Malus Law Calculator

Use this Malus Law Calculator to find how much linearly polarized light gets through a polarizer (analyzer) at a given angle, or solve for the angle or the starting intensity. It also supports unpolarized input and a simple non-ideal polarizer model with leakage.

How to use our Malus Law Calculator

1. Choose what to solve for in "Solve for" (transmitted intensity I, angle theta, or initial intensity I0).
2. Enter the known intensity value(s) using the same unit for I and I0 (for example W/m^2).
3. Enter the angle theta as the angle between the light's polarization direction and the analyzer transmission axis (0 deg = aligned, 90 deg = crossed).
4. Open "Advanced options" if you want radians instead of degrees, unpolarized-light handling, a non-ideal polarizer model, or different number formatting.
5. If you pick "Unpolarized" in Advanced options, remember the first ideal polarizer transmits about half on average, so results will be lower than the already-polarized case.
6. If you pick the non-ideal model, set "Max transmission" (percent at 0 deg) and "Extinction ratio" (must be greater than 1).
7. Click "Calculate".
8. Read "Transmitted fraction" and "Percent transmitted" to quickly see how much light gets through compared to I0.
9. If solving for theta, check both listed angle solutions (0 to 180 degrees or 0 to pi radians) and pick the one that matches your setup.
10. If you get an error about "no physical solution", adjust inputs so the intensity is within the model limits (ideal: 0 to I0; non-ideal: between the minimum and maximum transmission).

Definitions

Intensity (I): How much light power hits each area (power per area). The calculator treats it as a number; units can be W/m^2 or any consistent unit.

Initial intensity (I0): The intensity before the analyzer (the polarizer you rotate to change the transmitted light).

Polarizer: An optical filter that only lets through light polarized along one direction (its transmission axis). [2]

Analyzer: The second polarizer used to measure or control the polarization by rotating it relative to the incoming polarization.

Polarized light: Light where the electric field mostly vibrates in a particular direction. [3]

Angle theta: The angle between the incoming polarization direction and the analyzer transmission axis (not an angle of incidence). [1]

Extinction ratio: For a non-ideal polarizer, the ratio of maximum transmitted intensity (aligned) to minimum transmitted intensity (crossed). Bigger means better blocking.

Methodology

What the calculator is modeling

Malus's law says the transmitted intensity after an ideal linear polarizer depends on the square of cosine of the angle between the light's polarization direction and the polarizer (analyzer) axis. [1]

Step 1: Convert the angle to radians for trig

θrad = θdeg × (π/180)

If you choose radians in Advanced options, the calculator uses your input as θ_rad directly.

Step 2: Compute the transmission factor

Ideal polarizer model (default):

f = cos(θrad)2

Unpolarized input option: If the light is unpolarized before a first ideal polarizer, the average intensity after that first polarizer is half of the incoming value (because the average of cos^2 over all directions is 1/2). [1]

f = (1/2) × cos(θrad)2

Non-ideal polarizer model: The curve keeps the cos^2 shape but includes a maximum transmission and a minimum leakage (not zero at 90 degrees).

Tmax = (Max transmission percent)/100

Tmin = Tmax / (Extinction ratio)

f = Tmin + (Tmax - Tmin) × cos(θrad)2

Step 3: Compute outputs for each solve mode

Mode A: Solve for transmitted intensity I

I = I0 × f

Mode B: Solve for initial intensity I0

I0 = I / f

If f is 0 (or extremely close to 0), the calculator blocks the result or shows N/A because dividing by near-zero would explode the answer.

Mode C: Solve for angle theta

First compute the required fraction r:

r = I / I0

Ideal model requires 0 <= r <= 1. Non-ideal requires T_min <= r <= T_max. If r is outside the allowed range, there is no physical solution.

Then map r to cos^2 and solve for the principal angle:

c = sqrt(r) (ideal)

c = sqrt((r - Tmin)/(Tmax - Tmin)) (non-ideal)

θ1 = arccos(c)

θ2 = π - arccos(c)

The calculator reports both solutions in the range 0 to π radians (or 0 to 180 degrees). This matches the fact that cos^2 repeats every 180 degrees, so angles θ and 180 - θ give the same transmitted intensity.

Other reported results

Transmitted fraction = I/I0

Percent transmitted = 100 × (I/I0)

Notes and sensitivity checks

The calculator may show a warning when θ is near 90 degrees (crossed), because small angle changes can cause large relative changes when transmission is very small. It may also warn when the non-ideal model is selected so you remember the minimum leakage is not zero.

Sources