Question

In a polarized light experiment, a setup similar to the one in Figure 31.23 is used. Unpolarized light with intensity \(I_{0}\) is incident on polarizer 1\. Polarizers 1 and 3 are crossed (at a \(90^{\circ}\) angle), and their orientations are fixed during the experiment. Initially, polarizer 2 has its polarizing angle at \(45^{\circ} .\) Then, at time \(t=0,\) polarizer 2 starts to rotate with angular velocity \(\omega\) about the direction of propagation of light in a clockwise direction as viewed by an observer looking toward the light source. A photodiode is used to monitor the intensity of the light emerging from polarizer \(3 .\) a) Determine an expression for this intensity as a function of time. b) How would the expression from part (a) change if polarizer 2 were rotated about an axis parallel to the direction of propagation of the light but displaced by a distance \(d

Short answer

Answer: The expression for the intensity of light emerging from polarizer 3 as a function of time is \(I_3 = \frac{I_0}{2} \cos^4{(45^\circ - \omega t)}\).

Step-by-step solution

Understand the polarizer setup

There are three polarizers: Polarizers 1 and 3 are fixed at a 90-degree angle (crossed) while polarizer 2 is initially at a 45-degree angle and starts rotating with angular velocity \(\omega\) about the direction of propagation of light in a clockwise direction as viewed by an observer looking toward the light source.

Determine the intensity after polarizer 1

Unpolarized light of intensity \(I_0\) is incident on polarizer 1. Only half of the light intensity passes through polarizer 1, meaning the new intensity after passing through polarizer 1 is: $$I_1 = \frac{I_0}{2}.$$

Determine the intensity after polarizer 2

As polarizer 2 starts rotating with an angular velocity of \(\omega\), its polarizing angle changes with time (\(\theta(t) = 45^\circ - \omega t\)). The intensity after passing through polarizer 2 can be found using Malus's Law: $$I_2 = I_1 \cos^2{\theta(t)}$$ Substitute the value of \(I_1\) and the expression for \(\theta(t)\): $$I_2 = \frac{I_0}{2} \cos^2{(45^\circ - \omega t)}.$$

Determine the intensity after polarizer 3

At this point, the light is already polarized, but polarizer 3 filters it more. Since polarizers 1 and 3 are crossed, we consider polarizer 3 as having a polarizing angle of 0 degrees, and the angle between polarizer 2 and polarizer 3 is \((45^\circ - \omega t)\). Apply Malus's Law again to find the intensity after polarizer 3: $$I_3 = I_2 \cos^2{(45^\circ - \omega t)},$$ Substitute the expression for \(I_2\) from step 3 and simplify: $$I_3 = \frac{I_0}{2} \cos^4{(45^\circ - \omega t)}.$$ This is the expression for the intensity of light emerging from polarizer 3 as a function of time.

Determine the change in the expression for the given rotation axis in part (b)

In part (b), polarizer 2 is rotating around an axis parallel to the direction of light propagation but displaced by a distance d. However, since the polarized light remains in the x-y plane, the axis of rotation will not affect Malus's Law. Thus, the expression for intensity after polarizer 3 remains unchanged: $$I_3 = \frac{I_0}{2} \cos^4{(45^\circ - \omega t)}.$$

Key concepts

Exploring Malus's Law

Malus's Law is a fundamental principle that dictates how the intensity of polarized light changes as it passes through a polarizing filter. When unpolarized light encounters a polarizer, the filter will only allow light waves aligned with its axis to pass through. This selective transmission creates polarized light, characterized by waves that oscillate in a single plane.

For an ideal polarizer, the relationship between the initial intensity, \(I_0\), of an unpolarized light source and the transmitted intensity, \(I\), after passing through the polarizer at an angle \(\theta\) to the polarization direction is expressed as:
\[I = I_0 \cos^2 \theta\]

This relationship is known as Malus's Law. It is crucial in the experiment described, where a rotating polarizer alters the angle \(\theta\) with time, leading to a time-dependent intensity variation behind the final polarizer.

Understanding Polarization of Light

Polarization of light is a phenomenon whereby certain light waves are filtered so that their electric field vectors oscillate along the same plane. In nature, light is usually unpolarized, which means its waves oscillate in multiple, random directions perpendicular to the direction of propagation.

In the given experiment with three polarizers, light starts as unpolarized at the first polarizer. Upon passing through it, the light becomes polarized, with its electric field vector vibrating in the plane parallel to the polarizer. The intensity of this partially polarized light is reduced according to the polarizer's orientation. As the second polarizer rotates, the angle of vibration of the electric field vector changes continuously, resulting in a dynamic alteration in the intensity of the transmitted light.

Polarization is essential for various applications, such as reducing glare in sunglasses, and in scientific instruments like the polarimeters used in this experiment.

Analyzing Angular Velocity

Angular velocity, denoted by the symbol \(\omega\), is a vector quantity that represents the rate at which an object rotates around an axis. It is a measure of the angle that an object, in this case, polarizer 2, sweeps out per unit of time. In the context of the experiment, angular velocity is described as the speed at which the rotational angle of the polarizer changes with time.

In the provided solution, polarizer 2 starts from a fixed angle and rotates with a constant angular velocity \(\omega\). The angular velocity is directly related to the rate of change of the polarized light's intensity due to the altering position of polarizer 2. The concept of angular velocity is pivotal because it informs us not just about the speed of rotation but also about the direction, which is crucial for predicting the behavior of the light's intensity over time.