The Michelson interferometer is used in a class of...

The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance $\Delta d,$ and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser $\left(\lambda_{\mathrm{Red}}=632.8 \mathrm{nm}\right)$ is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance $\Delta d$, a number $\Delta N_{\text {Red }}=6.000 \cdot 10^{4}$ red fringes and $\Delta N_{\text {unknown }}=7.780 \cdot 10^{4}$ fringes pass by the reference photodiode. a) Calculate the wavelength of the unknown laser. b) Calculate the displacement, $\Delta d$, of the movable mirror.

The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance $\Delta d,$ and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser $\left(\lambda_{\mathrm{Red}}=632.8 \mathrm{nm}\right)$ is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance $\Delta d$, a number $\Delta N_{\text {Red }}=6.000 \cdot 10^{4}$ red fringes and $\Delta N_{\text {unknown }}=7.780 \cdot 10^{4}$ fringes pass by the reference photodiode.
a) Calculate the wavelength of the unknown laser.
b) Calculate the displacement, $\Delta d$, of the movable mirror.

Key Concepts

Michelson Interferometer

A Michelson interferometer is an optical instrument that splits a beam of light into two paths, reflects them back, and recombines them to produce interference fringes. The pattern of fringes is highly sensitive to differences in optical path lengths, making the device useful for precise measurements of length, wavelength, and other optical phenomena.

Optical Path Difference

Optical path difference refers to the difference in the actual distance traveled by two coherent light beams within an interferometer. This difference, often affected by mirror displacement or differing refractive indices, directly influences the interference pattern by causing constructive or destructive interference based on the phase differences of the beams.

Interference Fringe Counting

Interference fringe counting is a method that relates the number of observed fringes in an interferometer to the physical displacement or changes in the optical path. Since each fringe corresponds to a change in the optical path difference equal to one wavelength, counting fringes allows for a precise quantitative connection between mirror displacement and wavelength.

Reference Laser Calibration

Using a reference laser with a known wavelength allows for calibration in measurement devices such as wavelength meters. By comparing the number of fringes produced by the reference laser to those produced by an unknown laser, one can determine the unknown wavelength. This calibration step is fundamental to ensuring accurate measurements in optical instrumentation.

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