Polarization & Polarization Filters: Videos & Practice Problems

Polarization refers to the orientation of the electric field in electromagnetic waves. Light can be polarized by passing through a polarizer, which allows only the component of light aligned with its transmission axis to pass, reducing intensity by half. When polarized light encounters another polarizer at an angle, the intensity is further modified according to Malus's Law: $I_2=I_{1}cos\left(\theta \right)²$. Understanding these principles is crucial for analyzing light behavior in various applications.

Introduction to Polarization

Polarization refers to the orientation of light waves, specifically the direction in which the electric field oscillates. In electromagnetic waves, the electric field oscillates along one axis while the magnetic field oscillates along a perpendicular axis. The polarization of a light wave is defined by the direction of the electric field's oscillation. For instance, if the electric field oscillates purely along the z-axis, we say the light is polarized along the z-axis.

To simplify the representation of polarization, we use polarization diagrams, which are depicted as double-headed arrows pointing in the direction of the electric field's oscillation. If the electric field oscillates at an angle, such as 30 degrees from the z-axis, this is indicated in the diagram with a double-headed arrow and a notation of the angle.

Unpolarized light, on the other hand, consists of electric fields oscillating in multiple random directions. Common examples include sunlight and light from incandescent bulbs. Unpolarized light can be represented in diagrams with multiple arrows pointing in various directions.

When unpolarized light passes through a polarizer, it becomes polarized. A polarizer is a filter that allows only the components of light that are parallel to its transmission axis to pass through, effectively blocking other components. This process results in a reduction of light intensity, which can be quantified by the one-half rule. According to this rule, the intensity of the transmitted polarized light (I) is given by the equation:

I = \frac{1}{2} I_0

Here, \(I_0\) represents the intensity of the unpolarized light before passing through the polarizer. For example, if the intensity of the unpolarized light is 100 watts per square meter, the intensity of the transmitted light after passing through the polarizer would be:

I = \frac{1}{2} \times 100 = 50 \text{ watts/m}^2

This demonstrates that the intensity of the light is halved upon polarization. Understanding these concepts is crucial for solving problems related to light behavior and polarization in various applications.

Unpolarized light with intensity of 6 W/m² is incident on a polarizer. If the polarizer's transmission axis is at an angle of 45° above the horizontal, draw a diagram of this system and find the intensity of transmitted light.

• A. 12 W/m²
• B. 6 W/m²
• C. 3 W/m²
• D. 0 W/m²

Answer: (C)

Multiple Polarizers & Malus's Law

When unpolarized light passes through a polarizer, it becomes polarized, and its intensity is reduced. Specifically, if the initial intensity of unpolarized light is \( I_0 \), after passing through the first polarizer, the intensity \( I_1 \) is given by the equation:

\( I_1 = \frac{1}{2} I_0 \)

For example, if \( I_0 = 100 \) watts per square meter, then \( I_1 = 50 \) watts per square meter after the first polarizer. This reduction occurs because only the component of light aligned with the polarizer's transmission axis is transmitted, while the rest is blocked.

When polarized light encounters a second polarizer, often referred to as an analyzer, the intensity is further modified based on the angle \( \theta \) between the transmission axes of the two polarizers. The intensity after the second polarizer, \( I_2 \), is calculated using Malus's Law:

\( I_2 = I_1 \cdot \cos^2(\theta) \

In this context, \( I_1 \) is the intensity after the first polarizer, and \( \theta \) is the angle between the first and second polarizer's axes. It is crucial to determine the correct angle for \( \theta \), which is the angle between the two polarizers, not the individual angles relative to a reference line.

For instance, if the first polarizer is at 30 degrees and the second is horizontal (0 degrees), the angle between them is \( 90^\circ - 30^\circ = 60^\circ \). Thus, to find \( I_2 \), you would calculate:

\( I_2 = 50 \cdot \cos^2(60^\circ) \

Since \( \cos(60^\circ) = \frac{1}{2} \), it follows that \( \cos^2(60^\circ) = \frac{1}{4} \). Therefore:

\( I_2 = 50 \cdot \frac{1}{4} = 12.5 \) watts per square meter.

In summary, when dealing with multiple polarizers, always apply the one-half rule for the first polarizer when starting with unpolarized light, and use Malus's Law for subsequent polarizers with the appropriate angle between their axes. This systematic approach ensures accurate calculations of light intensity through a series of polarizers.

Horizontally polarized light is incident on a polarization filter. The initial intensity of the light is 0.55 W/m², but is then reduced to 0.40W/m² after passing through the filter. Calculate the angle of the transmission axis of the polarization filter with respect to the horizontal.

• A. 43.34 degrees (0.756 radians)
• B. 37.69 degrees (0.658 radians)
• C. 31.48 degrees (0.549 radians)
• D. Undefined

Answer: (C)

Example 1

In this problem, we explore the behavior of unpolarized light as it passes through multiple polarizers. Initially, sunlight is unpolarized with an average intensity of 1350 watts per square meter. When this light encounters two polarizers oriented at 90 degrees to each other, we can determine the intensity after passing through both filters.

To analyze the situation, we first apply the one-half rule for the first polarizer. The intensity after the first polarizer, denoted as \( I_1 \), is calculated as:

\( I_1 = \frac{1}{2} I_0 = \frac{1}{2} \times 1350 = 675 \, \text{watts/m}^2 \)

Next, we consider the second polarizer. Since the angle between the first and second polarizers is 90 degrees, we use the cosine squared rule to find the intensity after the second polarizer, \( I_2 \):

\( I_2 = I_1 \cdot \cos^2(\theta) \)

Here, \( \theta \) is the angle between the two polarizers, which is 90 degrees. Thus:

\( I_2 = 675 \cdot \cos^2(90^\circ) = 675 \cdot 0 = 0 \, \text{watts/m}^2 \)

This result indicates that no light passes through when two polarizers are oriented at 90 degrees to each other, a phenomenon known as cross polarization.

In the second part of the problem, we introduce a third polarizer positioned between the first two, with its transmission axis at 30 degrees to the horizontal. We need to recalculate the intensities through all three polarizers.

After the first polarizer, the intensity remains the same:

\( I_1 = 675 \, \text{watts/m}^2 \)

For the second polarizer, we need to determine the angle between the first and second polarizers, which is \( 90^\circ - 30^\circ = 60^\circ \). Thus, we calculate:

\( I_2 = I_1 \cdot \cos^2(60^\circ) = 675 \cdot \left(\frac{1}{2}\right)^2 = 675 \cdot \frac{1}{4} = 168.75 \, \text{watts/m}^2 \)

Finally, we calculate the intensity after the third polarizer. The angle between the second and third polarizers is \( 30^\circ - 0^\circ = 30^\circ \). Therefore:

\( I_3 = I_2 \cdot \cos^2(30^\circ) = 168.75 \cdot \left(\frac{\sqrt{3}}{2}\right)^2 = 168.75 \cdot \frac{3}{4} = 126.56 \, \text{watts/m}^2 \)

In conclusion, after passing through all three polarizers, the final intensity of the light is approximately 126.56 watts per square meter, demonstrating that the introduction of the third polarizer allows some light to pass through, contrary to the case with only two polarizers at 90 degrees.

Here’s what students ask on this topic:

What is polarization in light?

Polarization in light refers to the orientation of the electric field in an electromagnetic wave. In polarized light, the electric field oscillates in a specific direction or plane. For example, if the electric field oscillates purely along the z-axis, the light is said to be polarized along the z-axis. Unpolarized light, such as sunlight or light from a bulb, has electric fields oscillating in multiple random directions. Polarization can be achieved by passing light through a polarizer, which filters out all components of the light except those aligned with its transmission axis.

How does a polarizer work?

A polarizer works by allowing only the component of light that is aligned with its transmission axis to pass through. When unpolarized light, which has electric fields oscillating in multiple directions, passes through a polarizer, only the component of the light that is parallel to the transmission axis of the polarizer is transmitted. The other components are absorbed or blocked. This process reduces the intensity of the light by half, as described by the equation I = (1/2)I₀, where I₀ is the initial intensity of the unpolarized light.

What is Malus's Law?

Malus's Law describes the intensity of polarized light after it passes through a second polarizer, known as an analyzer, which is oriented at an angle θ to the first polarizer. The law is given by the equation I₂ = I₁cos²(θ), where I₁ is the intensity of the light after the first polarizer, and θ is the angle between the transmission axes of the two polarizers. This law shows that the intensity of the transmitted light depends on the cosine squared of the angle between the polarizers.

What happens to the intensity of light when it passes through multiple polarizers?

When light passes through multiple polarizers, its intensity is reduced at each stage according to specific rules. Initially, unpolarized light passing through the first polarizer has its intensity reduced by half, as described by I = (1/2)I₀. If the light then passes through a second polarizer oriented at an angle θ to the first, the intensity is further reduced according to Malus's Law: I₂ = I₁cos²(θ). This process continues for each subsequent polarizer, with the intensity being modified based on the angle between the transmission axes of the polarizers.

What is the difference between polarized and unpolarized light?

Polarized light has its electric field oscillating in a specific direction or plane, whereas unpolarized light has electric fields oscillating in multiple random directions. Polarized light can be produced by passing unpolarized light through a polarizer, which filters out all components of the light except those aligned with its transmission axis. Common sources of unpolarized light include sunlight and light from bulbs, while polarized light is often used in applications like sunglasses, camera filters, and LCD screens to reduce glare and improve visibility.