Two identical horizontal sheets of glass have a thin film of...

Two identical horizontal sheets of glass have a thin film of air of thickness $t$ between them. The glass has refractive index $1.40 .$ The thickness $t$ of the air layer can be varied. Light with wavelength $\lambda$ in air is at normal incidence onto the top of the air film. There is constructive interference between the light reflected at the top and bottom surfaces of the air film when its thickness is $650 \mathrm{nm} .$ For the same wavelength of light the next larger thickness for which there is constructive interference is $910 \mathrm{nm}$. (a) What is the wavelength $\lambda$ of the light when it is traveling in air? (b) What is the smallest thickness $t$ of the air film for which there is constructive interference for this wavelength of light?

Key Concepts

Interference Order and Constructive Interference Condition

For interference to be constructive, the phase difference between the two reflecting waves must be an integer multiple of 2?. In cases where one reflection involves a ? phase shift, the condition for constructive interference is modified to 2t = (m + 1/2)?, where m is a nonnegative integer (the order of interference) and ? is the wavelength in the film (or in air if the film is air). This condition relates the film thickness, the wavelength of light, and the order of the interference pattern.

Optical Path Difference

The optical path difference (OPD) between two interfering waves is the difference in the distances they travel, multiplied by the refractive index of the medium. In thin film interference, the OPD is typically 2t (twice the film thickness) for normal incidence, to which any phase changes upon reflection must be added. This OPD determines whether the light waves will interfere constructively or destructively.

Thin Film Interference

Thin film interference is the phenomenon where waves reflected from the top and bottom surfaces of a thin layer interfere with each other. This interference is highly sensitive to the film’s thickness, the wavelength of light, and the refractive indices of the media involved. It is the basis for colorful patterns seen in films, oil slicks, and other layered structures.

Phase Change upon Reflection

When light reflects from the boundary between two media of different refractive indices, it may acquire a phase shift. In particular, reflection from a medium of higher refractive index introduces a phase change of ? (half a wavelength), while reflection from a lower refractive index does not introduce a phase shift. This phase change is crucial in determining the conditions under which constructive or destructive interference occurs.

Please give Ace some feedback