Michelson fringes measure doubled paths

Let each arm length be measured from the beamsplitter to its mirror. In the simplest air-path geometry the returning fields have traveled twice those lengths, so their optical path difference is twice the arm-length difference. Moving one mirror along its arm by d changes the optical path difference by 2d. The phase shift is the optical path change divided by the vacuum wavelength, multiplied by a full cycle in radians.

Beamsplitter reflection phases can add a constant offset, and transparent plates can add unequal material dispersion. These affect the location of a chosen bright fringe, especially for broadband light. They do not remove the round-trip factor for a translating mirror in a uniform air arm. In a medium of constant refractive index n, the corresponding path increment is 2nd at normal incidence. A varying or dispersive index must be included through optical path rather than mechanical distance alone.

The returning beams can be described using the virtual images of the two mirrors. When those effective mirror planes are parallel, rays with the same inclination have equal path difference. A lens maps inclination to radius in its focal plane, producing equal-inclination rings. For effective separation h, the ideal angular path difference is 2h cos θ. Expanding the cosine at small angle explains why a broad beam can contain several rings even when its central ray is carefully adjusted.

A slight tilt between the effective mirror planes instead creates an air-wedge geometry. Near the mirror-image plane, local effective separation varies with transverse position, producing equal-thickness fringes that are approximately straight for a small uniform tilt. Saying fringes are localized means their contrast is concentrated near a particular observation plane for an extended source. It does not mean interference exists only on a material surface or that the same viewing plane suits every alignment.

For a broad spectrum, different wavelengths usually place maxima at different mirror settings. At a large delay their interference terms average toward zero. Near matched optical paths they agree sufficiently to produce a visible envelope, often with a central white-light region and colored neighbors. This temporal localization in delay is distinct from spatial fringe localization in an observation plane. Equal geometric arms alone may fail to match the group delays if one arm traverses extra dispersive glass.

The lab can test the phase part of a monochromatic Michelson scan. Reset and set both controls to phase zero degrees and ratio one, then predict constructive addition. Reset again and set phase 180 degrees and ratio one, predicting cancellation; this phase change corresponds to a mirror shift of one quarter wavelength in the ideal air-arm model. Both component waves in the instrument travel in the same direction. It has no mirrors, angular rings, plate dispersion, or bandwidth envelope.