Physics 415 Supplemental Problems

PHYSICS 415/416 SUPPLEMENTAL PROBLEMS Part 5

5A. (Interferometers) Consider two Fabry-Perot interferometers whose mirrors are separated by 1 mm. They are used to produce circular fringes for a HeNe laser’s output which is mainly at 632.8 nm. Let’s concentrate on the central portion of the pattern. We are going to compare the fringe patterns from these interferometers and that produced by a Michelson interferometer.

(a) What is the approximate value of m (the order number) for the center of the fringe pattern where we can assume that the transmission angle of the light is approximately zero (t = 0)? (b) Find the coefficients of finesse (F) for the F-P interferometers if the first one uses mirrors with r = 0.30 and the second with r = 0.95. (c) Using a computer, plot the fringe profiles for the above F-P interferometers.. To do this, plot the transmittance (T) versus the phase difference () from  = m2 to  = (m+3)2. On the same graph, plot the fringe profile for a Michelson interferometer. Use Eq. (8-2) and plot I/4Io. Your final plot should resemble Fig. 8-9 in the text book. (Note: Dividing the Michelson intensity by 4 just rescales the intensity ratio so that a fringe maximum has a value of one on the vertical axis, consistent with the F-P fringes. The actual brightness of a particular fringe profile depends on the source and optics used to collect the light and project the fringe pattern.) (d) For each fringe profile, calculate the fringe visibility. You can use values directly from your plots to calculate the visibility. (e) For each fringe profile, calculate the finesse (F ) graphically. Recall that the finesse is graphically defined to be the ratio of the phase difference between consecutive bright fringes to the phase width of one of the fringe peaks at half of the maximum intensity ratio:

 2 F   1/ 2 1/ 2

Measure 1/2 from the plot for each profile. (Note: You may not be able to do this for one of the Fabry-Perot interferometers. If not, explain why.) (f) For the Fabry-Perot fringe profiles, calculate the finesse using the equation derived by defining the finesse to be the ratio of the free spectral range to the minimum detectable wavelength difference:

FSR F    F 2 ()min Compare these finesse values to the ones found from your plot. (g) Compare the 3 interferometers you just analyzed: F-P with reflection coefficient of 0.30, F-P with coefficient of 0.95, and Michelson. Which interferometer has the largest finesse? Which interferometer has the highest fringe visibility? Which interferometer would be best in measuring small differences in wavelength?