps give the phase at every pixel, and the surface map follows after unwrapping. The result is reported as peak-to-valley and rms wavefront error, often with a fit to Zernike polynomials that separates power, astigmatism and higher-order terms. The surface error is half of the reflected wavefront error at normal incidence. Wavelength-shifting instruments tune the laser instead of moving the reference.

Accuracy and calibration

The measurement is relative to the reference, so the reference flat sets the accuracy floor. References are typically specified at λ/20 P-V, 31.6 nm, or better. Absolute flatness can be found without a better reference by the three-flat test, which measures three flats against each other in several orientations and solves for each surface. Air turbulence in the gap and in the beam path, vibration and temperature gradients in the test part limit repeatability, and averaging many phase maps reduces the random part.

Coherence and parasitic fringes

With a laser source, fringes appear even for cavities of tens of centimeters, because the coherence length of a single-mode helium-neon laser greatly exceeds the path difference. The same coherence makes every stray reflection interfere. When the test part is a parallel plate, its back surface returns a third beam and the image shows superposed fringe sets; a wedged part, index-matching fluid or petroleum jelly on the back surface, or a wavelength-shifting algorithm that separates cavities by their length, removes the effect. Reflectivity also matters: a bare glass test surface (about 4%) against an uncoated reference gives high fringe visibility, while a metal-coated or high-reflectance surface needs an attenuating filter in the cavity or a higher-reflectance reference to keep the contrast usable. When both surfaces are highly reflective, multiple reflections sharpen the fringes into a multiple-beam pattern, which complicates phase-shifting analysis.

Relation to other instruments

A test plate laid directly on a polished surface under a monochromatic lamp is a contact Fizeau interferometer, and its fringes are Newton's rings when the surfaces differ in curvature. A wavemeter of the Fizeau type uses a fixed air wedge in the opposite sense: the gap is known and the fringe period gives the wavelength. Twyman-Green and Michelson interferometers separate the reference and test arms, which allows the arm lengths to be matched for a short-coherence source and components to be tested in transmission, but forgoes the common-path stability.

Common questions

What does one fringe mean in a Fizeau interferometer?

One fringe corresponds to a change of λ/2 in the gap, which is λ/2 of surface height at normal incidence: 316.4 nm at 632.8 nm.

What is the difference between a Fizeau and a Michelson interferometer?

A Fizeau interferometer has a reference surface in the test beam itself, so both beams share almost one path. A Michelson splits the beam into two separate arms and recombines them, so the arms see different optics and environment.

References: D. Malacara (ed.), Optical Shop Testing, 3rd ed. (Wiley, 2007); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); E. Hecht, Optics, 5th ed. (Pearson, 2017).