Diffraction by a slit

When monochromatic light of wavelength $\lambda$ leaves a small source and arrives as a plane wave at a slit of width $b$, the diffracted light leaving the slit forms a pattern in space. As a function of angle $\theta$ the intensity is given by

$$I(\theta) = I_0 \left(sin\left(\beta\right) / \beta \right)^2$$ (81)

where $\beta$ is given by

$$\beta = \pi b sin\left(\theta\right) / \lambda$$ (82)

If the light from the slit is observed far from the slit so that $\theta$ is small, $\theta$ can be calculated from

$$\theta = (x - x_0) / R$$ (83)

where $x$ is the transverse distance from the center at $x_0$, and $R$ is the distance away from the slit. The geometry is illustrated in Figure 8.1

Figure 8.1: Light from a mercury lamp passes through a monochromator that separates the different wavelengths and then to a test slit of width b. The light is diffracted by a slit or straight edge, and the measured with a microscope or CCD camera.

Accordingly, the intensity will be zero where

$$\beta = m \pi$$ (84)

$$(m = 1, 2, 3, ...)$$

since for these values $sin(\beta)$ will be zero. By differentiating the expression for the intensity, you can also show that maxima occur where

$$tan(\beta) = \beta$$ (85)

Solutions to this transcendental equation can be found with a calculator. They will be nearly equal to

$$\beta = (2m-1) \pi /2$$ (86)

$$(m = 1, 2, 3, ...)$$

except for the obvious one, where $\beta$ is zero, corresponding to the center of the pattern.