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.

Coherence 3 - The Michelson interferometer

The Michelson interferometer (shown below) is an excellent setup for illustrating the concept of longitudinal coherence.  We will simplify the formalism of the previous post down to three essential ingredients:

• The beam consists of wave pulses of length l (corresponding to ξ_l in post 2).
• They are "split in half"; each half goes through one of the arms.
• The "twin" pulses arrive at the detector with a shift ΔL.

Coherence 2 - Time coherence

Time coherence is directly linked to the spectral width. Indeed, a perfectly monochromatic signal is coherent at all times: a maximum now ensures a maximum \(N\) periods later. Not so for a polychromatic source: two oscillations of slightly different frequency are in phase at \(t=0\) and become out of phase \(N\) periods later, as shown below.

Image from [BW].

This defines the coherence time (and the longitudinal coherence length \(\xi _l\)), via:
\begin{equation}
\label{shift}
\xi _l = N \lambda = (N - \frac{1}{2})(\lambda + \Delta \lambda) \Rightarrow N \simeq \frac{\lambda}{2 \Delta \lambda}\Rightarrow \xi _l \simeq \frac{\lambda^2}{2 \Delta \lambda}
\end{equation}

In conclusion, the larger the spectral width \(\Delta \lambda\), the shorter the longitudinal correlation length \(\xi _l\). Once again, the descriptions in the time and frequency domains are complementary.

In the next installment, we will look at time coherence in more detail, using the Michelson interferometer.

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 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.

• 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).