(II) A Michelson interferometer can be used to determine the...

(II) A Michelson interferometer can be used to determine the index of refraction of a glass plate. A glass plate $(thickness t)$ is placed on a platform that can rotate. The plate is placed in the light's path between the beam splitter and either the fixed or movable mirror, so that its thickness is in the direc- tion of the laser beam. The platform is rotated to various angles, and the number of fringes shifted is counted. It can be shown that if $N$ is the number of fringes shifted when the angle of rotation changes by $\theta$ , the index of refraction is $n=(2 t-N \lambda)(1-\cos \theta) /[2 t(1-\cos \theta)-N \lambda]$ where $t$ is the thickness of the plate. The accompanying Table shows the data collected by a student in determining the index of refraction of a transparent plate by a Michelson interferometer. $$\begin{array}{|l|l|l|l|l|l|}\hline N & {25} & {50} & {75} & {100} & {125} & {150} \\ \hline \theta(\text { degree) }& {5.5} & {6.9} & {8.6} & {10.0} & {11.3} & {12.5} \\ \hline\end{array}$$ In the experiment $\lambda=632.8 \mathrm{nm}$ and $t=4.0 \mathrm{mm}$ . Deter- mine $n$ for each $\theta$ and find the average $n .$

(II) A Michelson interferometer can be used to determine the
index of refraction of a glass plate. A glass plate $(thickness t)$
is placed on a platform that can rotate. The plate is placed in
the light's path between the beam splitter and either the
fixed or movable mirror, so that its thickness is in the direc-
tion of the laser beam. The platform is rotated to various
angles, and the number of fringes shifted is counted. It can
be shown that if $N$ is the number of fringes shifted when the
angle of rotation changes by $\theta$ , the index of refraction
is $n=(2 t-N \lambda)(1-\cos \theta) /[2 t(1-\cos \theta)-N \lambda]$ where
$t$ is the thickness of the plate. The accompanying Table shows
the data collected by a student in determining the index of
refraction of a transparent plate by a Michelson interferometer.
$$\begin{array}{|l|l|l|l|l|l|}\hline N & {25} & {50} & {75} & {100} & {125} & {150} \\ \hline \theta(\text { degree) }& {5.5} & {6.9} & {8.6} & {10.0} & {11.3} & {12.5} \\ \hline\end{array}$$
In the experiment $\lambda=632.8 \mathrm{nm}$ and $t=4.0 \mathrm{mm}$ . Deter-
mine $n$ for each $\theta$ and find the average $n .$

Key Concepts

Index of Refraction Measurement

The index of refraction is a fundamental property of a material that quantifies how much it slows down light compared to a vacuum. Using interferometric techniques, such as those involving a Michelson interferometer, the index of refraction can be determined by relating the fringe shifts (or phase changes) to the change in optical path length introduced by a material. This method provides a precise way to measure the optical properties of transparent media.

Optical Path Length and Phase Change

Optical path length is the product of the physical path length that light travels and the refractive index of the medium it traverses. Changes in the optical path length result in phase shifts between interfering light beams, which are observed as shifts in the interference fringes. Understanding this concept is critical when using interferometric methods to measure minute changes in position, thickness, or refractive properties of materials.

Michelson Interferometer

A Michelson interferometer is an optical instrument that splits a beam of light into two paths, reflects the beams back, and then recombines them to produce interference patterns. This setup is highly sensitive to changes in path length, which makes it useful for precise measurements of distances, refractive indices, and even small physical changes in the apparatus. Its design and operation are foundational in the study of wave optics and experimental physics.

Interference Fringe Pattern

Interference fringes are the alternating light and dark bands created when two or more light waves superimpose and interfere with one another. The positions and movements of these fringes are extremely sensitive to changes in the optical path lengths, providing a visual means to measure variations in parameters like distance or refractive index. Analyzing fringe shifts allows experimenters to deduce quantitative information about the optical system.

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