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Polarization

Polarization is the shape traced by an electromagnetic wave's electric-field vector in the plane perpendicular to its direction of travel. It is not color, brightness, or a cosmetic rendering mode. For a wave travelling in the +z direction, the useful state is two perpendicular components—usually x and y—plus their amplitudes and relative phase.

Think like a programmer

Represent the field as two synchronized signals, not a string such as "circular". Store amplitudes and phase difference, sample both at the same time, then classify the resulting trajectory. That representation makes the special cases testable and leaves room for a later optical element to transform the state.

Model checklist

Inputs
Propagation direction, transverse basis vectors, x and y amplitudes, angular frequency ω, phase difference δ, and sample time.
State
Two transverse electric-field components and their relative phase; optionally a normalized intensity.
Rule
Advance both sinusoidal components with the same angular frequency and their stored phase offset.
Output
A point or trajectory in the transverse plane, plus linear, circular, or elliptical classification.
Check
A zero component gives a line; equal components with δ = ±π/2 give a circle; a common phase shift does not change the polarization shape.

One compact ideal model is

\[E_x(t)=A_x\cos(\omega t),\qquad E_y(t)=A_y\cos(\omega t+\delta)\]

At a fixed position, plot E_x(t) horizontally and E_y(t) vertically as time advances. If δ = 0 or π, both components reach their extrema together and the tip stays on a straight line: linear polarization. If A_x = A_y and δ = +π/2 or −π/2, the tip rotates at constant radius: circular polarization. Every ordinary mismatch of amplitude or phase produces an ellipse. The sign convention determines whether a particular observer calls the rotation right- or left-handed, so code must document its viewing direction rather than attaching a universal label.

type PolarizationInput = { xAmplitude: number; yAmplitude: number; phase: number; omega: number };

function samplePolarization(input: PolarizationInput, time: number) {
  const phaseNow = input.omega * time;
  return {
    ex: input.xAmplitude * Math.cos(phaseNow),
    ey: input.yAmplitude * Math.cos(phaseNow + input.phase),
  };
}

The first checks should target limiting cases, not a rendered ellipse. With yAmplitude = 0, every sample must have ey = 0. With equal amplitudes and a quarter-cycle phase difference, ex² + ey² should remain constant up to floating-point tolerance. Adding the same extra phase to both components only chooses a different point at which to start sampling; it should not alter the ellipse. A normalized intensity proxy often scales as A_x² + A_y² in this simple basis, but intensity alone discards the relative phase that makes polarization meaningful.

Polarization state inspector

Change the y-component amplitude and its phase delay. The trace is the electric-field tip at one location, with x and y axes as its transverse basis.

Electric-field tip trajectory
Current component sample
Transverse component axes

x amplitude 1.0; y amplitude 1.0; relative phase 90°; ideal state: circular polarization.

The text result identifies the field components, relative phase, and ideal trajectory; the blue trace only makes that two-component state visible.

Try this experiment

Prediction: A quarter-cycle delay between equal transverse components makes the field tip rotate, while the same delay between unequal components makes an ellipse.

Sample A_x = A_y = 1 at δ = 0 and then at δ = π/2. Inspect the points at t = 0, T/4, T/2, and 3T/4, where T = 2π/ω. Then set A_y = 0.5 without changing the phase and state which regression check distinguishes the circle from the ellipse.

Polarizers are filters on this vector state. An ideal linear polarizer passes the component along its transmission axis and rejects the perpendicular component. A more complete simulation can encode that as a projection onto a unit axis; two crossed ideal polarizers then have zero transmitted field in the ideal model. Wave plates instead introduce a relative phase delay, which is why they can turn a suitable linear state into a circular one. These are transformations of components, not recoloring operations.

Where this model breaks

This lesson assumes a monochromatic plane wave, a fixed transverse basis, coherent components, and ideal optical elements. Real light can be partially polarized, broadband, scattered, depolarized, or spatially varying. Detector response, material absorption, birefringence, finite apertures, and quantum photon polarization require extra state and measurement models. The ellipse is also a local field description; it is not a literal path travelled by a photon.

Summary

Polarization is two transverse field components plus their relative phase. Sample those components directly, use limiting-case assertions to classify the trajectory, and treat polarizers and wave plates as state transformations. A brightness or color value cannot carry the phase information this model needs.

Glossary

Self-check

  1. Which values must be stored to reproduce a polarization ellipse?
  2. Why does equal component amplitude alone not guarantee circular polarization?
  3. What test detects the ideal circular special case without inspecting a drawing?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Polarization, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Maxwell’s Equations and EM Waves, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Polarization as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Polarization into a test

Model polarization from explicit transverse field components and their relative phase.

  1. Name the inputs and units that the maxwell’s equations and em waves model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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