Diffraction Grating Vs Double-Slit: Small angle Approx.

[Beth N]

Homework Statement

In double slit interference , the angle between the center bright fringe and the path length (aka the angle used to find the path length difference) can be approximate as a small angle. However, we cannot assume the angles of bright fringes due to diffraction gratings are small. Why? What is the difference? What angles are they exactly?

Homework Equations

$L= tan {\theta_m}$
$dsin {\theta_m}=m\lamda$

The Attempt at a Solution

I think in double slit interference, the angle can be approximated as small because we assume the distance between the 2 slits are very small compared to the distance between the slits and the screen opposite from them. Can't see why diffraction grating is different?

 

[Charles Link]

A hint is that it depends on the size $d$ which is often many times the wavelength for the case of just two slits, but typically is on the order of a wavelength for the spacing of the slits or lines in a diffraction grating. That isn't always the case, but it seems to be the assumption that you need to make here. (Edit: See the 3rd paragraph below for additional comments on this). $\\$ An item of interest here is the intensity peaks from a multi-slit or multi-line grating are narrower in the resulting $\Delta \theta$ than the intensity peaks from a two-slit pattern, where for a grating with many slits or lines the intensity peak can be very narrow in the angular spread $\Delta \theta$. You can see this in the figures above where for even 5 slits, the intensity peaks are narrower than for the two-slit case in the appearance on the screen. For the two slit case, each of the peaks is a wide blob. For the 5 slit case, the bright red region of each intensity peak is a narrower stripe. This is not why the small angle approximation can not be used for a grating though. $\\$ The question doesn't seem to be what I would call a very accurate question. In addition, the more important feature of the difference between the double-slit and multi-slit or multi-line grating result is the more narrow intensity peaks that result. The "small angle" approximation does still apply for both the double slit and multi-slit or multi-line grating at small angles, and starts to break down as $\theta$ gets larger for both cases. For this reason, I think the question really misses the boat.