9.2 Diffraction grating

Diffraction grating

Consider the case of N slit diffraction, We have $$E_1=\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(k{R}_1-\omega t)}$$ $$E_2=\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(k{R}_2-\omega t)}$$ $$\text{.}$$ $$\text{.}$$ $$\text{.}$$ $$E_N=\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(k{R}_N-\omega t)}$$ So we can just follow the steps of the two slit case and extend them and get (using $R_N=R-(N-1)d\sin \theta$ ) $${\displaystyle \begin{array}{c}E=\sum _{n=1}^N{E}_N\\ =\sum _{n=1}^N\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-2(n-1)\alpha -\omega t)}\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }\sum _{n=1}^N{e}^{i(kR-2(n-1)\alpha -\omega t)}\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\sum _{n=1}^N{e}^{-i2(n-1)\alpha }\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha }\end{array}}$$ This is the same geometric series we dealt with before $$\sum _{n=0}^{N-1}{x}^n=\frac{1-x^N}{1-x}$$ so $${\displaystyle \begin{array}{c}E=\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha }\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\frac{1-e^{-i2N\alpha }}{1-e^{-i2\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\frac{e^{-iN\alpha }}{e^{-i\alpha }}\frac{e^{iN\alpha }}{e^{i\alpha }}\frac{1-e^{-i2N\alpha }}{1-e^{-i2\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-\omega t)}\frac{e^{-iN\alpha }}{e^{-i\alpha }}\frac{e^{iN\alpha }-e^{-iN\alpha }}{e^{i\alpha }-e^{-i\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{\sin \beta }{\beta }{e}^{i(kR-(N-1)\alpha -\omega t)}\frac{\sin N\alpha }{\sin \alpha }\end{array}}$$

Notice that this just ends up being multisource interference multiplied by single slit diffraction.

Squaring it we see that: $$I(\theta )=I_0\frac{{\sin }^2\beta }{{\beta }^2}\frac{{\sin }^{2}N\alpha }{{\sin }^2\alpha }$$

Principal maxima occur when $$\frac{\sin N\alpha }{\sin \alpha }=N$$ or since $\alpha =kd(\sin \theta )/2$ $$kd\sin \theta =2n\pi n=0,1,2,3$$ or $$\frac{2\pi }{\lambda }d\sin \theta =2n\pi$$ or $$\sin \theta =\frac{n\lambda }{d}$$

and just like in multisource interference minima occur at $$\sin \theta =\frac{n\lambda }{Nd}n=1,2,3\dots \frac{n}{N}\ne integer$$ A diffraction grating is a repetitive array of diffracting elements such as slits or reflectors. Typically with N very large (100's). Notice how all butthe first maximum depend on $\lambda$ . So you can use a grating for spectroscopy.