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Chapter 34: Problem 30

Sometimes thin films are used as filters to prevent certain colors from entering a lens. Consider an infrared filter, designed to prevent 800.0 -nm light from entering a lens. Find the minimum film thickness for a layer of \(\mathrm{MgF}_{2}\) \((n=1.38)\) to prevent this light from entering the lens.

Short Answer

Expert verified
Answer: The minimum thickness of the magnesium fluoride film is 435nm.

Step by step solution

01

Identify the Conditions for Destructive Interference

The film causes destructive interference for a particular wavelength when the total phase difference between the light waves reflecting from the upper and lower surfaces of the film is a half-integer multiple of the wavelength in air. In this problem, we want the destructive interference for 800nm light. The phase difference is given by \(2nt \cos{\theta_2} - m\lambda\), where \(n\) is the index of refraction of the film, \(t\) is the thickness of the film, \(\lambda\) is the wavelength in air, \(\theta_2\) is the angle of refraction inside the film, and \(m\) is an integer (usually 0 for normal incidence).
02

Determine the Angle of Refraction

Since the angle of incidence is not given in the problem, we can assume normal incidence. For normal incidence, the angle of incidence, \(\theta_1 = 0^\circ, \) and according to Snell's law, the angle of refraction \(\theta_2\) will also be \(0^\circ\). Therefore, the term \(\cos{\theta_2}\) will be 1.
03

Calculate the Required Thickness for Destructive Interference

We must find the minimum thickness \(t\) such that \(2nt - m\lambda = \dfrac{(2m-1)\lambda}{2}\), where \(m = 1, 2, 3, ...\). For the minimum thickness, we can choose \(m = 1\). Now, we can substitute the values of the variables: \(2 \times 1.38 \times t - 800.0 \times 10^{-9} = \dfrac{(2 - 1)800.0 \times 10^{-9}}{2}\) Solve for \(t\): \(2 \times 1.38 \times t = 800.0 \times 10^{-9} + \dfrac{800.0 \times 10^{-9} }{2}\) \(t = \dfrac{1200.0 \times 10^{-9}}{2 \times 1.38}\) \(t \approx 4.35 \times 10^{-7}\,\text{m}\)
04

Express the Answer in Nanometers

To express the answer in nanometers, multiply by \(10^{9}\): \(t \approx 4.35 \times 10^{-7}\,\text{m} \times 10^{9} = 435\,\text{nm}\) Therefore, the minimum thickness of the magnesium fluoride film that can prevent 800nm light from entering the lens is 435nm.

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Most popular questions from this chapter

The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance \(\Delta d,\) and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser \(\left(\lambda_{\mathrm{Red}}=632.8 \mathrm{nm}\right)\) is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance \(\Delta d\), a number \(\Delta N_{\text {Red }}=6.000 \cdot 10^{4}\) red fringes and \(\Delta N_{\text {unknown }}=7.780 \cdot 10^{4}\) fringes pass by the reference photodiode. a) Calculate the wavelength of the unknown laser. b) Calculate the displacement, \(\Delta d\), of the movable mirror.

In a single-slit diffraction pattern, there is a bright central maximum surrounded by successively dimmer higher-order maxima. Farther out from the central maximum, eventually no more maxima are observed. Is this because the remaining maxima are too dim? Or is there an upper limit to the number of maxima that can be observed, no matter how good the observer's eyes, for a given slit and light source?

In Young's double-slit experiment, both slits were illuminated by a laser beam and the interference pattern was observed on a screen. If the viewing screen is moved farther from the slit, what happens to the interference pattern? a) The pattern gets brighter. b) The pattern gets brighter and closer together. c) The pattern gets less bright and farther apart. d) There is no change in the pattern. e) The pattern becomes unfocused. f) The pattern disappears.

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