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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

You are given a slide with two slits cut into it and asked how far apart the slits are. You shine white light on the slide and notice the first-order color spectrum that is created on a screen \(3.40 \mathrm{m}\) away. On the screen, the red light with a wavelength of 700 nm is separated from the violet light with a wavelength of 400 nm by 7.00 mm. What is the separation of the two slits?
When a double slit is illuminated with light of wavelength \(510 \mathrm{nm},\) the interference maxima on a screen \(2.4 \mathrm{m}\) away gradually decrease in intensity on either side of the 2.40 -cm-wide central maximum and reach a minimum in a spot where the fifth-order maximum is expected. (a) What is the width of the slits? (b) How far apart are the slits?
White light containing wavelengths from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) is shone through a grating. Assuming that at least part of the third-order spectrum is present, show that the second-and third-order spectra always overlap, regardless of the slit separation of the grating.
Two slits separated by \(20.0 \mu \mathrm{m}\) are illuminated by light of wavelength \(0.50 \mu \mathrm{m} .\) If the screen is \(8.0 \mathrm{m}\) from the slits, what is the distance between the \(m=0\) and \(m=1\) bright fringes?
A thin layer of an oil \((n=1.60)\) floats on top of water \((n=1.33) .\) One portion of this film appears green \((\lambda=510 \mathrm{nm})\) in reflected light. How thick is this portion of the film? Give the three smallest possibilities.
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