29 October 2013

Coherence 4 - Space coherence

The Michelson setup is an amplitude-splitting interferometer, and hence it is well adapted for time coherence measurements since the two beams originate in the same point \(P_1 = P_2\) and the time shift \(\tau\) is simply given by the difference in arm length. To measure the space coherence, we need a wavefront-splitting interferometer, with the two beams originating in different points \(P_1 \neq P_2\) but at the same time: \(\tau = 0\). The simplest example is the Young interferometer.

The Young setup can be used to measure the transverse coherence length \(\xi _t\) at a distance \(L\) from the primary (incoherent) source. Using the criterion of fringe visibility, we define \(\xi _t\) as the smallest distance \(d\) between pinholes for which the fringe pattern vanishes.

We will further simplify the problem by considering only the edges of the primary source, which is now reduced to two point sources \(S_1\) and \(S_2\), a distance \(a\) apart. Each of them creates on the screen its own interference pattern (red for \(S_1\) and green for \(S_2\)). When \(d\) is small, the two patterns are almost in phase and the visibility is reinforced1. As \(d\) increases, their shift increases until the maximum of one pattern coincides with the minimum of the other (see the Figure).

The phase shift between the two patterns can be easily determined as: \(\delta = a d /L \). The condition above then corresponds to: \( \displaystyle \delta = \frac{\lambda}{2} \Rightarrow \xi_t(L) = \frac{L \lambda}{2 a}\). The conclusion is that \(\xi_t\) increases linearly with the distance! Although we started with \(\xi_t = 0\), the radiation acquires an increasing degree of coherence as it propagates.

This gives us a simple recipe for obtaining coherent radiation from an incoherent source (at the price of severe intensity loss): place at a distance \(L\) from the primary (incoherent) source an iris smaller than \(\xi_t(L)\): the latter constitutes a (secondary) coherent source. Dennis Gabor used this phenomenon to create holograms using the light of a mercury lamp passed through a pinhole 3 ┬Ám in diameter 2.

Two comments are in order:
  • Even for \(a d /L =\lambda/2\), when only considering the two extreme sources \(S_1\) and \(S_2\) the fringes are not completely smeared out. However, when we take into account all points in-between the visibility does go to zero.
  • When \( d\) increases beyond this minimum value, the pattern is again visible, but with a much weaker contrast (which is an oscillating function of \( d\), with a rapidly decreasing envelope).
For more details see [BW] (§ 10.4).

 In this simplified setup, we were able to determine the evolution of \(\xi_t\) along the beam path, a concise way of quantifying the propagation of coherence. In the next post we will discuss a more general way of describing this propagation, namely the Van Cittert–Zernike theorem.

1. Note that we need to sum the intensity of the two patterns, since there is no coherence between the two sources and the cross-term averages to zero.
2. Dennis Gabor - Nobel Lecture: Holography, 1948-1971. Nobelprize.org. Nobel Media AB 2013. Web. 12 Oct 2013.

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