Grating and Spectra

Diffraction by a Double Slit

Interference vs Diffraction

\[\begin{align*} &\boxed{I(\theta)=I_0\left(\frac{\sin\alpha}{\alpha}\right)^2 \cos^2\beta}\\ &\beta=\frac{\delta_2}{2}=\frac{\pi}{\lambda}d\sin\theta\\ &\alpha=\frac{\pi}{\lambda}a\sin\theta \end{align*}\]

• Let \(a\rightarrow 0\), then \(\alpha \rightarrow 0\) and \(\frac{\sin\alpha}{\alpha}\rightarrow 1\), the equation just describes the interference pattern.
• Let \(d\rightarrow 0\), then \(\beta \rightarrow 0\) and \(\cos^2\beta\rightarrow 1\), the equation just describes the diffraction pattern.

The Definition of Interfrence and Diffraction

If the combining waves originate from a small number of elementary coherent sources (as in a double-slit experiment with \(\alpha \ll \lambda\) ), we call the process interference.

If the combining waves originate in a single wavefront (as in a single-slit experiment), we call the process diffraction.

Diffraction Grating

Objects that emit and absorb light is the diffraction grating, which has a much greater number \(N\) of slits, often called rulings.

Mutiple Slits with Monochromatic Light

The phase difference between adjacent slits is:

\[\begin{align*} \delta_N =\frac{2\pi d}{\lambda}\sin\theta \end{align*}\]

This case is for \(N=4\)

\[\begin{align*} \tilde{E}_{\theta}=E_0\left( e^{-3i\frac{\delta_4}{2}} + e^{-i\frac{\delta_4}{2}} + e^{i\frac{\delta_4}{2}} + e^{3i\frac{\delta_4}{2}}\right) \end{align*}\]

Pattern: With monochromatic (red) light incident on a diffraction grating (with a large number \(N\)), we can see on a viewing screen very narrow (and so are called lines ).

These lines correspond to

\[\begin{align*} &\delta_N=\frac{2\pi d}{\lambda}\sin\theta=2\pi m\\ &\text{or } \sin\theta = \frac{m \lambda}{d} \quad\text{ for }m = 0,1,2,\cdots\text{ (maximum lines)} \end{align*}\]

They are separated by relatively wide dark regions.

Width of the Lines

A grating's ability to resolve (separate) lines of different wavelengths depends on the linewidth. The half-width of the central line \(\Delta \theta_{hw}\) is determined by the first minimum in intensity , at which the \(N\) rays from the \(N\) slits of the grating cancel one another.

The first minimum occurs where the phase difference bewteen the adjacent slits is

\[\begin{align*} &Nd\sin\Delta\theta_{hw} = \lambda\\ \text{or }& \Delta\theta_{hw}\approx \sin\Delta\theta_{hw}=\frac{\lambda}{Nd} \end{align*}\]

X-Ray Diffraction

Atomic Grating

Suppose we would like to use the visble light \((\lambda \simeq 5.5 \times 10^{-7} m)\) to study the diffraction. The first-order mximum \((m=1)\) would occur at

\[\begin{align*} \sin\theta=\frac{m\lambda}{2d}=2750 \gg 1 \end{align*}\]

This means that we wouldn't observe the first-order maxima. Therefore, we need waves with much shorter wavelength \((\lambda \approx d)\) , that is, X rays.

The maxima turn out to be in directions as if the X rays were reflected by a family of crystal planes that extend through the atoms within the crystal(晶体) and that contain regular arrays of the atoms.

Bragg's law :

the intensity maxima for X-ray diffraction is

\[\begin{align*} 2d\sin\theta=m\lambda \end{align*}\]

where \(m=1,2,3,\dots\) is the order number of an intensity maximum. A monochromatic X-ray beam can be used to determine the geometrical structure of a crystal.