Fourier and Physical Optics
Homework 3

Problem 1: optical coherence tomography

OCT is a powerful technique used for medical imaging. A slideshow tutorial introduction by David Sampson is available on the Google Drive.

a. In the time-domain approach, light back-reflected from the sample is interfered with a reference beam in a Michelson interferometer as the reference beam is scanned. The light reflected from the sample will interfere with the reference over the coherence length (e.g. see slide 20) Simulate a TD-OCT trace such as shown as the “detector output” on slide 20. Include 3 sample reflector positions.  
Hint: for this problem, make use of the relation we derived with the autocorrelation theorem, i.e., that the FT of oscillating part of the field autocorrelation yields the power spectrum. To use this in reverse, you’ll want to take the real part of the inverseFT of the spectrum.

b. In the spectral domain approach, the light returning from the sample is interfered with the reference in the spectrometer, along the lines of spectral interferometry. Construct an interferogram for a similar sample used for part (a). Explain how the distance information for the object can be obtained from this interferogram.

problem 2: Periodic functions

a. Derive the Fourier transform of a periodic triangular wave, defined in one period as  f(t)=|t|, (-(T/2)<t<(T/2)). Hint: use the array theorem, and you can make use of the FT of the triangle pulse from HW1.

b. As a periodic function, you can also calculate a Fourier series for this waveform. How can the two results be related?

Problem 3: mode locked laser pulse train

A “mode-locked” laser puts out a train of pulses that are equally spaced, with a pulse spacing that is equal to the round-trip time (Subscript[T, RT])=2 L/c  of the pulse in the laser resonator (of length L). For example, if the pulse is Gaussian with a 1/e field half-width of (Subscript[τ, p]), we can represent the laser output by E(t)∝comb(t/(Subscript[T, RT]))⊗exp[-(t^2)/((Subscript[τ, p])^2)]. 
In this problem you may do the calculations by any method you prefer, but it is probably most straightforward to use the transform relations.

a. Calculate the spectral intensity (((|FT{E(t)}|))^2) for this beam.

b. From the relationship between the round-trip time and the spacing between the frequency comb lines, how short would the laser cavity need to be in order to resolve the comb lines with a good spectrometer (resolution ~0.1nm)? Assume a central wavelength of 800nm.

c. Suppose we use a fast optical shutter to select a burst of 10 pulses. Calculate and describe the change in the output spectrum. You can model the shutter with a rect( ) function, and assume that the edge of the shutter doesn’t coincide with any of the pulses.

d. Illustrate your discussions from part (c) with plots using different choices of (Subscript[T, RT]) and (Subscript[τ, p]).