Question

Instead of an anti reflective coating, suppose you wanted to coat a glass surface to enhance the reflection of visible light. Assuming that $1<n_{\text {coating }}<n_{\text {glass }},$ what should the minimum thickness of the coating be to maximize the reflected intensity for wavelength \(\lambda ?\)

Short answer

Answer: The minimum thickness of the coating is given by the formula \(t = \dfrac{\lambda}{2n_2}\), where \(t\) is the thickness, \(\lambda\) is the wavelength of light, and \(n_2\) is the refractive index of the coating material.

Step-by-step solution

Identify the conditions for constructive interference

Consider that a beam of light of wavelength \(\lambda\) is incident on the coating. The light will partially refract and partially reflect. The reflected waves from the top and bottom surfaces of the coating will interfere with each other. We want to find the minimum thickness \(t\) of the coating that maximizes this interference for a given wavelength. For constructive interference, the path difference between the two reflected waves must be an integer multiple of the wavelength (\(m\lambda\)), where \(m\) is an integer.

Calculate the path difference

We will now calculate the path difference. Let \(n_2\) be the refractive index of the coating and \(t\) be its thickness. For simplicity, assume that the light is incident normally on the coating surface. The path difference between the two reflected waves can be calculated as \(2n_2t\), where \(n_2\) accounts for the refractive index and \(t\) is the thickness of the coating.

Set up the constructive interference condition

For constructive interference, the path difference must be an integer multiple of the wavelength: \(2n_2t = m\lambda\)

Solve for the minimum thickness

We're asked to determine the minimum thickness that leads to maximum reflection. This occurs when the path difference is equal to one wavelength, which corresponds to \(m = 1\). Therefore, we can rearrange the previous equation to find the thickness \(t\): \(t = \dfrac{\lambda}{2n_2}\) This is the minimum thickness of the coating that will maximize the reflected intensity for the given wavelength \(\lambda\). Note that this thickness depends on both the wavelength of light and the refractive index of the coating material.