# Coherence

Coherence is the most important and unique characteristic of laser light.  Non-laser light sources have a low degree of coherence, while laser light exhibits a high degree of coherence.  What is coherence?  A coherent light wave evolves predictably in time and space.  We’ve been typically writing waves of the form

$\displaystyle E = A \cos{(k z - \omega t + \phi)}$

As long as $A$, $k$, $\omega$, and $\phi$ are constants, this wave is perfectly coherent.  The relationship between the wave at any one point in time and space and any other point in time in space is entirely predictable and can be calculated.

So what is an incoherent or partly coherent light wave?  Suppose that the ‘initial phase’ $\phi$ in the wave above varies in time in an unpredictable fashion.  For example, it may make unpredictable jumps at random intervals.  This is diagrammed below:

The more frequently these random phase jumps occur, the lower the degree of coherence of the light wave.  Incoherent light is produced by random spontaneous emission events, where excited atoms or molecules each independently emit photons whose phase is random relative to one another.  Most light sources (besides lasers) rely upon spontaneous emission to produce light, and therefore produce incoherent light.

A common analogy to coherence is the difference between a column of marching soldiers vs. a randomly milling crowd.  If you observe the soldiers for a short time, you can predict where each soldier will be located at a later time.  The randomly milling crowd, on the other hand, is difficult to predict.

A high degree of coherence means interference will be easy to produce and observe.  If you consider the two-slit experiment we discussed, the interference fringes on the screen rely upon the waves having predictable phases.  If a point on the screen is exactly one wavelength further from the bottom slit than the top slit, the difference in phase between the two light beams reaching that point will always be $2 \pi$, and they will always constructively interfere, as long as the light is coherent.  If, on the other hand, the wave from the bottom slit takes a random phase jump in the time interval between the arrival of the top wave and the bottom wave, the type of interference that will occur is unpredictable and will fluctuate in time.  In this case, when we look at the time-average intensity on the screen, the bright and dark spots are smeared out.  Below is an animation where the two-slit experiment is reproduced with highly incoherent light (frequent random phase jumps).  As you can see, the intensity pattern on the screen does not show clearly defined, predictable fringes.

decoherence

Let’s remind ourselves of the expression for the time-average intensity on the screen for the two-slit experiment:

$\displaystyle I_{avg} = \frac{C}{N T} \int_0^{NT} 4 A^2 \cos^2{\left( \frac{k (r_1 + r_2)}{2} - \omega t + \frac{\phi_1 + \phi_2}{2}\right) } \cos^2{\left( \frac{k (r_1 - r_2)}{2} + \frac{\phi_1 - \phi_2}{2}\right)} dt$

Let’s selectively use the cosine double-angle formula on the second $\cos^2$ only:

$\displaystyle I_{avg} = \frac{C}{N T} \int_0^{NT} 2 A^2 \cos^2{\left( \frac{k (r_1 + r_2)}{2} - \omega t + \frac{\phi_1 + \phi_2}{2}\right) }$

$\displaystyle [\cos{( k (r_1 - r_2) + (\phi_1 - \phi_2))} + 1] dt$

The second cosine contains $\phi_1 - \phi_2$, which will depend on time for incoherent light.  (In most cases it will vary much more slowly in time than $\omega t$, although not for the particular case shown in the animation above.)  In fact, if the difference $\phi_1 - \phi_2$ is a completely random function of time, the cosine containing that difference is equally likely to be positive or negative, so that its integral over time is zero.  Then we’re left with

$\displaystyle I_{avg} = \frac{C}{N T} \int_0^{NT} 2 A^2 \cos^2{\left( \frac{k (r_1 + r_2)}{2} - \omega t + \frac{\phi_1 + \phi_2}{2}\right) } dt$

$\displaystyle = C A^2$

So in this extreme case of zero coherence, the intensity on the screen will be the same at all points.  For the completely coherent case, the intensity on the screen included bright fringes with intensity $2 C A^2$ and dark spots with 0 intensity.  As you might guess, the intensity integrated over the area of the screen must be the same in both the incoherent and the coherent case, because total power must be conserved.

For the incoherent case, the initial phase $\phi$ was different for light arriving at the screen from slit 1 vs. slit 2.  There are two broad ways that this difference could arise:

1.  The incoming wave to the left of the slits had random phase jumps along the ‘wavefront’; i.e. perpendicular to its direction of propagation.  This is a lack of spatial coherence, and would directly lead to a difference in initial phase at the two slits.

2.  The incoming wave to the left of the slits had random phase jumps in time.  This is a lack of temporal coherence.  How does this result in differences between $\phi_1$ and $\phi_2$?  For most points on the screen, the distances from the slits to the screen, $r_1$ and $r_2$, are different.  Consider a point on the screen for which $r_1 > r_2$.  In this case, the light arriving at the point on the screen must have departed slit 1 earlier than it departed slit 2.  (Similar to having two trains arrive in Atlanta at the same time from New York and Seattle: the Seattle train must have departed earlier.)  If there was a random phase jump in the incoming (source) wave after wave 1 departed but before wave 2 departed, it would lead to a difference between $\phi_1$ and $\phi_2$.

Clearly, in this example, the average amount of time between random phase jumps is important.  This is related to the coherence time of the light wave $t_c$, which is the time over which the light maintains its coherence (predictable phase relationships).  We could also define a coherence length, which is $l_c = c t_c$.   A completely coherent light wave would have infinite coherence time, but in practice no light wave is completely coherent; all light waves have finite coherence time.  Coherent laser light has a much longer coherence time than incoherent lightbulb light.

As the coherence time decreases, the phase evolves in time more rapidly (via phase jumps or otherwise), which means that the spectral linewidth of the light wave will increase, as we are deviating more strongly from a perfect single harmonic.  Thus we have the relationship

$\displaystyle \Delta \nu = \frac{1}{t_c}$

There are other ways to quantify the coherence as well.  As you’ve seen, interference is strongest when the coherence time is long.  Therefore, measuring the strength of the interference observed on a screen in the two-slit experiment or in an interferometer (we’ll learn more about those later) is a proxy for the degree of coherence.  The strength of interference is quantified by the fringe visibility, $V$:

$\displaystyle V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$

where $I_{max}$ and $I_{min}$ are the maximum and minimum intensity measured on the screen, respectively.  For the two-slit experiment, in the coherent case the bright spots on the screen had $I_{max} = 2 C A^2$, while the dark spots had $I_{min} = 0$, so the fringe visibility was 1.  In the completely incoherent case, we have a uniform intensity on the screen, so $I_{max} = I_{min} = C A^2$, and the fringe visibility is zero.

Finally, the mathematical definition of the degree of (temporal) coherence $g(\tau)$ is the autocorrelation of the electric field divided by the time-average intensity.  In terms of the phasor electric field, it is

$\displaystyle g(\tau) = \frac{\lim_{N \to \infty} \int_0^{NT} E^*(t) E(t+\tau) dt }{\lim_{N \to \infty} \int_0^{NT} E^*(t) E(t) dt}$

The magnitude of the degree of temporal coherence is unity for small time displacements $\tau$, and drops to zero for time displacements much greater than the coherence time $\tau >> t_c$.

*Give some interpretation of degree of coherence and tau.  Comparison to experiment with interferometer