5.4 Single-Slit Diffraction and Intensity Distribution

Single-slit diffraction

Single-slit diffraction happens when light passes through a narrow opening, causing waves to spread out and interfere with each other. The resulting pattern has a bright central spot flanked by alternating dark and bright fringes that get progressively dimmer. Understanding this pattern and its intensity distribution is essential for predicting the behavior of optical instruments and their resolution limits.

Fundamentals of single-slit diffraction

Huygens' Principle is the key idea here. You treat every point along the slit as its own tiny source of secondary wavelets. These wavelets spread out in all directions and overlap, and where they meet they interfere constructively (bright regions) or destructively (dark regions). That interference is what builds the entire diffraction pattern.

• The pattern consists of a central maximum (the brightest spot) flanked by alternating dark and bright fringes of decreasing intensity.
• The slit width relative to the wavelength determines how much the light spreads. A narrower slit produces a wider diffraction pattern, and a wider slit produces a narrower one.
• You can observe diffraction in everyday situations: shadows cast by narrow objects, light squeezing through a barely open door, or water waves passing through a gap in a barrier.

Intensity distribution in single-slit diffraction

The brightness across the pattern is far from uniform. The central maximum dominates, containing about 84% of the total light intensity. The side maxima are much dimmer and fade quickly.

The intensity at any angle $\theta$ is given by:

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

where

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

and $I_0$ is the peak intensity at the center.

A few things to notice about this function:

• At $\theta = 0$, $\beta \to 0$ and the sinc function equals 1, so $I = I_0$. That's the central maximum.
• The pattern is symmetric about the central axis.
• The intensity of the $m$th-order secondary maximum falls off roughly as $1/m^2$, so the first side maximum is already only about 4.5% of $I_0$, and they drop fast from there.
• The angular half-width of the central maximum is approximately $\theta \approx \lambda / a$, so the full angular width is $\Delta\theta \approx 2\lambda / a$.

Applications and observations

• Spectrometers and monochromators rely on diffraction to separate wavelengths of light for analysis.
• Resolution limits: Diffraction sets a fundamental cap on how fine a detail microscopes and telescopes can resolve. This is why simply increasing magnification doesn't always help.
• X-ray diffraction uses the same principles to probe crystal structures at the atomic scale.
• Diffraction effects also matter in antenna design and acoustic engineering, anywhere waves pass through or around openings.

Minima positions in single-slit diffraction

Conditions for destructive interference

Dark fringes appear at angles where waves from different parts of the slit cancel each other completely. The standard approach is to mentally divide the slit into two equal halves. For every point in the top half, there's a corresponding point in the bottom half. If the path difference between each such pair equals $\lambda/2$, every wavelet finds a partner that cancels it, and you get total destructive interference at that angle.

This cancellation condition generalizes: minima occur whenever the total path difference across the full slit width equals a whole number of wavelengths (excluding zero, since that's the central maximum).

Derivation of the minima equation

Here's how to derive the condition step by step:

1. Consider light arriving at a distant screen at angle $\theta$ from the central axis.
2. The path difference between wavelets from the very top and very bottom of the slit is $a \sin \theta$, where $a$ is the slit width.
3. Divide the slit into two equal halves (each of width $a/2$). Pair each point in the top half with the corresponding point in the bottom half. The path difference for each pair is $(a/2)\sin\theta$.
4. Complete cancellation occurs when this path difference equals $\lambda/2$, giving $a \sin\theta = \lambda$ for the first minimum.
5. You can repeat this argument by dividing the slit into 4, 6, 8... equal strips, which yields the general condition:

$a \sin \theta = m\lambda, \quad m = \pm 1, \pm 2, \pm 3, \ldots$

Note that $m = 0$ is excluded because that corresponds to the central maximum, not a minimum.

Small-angle approximation: When $\theta$ is small (which it often is in practice), $\sin\theta \approx \theta$ (in radians), so the equation simplifies to $\theta_m \approx m\lambda / a$.

Practical applications of the minima equation

• Predicting dark fringe locations: Plug in $\lambda$, $a$, and $m$ to find exactly where each dark fringe falls.
• Measuring slit width: If you observe the diffraction pattern and measure the angles to the minima, you can solve for $a$. This is a common lab technique.
• The same physics applies to sound waves passing through doorways and water waves passing through harbor openings.

Central maximum and minima positions

Calculating central maximum width

The central maximum stretches from the first minimum on one side to the first minimum on the other. Since the first minima are at $m = \pm 1$:

1. From $a \sin\theta = \lambda$, the first minimum is at $\theta_1 = \arcsin(\lambda/a)$.
2. The full angular width of the central maximum is $\Delta\theta = 2\arcsin(\lambda/a)$, which for small angles simplifies to $\Delta\theta \approx 2\lambda/a$.
3. On a screen at distance $L$, the linear width of the central maximum is approximately:

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

The inverse relationship between slit width and central maximum width is worth remembering: halve the slit width, and the central bright fringe doubles in size.

Determining angular positions of minima

For higher-order minima:

$\theta_m = \arcsin\!\left(\frac{m\lambda}{a}\right), \quad m = \pm 1, \pm 2, \pm 3, \ldots$

• For small angles, use $\theta_m \approx m\lambda/a$.
• The spacing between adjacent minima increases slightly at higher orders because $\arcsin$ is nonlinear. At small angles the minima look roughly evenly spaced, but at larger angles they spread apart.
• The maximum observable order is limited by the requirement that $\sin\theta_m = m\lambda/a \leq 1$, so $m_{\max} = \lfloor a/\lambda \rfloor$.

Example: A slit of width $a = 5.0 \times 10^{-5}$ m illuminated by $\lambda = 500$ nm light. The first minimum is at $\theta_1 = \arcsin(500 \times 10^{-9} / 5.0 \times 10^{-5}) = \arcsin(0.01) \approx 0.57°$. On a screen 2.0 m away, the central maximum has a linear width of about $w \approx 2(2.0)(500 \times 10^{-9})/(5.0 \times 10^{-5}) = 0.04$ m, or 4.0 cm.

Practical considerations

• These relationships are central to understanding resolution limits in cameras, telescopes, and microscopes.
• In spectroscopy, knowing where minima fall helps identify wavelengths from observed patterns.
• The same math applies when designing any system where wave diffraction matters, from fiber optics to acoustic enclosures.

Factors influencing intensity distribution

Mathematical description of intensity distribution

The full intensity formula again:

$I(\theta) = I_0 \left(\frac{\sin \beta}{\beta}\right)^2, \quad \beta = \frac{\pi a \sin \theta}{\lambda}$

• Minima occur where $\sin\beta = 0$ but $\beta \neq 0$, i.e., $\beta = m\pi$, which recovers $a\sin\theta = m\lambda$.
• Secondary maxima fall roughly halfway between minima, near $\beta \approx (m + \tfrac{1}{2})\pi$. Their peak intensities are approximately $I_0 / [(m + \tfrac{1}{2})\pi]^2$. For the first secondary maximum ($m = 1$), that's about $I_0/22 \approx 4.5\%$ of the central peak.
• $I_0$ sets the overall brightness but doesn't change the relative shape of the pattern.

Physical parameters affecting the diffraction pattern

One subtlety: the edges of the slit matter too. Sharp, well-defined edges produce the cleanest diffraction patterns. Rough or rounded edges introduce extra scattering that blurs the fringes.

Practical implications

• Telescope and microscope designers must account for diffraction when choosing aperture sizes; the Rayleigh criterion for resolution comes directly from single-slit (and circular-aperture) diffraction.
• Diffraction-based devices like phase masks and diffractive optical elements are engineered by carefully controlling slit geometry and wavelength relationships.
• The same intensity distribution physics applies beyond visible light: electron microscopy, neutron diffraction, and radio-wave antenna design all use these principles.