A transparent film of thickness \(250 \mathrm{nm}\) and index of refraction of 1.40 is surrounded by air. What wavelength in a beam of white light at near- normal incidence to the film undergoes destructive interference when reflected?

Understanding the concept of optical path difference (OPD) is essential when studying the behavior of light waves as they travel through different mediums. OPD is the difference in the distance travelled by two waves when journeying from one point to another, taking into account the varying speeds as they pass through substances of different refractive indices.

OPD is crucial for predicting the patterns of interference as it determines how waves will combine when they meet; either constructively, resulting in increased amplitude, or destructively, leading to cancellation, depending on the phase relationship between them. In the context of thin films, like the film of thickness in our exercise, light waves reflecting off the top and bottom surfaces can interfere with one another.

The mathematical description of OPD for thin films is given by \( OPD = 2 \cdot n \cdot d \), where \(n\) is the index of refraction of the film, and \(d\) is its thickness. This formula accounts for the light traveling down and up through the film, essentially covering twice the thickness of the film, and also incorporates the slowing factor due to the film’s index of refraction.

The index of refraction, often denoted as \( n \), is a fundamental property of materials that quantifies how much they slow down light passing through them compared to the speed of light in a vacuum. This slowing effect is why we observe phenomena like bending of light when it enters water at an angle, an occurrence explained by Snell's Law.

Each material has a unique index of refraction that can change with factors like wavelength and temperature. The higher the index, the more the light will slow down, which influences the optical path length—a product of physical path length and the index of refraction. In our exercise, the index of refraction is given as 1.40 for the thin film, implying that light slows down 1.40 times more than it would in a vacuum as it travels through the film.

This property is integral to many applications ranging from corrective lenses in eyeglasses, which bend light to aid vision, to the design of anti-reflective coatings used in optical devices, connecting directly to how thin film interference operates.

Thin-film interference is a fascinating application of the principles of wave interference, where we observe bright and dark bands as a result of constructive and destructive interference. When light encounters a thin film, some of it reflects off the top surface while the rest enters the film, reflects off the bottom surface, and emerges to interfere with the light reflected from the top.

The colors we observe are due to different wavelengths of light meeting this interference criterion under different conditions. Hence, destructive interference, as in our problem scenario, occurs when the phase shift caused by reflection and the optical path difference leads to two light waves being completely out of phase.

Destructive interference in thin films can be used to calculate which wavelength of light will be 'cancelled out' upon reflection. For a film in air, such as soap bubbles or oil slicks, this effect is responsible for their iridescent appearance. In lenses and glasses, this principle is applied to reduce unwanted reflections by adding a thin anti-reflective coating. The thickness and index of the film are deliberately chosen to ensure that certain wavelengths undergo destructive interference, thus diminishing glare.