Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Electromagnetism → Electromagnetic wave propagation ↓


Polarization and Jones calculus


In the study of electromagnetic wave propagation, the concept of polarization plays a key role. Polarization describes the orientation of the electric field vector of an electromagnetic wave, and understanding this concept is crucial for analyzing the interactions between these waves and matter. When it comes to a mathematical framework for describing polarization changes as light passes through different mediums, Jones calculus is an indispensable tool.

Understanding polarization

Polarization refers to the geometric orientation of the oscillation of the electric field vector in a light wave. Imagine a light wave propagating in space: the electric field, the magnetic field, and the direction of propagation are all perpendicular to each other. When discussing polarization we focus on the direction of the electric field.

Common types of polarization include:

  • Linear polarization: The electric field oscillates in a single direction perpendicular to the direction of propagation.
  • Circular polarization: The electric field moves in a circular motion as it moves forward, forming a helix in the direction of propagation.
  • Elliptical polarization: This is the more common form of polarization, where the electric field describes an ellipse in any plane perpendicular to the direction of travel.

Visual example: linear and circular polarization


    
    
    I
    Linear




    
    
    I
    Circular

Jones calculus

The Jones calculus provides a powerful algebraic description of polarization. It enables us to represent polarized light and its transformation through optical elements using matrices. This method uses so-called Jones vectors and Jones matrices.

Jones vectors

The Jones vector describes the state of polarization of a light wave. For linearly polarized light, the Jones vector can be represented simply in terms of its x and y components:

| J > = | E_x | + i | E_y |

Here, E_x and E_y are the amplitudes of the wave in the x and y directions. The vector describes the standard polarization state. Complex numbers play a role because they can express not only the magnitude of the electric field components, but also the phase difference.

Jones matrices

To express optical elements using Jones calculus, we use Jones matrices. Each matrix can describe how an optical element affects the polarization of light passing through it.

The action of an optical element is given by its Jones matrix M When light with a polarization described by the Jones vector | J_in > passes through, the resulting polarization is

| J_out > = M | J_in >

Examples of Jones matrices include:

  • Polarizer: A polarizer parallel to the x-axis can have the Jones matrix: 
    M = | 1 0 | | 0 0 |
  • Quarter-wave plate: This element produces a phase difference of 90°. 
    M = 1/sqrt(2) | 1 i | | i 1 |

Using Jones calculus to analyze polarization

Suppose we have vertically polarized light that enters first through a linear polarizer and then a quarter wave plate. First, we need to represent the vertically polarized light using the Jones vector:

| J_in > = | 0 | | 1 |

Next, we apply the Jones matrix to the linear polarizer and then to the quarter-wave plate. Suppose that the polarizer transmits only vertical components:

| J_1 > = | 0 0 | | 0 | | 0 1 | | 1 |

Then apply the quarter-wave plate matrix:

| J_out > = 1/sqrt(2) | 1 i | | 0 | | i 1 | | 1 |

Visual example: polarization change

Consider the initial state and the displayed changes:


    
    j_in




    
    J_1




    
    j_out

Effect of polarization in applications

Polarization is not just an academic concept; it plays a vital role in many technologies and industries. Here are some major applications:

  • Optical instruments: Many optical instruments, such as microscopes and telescopes, use polarizers to improve contrast and image quality.
  • Communications: Polarization is used in antennas and satellite dishes to optimize signal transmission and reduce interference.
  • Display technology: Liquid crystal displays (LCDs) rely heavily on the control of light polarization.

Example: Polarization in LCD screens

LCD screens work by changing the polarization state of light. Such screens control the light passing through a liquid crystal located in the middle of a polarizing filter. By controlling the polarization state, different parts of the screen either block or allow light to pass through, forming an image.


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Polarization and Jones calculus

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