Geometrical Optics: Explaining the Effects of Small Wavelengths

[Anish Joshi]

TL;DR

Read that Ray Optics explains reflection, refraction, formation of shadows, as the wavelength of light is negligible compared to day-to-day objects. Want to understand why Exactly?

Read this in my textbook:-
The reason Geometrical optics works in case of formation of shadows, reflection and rarefaction is that the wavelength of light is much smaller compared to the reflecting/refracting surfaces as well as shadow causing objects that we use in day-to-day life.

I understand how the wavelength is negligible compared to the size of the object. But I want to understand the mechnaism - why and how does the small wavelength matter here?

First question here, so I apologize if there are any mistakes!

 

[DaveE]

OK, then you'll want to study "Diffraction" which describes how waves propagate through/past restrictions (like the diameter of a lens). You can also study "Huygens Principle". Also look for videos of diffraction of water waves in a wave tank. That's classically very similar to what light does.

Basically, shadows (or light beams) always have fuzzy edges. If the object is small enough all you get is fuzz. If you have a big object, no one cares about a relatively small amount of fuzziness at the edges.

 

[vanhees71]

Ray optics can be derived from wave optics (i.e., Maxwell's equations) by doing the so-called eikonal approximation (also known as WKB approximation in the context of quantum-mechanical wave mechanics). A good treatment can be found in Sommerfeld, Lectures on Theoretical Physics, vol. 4 (Optics).

 

[tech99]

TL;DR Summary:

I understand how the wavelength is negligible compared to the size of the object. But I want to understand the mechnaism - why and how does the small wavelength matter here?

To understand an effect of using a shorter wavelength, consider a parabolic reflector in a 1m diameter searchlight. If we can use a very tiny light source, such as a carbon arc, we can obtain a beam which is almost parallel - it is mainly controlled by the geometry and the ray diagram. But now use the same reflector for microwaves, wavelength say 3 cm. The beam now widens to approx 2 degrees. If we increase the wavelength to 30cm it now becomes 20 degrees. When we move a detector away from the axis of the beam, we lose signal because there is a phase error in the radiation from opposite sides of the reflector. When the wavelength is short, this phase error is greater for a given angle off beam. At the longer wavelengths, the geometry no longer gives a reliable answer for the beamwidth.