What is a grating? Define grating element?

A diffraction grating is a plane glass plate with a large number of equidistant, parallel, very fine slits (ruled lines); the grating element is \(e + d\), the distance between corresponding points of two adjacent slits, where \(e\) is the width of a slit and \(d\) the width of an opaque line.

If there are \(N'\) lines per cm, then \(e + d = \dfrac{1}{N'}\) cm. For example, 5000 lines/cm gives \(e + d = 2\times10^{-4}\) cm.

Fraunhofer diffraction at a grating

A parallel beam of monochromatic light of wavelength \(\lambda\) falls normally on a grating with \(N\) slits. The light diffracted at angle \(\theta\) is focused by a convex lens on a screen.

Step 1: one slit. From single-slit Fraunhofer diffraction, the resultant amplitude from one slit in direction \(\theta\) is

\[A = A_0\frac{\sin\alpha}{\alpha},\quad \alpha = \frac{\pi e\sin\theta}{\lambda}\]

Step 2: N slits. Waves from corresponding points of adjacent slits have path difference \((e+d)\sin\theta\), i.e. phase difference \(2\beta\), where

\[\beta = \frac{\pi(e+d)\sin\theta}{\lambda}\]

So we must add \(N\) waves of equal amplitude \(A\) with phases \(0, 2\beta, 4\beta, \dots, 2(N-1)\beta\). Adding them (vector polygon method, or summing the geometric series of \(e^{i2k\beta}\)) gives the resultant amplitude

\[R = A\,\frac{\sin N\beta}{\sin\beta}\]

Step 3: intensity. Since \(I \propto R^2\),

\[I = I_0\left(\frac{\sin\alpha}{\alpha}\right)^2\left(\frac{\sin N\beta}{\sin\beta}\right)^2\]

The first factor is the single-slit diffraction pattern; the second is the interference among the \(N\) slits.

Results

• Principal maxima: when \(\sin\beta = 0\), i.e. \(\beta = \pm n\pi\), the ratio \(\sin N\beta/\sin\beta \rightarrow \pm N\), so \(I = I_0(\sin\alpha/\alpha)^2N^2\), very bright. Condition: \((e+d)\sin\theta = n\lambda\), \(n = 0, 1, 2, \dots\) (grating equation).
• Minima: when \(\sin N\beta = 0\) but \(\sin\beta \ne 0\): \(N(e+d)\sin\theta = m\lambda\), \(m \ne 0, N, 2N, \dots\). So there are \(N - 1\) minima between two principal maxima.
• Secondary maxima: \(N - 2\) weak maxima between adjacent principal maxima. As \(N\) increases, they become negligible and the principal maxima become sharper and brighter.