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Chapter 35: Problem 37

The rhinestones in costume jewelry are glass with index of refraction \(1.50 .\) To make them more reflective, they are often coated with a layer of silicon monoxide of index of refraction 2.00 . What is the minimum coating thickness needed to ensure that light of wavelength \(560 \mathrm{nm}\) and of perpendicular incidence will be reflected from the two surfaces of the coating with fully constructive interference?

Short Answer

Expert verified
The minimum coating thickness is 70 nm.

Step by step solution

01

Understanding Constructive Interference

To achieve fully constructive interference for the reflected light, the path difference between the reflected light beams from the top and bottom surfaces of the coating should be equal to an integer multiple of the wavelength in the medium. Here, we're looking for the minimum thickness, which corresponds to the first-order constructive interference.
02

Determine Effective Wavelength Inside Silicon Monoxide

The wavelength of light inside the silicon monoxide layer is shorter than in air because of the refractive index. It can be calculated using:\[ \lambda_{n} = \frac{\lambda_0}{n} \]where \( \lambda_0 = 560\, \text{nm} \) is the wavelength in air, and \( n = 2.00 \) is the refractive index of silicon monoxide. Substituting, we get:\[ \lambda_{n} = \frac{560}{2.00} = 280\, \text{nm} \]
03

Calculate Minimum Thickness Using Path Difference

For constructive interference, the path difference must be an integer multiple (while not causing additional phase shifts) of the wavelength inside the medium. Since we're aiming for the first minimum thickness:\[ 2 t = \frac{\lambda_{n}}{2} \]which simplifies to:\[ t = \frac{\lambda_{n}}{4} \]Substituting \( \lambda_{n} = 280 \text{ nm} \), we find:\[ t = \frac{280}{4} = 70 \text{ nm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Index of Refraction
The index of refraction, often denoted as "n," is a measure of how much a ray of light slows down as it passes through a material. In essence, it compares the speed of light in a vacuum to the speed in another medium.

Some important points to remember about the index of refraction are:
  • The higher the index, the slower the light travels through that material.
  • Materials with a higher index of refraction are often more optically dense.
  • When light passes from a medium with a lower refractive index to one with a higher refractive index, it bends towards the normal line (the perpendicular line to the surface).
In the problem above, the rhinestones are coated with silicon monoxide, which has an index of refraction of 2.00. This is higher than that of the rhinestones themselves. This difference is key to manipulating light's behavior at the interface, like enhancing reflection through techniques such as creating constructive interference.
Constructive Interference
Constructive interference occurs when two or more waves combine to produce a wave of larger amplitude. In optics, this is essential for technologies like anti-reflective coatings and enhancing reflections.

Key aspects of constructive interference are:
  • The waves must be "in phase," meaning their peaks and troughs must align perfectly.
  • When light waves reflect from surfaces within a thin film, the reflected waves can interfere constructively.
  • For constructive interference, the path difference between the reflected waves must be an integer multiple of the effective wavelength within the medium.
Recognizing patterns of constructive interference can lead to effective design of materials, like in this exercise where the coating thickness is tuned to achieve maximum reflection for specific light wavelengths.
Wave Interference
Wave interference is a phenomenon that occurs when two or more waves overlap in space, resulting in a new wave pattern. The two main types of interference are constructive and destructive interference.

Here's what to know about wave interference:
  • Constructive interference happens when waves align to enhance each other, resulting in increased amplitude.
  • Destructive interference occurs when waves are out of phase, causing them to cancel each other out.
  • Interference is influenced by factors like the wavelength of the waves, the medium they travel through, and their phase relationship.
In the context of the exercise, the coating thickness affects how light waves interfere with each other upon reflection. Proper manipulation of these variables can lead to desirable optical effects like higher reflection efficiency.
Wavelength in Medium
The wavelength of light in a medium is different from its wavelength in a vacuum. When light enters a medium other than vacuum, its speed changes, which in turn alters its wavelength.

Important points about wavelength in a medium include:
  • The wavelength in a medium, denoted by \( \lambda_{n} \), is calculated by dividing the wavelength in vacuum \( \lambda_0 \) by the refractive index \( n \) of the medium: \( \lambda_{n} = \frac{\lambda_0}{n} \).
  • A higher index of refraction results in a shorter wavelength inside the medium, as light slows down.
  • This change is crucial for determining optical properties like interference patterns.
In our problem, knowing the wavelength of light within the silicon monoxide layer was essential for calculating the minimum thickness needed for constructive interference. By understanding how the index of refraction modifies wavelength, we can predict and control other properties, like the interference of light waves.

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Most popular questions from this chapter

Find the sum y of the following quantities: $$y_{1}=10 \sin \omega t \quad \text { and } \quad y_{2}=8.0 \sin \left(\omega t+30^{\circ}\right)$$

In a double-slit arrangement the slits are separated by a distance equal to 100 times the wavelength of the light passing through the slits. (a) What is the angular separation in radians between the central maximum and an adjacent maximum? (b) What is the distance between these maxima on a screen 50.0 cm from the slits?

A double-slit arrangement produces bright interference fringes for sodium light (a distinct yellow light at a wavelength of \(\lambda=589 \mathrm{nm}\) ). The fringes are angularly separated by \(0.30^{\circ}\) near the center of the pattern. What is the angular fringe separation if the entire arrangement is immersed in water, which has an index of refraction of \(1.33 ?\)

Three electromagnetic waves travel through a certain point \(P\) along an \(x\) axis. They are polarized parallel to a \(y\) axis, with the following variations in their amplitudes. Find their resultant at \(P\). $$\begin{array}{l} E_{1}=(10.0 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t\right] \\ E_{2}=(5.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t+45.0^{\circ}\right] \\ E_{3}=(5.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t-45.0^{\circ}\right] \end{array}$$

57 through 68 Transmission through thin layers. In Fig. \(35-43,\) light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3 . (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray \(r_{3}\) (the light does not reflect inside material 2 ) and \(r_{4}\) (the light reflects twice inside material 2). The waves of \(r_{3}\) and \(r_{4}\) interfere, and here we consider the type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table \(35-3\) refers to the indexes of refraction \(n_{1}, n_{2},\) and \(n_{3},\) the type of interference, the thin-layer thickness \(L\) in nanometers, and the wavelength \(\lambda\) in nanometers of the light as measured in air. Where \(\lambda\) is missing, give the wavelength that is in the visible range. Where \(L\) is missing, give the second least thickness or the third least thickness as indicated. Table 35-3 Problems 57 through 68: Transmission Through Thin Layers. See the setup for these problems. $$\begin{array}{llllllll} & n_{1} & n_{2} & n_{3} & \text { Type } & L & \lambda \\ \hline 62 & 1.68 & 1.59 & 1.50 & \max & 2 n \mathrm{~d} & 342 \end{array}$$
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