Physics 6. Part V: Optics 6.2. Wave Optics 6.2.4. Single-Slit Diffraction

6.2.4. Single-Slit Diffraction

Table of Contents

Why a Single Slit Produces a Pattern

When light passes through a very narrow slit, it does not simply continue in a straight geometric beam. Instead, it spreads out and forms a pattern of bright and dark regions on a screen. This effect is called single-slit diffraction.

Single-slit diffraction is a wave effect. Different parts of the slit act like sources of waves, and these waves interfere with one another. At some angles they reinforce each other, producing bright regions. At other angles they cancel, producing dark regions.

This phenomenon becomes especially noticeable when the slit width is comparable to the wavelength of the light. If the slit is very wide compared with the wavelength, diffraction is much weaker and the spreading is small.

Geometry of the Diffraction Pattern

Consider a slit of width $a$, illuminated by monochromatic light of wavelength $\lambda$. A screen is placed far away, or a lens is used so that we observe the far-field pattern. The center of the pattern is a wide bright band called the central maximum. On both sides of it are weaker bright fringes separated by dark minima.

The dark minima are the most important starting point, because their positions are easy to calculate.

Rendering scientific sketch…

Single-slit diffraction geometry

Condition for Dark Fringes

To find a dark fringe, imagine dividing the slit into two equal halves. For a certain angle $\theta$, light from one half can arrive exactly out of phase with light from the corresponding point in the other half. Then the two halves cancel each other.

The path difference across the full slit is $a \sin\theta$. The first minimum occurs when this equals one wavelength:

$$
a\sin\theta = \lambda
$$

More generally, the minima occur at

$$
a\sin\theta = m\lambda
\qquad \text{for } m = 1,2,3,\dots
$$

These values give the dark bands on both sides of the center. The pattern is symmetric, so minima also occur at negative angles.

For single-slit diffraction minima,
$$
a\sin\theta = m\lambda, \qquad m=1,2,3,\dots
$$
where $a$ is the slit width, $\lambda$ is the wavelength, and $\theta$ is the angle from the central axis.

Small-Angle Form

If the screen is far away and the angles are small, we may use the approximation

$$
\sin\theta \approx \tan\theta \approx \theta
$$

If $y_m$ is the distance from the center of the screen to the $m$th minimum, and the screen is a distance $L$ from the slit, then

$$
\tan\theta \approx \frac{y_m}{L}
$$

So the minima are approximately at

$$
y_m \approx \frac{m\lambda L}{a}
$$

This is often the most useful formula in experiments.

For small angles, the position of the $m$th minimum on a screen is
$$
y_m \approx \frac{m\lambda L}{a}
$$
This approximation works when $y_m \ll L$.

Width of the Central Maximum

The central bright fringe extends from the first minimum on one side to the first minimum on the other side. Since the first minima are at

$$
y_1 \approx \frac{\lambda L}{a}
$$

the full width of the central maximum is approximately

$$
2y_1 \approx \frac{2\lambda L}{a}
$$

This shows an important feature of diffraction. A narrower slit gives a wider diffraction pattern. That means reducing $a$ increases the spreading.

A smaller slit width produces greater diffraction spreading.
Since
$$
y_m \propto \frac{1}{a},
$$
the pattern becomes wider when $a$ becomes smaller.

Bright Fringes and Intensity

The central maximum is the brightest part of the pattern. The secondary bright fringes on the sides are much weaker. Their exact positions are not given by such a simple formula as the minima.

The intensity distribution for single-slit diffraction is

$$
I(\theta) = I_0 \left(\frac{\sin \beta}{\beta}\right)^2
$$

where

$$
\beta = \frac{\pi a \sin\theta}{\lambda}
$$

Here, $I_0$ is the maximum intensity at the center, where $\theta = 0$.

The minima occur when $\sin\beta = 0$, except at $\beta = 0$. This gives

$$
\beta = m\pi
$$

which leads again to

$$
a\sin\theta = m\lambda
$$

Features of the Pattern

Single-slit diffraction has a characteristic appearance that distinguishes it from other interference patterns. The central maximum is much wider than the side maxima. Also, the side maxima decrease in intensity as they move away from the center.

The table below summarizes the main features.

Feature	Single-slit diffraction pattern
Center	Brightest region
Central maximum width	Twice the distance from center to first minimum
Side maxima	Much weaker than central maximum
Minima condition	$a\sin\theta = m\lambda$
Effect of increasing slit width	Pattern becomes narrower
Effect of increasing wavelength	Pattern becomes wider

Physical Interpretation

Why does a narrow slit spread the wave more strongly? A wave confined to a small opening cannot remain tightly confined afterward. The wave emerging from the slit bends into the region behind the barrier, and the narrower the opening, the stronger this effect.

This is one of the key differences between wave behavior and simple ray behavior. Geometrical optics works well when apertures are much larger than the wavelength. Diffraction becomes important when the opening size is not much larger than $\lambda$.

Example Formula Use

Suppose light of wavelength $\lambda = 600 \text{ nm}$ passes through a slit of width $a = 0.30 \text{ mm}$, and the screen is $L = 2.0 \text{ m}$ away. The first minimum is at

$$
y_1 \approx \frac{\lambda L}{a}
$$

Substitute the values in SI units:

$$
\lambda = 6.0 \times 10^{-7}\text{ m}, \qquad
a = 3.0 \times 10^{-4}\text{ m}
$$

Then

$$
y_1 \approx \frac{(6.0 \times 10^{-7})(2.0)}{3.0 \times 10^{-4}}
= 4.0 \times 10^{-3}\text{ m}
$$

So the first minimum is about

$$
y_1 = 4.0 \text{ mm}
$$

from the center. The central maximum width is about

$$
2y_1 = 8.0 \text{ mm}
$$

Visual Form of the Pattern

A simple sketch of the intensity pattern is shown below.

Rendering scientific sketch…

Intensity pattern for single-slit diffraction

Practical Importance

Single-slit diffraction matters whenever waves pass through small openings. It is important in optics, microscopy, imaging systems, and many measurement techniques. It also places limits on how sharply beams can be confined and how well details can be resolved.

Even though the mathematics can become more advanced, the main idea is simple. A slit does not merely let light through. It also reshapes the wave, producing a distinctive diffraction pattern.

Key results for single-slit diffraction:
$$
a\sin\theta = m\lambda, \qquad m=1,2,3,\dots
$$
$$
y_m \approx \frac{m\lambda L}{a}
$$
$$
\text{central maximum width} \approx \frac{2\lambda L}{a}
$$
Narrower slit, wider diffraction pattern.

Final Picture

Single-slit diffraction is the spreading and interference of light after passing through one narrow opening. The central bright region is the widest and strongest part of the pattern. Dark fringes occur at angles determined by the slit width and wavelength. This makes single-slit diffraction one of the clearest demonstrations that light behaves as a wave.

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