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Consider the case of N slit diffraction, We have $${E}_{1}=\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(k{R}_{1}-\omega t)}$$ $${E}_{2}=\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(k{R}_{2}-\omega t)}$$ $$\text{.}$$ $$\text{.}$$ $$\text{.}$$ $${E}_{N}=\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(k{R}_{N}-\omega t)}$$ So we can just follow the steps of the two slit case and extend them and get (using ${R}_{N}=R-(N-1)d{\mathrm{sin}}\theta $ ) $$\begin{array}{c}E=\sum _{n=1}^{N}{E}_{N}\\ =\sum _{n=1}^{N}\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-2(n-1)\alpha -\omega t)}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}\sum _{n=1}^{N}{e}^{i(kR-2(n-1)\alpha -\omega t)}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\sum _{n=1}^{N}{e}^{-i2(n-1)\alpha}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha}\end{array}$$ This is the same geometric series we dealt with before $$\sum _{n=0}^{N-1}{x}^{n}=\frac{1-{x}^{N}}{1-x}$$ so $$\begin{array}{c}E=\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\frac{1-{e}^{-i2N\alpha}}{1-{e}^{-i2\alpha}}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\frac{{e}^{-iN\alpha}}{{e}^{-i\alpha}}\frac{{e}^{iN\alpha}}{{e}^{i\alpha}}\frac{1-{e}^{-i2N\alpha}}{1-{e}^{-i2\alpha}}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-\omega t)}\frac{{e}^{-iN\alpha}}{{e}^{-i\alpha}}\frac{{e}^{iN\alpha}-{e}^{-iN\alpha}}{{e}^{i\alpha}-{e}^{-i\alpha}}\\ =\frac{{\epsilon}_{L}a}{R}\frac{{\mathrm{sin}}\beta}{\beta}{e}^{i(kR-(N-1)\alpha -\omega t)}\frac{{\mathrm{sin}}N\alpha}{{\mathrm{sin}}\alpha}\end{array}$$
Notice that this just ends up being multisource interference multiplied by single slit diffraction.
Squaring it we see that: $$I(\theta )={I}_{0}\frac{{{\mathrm{sin}}}^{2}\beta}{{\beta}^{2}}\frac{{{\mathrm{sin}}}^{2}N\alpha}{{{\mathrm{sin}}}^{2}\alpha}$$
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 $\alpha =kd({\mathrm{sin}}\theta )/2$ $$kd{\mathrm{sin}}\theta =2n\pi \text{\hspace{1em}\hspace{1em}}n=0,1,2,3$$ 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 $${\mathrm{sin}}\theta =\frac{n\lambda}{Nd}\text{\hspace{1em}\hspace{1em}}n=1,2,3\dots \text{\hspace{1em}}\frac{n}{N}\ne integer$$ 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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