# Is this wave spatially coherent?

Is the following wave (the solid lines representing the intensity maxima) spatially coherent?

I've been seeing different, seemingly contradictory definitions of spatial coherence.

Some places define seem to be defining spatially incoherent light as light that has "spatially varying temporal coherence". I.e. the Wavefront (defined as the surface of constant phase) shape changes from Wavefront to Wavefront. From that definition, this would be spatially coherent because while the Wavefront is distorted, the Wavefront doesn't change over time.

Other places seem to define it as measuring phase correlation perpendicular to the direction of propagation of the wave. In this case I wouldn't consider this example wave to be spatially coherent because if you move horizontally along the wave, the phase changes rapidly. (I could have picked a more noisy variation in this direction, but I picked a sine wave for simplicity.)

• just to double check, in the image the lines are intensity (as you say) or field maxima? – scrx2 Dec 18 '15 at 15:18
• That should be essentially the same thing, no? – aquirdturtle Dec 19 '15 at 21:11
• No, the principle is different. Imagine for simplicity the red lines are straight. If you show intensity maxima you have basically some interference fringes. If you show the field, the intensity is constant. – scrx2 Dec 20 '15 at 13:29
• If you look at the instantaneous field (not time-averaged) of a plane wave, which is what I'm suggesting, the intensity is most certainly NOT constant, but varies just as the field varies with maxima at both the maxima and minima (negative maxima) of the field. So the only difference is the wavelength of the light, which is irrelevant for my question. – aquirdturtle Dec 21 '15 at 1:46
• Ok, it's a matter of definition, I'm used to think the intensity as the average over a period of the modulus squared of the field. This definition is based on experience: if you place a detector in your wave, it measures a constant signal, because no detector is able to see variations of "intensity" (in your definition) in the order of $10^{16}$ Hz. – scrx2 Dec 21 '15 at 11:00

I assume in your figure the red lines represent the phase profile of the wave (the points where the phase is constant), and not the intensity. And that the wave propagates from top to bottom.

Is the following wave spatially coherent? Yes.

If those lines were straight, the wave would be a plane wave, perfectly coherent in time and space. I consider then your plot as a propagating wave with a periodically undulated phase profile. Again, it's perfectly coherent in time and space.

The definition of coherence which I consider accepted says that a field is coherent where (when) there is a fixed phase relation between the electric field in different locations (times).

Mathematically this is described by the cross correlation of the field at two points in space or time.
Experimentally, we measure the intensity $|E|^2$, not the field $E$, so coherence is tested using interference as follows:

Spatial coherence: you measure the intensity fringe contrast (max-min) in the double slit experiment, as a function of the distance between the slits. The maximum distance between the slits at which there is still some interference is the spatial coherence length of the wavefront (the contrast depends on the phase correlation between two different points in space).

Temporal coherence: in an interferometer with 2 different arm lengths (e.g. Michelson) each point, or beam, of the wavefront interferes with a delayed copy of itself. The maximum delay (difference between the lengths of the 2 arms) at which there is still some interference is the temporal coherence length of the field.

Applying the above 2 criteria to your wave, you can see that it's both time and space coherent. This will hold true if you distort the red lines even more (you can make them random), IF they remain one identical to the others and equispaced (like the figure below, left). To decrease the temporal coherence, you can make the distance between the lines, i.e. the wavelength, more randomly changing (fig. below, middle). In this way, the phase difference $\Delta \phi$ between the field $E(x_o,t)=P1$ and the field $E(x_o,t+\tau)=P'1$ is not constant anymore, and the spectrum is not a delta but a broader peak. To decrease the spatial coherence, while keeping the temporal one, you can change the wavelength across the wavefront (fig. below right).

Probably a useful way to see if a wave is coherent is to consider the local wavelength $\lambda(x,t)$, and asking if it constant in $t$ (vertical direction in the figure) and / or in $x$ (horizontal direction in the figure), etc.

• I like this answer & definition, but could you more specifically address the other possible definition of spatial coherence? This definition also seems like a natural definition of spatial coherence, and I'd like to better understand why it's not useful / used. – aquirdturtle Mar 29 '16 at 17:18

What you have is a realization of a random process. Calling a realization coherent or not-coherent is a mistake, indeed I believe this mistake was made once on Wikipedia.

The traditional definition of spatial coherence asks how related the values at one detector are to the values at another detector. We then assume homogeneity which means that the relation doesn't change spatially and can write down the typical length of this relation which is often converted into an area taking the value as a radius.

To answer your question, spatial coherence can be measured by two probes along the wave-front but without seeing more realizations of the wave we really can't say anything about its coherence!

Ps. If you are using one probe and looking at different times that would be temporal coherence, which sounds like what you imagine for the perpendicular case.

• What exactly are you calling a random process? My wave? The Measurement? I'm pretty sure that it's fair to consider the coherence or non-coherence of the given wave. – aquirdturtle Dec 14 '15 at 7:19
• The 'field' is intrinsically random and you drew one possible realization. Notice that the squiggles you drew could be due to a 100% coherent source that scattered off some kind of grid. As you noted, its hard to understand the coherence of a bunch of squiggles (because thats not conceptually correct). – Mikhail Dec 14 '15 at 7:44