Why does small angle approximation work for Young's double slit and not for single/multiple slits?

Question

I understand that for a double slit experiment the equation is $\lambda=xd/L$, which has come from the small angle approximation of $d\sin\theta = n\lambda$ by assuming that $\sin\theta = (x/L)$ and therefore can be subbed in. I don't get why this doesn't work for both single slit diffraction and especially multiple slit diffraction. Can anyone explain?

Answer 1

To add some numbers to @PhilipWood's correct answer, a typical diffraction grating has around 250 lines per mm. In other words, the distance between the "slits" is $d=0.004$ mm $= 4\times 10^{-6}$m.

Suppose you were doing an experiment with a green laser ($\lambda \approx 500$ nm = $5 \times 10^{-7}$ m). The grating equation tells us that the maxima appear at $$\sin{\theta} = \frac{n \lambda}{d} = 0.125\, n.$$

Plotting $\theta$ and $\sin{\theta}$ for different orders, we'd get a graph like the one below, and you should be able to see that for values of $n > 4$, the approximation stops being very faithful!

Another way to see this is to realise that if we could write

$$\sin \theta = \frac{x}{L},$$ then $x$ would never be greater than $L$. However, anyone who's actually done an experiment with a diffraction grating and a laser will be able to tell you that it's not very hard to have $x > L$ for large $n$.

Answer 2

It's not because of the multiple slits in the grating, but because the slits are much closer together than Young's slits. This means that for the grating the non-zero-order constructive interference 'fringes' are at much greater angles, $\theta$, to the normal than for Young's fringes. Whereas for Young's fringes the angles are small enough to use the approximation, $$\sin \theta \approx \theta \approx \tan \theta=\frac xL,$$ for the grating the angles are too large for the approximation to be applicable.