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Chapter 26: Problem 19

The lenses of a particular set of binoculars have a coating with index of refraction \(n=1.38,\) and the glass itself has \(n=1.52 .\) If the lenses reflect a wavelength of \(525 \mathrm{nm}\) the most strongly, what is the minimum thickness of the coating?

Short Answer

Expert verified
The minimum thickness of the coating is approximately 95.11 nm.

Step by step solution

01

Identify the Relevant Formula

In thin film interference, the condition for constructive interference in reflected light when the film is on a substrate with higher refractive index is: \(2nt = m\lambda\), where \(n\) is the refractive index of the film, \(t\) is the thickness of the film, \(m\) is the order of interference (an integer), and \(\lambda\) is the wavelength of light in vacuum. We use \(m = 1\) for the minimum thickness.
02

Adjust the Wavelength for the Medium

Find the effective wavelength of the light within the coating by dividing the vacuum wavelength by the index of refraction of the coating: \(\lambda_n = \frac{\lambda}{n} = \frac{525 \text{ nm}}{1.38} \approx 380.43 \text{ nm}\).
03

Solve for the Thickness

Using the formula \(2nt = (m + \frac{1}{2})\lambda\) for constructive interference with a coating on a denser substrate (since \(n_2 > n_{1,coating}\): \(t = \frac{(m + \frac{1}{2})\lambda_n}{2n}\). Substitute \(m = 0, n = 1.38,\) and \(\lambda = 525 \text{ nm}\): \[ t = \frac{(0 + \frac{1}{2}) \cdot 525 \text{ nm}}{2 \times 1.38} \approx 95.11 \text{ nm} \]

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Key Concepts

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

Refractive Index
The refractive index is a numerical value that indicates how much light slows down when passing through a medium. It's calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium. For example, if the refractive index of the coating on a lens is 1.38, this means light travels 1.38 times slower in the coating than it would in a vacuum.
This concept is crucial in optical technologies, such as lenses and coatings, as it affects the bending and reflection of light. When light enters a new medium at an angle, its path bends according to the refractive index, leading to various optical phenomena.
  • A higher refractive index means more bending of light.
  • Different materials have different indices, influencing how they are used in optics.
Constructive Interference
Constructive interference is a principle in wave physics where two or more waves overlap to produce a wave with a greater amplitude. In the context of thin film interference, it occurs when the path difference between reflected light waves is an integer multiple of the wavelength. This results in amplified reflected light at certain wavelengths.
In the case of the binocular lens coating, the condition for constructive interference gives the lens its anti-reflective properties by canceling specific wavelengths.
The formula for constructive interference in a thin film with a higher substrate index is:
  • For a film coating, use: \(2nt = m\lambda\), where \(m\) is the order.
  • This formula helps determine film thickness for minimal reflective loss.
  • Correct thickness can virtually eliminate reflections at specific wavelengths.
Binocular Lens Coating
Binocular lens coatings are applied to surfaces to improve image clarity and reduce glare. They are designed using thin film physics principles to enhance light transmission while minimizing reflections. By selectively interfering with certain wavelengths, coatings optimize the visual experience.
The exercise problem's lens coating uses constructive interference to target a wavelength of 525 nm, common in green light, making the reflection of this color minimal and improving optical quality.
  • Lens coatings use physics to boost contrast and color fidelity.
  • They work by carefully designing thickness and materials, considering the refractive index.
  • Binoculars benefit from these coatings, offering clearer, brighter images.

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Most popular questions from this chapter

Coherent light from a sodium-vapor lamp is passed through a filter that blocks everything except for light of a single wavelength. It then falls on two slits separated by \(0.460 \mathrm{~mm}\). In the resulting interference pattern on a screen \(2.20 \mathrm{~m}\) away, adjacent bright fringes are separated by \(2.82 \mathrm{~mm}\). What is the wavelength of the light that falls on the slits?

A plate of glass \(9.00 \mathrm{~cm}\) long is placed in contact with a second plate and is held at a small angle with it by a metal strip \(0.0800 \mathrm{~mm}\) thick placed under one end. The space between the plates is filled with air. The glass is illuminated from above with light having a wavelength in air of \(656 \mathrm{nm}\). How many interference fringes are observed per centimeter in the reflected light?

The lead architect on the design team of a new skyscraper decides that the outer surface of the building's windows needs to be coated with a reflective layer in order to lower airconditioning costs. \(\mathrm{MgF}_{2}\) is chosen for the reflective layer, which has a refractive index \(n=1.38 .\) Assume that the refractive index of the window glass is 1.5 and that the reflective coating should work optimally at the peak of the sunlight spectrum, \(500 \mathrm{nm}\). (a) What is the minimum thickness of film that is needed? (b) Suppose the layer needs to be thicker than \(200 \mathrm{nm}\) in order to hold up to the weather. What other thicknesses would also work? Give only the three thinnest ones.

A reflecting telescope is used to observe two distant point sources that are \(2.50 \mathrm{~m}\) apart with light of wavelength \(600 \mathrm{nm} .\) The telescope's mirror has a radius of \(3.5 \mathrm{~cm} .\) What is the maximum distance in meters at which the two sources may be distinguished?

A researcher measures the thickness of a layer of benzene \((n=1.50)\) floating on water by shining monochromatic light onto the film and varying the wavelength of the light. She finds that light of wavelength \(575 \mathrm{nm}\) is reflected most strongly from the film. What does she calculate for the minimum thickness of the film?
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