12 October 2013

Coherence 1 - Introduction

This is the first in a series of posts where I'll try to give a simplified account of optical coherence, based on a couple of lectures I presented at the HERCULES school a few years ago. The subject is quite complex and those looking for a more rigorous and complete text should check the references below.
  • What is coherence?
Let us start by putting together the "diffuse knowledge" one might have about coherence:
-  Laser light is coherent, while that emitted by thermal sources is incoherent.
-  Coherence is related to the presence (or visibility) of interference fringes.
-  Coherence "decreases as the wavelength λ decreases". This statement needs some elaboration, but it is true that is much harder to achieve coherence in the X-ray range than in the optical one.
A first attempt at a definition would be:

"Coherence" is the extent to which a field maintains a constant phase relation over time or across space.
  • Why should we care?
Coherence is a fundamental property of light (and waves in general). Whenever we describe a phenomenon related to wave propagation we need to make an (explicit or implicit) assumption about the coherence of that field. Undergrad level treatments of physical optics and X-ray crystallography generally assume perfect coherence. Knowing the limitations of these approaches is essential for a deeper understanding of the underlying phenomena. Moreover, the concept of coherence is fundamental for modern applications such as holography.
  • Definition
The mutual coherence function of the wave field V(r,t) - taken as scalar for simplicity - is defined as:
\begin{equation}
\begin{split}
\Gamma _V (P_1, P_2, \tau) &= \left \langle V(P_1,t+\tau) \tilde{V}(P_2,t) \right \rangle \\
& \quad \text{with} \quad \left \langle F(t) \right \rangle = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T F(T) \, \text{d}t
\label{Gam}
\end{split}
\end{equation}
where ~ stands for the complex conjugate; we need to integrate over long times T, since detectors are very slow compared to the radiation frequency (for the visible range), so any measurement will be averaged over a large number of periods.

Equation (1) by itself is not very eloquent, but its meaning should become more clear in the following posts, through simplifications and examples.

A first observation is that the coherence function Γ resembles an intensity. Indeed, when P1=P2 and τ=0, it is simply the average intensity at that point. We can thus renormalize by the intensity to obtain a degree of coherence γ :
\begin{equation}
\gamma _V (P_1, P_2, \tau) \equiv \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{\Gamma _V (P_1, P_1, 0)}\sqrt{\Gamma _V (P_2, P_2, 0)}} = \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{I(P_1)}\sqrt{I(P_2)}}
\label{degree}
\end{equation}The coherence function contains all the information we might want. However, it is quite cumbersome. In practice, it is often convenient to reduce it to only two parameters, a time (or longitudinal) coherence length, which is roughly the decay length of ΓV with τ, and a space (or transverse) coherence length, quantifying the loss of coherence as P2 moves away from P1. We will look at longitudinal coherence in the next post.
  • References
[BW]   - M. Born and E. Wolf, Principles of Optics (7th ed.),
                 Cambridge Univ. Press (1999).
[MW]  - L. Mandel and E. Wolf, Optical Coherence and Quantum Optics,
                 Cambridge Univ. Press (1995).
[G]      - J. Goodman, Introduction to Fourier Optics (3rd ed.)
                Roberts & Co Publishers (2005).
[W]    - E. Wolf, Introduction to the Theory of Coherence and Polarization of Light,
               Cambridge Univ. Press (2007).

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