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.

Step by step solution

01

a) Calculate the wavelength of the unknown laser.

First, let's use the formula for the number of fringes formed in an interferometer, which is given by: Number of fringes, \(\Delta N = \frac{2 \Delta d}{\lambda}\) Here, \(\Delta d\) is the distance the movable mirror is displaced, and \(\lambda\) is the wavelength of the laser. We know the reference laser's wavelength and the number of fringes produced, so let's rearrange this formula and find the displacement \(\Delta d\) for the reference laser: \(\Delta d = \frac{\Delta N_{Red} * \lambda_{Red}}{2}\) Plug the given values to calculate \(\Delta d\):

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