The movable mirror of a Michelson interferometer is attached to one end of a thin metal rod of length 23.3 mm. The other end of the rod is anchored so it does not move. As the temperature of the rod changes from \(15^{\circ} \mathrm{C}\) to \(25 \mathrm{C},\) a change of 14 fringes is observed. The light source is a He Ne laser, \(\lambda=632.8 \mathrm{nm}\). What is the change in length of the metal bar, and what is its thermal expansion coefficient?

A Michelson interferometer is a precision instrument used to measure the wavelength of light, small distances, and changes in refractive index. It operates on the principle of interference, which occurs when two or more waves overlap to produce a new wave pattern. The interferometer splits a single light source into two beams, which travel different paths: one reflects off a fixed mirror, while the other reflects off a movable mirror. When these two beams recombine, the difference in their path lengths creates constructive or destructive interference, resulting in an interference pattern of bright and dark fringes.

The fringes are very sensitive to variations in the path length, such as those caused by a change in temperature affecting the length of the rod to which the movable mirror is attached. In the context of the exercise, the change in the rod length due to thermal expansion alters the optical path length enough to result in a observable shift in the fringe pattern. Since the device can detect half-wavelength shifts in the path length, even extremely small changes can be measured accurately, making the Michelson interferometer an indispensable tool in various scientific and engineering fields.

Optical path length (OPL) represents the distance light travels in a medium, adjusted for the medium's refractive index, which describes how much the medium slows the light relative to vacuum. In other words, it is the product of the physical path length and the refractive index of the medium through which the light propagates. In vacuum, the refractive index is 1, so the optical path length equals the geometric path length.

In the Michelson interferometer exercise, the term 'optical path length' denotes the distance travelled by the light beam from the beam splitter to the movable mirror and back again. When the metal rod expands or contracts due to temperature changes, this distance changes, affecting the interference pattern. Notably, the optical path length change is double the actual physical change in the rod length because the light travels to the mirror and then back to the point of recombination.

The Helium-Neon (He-Ne) laser is a type of gas laser where the primary active laser medium is a mixture of helium and neon. It emits a characteristic red light, with its most common wavelength being about 632.8 nanometers (nm). This wavelength falls in the visible range of the electromagnetic spectrum and is a fundamental parameter for various precise measurements as it serves as a standard for interference, diffraction, and spectroscopy experiments.

In our given scenario, the wavelength of the He-Ne laser acts as a ruler to measure changes in the optical path length within the Michelson interferometer. By knowing the laser's precise wavelength, the observed changes in the interference pattern can be converted into detailed measurements of physical changes, such as the linear thermal expansion of the metal rod. The wavelength's conversion into meters, which ensures that all measurements are in standard SI units, is essential for the precise calculation of these small physical changes and their related coefficients.