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.

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.

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.

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 Γ

_{V} 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.

[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).