How Does the Diffraction Limit Change If the Wavelength Observed Increases?

If the wavelength observed increases, the diffraction limit becomes larger, meaning the ability to resolve fine details decreases.

Understanding the Diffraction Limit

The diffraction limit is a fundamental constraint on the resolution of optical instruments like telescopes and microscopes. It arises because light waves spread out when they pass through an aperture (like a lens or a slit), a phenomenon known as diffraction. This spreading causes point sources of light to appear as blurred patterns (Airy disks) rather than sharp points.

The ability to distinguish between two nearby objects is limited by the extent of this spreading. The diffraction limit is often quantified by the minimum angular separation ($\theta$) between two objects that can be resolved, typically given by the Rayleigh criterion:

$\theta \approx 1.22 \frac{\lambda}{D}$ (for a circular aperture)

Where:

• $\theta$ is the minimum resolvable angle (in radians)
• $\lambda$ is the wavelength of light
• $D$ is the diameter of the aperture

For a simple slit, the formula is similar but slightly different:

$\theta \approx \frac{\lambda}{D}$ (for a slit of width D)

This minimum angle represents the diffraction limit on resolution. A smaller value of $\theta$ means better resolution (you can distinguish objects that are closer together). A larger value of $\theta$ means poorer resolution (you need a larger separation to distinguish objects).

The Impact of Increasing Wavelength

Looking at the formulas above, it's clear that the minimum resolvable angle $\theta$ is directly proportional to the wavelength $\lambda$.

• If the wavelength ($\lambda$) increases, the minimum resolvable angle ($\theta$) increases.
• A larger minimum resolvable angle ($\theta$) means that objects need to be further apart to be distinguished.

Therefore, increasing the wavelength worsens the resolution limited by diffraction.

In simple terms: Longer wavelengths of light diffract more when passing through an aperture of a given size. This greater spreading makes it harder to distinguish between closely spaced points of light.

Practical Implications and the Reference

This relationship has significant practical implications in various fields:

• Microscopy: To see finer details, microscopes often use shorter wavelengths (like UV light or electron beams in electron microscopes) because they offer better resolution than visible light.
• Telescopy: Radio telescopes, which observe much longer wavelengths than optical telescopes, often require very large dishes or arrays of dishes (like interferometers) to achieve comparable resolution to smaller optical telescopes. The larger diameter ($D$) helps counteract the effect of the larger wavelength ($\lambda$) on the resolution ($\theta \approx \lambda/D$).
• Imaging Systems: Any imaging system limited by diffraction will see its resolution decrease if it switches to observing at longer wavelengths.

The provided reference highlights an extreme case of this principle: "If the wavelength is much larger than the width of a slit, again, no diffraction pattern will be observed. However, the slit now acts as a point source, i.e. the narrow opening becomes the source of a new wave (Huygen's principle)." This scenario describes what happens when the wavelength is so large relative to the aperture that the light simply spreads out in a broad, spherical wave from the opening, much like a single point source would. While "no diffraction pattern" might imply a lack of distinct maxima and minima, it represents a state of maximum diffraction spreading, where any potential fine detail is completely lost because the light from the aperture has spread out so broadly. This is consistent with the idea that a significantly larger wavelength drastically increases the diffraction angle and reduces the ability to resolve any structure.

Summary Table

Here's a quick summary of the relationship:

Increasing the wavelength directly leads to a larger diffraction limit angle, impairing the instrument's ability to resolve fine details.