Polarization and Snell's Law in Electromagnetic Waves: Linear, Circular, and Elliptical Polarization

Short Summary

In this MIT OpenCourseWare session, the lecturer revisits Snell's law and boundary conditions to understand refraction, then dives into polarization, showing how two orthogonal electric field components combine to form linear, circular, and elliptical polarization. The talk includes practical demonstrations with polarizers and an optical fiber setup, connecting theory to everyday optical phenomena and technologies.

• Snell's law and boundary conditions fix the relationships between incident, refracted angles, and wave frequencies.
• Polarization states arise from the superposition of two orthogonal field components with phase and amplitude differences.
• Polarizers select polarization components, changing transmitted intensity via cos^2 theta dependence.
• Applications include optical fibers and the quantum-inspired discussion of single-photon polarization experiments.

Medium Summary

This lecture builds on geometrical optics to develop a deeper picture of electromagnetic wave propagation across interfaces, emphasizing polarization as a superposition of two orthogonal electric field components. The instructor begins by reviewing two-material boundaries, where the projection of the wave vector on the boundary direction must be continuous and the frequency must be the same on both sides. Using these conditions, the magnitudes of the wave vectors are fixed by refractive indices, leading to Snell's law in the form N sin theta = N' sin theta', and the requirement that the Y-component of the wave vector remains continuous across the boundary. The discussion then shifts to polarization, introducing a compact matrix language for the electric field vector with two components, E_x and E_y, propagating along z. The field can be written as the real part of a two-component vector psi times e^{i(kz - omega t)}, allowing a clean description of polarization states through phase and amplitude relations. “It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. But if it doesn't agree with experiment, it's wrong.” - Feynman

Polarization Fundamentals: Linear, Circular, and Elliptical

The lecturer shows that two plane waves with identical frequency but different polarization directions combine to yield a resultant electric field that traces a path in the XY plane as time evolves. If the two components are in phase and have equal amplitudes, the tip of E traces a line, i.e., linear polarization. If the two components have a 90-degree phase difference and equal amplitudes, a circle is traced, which is circular polarization; unequal amplitudes yield an ellipse, i.e., elliptical polarization. A representation in matrix form clarifies how the Z vector, with components for the X and Y directions, encapsulates these relationships. The discussion also demonstrates how varying the phase difference between the components (delta phi) creates a continuum of polarization states between linear, circular, and elliptical. “The rainbow is actually really related to Snell's law.” - Lecturer

Polarizers, Easy Axes, and Practical Implications

Next, the talk introduces polarizers with defined easy axes. A polarizer projects the incident field onto its axis, leaving only the parallel component and reducing the transmitted intensity by cos^2 theta. This concept is connected to real-world demonstrations, such as reading a computer screen through different polarizers, and extends to devices like optical fibers where total internal reflection and internal reflections govern light propagation. The lecturer then ties these ideas to quantum questions, including a thought experiment about single photons and polarization, hinting at the wave-particle duality that underpins much of quantum optics. Finally, the talk foreshadows topics like quarter-wave plates and how polarization can be engineered for communication and analysis. “God does not play dice with the universe.” - Einstein

Closing Notes and Visual Demos

The session closes with practical demonstrations of polarization effects using everyday displays and polarizers to illustrate the selective transmission of polarized light. The lecturer reflects on the connection between polarization and everyday technology, such as fiber-optic communication, and notes that future lectures will cover wave plates and the generation of polarization states. The discussion emphasizes the power of combining simple boundary conditions with vector field descriptions to understand complex light-mield behavior, and it invites students to explore the rich landscape where classical electromagnetism meets quantum phenomena.