Wave Optics: Polarization of Light Waves Study Guide

Definition of Polarization: The polarization of an electromagnetic (EM) wave refers specifically to the direction of its electric field ( $\vec{E}$ ).
Wave Propagation Schematic:
    * In a standard schematic of a polarized electromagnetic wave, the wave propagates in the $x$ -direction.
    * The electric field vector ( $\vec{E}$ ) vibrates within the $xy$ -plane.
    * The magnetic field vector ( $\vec{B}$ ) vibrates within the $xz$ -plane.
Specific Polarization Examples:
* Polarization in the $z$ -direction: The electric field vector oscillates exclusively along the $z$ -axis.
* Polarization in the $y-z$ plane: The electric field can be oriented at a specific angle (e.g., $60^{\circ}$ ) with respect to the $y$ -axis.

Unpolarized Light:
    * Atomic Origin: Each individual atom produces a wave with its own specific orientation of the electric field ( $\vec{E}$ ).
    * Probability distribution: All directions of the electric field vector ( $\vec{E}$ ) are equally probable.
    * Orientation: These vectors lie in a plane that is perpendicular to the direction of wave propagation.
    * Visual Representation: Viewed along the direction of propagation, unpolarized light appears as a set of radial vectors pointing in all possible directions within the plane.
Linearly Polarized Light:
* Condition: A wave is defined as linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point.
* Directionality: In a vertically polarized beam, the electric field vector vibrates only in the vertical direction.
Summary of Differences:
* Polarized Light: Electric fields are all oriented in the same direction.
* Unpolarized Light: Electric fields are oriented in random directions.

Mechanism: Selective absorption is the most common technique used to polarize light. It utilizes materials that transmit waves whose electric field vectors vibrate in a plane parallel to a specific direction, while simultaneously absorbing waves whose electric field vectors vibrate in directions perpendicular to that specific direction.
Polaroid Material: Invented by E.H. Land, this material polarizes light through selective absorption.
Experimental Setup:
* Polarizer: The first polarizing sheet encountered by an unpolarized beam.
* Analyzer: A second polarizing sheet placed after the polarizer to control or measure the intensity of the light.
Transmission Principle: The light exiting a polarizer is polarized in the exact same direction as the polarizer's transmission axis.

Intensity Units: The SI unit for intensity ( $I$ ) is watts per meters squared ( $W/m^{2}$ ).
Unpolarized Light through a Polarizer:
* When an unpolarized beam of initial intensity ( $I_{0}$ ) passes through a polarizer, the transmitted intensity is exactly half of the initial intensity.
* Formula: $I = \frac{1}{2} I_{0}$ .
Polarized Light through an Analyzer (Malus’ Law):
    * If a polarized beam with intensity ( $I_{0}$ ) encounters a polarizing sheet oriented at an angle ( $\theta$ ) relative to the direction of polarization, the transmitted intensity is given by:
    * Formula: $I = I_{0} \cos^{2}(\theta)$ .
    * This law applies to any two polarizing materials with transmission axes at an angle ( $\theta$ ) to one another.
Combined Formula: For an unpolarized beam passing through a polarizer and then an analyzer, the final intensity is:
* Formula: $I = \frac{1}{2} I_{0} \cos^{2}(\theta)$ .

Scenario: Unpolarized light with initial intensity ( $I_{b}$ ) is incident upon three polarizers.
    * Polarizer 1: Vertical transmission axis.
    * Polarizer 2: Transmission axis rotated $30.0^{\circ}$ relative to the first.
    * Polarizer 3: Transmission axis rotated $75.0^{\circ}$ relative to the first.
Part (a): Intensity after the second polarizer ( $I_{2}$ ):
    * Calculation: $I_{2} = (\frac{1}{2} I_{b}) \cos^{2}(30.0^{\circ})$
    * Since $\cos(30.0^{\circ}) = \frac{\sqrt{3}}{2}$ , then $\cos^{2}(30.0^{\circ}) = \frac{3}{4}$ .
    * Result: $I_{2} = (\frac{1}{2} I_{b}) \times (\frac{3}{4}) = \frac{3}{8} I_{b}$ .
Part (b): Intensity after the third polarizer ( $I_{3}$ ):
    * The angle ( $\theta$ ) between the second and third polarizer is $75.0^{\circ} - 30.0^{\circ} = 45.0^{\circ}$ .
    * Calculation based on transcript logic: $I_{3} = I_{2} \cos^{2}(45.0^{\circ})$ .
    * Using the value from part (a): $I_{3} = (\frac{3}{8} I_{b}) \cos^{2}(45.0^{\circ}) = (\frac{3}{8} I_{b}) \times (\frac{1}{2})$ .
    * Result: $I_{3} = \frac{3}{16} I_{b}$ .
    * Note: Detailed solution slides show specific intermediate values where $I_{3} = \frac{3}{16} I_{b}$ was derived using $\cos^{2}(45.0^{\circ}) = 0.5$ .

Polarization by Reflection:
    * When unpolarized light reflects from a surface, the resulting light can be completely polarized, partially polarized, or unpolarized, depending on the angle of incidence.
    * Angle Effects:
        * At $0^{\circ}$ (normal incidence) or $90^{\circ}$ (grazing incidence), the reflected beam remains unpolarized.
        * Between $0^{\circ}$ and $90^{\circ}$ , state of polarization varies.
        * There is one particular angle at which the reflected beam is completely polarized.
    * Application: Polaroid sunglasses with vertical transmission axes are used to reduce glare, as light reflecting from horizontal surfaces (like water or roads) tends to be horizontally polarized.
Polarization by Scattering:
    * Scattering occurs when light hits particles and electrons in the medium absorb and reradiate part of the light.
    * Atmospheric Example:
        * Observer at Right Angles: Sunlight reaching an observer looking at a $90^{\circ}$ angle relative to the Sun is polarized. This observer sees more blue light than red light.
        * Observer Looking Toward the Sun: Sees unpolarized light that contains more red light than blue light.

Core Definition: Polarization is the direction of the electric field vector within an EM wave.
Polarizer Function: Transmits only the component of the electric field that is parallel to the polarizer’s transmission axis.
Intensity Rules:
* Initially unpolarized beam: $I = \frac{1}{2} I_{0}$ .
* Initially polarized beam at angle (Malus' Law): $I = I_{0} \cos^{2}(\theta)$ .
Scattering Summary: Atmospheric light is polarized when viewed at right angles to the position of the Sun.
Reflection Summary: Light reflecting from a horizontal surface becomes partially polarized in the horizontal direction.