Show that the angular full width at half maximum of the central peak in a double-slit interference pattern is given by ∆θ = λ/2d if λ ≪ d.
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34. Wave Optics
Diffraction with Huygen's Principle
Problem 63
Textbook Question
A series of polarizers are each rotated 10° from the previous polarizer. Unpolarized light is incident on this series of polarizers. How many polarizers does the light have to go through before it is 1/6 of its original intensity?

1
Start by recalling Malus's Law, which states that the intensity of light passing through a polarizer is given by: , where is the initial intensity, is the angle between the light's polarization direction and the polarizer's axis, and is the transmitted intensity.
For unpolarized light incident on the first polarizer, the intensity is reduced by half. Thus, after the first polarizer, the intensity becomes .
For each subsequent polarizer, the transmitted intensity is reduced according to Malus's Law. If the angle between each polarizer is , the intensity after the second polarizer is . Similarly, after the third polarizer, the intensity is , and so on.
Generalize the formula for the intensity after polarizers. The intensity after the -th polarizer is given by: . Here, accounts for the number of additional polarizers after the first one.
Set the final intensity equal to and solve for . This gives the equation: . Simplify and solve for by taking the logarithm of both sides to isolate .

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Malus's Law
Malus's Law states that when polarized light passes through a polarizer, the intensity of the transmitted light is proportional to the cosine square of the angle between the light's polarization direction and the polarizer's axis. Mathematically, it is expressed as I = I0 * cos²(θ), where I0 is the initial intensity, I is the transmitted intensity, and θ is the angle between the light's polarization direction and the polarizer.
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Multiple Polarizers & Malus's Law
Unpolarized Light
Unpolarized light consists of waves vibrating in multiple planes perpendicular to the direction of propagation. When unpolarized light encounters a polarizer, it becomes polarized, with its intensity reduced according to the angle of the polarizer. The first polarizer will reduce the intensity of unpolarized light to half of its original value, regardless of the angle.
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Ray Nature of Light
Intensity Reduction through Multiple Polarizers
When light passes through multiple polarizers, the intensity reduction is cumulative. Each polarizer further reduces the intensity based on the angle between the light's polarization direction and the polarizer's axis. For a series of polarizers rotated by a constant angle, the intensity after n polarizers can be calculated using the formula I = I0 * (1/2) * cos²(θ)^(n-1), where θ is the angle between successive polarizers.
Recommended video:
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Multiple Polarizers & Malus's Law
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