What is a 'good' coherence/correlation function for multimode spectra?

Question

Following up on the question here and this answer in particular, I would like to pose the following question:

How to derive a formula for the coherence length of a multispectral source, such as a common multimode laser diode?

Imagine a light source with a spectrum that consists of many peaks of certain line widths. The peaks are not necessarily equidistant and the shape might vary from one peak to the next. This is the general case; if it aids the calculations/derivation, one may also assume equidistant spacing between peaks (a constant free spectral range), constant peak widths and heights, and even discrete frequencies (e.g. a sequence of delta functions). But the ultimate goal is to arrive at a general expression for the coherence length with the spectral distribution as a free parameter.

Example to get started: Let's assume a spectral distribution $f(\omega)$ of $N$ equidistant peaks with equal shape/width/height,

$$f(\omega) = \sum_{n=1}^N f_{\omega_n}(\omega) \tag{1}$$

with $f_{\omega_n}(\omega)$ representing e.g. Lorentzian or Gaussian functions (not sure what it would be in the case of an ideal laser resonator) centered at wavelengths $\omega_n$.

Furthermore, let's approximate the electromagnetic waves as scalar waves. With equation $(1)$, we get

$$\psi(t) = \psi_0 \int d\omega \ f(\omega) e^{-i \omega t} \tag{2}$$

for the field oscillation at the source, which has a complex waveform with spectral distribution $f(\omega)$. From this source, a scalar wave propagates and enters the Michelson interferometer,

$$\psi_{in}(x, t) = \psi_0 \int d\omega \ f(\omega) e^{i(kx - \omega t)} \tag{3}$$

To write down all frequency dependencies explicitly and simplify expressions, let's use $k = \frac{\omega}{c}$ and set $c \equiv 1$ from here on:

$$\psi_{in}(x, t) = \psi_0 \int d\omega \ f(\omega) e^{i \omega (x - t)} \tag{3}$$

In the interferometer, the wave is split by a 50:50 beam splitter into two waves of equal intensities

$$\psi_A(x, t) = \frac{\psi_0}{2} \int d\omega \ f(\omega) e^{i \omega (x - t)} \tag{4}$$ $$\psi_B(x, t) = \frac{\psi_0}{2} \int d\omega \ f(\omega) e^{i \omega (x - t)} \tag{5}$$

Both waves propagate for certain distances $x_A$ and $x_B$ along their respective interferometer arms. The difference in propagation distance $\Delta x = x_A-x_B$ traveled by $\psi_A$ and $\psi_B$ is called optical path length difference (OPD) and is adjustable by translating one of the mirrors along the beam path.

This OPD results in an accumulation of relative phase shift $\Delta\phi = \omega \Delta x = \omega (x_A-x_B)$ between $\psi_{A}$ and $\psi_{B}$, which recombine at the beam splitter and propagate from there on as a superposition

$$\psi_{out}(x, \Delta x, t) = \psi'_{A} + \psi'_{B} = \frac{\psi_0}{2} \int d\omega \ f(\omega) ( e^{i \omega (x - t)} + e^{i \omega (x + \Delta x - t)} ) \tag{6}$$

This superposition $\psi'_{A} + \psi'_{B}$ results in interference patterns - in this case, bright linear fringes - that can be observed on a screen. Since observed or measured intensities of electromagnetic fields are proportional to the squared modulus of the field amplitudes, we need to calculate

$$I(x, \Delta x, t) \propto |\psi_{out}(x, \Delta x, t)|^2 = | \frac{\psi_0}{2} \int d\omega\ f(\omega) ( e^{i \omega (x - t)} + e^{i \omega (x + \Delta x - t)} ) |^2 \tag{7}$$

Now comes the interesting part:

• For a single wavelength, i.e. a monochromatic source, the fringe contrast of the interference pattern will monotonically decay when increasing OPD $\Delta x$ further and further (by translating one of the mirrors), as more and more phase correlation is lost. Upon reaching a maximal OPD, $\Delta x_{max}$, no more fringes will be visible (only a homogeneous intensity distribution). This $\Delta x_{max}$ corresponds to the coherence length of the source, characterizing its temporal coherence property, and one can directly derive the spectral line width from it.

• In the case of multiple wavelengths, however, the fringe contrast will not simply decay with increasing OPD, but also oscillate as a function of $\Delta x$: Fringes will disappear and reappear periodically when increasing the OPD further and further. In other words, the function $I(x, \Delta x, t)$ has an oscillatory envelope depending on the spectral distribution $f(\omega)$ of the light source. Importantly: We can no longer deduce the coherence length or spectral width directly from $\Delta x_{max}$, since this quantity seems no longer well-defined!

To visualize the problem, consider the following experimentally measured fringe visibility envelope of a 787 nm multi-mode laser diode using a Michelson interferometer:

Note that our light source still has certain coherence properties that can be measured and characterized! As an extreme example, just imagine an ideal frequency comb, i.e. a multimodal spectrum consisting of perfect delta peaks with zero phase noise - obviously, such would be an extremely coherent source! It's just that the situation is now more complex than with a single-mode or even single-frequency source:

Simply taking e.g. the first minimum in fringe visibility to compute a single coherence length $L_{coh}$ would result in a "wrong" value, since you'll see fringes reappearing way beyond $L_{coh}$.

Also, isn't it perfectly possible to have peaks in fringe visibility appearing for larger OPDs that are much broader than the peak at zero OPD?

This begs the question: Is there a concise mathematical concept/expression that fully describes this oscillatory envelope in fringe visilibity, and is it still possible to derive a meaningful "coherence length functional" from it that adequately describes the temporal coherence property of general, multispectral sources?

Note: It should be clear at this point that we're no longer looking for a single number $L_{coh}$, but rather a statistical functional or even something like a classifier in the machine learning sense.

See also e.g. this question and the answers and discussions therein, which, despite starting off in the right direction, do not arrive at a conclusive result.

Here is a small "simulation" to further motivate the question and also demonstrate some of the gaps in understanding. Consider, for instance, a simple superposition of 10 cosine functions with different frequencies:

(Wolfram Alpha: 1st plot, 2nd plot)

If each cosine function is interpreted as the interference fringe pattern generated by a single frequency, it is easy to explain oscillating fringe contrast envelopes by a simple superposition. This pattern, however, extends to $\pm\infty$ without any decay in fringe contrast, i.e. the fringe visibility function does not have a compact support as in the experimental data shown above or, in fact, for any realistic case. So already the reproduction of such experimental data - i.e. oscillating fringe visibility functions with compact support - is by all means not a trivial problem and requires a continuum description w.r.t. frequencies and phases, Fourier theory etc.

Another "disturbing fact", to illustrate the whole problem from a different angle: Note that a superposition of two waves with different frequencies can actually never produce static interference fringes, as explained in this answer. Therefore, the mere fact that a superposition of waves with a sufficiently large number of frequencies suddenly leads to the emergence of visible fringes is actually quite astonishing! And the question

"What are the exact conditions for the emergence of static interference fringes from chaos?"

is actually complementary to asking

"What is a meaningful coherence length functional for multispectral sources?"

Answer 1

"the ultimate goal is to arrive at a general expression for the coherence length with the spectral distribution"

Then, the answer is very simple. The "spectral distribution" is the intensity as a function of frequency. Even better, we express the spectrum as a function of the wave number $k$ (cycles per unit length, spatial frequency), i.e. $S(k)$.

From the spectrum, it is immediate to calculate the correlation function $C(\Delta x)$, which is the result of the interferometric measurements. Indeed, $C(\Delta x)$ is the Fourier transform of the spectrum $S(k)$. Conversely, the spectrum $S(k)$ is the anti-transform of $C(\Delta x)$.

Then, from the correlation, it is also easy to calculate the coherence length. I assume that you mean the HMFW of the correlation function. We take the maximum of $C_{max}=C(\Delta x)$. Then we calculate one half of it. And finally we look for the $\Delta x$ such that $C(\Delta x)=C_{max}/2$. Then $2 \Delta x$ is the HMFW of the correlation function, one of the possible definitions of coherence length.

If there are bumps, it is possible that $C(\Delta x)$ equals $C_{max}/2$ in multiple points. Then, you should take the first (minimum) $\Delta x$. Why? Because this definition ensures that, within the coherence length, the correlation remains more than $C_{max}/2$.

Of course, you can define the coherence length in other ways. For example, as the width of the envelope. Or the variance, considering $C(\Delta x)$ as a distribution. Every time you say that the coherence length is, say, 4 cm, you should always say if you mean the HMFW or something else. The definition of "coherence length" depends on you, not on the multi- or single-mode laser.

EDITED AND ADDED

Maybe we can rephrase the question in a different way. Observing eq.1 of the question, I generalize saying that the power spectrum is the sum of equally spaced peaks (single modes), each having a width $\delta k$ (where $k$ is the wave number), spaced by $1/L$, and with an envelope of width $\Delta k$.

The presence of $N$ modes with same height would be represented by a flat-top envelope, quite unusual, so I will think of a more usual bell-shaped envelope.

Then, the correlation function (representing the visibility) is just the Fourier transform of the power spectrum. How are the three numbers describing the correlation function?

1. $\Delta x = 1/\delta k$ is the width of the envelope of the correlation function;

2. $\delta x = 1/\Delta k$ is the width of the central peak of the correlation function;

3. $L$ is the period of oscillation of the repeated peaks of the correlation function.

Here, I neglected the factors, since they depend on the exact shape of the bell-shaped curve and can vary.