Diffraction Gratings — Question 1

Question 1

A finite grating is a geometric sum of equally rotated phasors

N identical narrow slits contribute equal amplitudes $E_1e^{ij\delta}$, where $j=0,\ldots,N-1$ and $\delta=2\pi d\sin\theta/\lambda$. Ignore the slowly varying individual-slit envelope.

1. Sum the phasors and derive the array intensity factor.

2. Evaluate the removable limit at a principal maximum and the peak increase if N doubles with each slit unchanged.

Question 1 - Solution

A finite grating is a geometric sum of equally rotated phasors

(a) Method and calculation

$$E_N=E_1\frac{1-e^{iN\delta}}{1-e^{i\delta}}=E_1e^{i(N-1)\delta/2}\frac{\sin(N\delta/2)}{\sin(\delta/2)}.$$$$\boxed{I_N=I_1\left[\frac{\sin(N\delta/2)}{\sin(\delta/2)}\right]^2}.$$

(b) Interpretation and further calculation

$$\delta\to2\pi m:\quad\boxed{I_{\mathrm{principal}}=N^2I_1}.$$$$N\to2N\Longrightarrow\boxed{I_{\mathrm{principal}}\to4I_{\mathrm{principal}}}.$$

Physical check

The zero-over-zero form at a principal order represents complete phase alignment rather than a dark direction.