When viewing a piece of art that is behind glass, one often...

When viewing a piece of art that is behind glass, one often is affected by the light that is reflected off the front of the glass (called glare), which can make it difficult to see the art clearly. One solution is to coat the outer surface of the glass with a film to cancel part of the glare. (a) If the glass has a refractive index of 1.62 and you use $\mathrm{TiO}_{2}$, which has an index of refraction of 2.62 , as the coating, what is the minimum film thickness that will cancel light of wavelength $505 \mathrm{nm} ?$ (b) If this coating is too thin to stand up to wear, what other thickness would also work? Find only the three thinnest ones.

Key Concepts

Optical Path Length

Optical path length is the product of the physical thickness of a material and its refractive index. In the context of thin film interference, it allows us to equate physical thickness with phase shift, facilitating the calculation of the interference conditions necessary for achieving phenomena like glare cancellation.

Phase Change on Reflection

When light reflects from an interface where it travels from a medium of lower refractive index to a medium of higher refractive index, it incurs a phase shift of ? (a half wavelength). This phase change is an essential factor in determining whether interference between reflected waves is constructive or destructive.

Thin Film Interference

Thin film interference arises from the superposition of light waves reflected from the outer and inner boundaries of a thin layer. The resulting interference—constructive or destructive—depends on the optical path difference between the waves, which is determined by the film's thickness, the wavelength of light, and the refractive indices of the materials involved.

Refractive Index

The refractive index of a material defines how much the speed of light is reduced inside the medium compared to vacuum. This parameter is critical in determining the wavelength of light within the film, as well as the phase changes upon reflection, all of which influence the interference conditions.

Quarter-Wave Coating

A quarter-wave coating is a design strategy in which the film's optical thickness is set to one quarter of the target wavelength. This specific thickness is chosen because it causes the light reflected from the top and bottom surfaces of the film to interfere destructively, thereby minimizing unwanted reflections such as glare.

Transcript

00:01 Here in this question, we know that the light through the film, light should undergo destructive interference condition.

00:11 And for destructive interference in the film, we can express the relation as 2 multiply by n, multiply by t equal to m multiply by lambda.

00:27 Now here n is the refractive index of the material, t is the thickness, m is the order of the fringe, and lambda is the wavelength.

00:39 So now comes to part a of this question.

00:43 So here we are going to find out the minimum thickness of the film that will cancel the light of wavelength 505 nanometer.

00:55 So here to calculate the minimum thickness, of the film we are using the above relation so for this we know that for minimum thickness the order of frames that is m must be equal to one so now we substitute 2 .62 for n 1 for m and 505 nanometer for lambda in the equation 2 multiply by n multiplied by t equal m multiply by lambda so we get 2 multiplied by 2 .62 multiply by t equals to 1 multiplied by 505 nanometer so now let us rearrange this for t so we get t equals to 1 multiplied by 500 5 nanometer so now let us rearrange this for t so we get t equals to 5 505 divided by 2 multiply by 2 .62.

02:20 So we get t equals to 96 .4 nanometer.

02:29 Thus, the minimum thickness of the film that will cancel light of wavelength 505 nanometer is equals to 96 .4 nanometer.

02:41 Now comes to part p of this question.

02:45 So here, to calculate the minimum thickness of the film for the three thinnest cases, we again use the relation to multiply by n, multiply by t equal to m multiply by lambda...

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