(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The sodium D-lines are a pair of narrow, closely spaced, approximately equal intensity spectral lines with a mean wavelength of approximately 589 nm. A Michelson interferometer is set up to study the D-lines from a sodium lamp. High contrast fringes are seen for zero pathlength difference between the two arms of the interferometer. The fringes disappear when the pathlength difference is increased to 0.29 mm.

a. What is the wavelength difference between the lines?

b. What would you expect to see if the pathlength difference were increased to 0.58 mm, assuming the spectral lines are very narrow?

c. If the spectral lines have approximately Gaussian shapes, with a width of 50 pm (taken between the points of the line shape where the intensity falls to e −1/2 of the peak intensity), what is the maximum fringe contrast (visibility) seen for a pathlength difference of around 4 mm?

2. Relevant equations

Fringe visibility: ## v = \frac{I_{max}-I_{min}}{I_{max}+I_{min}} ##

3. The attempt at a solution

a) The phase difference of the two beams increase by ##\pi## due to the pathlength difference, hence:

## \pi = 2\pi \delta x (\frac{1}{\lambda - \delta\lambda /2} - \frac{1}{\lambda + \delta\lambda /2} )##

giving ##\delta\lambda = 0.60nm##

b) We now have ##\delta x' = 2 \delta x##, hence ##\delta \phi = 2\pi## so the fringe pattern should be the same as initally.

c) This is the part where I get stuck. ##\delta x'' = 4 mm## gives ##\delta \phi '' \approx 13.8\pi##, meaning that the peaks neither exactly overlap or exactly overlap with each others minima. However, from this point on I am not sure how to find their separation in the "line shape". What is the "line shape" plotted as a function of? Considering that the width of the peaks in the line shape is given in units pm, I suppose it must be some sort of length. However, I am not sure what length, because the one on the screen is not general enough (it depends on the distance etc.).

Could somebody please give me some guidance on how to attack the last part of the problem? Thank you. :)

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# Homework Help: Fringe Visibility of Sodium Doublet in Michelson interferometer

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