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Chapter 25: Problem 69

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

Expert verified
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

01

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.
02

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.
03

Set up the constructive interference condition

For constructive interference, the path difference must be an integer multiple of the wavelength: \(2n_2t = m\lambda\)
04

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.

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Most popular questions from this chapter

A Michelson interferometer is set up using white light. The arms are adjusted so that a bright white spot appears on the screen (constructive interference for all wavelengths). A slab of glass \((n=1.46)\) is inserted into one of the arms. To return to the white spot, the mirror in the other arm is moved $6.73 \mathrm{cm} .$ (a) Is the mirror moved in or out? Explain. (b) What is the thickness of the slab of glass?
Two incoherent EM waves of intensities \(9 I_{0}\) and \(16 I_{0}\) travel in the same direction in the same region of space. What is the intensity of EM radiation in this region?
A grating \(1.600 \mathrm{cm}\) wide has exactly 12000 slits. The grating is used to resolve two nearly equal wavelengths in a light source: \(\lambda_{\mathrm{a}}=440.000 \mathrm{nm}\) and $\lambda_{\mathrm{b}}=440.936 \mathrm{nm}$ (a) How many orders of the lines can be seen with the grating? (b) What is the angular separation \(\theta_{b}-\theta_{a}\) between the lines in each order? (c) Which order best resolves the two lines? Explain.
Roger is in a ship offshore and listening to a baseball game on his radio. He notices that there is destructive interference when seaplanes from the nearby Coast Guard station are flying directly overhead at elevations of $780 \mathrm{m}, 975 \mathrm{m},\( and \)1170 \mathrm{m} .$ The broadcast station is \(102 \mathrm{km}\) away. Assume there is a \(180^{\circ}\) phase shift when the EM waves reflect from the seaplanes. What is the frequency of the broadcast?
A thin film of oil \((n=1.50)\) is spread over a puddle of water \((n=1.33) .\) In a region where the film looks red from directly above $(\lambda=630 \mathrm{nm}),$ what is the minimum possible thickness of the film? (tutorial: thin film).
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