A good quality camera "lens" is actually a system of lenses, rather than a single lens, but a side effect is that a reflection from the surface of one lens can bounce around many times within the system, creating artifacts in the photograph. To counteract this problem, one of the lenses in such a system is coated with a thin layer of material \((n=1.28)\) on one side. The index of refraction of the lens glass is \(1.68 .\) What is the smallest thickness of the coating that reduces the reflection at \(640 \mathrm{nm}\) by destructive interference? (In other words, the coating's effect is to be optimized for \(\lambda=640 \mathrm{nm} .)\)

The phenomenon of destructive interference is intrinsic to the nature of wave interactions. Imagine two waves traveling along the same path but reaching different points at varying times due to a thin film. A camera's lens, for instance, may exhibit reflections due to this interfering effect.

For such interference to be termed 'destructive,' the crest of one wave aligns with the trough of the other, effectively canceling each other out. This is highly desirable in photographic equipment to prevent unwanted light artifacts. The condition for this to occur is that the path difference between the two interfering waves must be an odd multiple of half their wavelength, mathematically expressed as \(2m+1) \frac{\lambda}{2}\), with \(m\) representing an integer.

Understanding optical path difference is crucial when analyzing thin-film interference. It's about how far one wave has traveled relative to another. In physical terms, it is defined by the physical thickness of the film and the refractive indices of the media it traverses.

When light passes through media with different indices of refraction, its speed changes, affecting the distance it travels over the same time period. This is why the optical path length is calculated by the product of the thickness of the film and the index of refraction. Notably, in our camera lens scenario, the optical path difference is a decisive factor in achieving the exact interference needed to minimize reflections.

The index of refraction, denoted by \(n\), is a measure of how much light bends when it enters a material from another medium, like air. It’s a dimensionless number representing the ratio of the speed of light in a vacuum to its speed in the material. In thin-film interference, different indexes between the film and the air or glass cause a phase shift in the reflected light.

This phase shift is pivotal because, for destructive interference to occur, the condition is that the optical path difference has to equal half a wavelength (or an odd multiple of it). The indices of refraction play a role in how the light wave's speed is altered as it moves through different materials, directly influencing the wavelength and phase of the wave.

When dealing with photography or optics, reflective light artifacts can be a significant nuisance. These artifacts, also known as ghost images, are unwanted visual effects typically caused by reflections within lens systems. They appear as light or dark spots, streaks, or halos in photographs, reducing image clarity and contrast.

In lenses, the solution is often a carefully engineered thin film with a specific thickness that causes destructive interference at particular wavelengths of light—usually those most problematic for the camera sensor or film. By customizing the thickness of the anti-reflective coating based on precise calculations involving destructive interference, one can significantly reduce the occurrence of these reflective light artifacts.