# 9.2 Diffraction grating

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We derive the interference patter for a diffraction grating.

## Diffraction grating

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

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

Squaring it we see that: $I\left(\theta \right)={I}_{0}\frac{{{\mathrm{sin}}}^{2}\beta }{{\beta }^{2}}\frac{{{\mathrm{sin}}}^{2}N\alpha }{{{\mathrm{sin}}}^{2}\alpha }$

Interference with diffractionfor 6 slits with $d=4a$

Interference with diffractionfor 6 slits with $d=4a$

Interference with diffractionfor10 slits with $d=4a$

Interference with diffractionfor10 slits with $d=4a$

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

and just like in multisource interference minima occur at ${\mathrm{sin}}\theta =\frac{n\lambda }{Nd}\text{ }n=1,2,3\dots \text{ }\frac{n}{N}\ne integer$ 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.

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