What is diffraction?

• It is the spreading of a wave when it encounters an obstacle or a slit.
• In this course, we primarily use the term to describe single-slit interference.
• There is no fundamental difference between diffraction and interference.
• Conventionally, the term diffration is used when the resulting wave arises from the interference of a large number or a continuium of sources.
• A single slit could be modeled as inifinitely many small slits lining up together.

For a single-slit of width $a$, destructive interference occurs at the angle give by:

where $m$ is an non-zero integer.

Compare with the double-slit inteference side by side:

where $m$ is any integer, including zero.

Comparison of two equations.

Single-slit	Double-slit
$a \sin\theta_m = m \lambda$	$d \sin\theta_m = m \lambda$
$a$: width of the slit	$d$: separation between two slits
minima for integer $m\neq 0$	maxima for integer $m$
$m=0$ gives the central maximum.	$m=0$ gives the central maximum.
central maximum twice as wide as other maxima	central maximum is approximately the same width as other maxima
central maximum much brighter than other maxima	central maximum is roughly the same brightness as other maxima (for very narrow slits)

Location of the dark fringes

Simple trigonometry tells us the vertical position of the $m$-th order dark fringe is: $$ \begin{eqnarray} \tan \theta_m &=& \frac{y_m}{L} \\ \Rightarrow y_m &=& L \tan \theta_m \end{eqnarray} $$ where $L$ is the separation between the viewing screen and the slit.

Applying the small angle approximation, we have: $$ \begin{eqnarray} y_m &=& L \tan \theta_m \approx L \sin \theta_m \tag{*} \end{eqnarray} $$

But we know the $m$-order minimum obeys: $$ \begin{eqnarray} a\sin\theta_m &=& m\lambda \\ \Rightarrow \sin \theta_m &=& m \frac{\lambda}{a} \end{eqnarray} $$ Put this into (*) gives: $$ \begin{eqnarray} y_m &\approx& L \sin \theta_m \\ &=& L (m \frac{\lambda}{a}) \\ \Rightarrow y_m &\approx& m \frac{\lambda}{a} L \end{eqnarray} $$ Remember $m\neq 0$ for the minima.

For single-slit, because the central maximum is so much brighter than all other maxima, we usually only care about the central maximum. The dark fringes on both sides of the central maximum is $m=\pm 1$: $$ \begin{eqnarray} \theta_1 &\approx& \sin \theta_1 = \frac{\lambda}{a} \\ y_1 &\approx& \frac{\lambda}{a} L \end{eqnarray} $$

The width $w$ of the central maximum is twice that of $y_1$: $$ w = 2 y_1 = 2\frac{\lambda}{a} L $$

$y_m$ without small angle approximation

• Diffraction grating is a barrier with multiple slits.
• As the number of slits $N$ increases, each maximum becomes narrower (width $\sim \frac{1}{N}$) while the intensity at the peaks goes up as $N^2$.
• Compared with double-slit, a diffraction grating with large $N$ produces much sharper and brighter fringes.

Locations of the maxima of a diffraction grating is the same as the double-slit, given by:

where $d$ is the separation between neighboring slits.

⚠ $d$ for diffration gratings could be made very small, giving relatively large value of $\theta$ ($\theta \sim 1rad$). Therefore it is better not to use the small angle approximations.