In a polarized light experiment, a setup similar to the one...

In a polarized light experiment, a setup similar to the one in Figure 31.22 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 $t=0 \mathrm{~s}$, 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 $d<R,$ where $R$ is the radius of the polarizer?

In a polarized light experiment, a setup similar to the one in Figure 31.22 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 $t=0 \mathrm{~s}$, 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 $d<R,$ where $R$ is the radius of the polarizer?

Key Concepts

Polarization of Light

This concept concerns the orientation of the electric field vector in an electromagnetic wave. In many optical experiments, controlling the polarization is crucial because various optical elements, such as polarizers, interact differently with light depending on its polarization state.

Malus's Law

Malus's Law is the fundamental rule that describes how the intensity of polarized light changes when it passes through a polarizing filter. The transmitted intensity is proportional to the cosine squared of the angle between the light's polarization direction and the transmission axis of the polarizer. This law is key to understanding and calculating light intensity in experiments involving multiple polarizers.

Successive Polarization Filters

When light encounters a series of polarizers, each filter transmits only the component of the light's electric field aligned with its own transmission axis. This step?by?step projection process is critical because even if two polarizers are crossed (i.e., oriented at 90° with respect to one another), the introduction of an intermediate polarizer at an angle can allow some light to be transmitted by altering the light’s polarization direction.

Dynamic Rotation of Polarizers

In experiments where a polarizer is rotated, the transmission axis changes with time, leading to a time-dependent projection of the light's electric field and therefore a time-dependent intensity. This dynamic shift is analyzed by incorporating the angular velocity of the polarizer into the cosine squared dependence described by Malus’s Law, resulting in expressions that model how the intensity varies as the polarizer rotates.

Rotational Axis Displacement

The placement of the rotation axis relative to the center of the polarizer affects the uniformity of the polarization transformation across the beam. When the rotation axis is displaced from the center, it introduces additional geometric considerations, such as spatial variations in the effective angle and possible beam displacement. However, if the polarizer retains its ideal polarization properties, the overall angular dependence in intensity, as dictated by Malus's Law, remains unchanged.

Transcript

00:01 For this problem on the topic of electromagnetic waves, in a polarized light experiment, we have unpolarized light with intensity i0 incident upon polarizer 1.

00:11 1 and 3 are crossed at 90 degrees and the orientations are fixed during the experiment.

00:16 Polarizer 2 has an angle of 45 degrees, and at time t is equal to 0, it starts to rotate with angular velocity omega about the direction of propagation of light in a clockwise direction, as an observer is doing it toward the light source.

00:30 If the intensity of the light emerging from polarizer 3 is monitored, we want to find an expression for this intensity as a function of time, and we want to know how this expression would change from a, if polarizer 2 would then rotate about an axis that is parallel to the direction of propagation of light, but displaced by distance d, which is less than r, where r is the radius of the polarizer.

00:54 Now, the intensity of light passing through the first polarizer, i1, is equal to the incident intensity, i -0, over 2 and the angle between the transmission axis of the first and second polarizer as a function of time.

01:07 We'll call it theta 1 comma 2 and this is equal to 45 degrees plus omega t, where t is the time in seconds and omega is in radiance per second.

01:21 The intensity of the light passing the second polarizer i2 is equal to i1 cosine squared of the 1 -2 which is a half i0 cosine squared of 45 degrees plus omega t so this is i2 as a function of t the angle between the transmission axis of the second and third polarizer as a function of time theta 2 -3 is equal to 45 degrees minus omega t now the intensity of the light the third polarizer i3 is equal to i2 times the cosine squared of theta 2 3 which is a half i 0 cosine squared of 45 degrees plus omega t times cosine squared 45 degrees minus omega t which we can write as a half i0 into cosine of 45 degrees plus omega t cosine of 45 degrees minus omega t all squared this is i t as a function of time now we can use the trigonometric identity that cosine u cosine v is equal to a half of cosine u plus v plus cosine u minus v and this gives us the cosine of 45 degrees plus omega t times the cosine of 45 degrees minus omega t equal to a half into the cosine of 45 degrees plus omega t plus 45 degrees minus omega t plus the cosine of 45 degrees plus omega t...

Please give Ace some feedback