🌊 Polarization of Light

🔬 EDU Lab Simulation Science

The Science of Light Polarization

Explore the physics, mathematics, and applications of one of the most fascinating properties of electromagnetic waves.

🌊 What is Polarization?

Polarization is the phenomenon of restricting the vibrations of a transverse wave to a particular direction perpendicular to the direction of propagation. It provides strong experimental evidence that light is transverse in nature.

In unpolarized light, the electric field vector vibrates randomly in all possible directions perpendicular to propagation. In polarized light, vibrations occur only in one definite direction.

Key insight: Only transverse waves can be polarized. Longitudinal waves (like sound in air) cannot be polarized because their vibrations are parallel to propagation.

📐 Malus' Law

When plane-polarized light passes through an analyzer, the transmitted intensity depends on the angle between the polarization direction and the analyzer's transmission axis.

Malus' Law
$$I = I_0 \cos^2\theta$$
Where: $I_0$ = intensity of incident polarized light, $I$ = transmitted intensity, $\theta$ = angle between polarization direction and analyzer axis. When $\theta = 0°$, $I = I_0$ (maximum). When $\theta = 90°$, $I = 0$ (crossed polaroids).
Two Polaroids with Unpolarized Light
$$I = \frac{I_0}{2}\cos^2\theta$$
When unpolarized light of intensity $I_0$ passes through a first polaroid, the intensity becomes $I_0/2$. The second polaroid (analyzer) at angle $\theta$ further reduces it by $\cos^2\theta$.

🔍 Brewster's Law

When unpolarized light hits a transparent surface at the Brewster angle, the reflected light becomes completely plane-polarized. At this angle, the reflected and refracted rays are perpendicular.

Brewster's Law
$$\mu = \tan\theta_B$$
Where: $\mu$ = refractive index of the medium, $\theta_B$ = Brewster angle (polarizing angle). For glass ($\mu = 1.5$), $\theta_B = \arctan(1.5) = 56.3°$. For water ($\mu = 1.33$), $\theta_B = 53.1°$.
At Brewster Angle
$$\theta_B + \theta_r = 90°$$
The reflected ray and refracted ray are perpendicular to each other. The reflected light is polarized perpendicular to the plane of incidence.

📊 Degree of Polarization

Degree of Polarization
$$P = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$$
For completely polarized light: $P = 1$ ($I_{min} = 0$). For unpolarized light: $P = 0$ ($I_{max} = I_{min}$). For partially polarized light: $0 < P < 1$.

⚡ Electric Field Equations

Linear Polarization
$$\vec{E} = E_0 \sin(\omega t - kz)\,\hat{x}$$
Circular Polarization
$$E_x = E_0 \sin(\omega t),\quad E_y = E_0 \cos(\omega t)$$
Phase difference $\delta = \pi/2$ between components of equal amplitude.
Elliptical Polarization (General)
$$E_x = A\sin(\omega t),\quad E_y = B\sin(\omega t + \delta)$$
Unequal amplitudes $A \neq B$ or arbitrary phase difference $\delta$. Linear and circular are special cases.

🧮 Malus' Law Calculator

Calculate transmitted intensity through two polaroids.

Interactive Calculator

200
30°
After First Polaroid (I₁ = I₀/2)
100.0 W/m²
After Analyzer (I = I₁ cos²θ)
75.0 W/m²
37.5% of original intensity transmitted
Intensity visualization:

🔍 Brewster Angle Calculator

Calculate the Brewster angle for different materials.

Brewster's Law: μ = tan θ_B

1.50
Brewster Angle
56.3°
At this angle, reflected light is completely plane-polarized
Refraction Angle
33.7°
θ_B + θ_r = 90° (reflected ⊥ refracted)

📊 Degree of Polarization Calculator

P = (I_max − I_min) / (I_max + I_min)

100
20
Degree of Polarization
P = 0.667
Partially polarized light (66.7%)

📐 Types of Polarization

TypeE-field PathConditionComponents
Linear (Plane) Straight line Single direction oscillation $E = E_0\sin(\omega t)$
Circular Circle Equal amplitudes, δ = π/2 $E_x = E_0\sin(\omega t)$, $E_y = E_0\cos(\omega t)$
Elliptical Ellipse Unequal amplitudes or δ ≠ π/2 $E_x = A\sin(\omega t)$, $E_y = B\sin(\omega t + \delta)$

Note: Elliptical polarization is the most general state. Linear polarization is a special case when $B = 0$ or $\delta = 0, \pi$. Circular polarization is a special case when $A = B$ and $\delta = \pm\pi/2$.

🔄 Right vs Left Circular Polarization

PropertyRight Circular (RCP)Left Circular (LCP)
Phase difference$\delta = +\pi/2$$\delta = -\pi/2$
E-field rotationClockwise (looking toward source)Counter-clockwise
HelicityPositiveNegative
Produced byλ/4 plate, fast axis at +45°λ/4 plate, fast axis at −45°

📏 Wave Plates

Wave PlatePhase RetardationEffect
Quarter-wave (λ/4)π/2 (90°)Linear → Circular (at 45°), Circular → Linear
Half-wave (λ/2)π (180°)Rotates linear polarization by 2θ
Full-wave (λ)2π (360°)No net effect (used for wavelength selection)

🔧 Methods of Producing Polarized Light

1. Selective Absorption (Polaroid Filters)

Materials like Polaroid sheets contain long-chain polymer molecules aligned in one direction. They absorb the electric field component parallel to the chains and transmit the perpendicular component.

Intensity After One Polaroid
$$I = \frac{I_0}{2}$$

2. Reflection (Brewster's Law)

At the Brewster angle, reflected light is completely plane-polarized perpendicular to the plane of incidence. The refracted light is partially polarized.

Brewster's Law
$$\mu = \tan\theta_B$$

3. Double Refraction (Birefringence)

Anisotropic crystals like calcite split unpolarized light into two rays:

  • Ordinary ray (O-ray): Obeys Snell's law, polarized perpendicular to the principal plane
  • Extraordinary ray (E-ray): Does not obey Snell's law, polarized in the principal plane

The two rays travel at different velocities due to different refractive indices ($n_o$ and $n_e$).

4. Scattering

Light scattered by small particles (Rayleigh scattering) becomes partially polarized. Maximum polarization occurs at 90° scattering angle. This is why the sky appears partially polarized — observable with polarized sunglasses.

💡 Applications of Polarization

  • Polarized sunglasses: Reduce glare by absorbing horizontally polarized light reflected from water, roads, and snow
  • LCD displays: Use two polarizers and liquid crystals to control light transmission pixel by pixel
  • Stress analysis (Photoelasticity): Polarized light reveals strain patterns in transparent materials under stress
  • Photography: Polarizing filters enhance contrast, suppress reflections, and deepen sky colors
  • 3D cinema: Left and right images use different circular polarizations, separated by polarized glasses
  • Optical communications: Polarization-division multiplexing doubles fiber capacity
  • Astronomy: Polarimetry reveals magnetic fields, dust properties, and scattering geometry
  • Microscopy: Polarized light microscopy identifies crystalline materials and mineral structures

❓ Frequently Asked Questions

Polarization means restricting vibrations to a specific direction perpendicular to propagation. In longitudinal waves (like sound in air), particles vibrate parallel to the direction of propagation — there's only one possible vibration direction, so there's nothing to "restrict." Transverse waves vibrate in a plane perpendicular to propagation, offering infinite possible vibration directions that can be filtered.

Unpolarized light has its electric field distributed equally in all directions perpendicular to propagation. A Polaroid transmits only the component along its transmission axis. For each random direction at angle $\theta$, the transmitted component is $E_0\cos\theta$, and intensity goes as $\cos^2\theta$. Averaging $\cos^2\theta$ over all angles from 0 to 2π gives exactly $1/2$. Therefore $I = I_0/2$.

Two crossed polaroids (90° apart) block all light. However, if you insert a third polaroid at 45° between them, light passes through! The first polaroid polarizes vertically. The middle one at 45° transmits $\cos^2(45°) = 1/2$ of that, now polarized at 45°. The final polaroid at 90° (horizontal) transmits $\cos^2(45°) = 1/2$ again. Final intensity: $I_0/2 \times 1/2 \times 1/2 = I_0/8$.

Sunlight is unpolarized. When it scatters off air molecules (Rayleigh scattering), the scattered light becomes partially polarized. Maximum polarization occurs at 90° from the sun. This is because the oscillating electrons in air molecules radiate preferentially perpendicular to their oscillation direction. Bees and some birds can detect sky polarization for navigation.

An LCD has two crossed polarizers with liquid crystal material between them. Without voltage, the liquid crystals twist the polarization by 90°, allowing light through both polarizers. When voltage is applied, the crystals align and stop rotating the polarization — light is blocked by the second polarizer. Each pixel is independently controlled, creating the image. This is why LCD screens appear dark when viewed through polarized sunglasses at certain angles.

Optical activity (or optical rotation) is the ability of certain substances to rotate the plane of polarization of linearly polarized light. Substances that rotate clockwise (looking toward the source) are dextrorotatory (+), and those rotating counter-clockwise are levorotatory (−). Sugar solutions, quartz, and many biological molecules exhibit optical activity. The rotation angle depends on concentration, path length, and wavelength.

📝 Solved Problems

Problem: Unpolarized light of intensity 200 W/m² passes through two ideal polaroids. The angle between their transmission axes is 30°. Find the final transmitted intensity.

Step 1: After first polaroid
$$I_1 = \frac{I_0}{2} = \frac{200}{2} = 100 \text{ W/m}^2$$
Step 2: Apply Malus' Law
$$I = I_1\cos^2(30°) = 100 \times \left(\frac{\sqrt{3}}{2}\right)^2 = 100 \times \frac{3}{4} = 75 \text{ W/m}^2$$

Answer: Final transmitted intensity = 75 W/m²

Problem: Plane-polarized light of intensity 50 W/m² is incident on an analyzer at 45° to the plane of polarization.

Malus' Law
$$I = I_0\cos^2(45°) = 50 \times \left(\frac{1}{\sqrt{2}}\right)^2 = 50 \times \frac{1}{2} = 25 \text{ W/m}^2$$

Answer: Transmitted intensity = 25 W/m²

Problem: Calculate the Brewster angle for water with refractive index 1.33.

Brewster's Law
$$\theta_B = \arctan(\mu) = \arctan(1.33) = 53.1°$$

At this angle, light reflected from the water surface is completely horizontally polarized. This is why polarized sunglasses (which block horizontal polarization) are so effective at reducing water glare.

Answer: Brewster angle = 53.1°

Problem: Unpolarized light of intensity I₀ passes through three polaroids at 0°, 45°, and 90°. Find the final intensity.

Step 1: After first polaroid (0°)
$$I_1 = \frac{I_0}{2}$$
Step 2: After second polaroid (45° from first)
$$I_2 = I_1\cos^2(45°) = \frac{I_0}{2} \times \frac{1}{2} = \frac{I_0}{4}$$
Step 3: After third polaroid (45° from second)
$$I_3 = I_2\cos^2(45°) = \frac{I_0}{4} \times \frac{1}{2} = \frac{I_0}{8}$$

Answer: $I = I_0/8$ — Remarkably, light passes through even though the first and last polaroids are crossed (90° apart)!

📚 Bibliography & References

Foundational textbooks, papers, and resources on light polarization and optics.

Textbooks
[1] Hecht, E. (2017). Optics (5th ed.). Pearson. ISBN: 978-0-133-97722-6.
[2] Born, M. & Wolf, E. (1999). Principles of Optics (7th ed.). Cambridge University Press. ISBN: 978-0-521-64222-4.
DOI: 10.1017/CBO9781139644181
[3] Saleh, B.E.A. & Teich, M.C. (2019). Fundamentals of Photonics (3rd ed.). Wiley. ISBN: 978-1-119-50687-1.
[4] Pedrotti, F.L., Pedrotti, L.M. & Pedrotti, L.S. (2017). Introduction to Optics (3rd ed.). Cambridge University Press. ISBN: 978-1-108-42826-2.
[5] Goldstein, D.H. (2011). Polarized Light (3rd ed.). CRC Press. ISBN: 978-1-4398-3041-3.
Key Papers
[6] Malus, É.L. (1809). Sur une propriété de la lumière réfléchie. Mémoires de physique et de chimie de la Société d'Arcueil, 2, 143–158.
[7] Brewster, D. (1815). On the laws which regulate the polarisation of light by reflexion from transparent bodies. Philosophical Transactions of the Royal Society, 105, 125–159.
DOI: 10.1098/rstl.1815.0010
[8] Stokes, G.G. (1852). On the composition and resolution of streams of polarized light from different sources. Transactions of the Cambridge Philosophical Society, 9, 399–416.
[9] Jones, R.C. (1941). A new calculus for the treatment of optical systems. Journal of the Optical Society of America, 31(7), 488–493.
DOI: 10.1364/JOSA.31.000488
Online Resources
[10] HyperPhysics. Polarization of Light. Georgia State University.
hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polarcon.html
[11] Khan Academy. Polarization of light and Malus' law.
khanacademy.org — Polarization of light
[12] MIT OpenCourseWare. 8.03 Physics III: Vibrations and Waves — Polarization.
ocw.mit.edu — Physics III
[13] Edmund Optics. Introduction to Polarization.
edmundoptics.com — Introduction to Polarization
Made by Sorin Zgura