Polarization Visualization
Interactive 2D and 3D demonstration of electromagnetic wave polarization. Explore how the amplitude ratio (ψ) and phase difference (δ) control linear, circular, and elliptical polarization patterns.
- ψ (Psi): Controls the amplitude ratio between x and y components (0° to 90°)
- δ (Delta): Sets the phase difference between components (-180° to 180°)
- Linear Polarization: Set δ = 0° with any ψ value
- Circular Polarization: Set ψ = 45° with δ = ±90°
- Elliptical Polarization: Most other parameter combinations
- 3D Mode: Toggle to view the wave propagating in space
- Animation: Click Start to see the wave oscillate over time
- Use preset buttons for common polarization states
📚 Physics Background
🌊 Electromagnetic Wave Polarization
Polarization describes the orientation and behavior of the electric field vector in an electromagnetic wave. In a plane electromagnetic wave propagating through space, the electric field E⃗ and magnetic field B⃗ oscillate perpendicular to each other and to the direction of propagation (typically the z-axis).
This tool visualizes the electric field component, which traces out various shapes as the wave propagates—from straight lines (linear) to circles (circular) to ellipses (elliptical).
📐 Mathematical Formulation
For a wave propagating in the +z direction, the electric field components are given by:
Ex(z, t) = E0x cos(kz - ωt)
Ey(z, t) = E0y cos(kz - ωt + δ)
Where:
- E0x and E0y are the amplitude magnitudes in the x and y directions
- k = 2π/λ is the wave number (λ = wavelength)
- ω = 2πf is the angular frequency (f = frequency)
- δ is the phase difference between the x and y components
- z is the position along the propagation direction
- t is time
At a fixed point in space (e.g., z = 0), the electric field vector's tip traces an ellipse in the x-y plane as time progresses.
🎯 Parameters Explained
Ψ (Psi) - Amplitude Ratio Angle
The angle ψ relates to the ratio of amplitudes of the two perpendicular electric field components:
tan(ψ) = E0y / E0x
- ψ = 0°: Only x-component (E0y = 0), horizontal linear polarization
- ψ = 45°: Equal amplitudes (E0x = E0y), prerequisite for circular polarization
- ψ = 90°: Only y-component (E0x = 0), vertical linear polarization
This parameter controls the shape of the polarization ellipse—specifically, how elongated or circular it is.
δ (Delta) - Phase Difference
The phase difference δ is the shift between the oscillations of the x and y components:
δ = φy - φx
- δ = 0° or ±180°: Components oscillate in phase or exactly out of phase → Linear polarization
- δ = +90°: y-component lags x-component by 90° → Right-hand circular (with ψ = 45°)
- δ = -90°: y-component leads x-component by 90° → Left-hand circular (with ψ = 45°)
- Other values: Create elliptical polarization with various orientations
This parameter controls the orientation and handedness of the polarization ellipse.
🔄 Polarization States and Physical Interpretation
🔸 Linear Polarization
Condition: δ = 0° or δ = ±180° (any ψ value)
Physical meaning: The electric field oscillates along a single direction in the x-y plane. The field vector traces a straight line as time evolves. The angle of this line with respect to the x-axis is determined by ψ.
When it occurs: Naturally from lasers, reflection off non-metallic surfaces at Brewster's angle, or transmission through polarizing filters (like polaroid sunglasses).
Applications: 3D cinema glasses, LCD displays, stress analysis, reducing glare.
⭕ Circular Polarization
Condition: ψ = 45° (equal amplitudes) AND δ = ±90°
Physical meaning: The electric field vector has constant magnitude but rotates uniformly in the x-y plane. The tip of the vector traces a circle. The direction of rotation determines handedness:
- Right-hand (RH): δ = +90°, field rotates clockwise when viewed from the direction the wave is approaching
- Left-hand (LH): δ = -90°, field rotates counter-clockwise
When it occurs: Passing linear polarized light through a quarter-wave plate at 45°, emission from certain astronomical sources, reflection from chiral molecules.
Applications: Satellite communications, radar systems, optical activity measurements, reducing signal degradation in fiber optics.
⬭ Elliptical Polarization
Condition: Any combination where δ ≠ 0°, ±90°, ±180° or ψ ≠ 45° with δ = ±90°
Physical meaning: The electric field vector traces an ellipse in the x-y plane. This is the most general case—linear and circular polarization are special cases of elliptical polarization. The ellipse has a specific orientation, aspect ratio (determined by ψ), and handedness (determined by sign of δ).
When it occurs: Most common in nature—partial reflection from surfaces, scattering in the atmosphere, propagation through birefringent crystals, or any system where phase and amplitude aren't perfectly controlled.
Applications: Ellipsometry (measuring thin film thickness and optical properties), polarimetry in astronomy and remote sensing, characterizing optical materials.
🔬 Related Optical Phenomena
Understanding polarization is essential for explaining various optical effects:
- Brewster's Angle: At this specific angle of incidence, reflected light is perfectly linearly polarized perpendicular to the plane of incidence. Used in polarizing beam splitters and laser cavities.
- Birefringence: In certain crystals (like calcite), light splits into two rays with different polarizations and velocities, creating double images. Used in wave plates and optical modulators.
- Malus's Law: The intensity of light passing through a polarizer is I = I₀ cos²(θ), where θ is the angle between the light's polarization and the polarizer axis.
- Faraday Effect: A magnetic field can rotate the plane of polarization—a magneto-optical effect used in optical isolators.
- Photoelasticity: Mechanical stress induces birefringence in normally isotropic materials, revealing stress patterns with polarized light.
🌟 The Stokes Parameters
In advanced optics and astronomy, polarization is often described using the Stokes parameters (I, Q, U, V), which provide a complete description of the polarization state:
- I: Total intensity
- Q: Difference between horizontal and vertical linear polarization
- U: Difference between +45° and -45° linear polarization
- V: Difference between right and left circular polarization
The degree of polarization is P = √(Q² + U² + V²) / I, ranging from 0 (unpolarized) to 1 (fully polarized).