# When deriving $d\sin\theta=n\lambda$ are the light rays approximately parallel or actually parallel?

In learning to derive the diffraction grating equation to form an interference pattern on a screen parallel to the diffraction grating, this kind of image is used extensively:

Then as normal, $$AB = d$$, $$AC = n\lambda$$ then $$d\sin\theta = n\lambda$$

This assumes the lines are parallel. However, in every derivation I've searched, it states this, but not if this is an approximation as my intuition states:

My possibly flawed understanding is that these 3 light rays shown in the picture would converge onto a single point and constructively interfere, creating a maxima, however this means that these light rays aren't parallel.

Is there something I'm missing? Are they parallel enough to ignore the small differences in angles, or are they actually parallel?

• The parallel lines come from the fact that it's a far-field approximation. As you can see, things get more complicated when you have a screen close to the slits because then things will depend on the exact distance. As an exercise, you can try choosing a wavelength and d separation and calculating for yourself how close a screen has to be to the slits for the far-field approximation to start breaking down. – Samuel Weir Jun 25 '18 at 18:28
• Consider that a typical diffraction grating has a few hundred lines per mm (spacing $d\lessapprox 10^{-5}\text{m}$) and that a typical diffraction demonstration has a diffraction pattern shown on the screen a few meters from the light source. The rays are thus generally "parallel enough" to be treated as parallel. – enumaris Jun 25 '18 at 19:01
• It was as I expected, but non of the literature made it clear, I thought I was missing something, thanks for the clarifications! :) – Skidushe Jun 25 '18 at 19:04

It depends.

If your viewing screen is "close" to the slits, the rays will not be parallel, as you have noted.

There are two ways to make the theory more directly applicable. One way is to move the viewing screen far from the slits, much farther than the distance between them. Then the rays are parallel for all practical purposes.

The second way is to interpose a lens between the slits an the viewing screen. Parallel rays emerging from the grating will then be imaged on the screen.

• Another point is that the next step often, and especially in a textbook presentation, is to take the small angle approximation $\sin \theta \approx. \theta$. That's not exact either, and depending on your overall setup and goals that might be the larger error. – Brick Aug 7 at 14:22

You are correct ─ this is an approximation.

Unless you introduce some kind of focusing optics to view the fringes, for the rays to interfere they need to reach the same point on the screen, and for this to happen, as you have noted, they need to be at slightly different angles.

However, it is important to keep in mind just what the differences are here: the inter-slit distances will be of the order of a few tens of microns, say, whereas the distance to the screen will be in the tens of centimeters or so. This means that the difference in angle for all of those rays will be of the order of $$\delta \theta \sim \frac{10\:\mu\rm m}{10\:\rm cm} \sim 10^{-4}.$$ While nonzero, this is generally much smaller than other effects that you'd need to worry about first (say, a careful understanding of the linewidth at the screen, and how it depends on the number of slits). If you want to go for absolute accuracy, of course, then you need to include it, but at textbook levels it is an approximation. It is an excellent approximation (i.e. so good that it doesn't even need much remarking on at that level of treatment) but an approximation nevertheless.

When a diffraction grating is set up on a spectrometer the light which in incident on the diffraction grating after emerging from the collimator is adjusted to be parallel and the telescope collecting the light emerging from the diffraction grating is adjusted to collect parallel light.

With a layer the emergent light which is incident on the diffraction grating is to a very good approximation parallel and if the viewing screen on which the resulting spectrum is observed is a reasonably long distance away from the grating then the light hitting the viewing screen can be assumed to be approximately parallel.

What you are observing is the result of what is called Fraunhofer diffraction whereas if the viewing screen is very close to the grating (ie the light hitting the viewing screen not being parallel) then it is termed Fresnel diffraction.