# Misunderstanding of the functioning of the reflective diffraction grating

Suppose we have a sawtooth diffraction grating, as depicted below:

where the angle $\beta$ is the angle of inclination of the 'teeth' of the grating with respect to the plane of the grating and incident plane monochromatic waves normal to the plane of the grating. I am supposed to determine the angle $\theta$ for which the interference pattern for one 'saw-tooth' has a maximum. The diagram in the mark-scheme is as follows:

where the points $A,B$ both belong to the same 'saw-tooth' and the distance between $A$ and $B$ is $d$. The path difference between the two waves, according to the mark-scheme is given by $\Delta = BF - AE = d \sin \beta - d \sin \theta$. My question might seem trivial, but why are the two (BF and AE) not equal? In other words, shouldn't the two parallel incident waves (incident at an angle $\beta$) be simply reflected from the face of the 'saw-tooth' at exactly the same angle, in accordance with the law of reflection? Why even bother defining $\theta$? What am I missing here?

• – Farcher Jun 27 '18 at 5:58

shouldn't the two parallel incident waves (incident at an angle β ) be simply reflected from the face of the 'saw-tooth' at exactly the same angle, in accordance with the law of reflection?

This is a useful approximation of what actually happens, but in reality the behavior of light is a little more complicated.

The operation of diffraction grating is based on the Huygens–Fresnel principle and the interference.

If we apply the H-F principle to your example, we'll have to assume that every point of the light-reflecting surface will produce its own hemispherical wave rather than just a ray.

The interference between all these waves will define the reflected light, which, in addition to the central maximum, corresponding to the "normal" reflection of a specular surface, will produce additional maxima and minima, as depicted on your diagram.

For $\theta=\beta$, we'll get the central maximum. To find other max and min points, we'll have to consider other $\theta$ values.

Your question concerning the law of reflection is a good one, and it still applies here. This is what's known as the 0th (zeroth) order, or specular, pattern. There are other orders, or reflection peaks, that occur at different angles due to diffraction and interference. I think your problem asks you to find those other orders as well. If you google diffraction grating I'm sure you'll find the analysis.

For the light to interfer constructively when propagating at an angle $\theta$, the rays from point $A$ and point $B$ must be in phase. Therefore $$BF-AE=d\sin\beta-d\sin\theta=n\lambda .$$ This leads to the grating equation $$\sin\beta-\sin\theta=\frac{n\lambda}{d} .$$ Here $n$ denotes the diffraction order. If $n=0$, then you have the zero-th diffraction order, for which $\beta=\theta$. Hence, basic reflection, no diffraction. For $n=1$, you have the first diffraction order, for which $\beta\neq\theta$.

The design of the grooves is to optimize the diffraction efficiency into a specific order, such as the first diffraction order.