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Unformatted text preview: recombine them (see the ﬁgure). If the two beams at the output are in phase, then they will constructively
interfere, and the signal observed, Iout , is maximized. If they destructively interfere, then Iout will be
minimized. E1 E2
E3 E5 E4 E5 Assume the refractive index of the air is 1, and the refractive index of the water is 1.33. The refractive index
of water is increased by 0.002 for each gram of sugar added to water.
(a) First assume one beam (solid line) passes through nothing but air, but the second beam (dotted line)
passes through a container of a sugar solution in water. Assume that containers are made of optically
transparent materials and the thickness of the container is d. thick containers. Write expressions for
the electric ﬁelds E1 through E5 at the points indicated on the graph. Answers can be left in terms of
the index of the solution ns and the wavevector k . Now ﬁnd the intensity Iout How does the intensity
change as the solution concentration increases?
(b) Assume the thickness of the container is 1 cm and the wavelength is 633 nm. How much sugar must
be added to the solution to make the Iout equal to zero? Where do you think the energy goes in this
(c) Now assume you have an identical container of 1 cm thickness in the ﬁrst beam but it’s ﬁlled with
water. How much sugar must be added to the solution in the container in the sensing arm to minimize
Iout now? Explain the diﬀerence in the sensitivity of the Mach-Zehnder in these two cases. E1 E2
E3 E5 E4 E5 (d) Consider using this device as a sensor for refractive index, and the only signal you have is an intensity
readout from a power meter placed at the output. You ﬁrst take a calibration with a solution that has 5 6.007 Spring 2011 Problem Set 7 a known amount of sugar dissolved. Then, your friend adds some sugar to the starting solution and
you are allowed to measure again the output intensity with this solution placed in the Mach-Zehnder
sensor. Assuming the change in index between this and the reference solution is small, what is the
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- Spring '11