Polarization & Waveplate Simulator
Rotate polarizers and waveplates to explore Linear, Circular, and Elliptical Polarization. Watch the electric field change and discover how quarter-wave plates, half-wave plates, and analyzer angles affect light transmission in real time.
Electric-field traces use a shared scale relative to the input field. Motion is slowed for illustration. This is the x–y plane; light propagates along +z, out of the screen.
*Relative to the power entering P2. All other power readings are relative to the original source. “No light” means transmission below the numerical tolerance of this ideal model.
The curve shows what an ideal analyzer would transmit at each angle, relative to source power. It remains a hypothetical scan when P2 is bypassed.
How the model works
This simulator uses complex Jones vectors for fully polarized monochromatic light. The source vector is [cos α, sin α]. Each ideal linear polarizer projects the electric field onto its transmission axis. A lossless waveplate introduces a relative phase delay between its fast and slow axes.
Malus’s law: Iout = Iin cos²(θ)
Quarter-wave retardance: δ = π/2 · Half-wave retardance: δ = π
θ in Malus’s law is the angle between a linear incident polarization and the polarizer axis. It does not describe an arbitrary elliptical input using a single orientation angle. An ideal analyzer transmits half the incident power of circularly polarized light at every angle.
The Jones convention here is E(t) = Re{J exp(−iωt)}; the slow-axis Jones component is multiplied by exp(+iδ). The plots state the propagation direction to make rotation unambiguous. No right/left-handed polarization labels are used.
A quarter-wave plate at 45° to linear input produces circular polarization; aligned axes leave it linear. A half-wave plate maps input orientation α to 2β − α, where β is its fast-axis angle. After an ideal transmitting analyzer, the output is always linear.
Assumptions: ideal polarizers with zero leakage, lossless waveplates at their design wavelength, normal incidence, and no depolarization, reflections or dispersion. Unpolarized and partially polarized sources are not modeled. These results illustrate ideal optics rather than a specific component’s measured performance.
Reference: Understanding Waveplates and Retarders — Edmund Optics.
Understanding Light Polarization
Why does rotating a polarizer make a beam brighter or darker? The answer lies in the direction of the light’s electric field. This polarization and waveplate simulator lets you rotate optical elements and see how the field and transmitted power change together.
Polarization describes the electric field’s behavior in the plane perpendicular to the direction of travel. If you would like to revisit the relationship between electric and magnetic fields, explore our electromagnetic spectrum simulator.
Linear, Circular and Elliptical Polarization
The animated traces show the path followed by the tip of the electric-field vector at one position as time passes. The motion is slowed down so you can inspect it.
- Linear polarization: the electric field oscillates along a fixed line.
- Circular polarization: two perpendicular field components have equal amplitudes and a quarter-cycle phase difference. Their combined field traces a circle.
- Elliptical polarization: the combined field traces an ellipse. Its shape depends on the relative amplitudes and phase of the components.
The source in this simulator is fully linearly polarized. Its incident power is normalized to 100%, allowing you to compare transmission without entering a power in watts.
How Polarizers Work: Malus’s Law
An ideal linear polarizer transmits the electric-field component along its transmission axis. For linearly polarized light, the transmitted intensity follows Malus’s law:
Here, θ is the angle between the incoming linear polarization and the polarizer’s transmission axis. Parallel alignment transmits 100% of the incident power in this ideal model. At 45°, transmission is 50%. At 90°, the polarizer blocks the light.
Select Two polarizers: Malus’s law and rotate the analyzer, P2. With the input and P1 aligned at 0° and no waveplate inserted, the analyzer scan follows a cos² curve. The Crossed polarizers: extinction preset places P2 at 90° for zero ideal transmission.
These percentages refer to linearly polarized incident light. Unpolarized light would require a different source model; it is not included in this simulator.
What Does a Quarter-Wave Plate Do?
A quarter-wave plate introduces a 90° phase difference between field components along its fast and slow axes. Unlike an ideal polarizer, an ideal waveplate changes polarization without reducing the total transmitted power.
Select Quarter-wave: linear to circular. The incoming polarization is horizontal and the plate’s fast axis is at 45°, giving equal field amplitudes along the two plate axes. The quarter-cycle phase difference produces circular polarization before the analyzer.
Rotate the fast axis away from 45° to produce elliptical polarization. When the input aligns with either plate axis, the output stays linear. The relevant angle is always the plate axis relative to the polarization reaching it.
How a Half-Wave Plate Rotates Polarization
A half-wave plate introduces a 180° phase difference. For linear input, it produces linear output with a new orientation determined by the fast-axis angle.
Here, β is the fast-axis angle and α describes the linear polarization orientation, with equivalent orientations separated by 180°. For horizontal input at 0°, a half-wave plate at 45° produces vertical output at 90°.
Try Half-wave: rotate by 90°. The analyzer is vertical, so the rotated light passes through. Bypass the waveplate and the same analyzer blocks the horizontal input.
Reading the Two Polarization Plots
Before analyzer shows the polarization after P1 and the waveplate. Final output shows what remains after P2. This distinction matters: circular or elliptical light becomes linear after passing through an ideal linear analyzer, provided some light is transmitted.
A circularly polarized beam sends 50% of its power through an ideal analyzer at every angle. That is why the analyzer scan is flat for the circular preset. If you bypass P2, the final plot retains the incoming polarization and the scan shows the hypothetical transmission if an analyzer were inserted.
Total transmission is measured relative to the original source. P2 transmission is measured relative to the light entering P2. For example, if P1 passes 50% of the source and P2 passes half of that remaining light, total transmission is 25%.
Polarization and Reflection
Polarization also affects reflection at optical surfaces. The s and p directions are defined relative to the plane of incidence. Use our Fresnel reflection calculator to compare their reflected and transmitted powers for different refractive indices and incidence angles.
Using the Model in a Laser Setup
The simulator uses Jones vectors for fully polarized monochromatic light, ideal polarizers and lossless waveplates at their design wavelength. Real optics have finite extinction ratios, reflection losses and retardance tolerances. Wavelength, temperature and incidence angle can also affect a real waveplate.
When planning a laser system, consider polarization control alongside the source and its electronics. A laser diode driver controls the electrical current supplied to the diode; polarizers and waveplates act on the emitted light. They serve different functions in the system.
For more interactive examples, visit our photonics simulators, or browse the photonics calculators for related optical calculations.