Coherence is broadly the ability for light to constructively/destructively interfere with itself.
Suppose we have a point in space denoted P, where two incident plane waves originating from sources S1 and S2 intersect.
We denote the total electric field intensity at this point as .
Therefore, we say that the total intensity at that point is
Expanding this out, we have
I=\langle(\vec E_1+\vec E_2)\cdot (\vec E_1^*+\vec E_2^*)\rangle=\langle |E_1|^2 + |E_2|^2+ 2\text{Re}(E_1\cdot E_2^*)\rangle = I_1+I_2+2\text{Re}\langle E_1E_2^*\rangle |
Evidently, the total intensity is the sum of the two intensities, plus a cross term. This is the "interference" term.
When no interference occurs, we say that the two beams are mutually incoherent.
We can now try to quantify this notion of coherence a bit more by focusing on the interference term.
Suppose that the two
There are two notions of coherence: Spatial and Temporal.
Temporal coherence asks: if I take a single point in space and look at the magnitude of the E field at t=0, do I have any information about what the E field will be at some later time t=tau? ie, does the E field at that point in space have a steady phase over time?
Spatial coherence asks: if I take two separate points in space, does knowing the E field at one point give any information about the E field at another point? ie, do two spatially separate points have the same phase relationship?
Spatial coherence has two aspects: transverse and longitudinal. this is only based off what your choice of those two points are.
We've already discussed transverse spatial coherence. This is what people typically mean when they say "spatial coherence". Essentially we ask how correlated the E field is between two points that are transverse to the direction of propogation.
Longitudinal spatial coherence asks how correlated the E field is between two points along the same direction of propogation. This is essentially just set by the coherence length of the source, which is actually determined by the coherence time of the source and thus is highly related to temporal coherence. For two points that are closer together than the coherence length (c*t_coherence), the E fields are highly correlated. For two points further than the coherence length, the E fields have no correlation.
Suppose we have some extended source emitting light and we have two downstream points P1 (fixed) and P2 (variable) being illuminated.
The mutual degree of coherence between these two points is equivalent to the intensity in the following scenario
A spherical wave converges to P1, passing through an aperture of the same shape and size as the extended source. it diffracts, and we find the complex amplitude at point P2.
Consider the case where the distance to the source is very large compared to the source size. Also assume we have a circular source.
This allows us to approximate the incoming spherical wave as a plane wave. We further approximate that the plane wave has normal incidence.
Then, the mutual degree of coherence between the central point P1 (lying at the 0th order diffraction maxima) and the off axis point P2,
This schematic here depicts incident light hitting a reflective grating
At the grating, we want the incident wavefront to have a transverse spatial coherence equal to or larger than the grating width because this maximizes the number of lines that are illuminated with spatially coherent light. This gives us the maximum spectral resolution since it makes the peaks sharper.
Lets try to quantify what we mean by "sharper" here.
Suppose we have a grating with some grating period
We also have a monochromatic light source with wavelength , at normal incidence, uniformly illuminating the grating with intensity
.
This means that there is an "incident electric field" on the grating, which we denote . Let x=0 be the center of the grating and have the grating extend infinitely outwards to the left and right.
We define the mutual intensity J as follows
We've seen this before, its just the interference term from earlier, also called the mutual coherence function or correlation function. When this value is high, we have high potential for interference and when it is close to 0 we have essentially no interference.
In the case where x1=x2=x, ie the diagonal, it just reduces to the intensity at x
OK now we take a step back and think from some first principles.
What we observe with a diffraction grating in the far field is essentially the spatial fourier transform of the incident electric field at the grating.
E(\theta)=C\int_{-\infty}^{+\infty}E_{out}e^{i\frac{2\pi}{\lambda}x\sin\theta}dx |
Where C is a constant prefactor which has the appropriate units. The term in the exponential is essentially the phase difference between a ray at x=0 and a ray at x=x. This is a limiting case of the Fresnel-kirchhof formula.
Anyways, E_out needs to be determined. The grating is essentially a series of opaque and non opaque strips, so we can describe its transmission as
t(x) = \begin{cases}
1 & \text{if } \mod(\lfloor x/\Lambda\rfloor,2)\neq0 \\
0 & \text{otherwise}
\end{cases} |
So lets say E_out(x) = E_in(x)t(x)
E(\theta)=C\int_{-\infty}^{+\infty}E_{in}(x)t(x)e^{i\frac{2\pi}{\lambda}x\sin\theta}dx |
The intensity is therefore
I(\theta)=\langle E^*(\theta)E(\theta)\rangle = \langle C^2\int_{-\infty}^{+\infty}E^*_{in}(x_1)t^*(x_1)e^{-i\frac{2\pi}{\lambda}x_1\sin\theta}dx_1 \int_{-\infty}^{+\infty}E_{in}(x_2)t(x_2)e^{i\frac{2\pi}{\lambda}x_2\sin\theta}dx_2\rangle
|
OK lets simplify this. First of all, move all the things that are not time dependent out, and combine the two integrals as they integrate independent quantities.
I(\theta)= C^2\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} t^*(x_1)t(x_2) \langle E^*_{in}(x_1)E_{in}(x_2)\rangle e^{i\frac{2\pi}{\lambda}(x_2-x_1)\sin\theta} dx_1dx_2 |
OK now notice that the only time dependent term is the E field part, so actually we can simplify the entire thing down using the mutual intensity we found earlier
I(\theta)= C^2\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}t^*(x_1)t(x_2)\cdot J(x_1,x_2)e^{i\frac{2\pi}{\lambda}(x_2-x_1)\sin\theta} dx_1dx_2 |
OK now lets figure out more details about the mutual coherence J.
First of all, as we've seen before, J is related to the complex degree of coherence as follows
\mu(x_1,x_2)=\frac{J(x_1,x_2)}{\sqrt{I(x_1)I(x_2)}}=\frac{J(x_1,x_2)}{I_0} |
The second step follows from the fact that we assume uniform incident intensity across the grating I_0
Now all we need to do is specify the complex degree of coherence between two points x_1 and x_2 on the grating!
For this, we can utilize whats called the "Gaussian-Schell Model". We know obviously that when x_1=x_2=x, then the complex degree of coherence should be 1 since a point is perfectly coherent to itself (if you know the value of E at x=0, then you know obviously the value of E at x=0)
For typical spatially incoherent sources, there is some typical "transverse spatial coherence width" where points closer than this are roughly spatially coherent and points further away are spatially incoherent.
From this, we create the following model for the complex degree of coherence!
\mu(x_1,x_2)=e^{-\frac{(x_2-x_1)^2}{2l_c^2}} |
This follows a gaussian distribution and will be our model of spatial coherence across the grating surface.
We now solve for J(x1,x2)!
Anyways, we can now plug it into the equation for I
I(\theta)= C^2\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}t^*(x_1)t(x_2)\cdot I_0e^{-\frac{(x_2-x_1)^2}{2l_c^2}} e^{i\frac{2\pi}{\lambda}(x_2-x_1)\sin\theta} dx_1dx_2 |
quite a hairy integral I'll try to simplify it a bit down now
I(\theta)= C^2I_0\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}t^*(x_1)t(x_2)\cdot e^{-\frac{\Delta x^2}{2l_c^2}} e^{i\frac{2\pi}{\lambda}\Delta x \sin\theta} dx_1dx_2 |
I'm sure theres a way to solve this explicitly, but I'm not quite that good at math and instead lets just look at the limiting cases and then do numerical simulation in between.
First case is for perfectly spatially coherent light. This means the transverse spatial coherence width is infinite, l_c=infinity, so then J(x1,x2)=I_0
I(\theta)= C^2I_0\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}t^*(x_1)t(x_2)\cdot e^{i\frac{2\pi}{\lambda}\Delta x \sin\theta} dx_1dx_2 |
I(\theta)= C^2I_0\cdot |\int_{\text{slits}}e^{i\frac{2\pi}{\lambda}x \sin\theta} dx|^2 |
blah blah blah some math later (fill in later)
We get the typical formula for perfect spatial coherent diffraction against N slits
We also check the fully incoherent limit, and that pops out that we should get uniform illumination in the far field.
OK I had claude solve this integral explictly so not only do we now know the limiting cases, we also analytically know the general solution.
I made a script that shows the resulting angular diffraction pattern for various incident transverse spatial coherence lengths and its shown below.
OK so here is the simulation.
The dashed orange line is what you would expect from the typical diffraction grating intensity profile which assumes perfectly spatially coherent light (ideal plane wave) incident on the grating.
As you can see when the coherence length is much larger than the slit, the resulting diffraction pattern approaches the perfect coherence limiting case.
One issue I had was the conservation of energy. Essentially when the coherence length was extremely short → 0, the total intensity would shrink. so obviously this is bad because thats a loss of conservation of energy.
Actually this is not an issue though! you see, what we computed was essentially the intensity contribution from interference effects. Even though the total intensity appears to go to zero, energy is conserved because this intensity is actually just showing up in the independent intensity sum part.
OK finally, lets go back to what we wanted to figure out initially.
Looking at the gif, we clearly see that as we increase our coherence length more and more, the "sharpness" of the peaks increases and approaches our ideal scenario.
The overall conclusion here is that we need to maximize our spatial coherence across the grating to give us the sharpest possible peaks.