Calculator Inputs
Choose a mode, enter the values, and submit. The result appears above this form.
Plotly Graph
The chart updates from the current calculation mode and values.
Example Data Table
| Mode | Input Values | Output | Interpretation |
|---|---|---|---|
| Malus law | I₀ = 100 W/m², θ = 30° | I = 75 W/m² | Three quarters of the intensity passes through. |
| Degree of polarization | Imax = 80, Imin = 20 | P = 0.6 or 60% | The wave is partially polarized. |
| Brewster angle | n₁ = 1.00, n₂ = 1.50 | θB ≈ 56.31° | Reflected light becomes perfectly polarized at this angle. |
| Ellipse analysis | Eₓ = 4, Eᵧ = 2, δ = 90° | Elliptical, axial ratio ≈ 2:1 | Unequal amplitudes with quarter phase shift form an ellipse. |
Formula Used
1) Malus law
I = I₀ cos²θ
This gives transmitted intensity after a plane polarizer when the incident wave is already linearly polarized.
2) Degree of polarization
P = (Imax − Imin) / (Imax + Imin)
This measures how strongly the wave is polarized from intensity extremes.
3) Brewster angle
tan θB = n₂ / n₁
At this incidence angle, the reflected wave is perfectly linearly polarized.
4) Polarization ellipse and Stokes parameters
S0 = Eₓ² + Eᵧ²
S1 = Eₓ² − Eᵧ²
S2 = 2EₓEᵧ cosδ
S3 = 2EₓEᵧ sinδ
ψ = 0.5 atan2(S2, S1)
χ = 0.5 asin(S3 / S0)
These relations classify the state as linear, circular, or elliptical.
How to Use This Calculator
- Select the calculation mode that matches your problem.
- Enter the known values in the visible input fields.
- Press Calculate to generate the result above the form.
- Review the numerical output and the Plotly graph.
- Download the current summary as CSV or PDF if needed.
- Use the example table and formulas section to verify your setup.
Answers to the Requested Theory Questions
Benefits of different wave polarization in electromagnetic wave
Linear polarization: useful for antennas, filters, glare reduction, and polarization-sensitive measurements.
Circular polarization: useful when transmitter and receiver orientation may change, such as satellites, GPS, and mobile links.
Elliptical polarization: models many real propagation cases and helps describe imperfect circular or mixed polarization states.
Linearly polarized electromagnetic wave equation
A common linearly polarized plane wave moving in the z direction is:
𝐄(z,t) = E₀ cos(kz − ωt + φ) x̂
Its electric field oscillates along one fixed direction, here the x-axis.
An unpolarized EM wave is incident on two orthogonal polarizers
For ideal polarizers, the first polarizer transmits half the incident intensity:
I₁ = I₀ / 2
The second polarizer is orthogonal to the first, so:
I₂ = I₁ cos²90° = 0
The final transmitted intensity is zero.
Can sound waves be polarized?
Ordinary sound waves in air cannot be polarized because they are longitudinal. Their particle motion is parallel to propagation. However, transverse mechanical waves in solids can show polarization because their motion can occur in different perpendicular directions.
Frequently Asked Questions
1) What is polarization of a wave?
Polarization describes the direction and behavior of oscillation in a transverse wave. For electromagnetic waves, it tells how the electric field vector changes with time and space.
2) What does Malus law calculate?
It calculates transmitted intensity when linearly polarized light passes through a polarizer at angle θ. The relation is I = I₀ cos²θ.
3) What is Brewster angle physically?
It is the angle of incidence where reflected light becomes perfectly linearly polarized. At that angle, the reflected and refracted rays are perpendicular.
4) How do I know if polarization is circular?
Circular polarization occurs when the two orthogonal field components have equal amplitudes and a phase difference of ±90 degrees. The field tip traces a circle.
5) How is elliptical polarization different from linear polarization?
Linear polarization has a fixed oscillation direction. Elliptical polarization has rotating field direction, and the field tip traces an ellipse because amplitudes or phase difference create unequal motion.
6) Why does unpolarized light lose half its intensity in one polarizer?
Unpolarized light contains all transverse directions equally. An ideal polarizer selects one component only, so the average transmitted intensity becomes half of the incident intensity.
7) Can this calculator help with antenna polarization ideas?
Yes. The same polarization concepts guide antenna alignment, mismatch loss, and link behavior. Linear, circular, and elliptical states all matter in communication systems.
8) Why can the angle input be negative or above 180 degrees?
Polarization formulas are periodic. The calculator still works because the physical transmission repeats with angle, especially through the cosine-squared dependence in Malus law.