Single Slit Diffraction and Resolution HL

Understanding how light waves bend around obstacles—a phenomenon called diffraction—is crucial for explaining why optical instruments have fundamental limits. In IB Physics HL, you move beyond geometric optics to explore wave optics, where single slit diffraction dictates the detail a telescope or microscope can reveal. Mastering this links the mathematics of interference patterns to the practical performance of all imaging systems, from your eye to the Hubble Space Telescope.

The Foundation: Single Slit Diffraction

When a plane wave of light passes through a narrow slit, it doesn't simply project a sharp image of the slit. Instead, the wavefront acts as a new set of point sources, and the light spreads out or diffracts. This spreading is most pronounced when the slit width $a$ is comparable to the wavelength $\lambda$ of the light. The resulting pattern on a distant screen is not a single bright line but a broad, bright central maximum flanked by alternating dark and bright fringes of decreasing intensity.

This occurs due to Huygens' principle, where every point on the wavefront within the slit acts as a source of secondary wavelets. These wavelets interfere with each other, constructively in some directions and destructively in others. The key to predicting the pattern is to consider the path difference between wavelets originating from different parts of the slit.

Calculating the Angular Position of Minima

The condition for complete destructive interference (a dark fringe or minimum) is derived by splitting the slit into two halves. Consider light diffracting at an angle $\theta$ from the central axis. The path difference between a wavelet from the top edge of the slit and one from the center is $(a/2)\sin \theta$. For these two wavelets to cancel, this path difference must be half a wavelength: $(a/2)\sin \theta =\lambda /2$. Simplifying gives the general condition for minima:

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

Here, $a$ is the slit width, $\lambda$ is the wavelength, $\theta$ is the angular position of the minimum from the central axis, and $m$ is the order number (note: $m=0$ is the central maximum, not a minimum). For the first minimum ($m=1$), $\sin {\theta }_1=\lambda /a$. This equation is fundamental: it tells you the angular spread of the central maximum. A smaller slit width $a$ produces a larger ${\theta }_1$, meaning the light diffracts more widely.

Worked Example: Monochromatic light of wavelength 600 nm passes through a slit of width 0.10 mm. Calculate the angular position of the first minimum.

1. Convert to consistent units: $\lambda =600\times 10^{-9}$ m, $a=0.10\times 10^{-3}$ m.
2. Apply the minima condition: $\sin {\theta }_1=\lambda /a=(600\times 10^{-9})/(0.10\times 10^{-3})=6.0\times 10^{-3}$.
3. Therefore, ${\theta }_1={\sin }^{-1}(6.0\times 10^{-3})\approx 0.34^{\circ }$.

The small angle approximation ($\sin \theta \approx \theta$ in radians) is often valid here, giving ${\theta }_1\approx 6.0\times 10^{-3}$ rad.

Analysing the Intensity Pattern

The intensity pattern from single slit diffraction is not a series of equally bright fringes like in a double-slit experiment. Instead, intensity falls off dramatically. The central maximum is the brightest and twice as wide (angularly) as any secondary maximum. The intensity $I$ at an angle $\theta$ is given by:

$$I=I_0{\left[\frac{\sin \beta }{\beta }\right]}^2\text{where}\beta =\frac{\pi a\sin \theta }{\lambda }$$

Here, $I_0$ is the intensity at the center ($\theta =0$). The function $(\sin \beta /\beta {)}^2$ describes the modulation. Minima occur precisely where $\sin \theta =m\lambda /a$, making $\beta =m\pi$ and $\sin \beta =0$. Secondary maxima occur approximately halfway between minima but are much dimmer. Slit width $a$ directly controls the pattern's scale: a wider slit ($a\gg \lambda$) produces a very narrow central maximum (minimal diffraction), while a narrower slit creates a broad, diffuse pattern.

The Rayleigh Criterion and Angular Resolution

Diffraction fundamentally limits an instrument's ability to distinguish two close objects. Consider a telescope observing two distant stars. Each star's image is not a point but a circular diffraction pattern (Airy disc) caused by the aperture's edge acting like a circular slit. When two such patterns are too close, they overlap and become indistinguishable.

The Rayleigh criterion provides a practical limit for resolution. It states that two point sources are just resolvable when the central maximum of one diffraction pattern coincides with the first minimum of the other. For a circular aperture of diameter $D$, the minimum angular separation ${\theta }_R$ (in radians) at which two objects can be resolved is:

$${\theta }_R=1.22\frac{\lambda }{D}$$

This is the minimum angular resolution. The factor 1.22 arises from the mathematics of a circular aperture. The smaller ${\theta }_R$, the better the resolving power of the instrument. This criterion applies to any optical system, including microscopes (where $D$ relates to the objective lens diameter) and your eye (where $D$ is the pupil diameter).

Worked Example: A space telescope has a primary mirror diameter of 2.4 m. For light of wavelength 500 nm, what is its theoretical angular resolution?

1. $D=2.4$ m, $\lambda =500\times 10^{-9}$ m.
2. ${\theta }_R=1.22\times (500\times 10^{-9})/2.4\approx 2.54\times 10^{-7}$ rad.
3. In arcseconds (1 rad ≈ 206265 arcsec): ${\theta }_R\approx 0.052$ arcseconds. This sets the fundamental limit of its detail.

Factors Affecting Resolving Power

The resolving power of an instrument, its ability to reveal fine detail, depends directly on the Rayleigh criterion formula ${\theta }_R=1.22\lambda /D$.

1. Aperture Diameter ($D$): Resolving power improves (θ_R decreases) with a larger aperture. This is why astronomical telescopes are built with the largest possible mirrors or lenses. Doubling the diameter halves the minimum resolvable angle, allowing the instrument to see twice the detail.
2. Wavelength of Light ($\lambda$): Resolving power improves with a shorter wavelength. This is a key advantage of electron microscopes, which use de Broglie wavelengths thousands of times shorter than visible light, achieving phenomenal resolution. In optics, using a blue filter ($\lambda \approx 450$ nm) provides better resolution than a red filter ($\lambda \approx 650$ nm) with the same aperture.
3. The Medium: The formula uses the wavelength in the medium. In oil immersion microscopy, oil reduces the wavelength of light (${\lambda }_{medium}={\lambda }_{vacuum}/n$), thereby improving resolution compared to air.

It's critical to distinguish resolution from magnification. Empty magnification enlarges the blurry diffraction patterns without revealing new detail. True useful magnification is limited by the diffraction limit set by ${\theta }_R$.

Common Pitfalls

Confusing the conditions for single-slit minima and double-slit maxima.

• Pitfall: Using $d\sin \theta =m\lambda$ (the double-slit maxima condition) for single-slit calculations.
• Correction: Remember, for a single slit, $a\sin \theta =m\lambda$ gives the locations of the minima (dark fringes). The central maximum is at $m=0$.

Misapplying the small-angle approximation.

• Pitfall: Using $\theta \approx \sin \theta \approx \tan \theta$ without checking if the angle is indeed small (often < 10° or ~0.17 rad).
• Correction: Always calculate $\sin \theta$ first. If the value is less than about 0.17, the approximation is valid. If not, you must use the inverse sine function to find $\theta$.

Forgetting the factor of 1.22 for circular apertures.

• Pitfall: Using the single-slit formula $\theta =\lambda /D$ for the resolution of a telescope or microscope.
• Correction: The Rayleigh criterion for a circular aperture is ${\theta }_R=1.22\lambda /D$. Omitting the 1.22 is incorrect and will lose you marks in an IB exam.

Equating higher magnification with better resolution.

• Pitfall: Believing that magnifying an image further will always allow you to see more detail.
• Correction: Resolution is fundamentally limited by diffraction (${\theta }_R$). Beyond a certain point, magnification only makes the blurred diffraction pattern bigger without separating details.

Summary

• Single slit diffraction produces a pattern with a broad central maximum and weaker secondary maxima, described by the minima condition $a\sin \theta =m\lambda$ for $m=\pm 1,\pm 2,...$.
• The slit width $a$ is inversely related to the diffraction angle: narrower slits cause greater spreading of the light wave.
• The Rayleigh criterion ${\theta }_R=1.22\lambda /D$ defines the minimum angular separation at which two point sources can be distinguished, setting the diffraction limit for any optical instrument.
• Resolving power improves with a larger aperture $D$ or a shorter wavelength $\lambda$, which minimizes ${\theta }_R$.
• Ultimate detail is limited by physical optics (diffraction), not just geometric optics (magnification); increasing magnification past the diffraction limit yields no new information.