Malus's Law Calculator
Calculate how much linearly polarized light passes through a polarizer using Malus's law. Enter the incident intensity and the angle between the polarization direction and the analyzer axis.
Find How Much Polarized Light Passes Through an Analyzer
Malus's Law Formula
Where:
- \(I_0\) — intensity of the linearly polarized light incident on the analyzer (W/m²)
- \(I\) — transmitted intensity after the analyzer (W/m²)
- \(\theta\) — angle between the light's polarization direction and the analyzer's transmission axis
The law states that a polarizer transmits only the component of the electric field parallel to its transmission axis, so the transmitted intensity scales with the square of the cosine of the angle between the two directions.
Malus's Law – Calculation Example
Example: linearly polarized light with an intensity of 100 W/m² passes through an analyzer whose transmission axis is rotated 30° relative to the light's polarization direction.
\[ I = I_0 \cos^2\theta = 100 \times \cos^2(30^\circ) \]
\[ \cos(30^\circ) \approx 0.8660, \quad \cos^2(30^\circ) = 0.75 \]
\[ I = 100 \times 0.75 = 75 \; \text{W/m}^2 \]
Result: the analyzer transmits 75 W/m², which is 75 % of the incident light; the remaining 25 W/m² is absorbed or blocked.
How to use the calculator
- Enter the Incident intensity (I₀) — the intensity of the linearly polarized light before the analyzer — and choose its unit: W/m², mW/cm², or W/cm².
- Enter the Angle between axes (θ) — the angle between the light's polarization direction and the analyzer's transmission axis — in degrees or radians. You can enter negative or values larger than 360°; only the cosine matters.
- Read the Transmitted light intensity, shown in all three intensity units, and the Transmission fraction and details, including cos θ, cos²θ, and the blocked intensity.
What is Malus's law?
Malus's law, published by Étienne-Louis Malus in 1809, describes how much linearly polarized light passes through a polarizing filter (often called an analyzer when it is used to examine polarized light). Light is an electromagnetic wave, and its electric field oscillates in a direction perpendicular to the direction of travel. In linearly polarized light, that oscillation happens along one fixed plane.
A polarizer only transmits the component of the electric field that lies along its transmission axis. If the light's polarization direction makes an angle θ with that axis, the field component that survives is \(E_0 \cos\theta\). Because intensity is proportional to the square of the field amplitude, the transmitted intensity becomes \(I_0 \cos^2\theta\) — Malus's law.
The angle dependence has some important special cases. At θ = 0° the axes are aligned and all the light passes through. At θ = 90° the polarizers are crossed and, ideally, no light passes at all — this is the principle behind adjustable sunglasses and LCD shutters. At θ = 45° exactly half the intensity is transmitted. Because of the squared cosine, the transmission is also periodic with a period of 180°, so 30° and 150° give the same result.
Malus's law is central to polarization optics and appears in physics courses alongside other wave phenomena such as double-slit interference and diffraction gratings. If you work with intensity on a logarithmic scale, the related sound intensity level calculator converts between ratios and decibels.
Assumptions, limitations, and practical tips
- Perfect polarizer: the law assumes an ideal analyzer that transmits 100 % of light polarized along its axis and blocks 100 % perpendicular to it. Real polarizers absorb 10–20 % of the aligned component, so measured values are somewhat lower than the ideal result.
- Fully polarized input: the incident light must be linearly polarized. If unpolarized light of intensity \(I_0\) enters a polarizer, exactly half is transmitted on average (\(I = I_0/2\)), regardless of the angle — you can then apply Malus's law to that first polarizer's output.
- Energy conservation: the blocked component is not lost; it is absorbed as heat (as in Polaroid film) or deflected (as in polarizing beam splitters). The calculator reports this blocked intensity as \(I_0 - I\).
- Angle periodicity: since \(\cos^2\theta\) has a period of 180°, any angle can be reduced modulo 180° without changing the result. Angles above 90° give the same transmission as their supplement (e.g. 120° behaves like 60°).
- Partial polarization: for partially polarized light, treat the input as the sum of a polarized part (which follows Malus's law) and an unpolarized part (which always contributes half of its intensity).
Reference table: transmission vs. angle
| Angle θ (degrees) | cos²θ | Transmitted fraction |
|---|---|---|
| 0° | 1.000 | 100 % |
| 15° | 0.933 | 93.3 % |
| 30° | 0.750 | 75 % |
| 45° | 0.500 | 50 % |
| 60° | 0.250 | 25 % |
| 75° | 0.067 | 6.7 % |
| 90° | 0.000 | 0 % |
Use this table to sanity-check your calculator inputs: the transmitted fraction should drop smoothly from 100 % to 0 % as θ goes from 0° to 90°.
Frequently Asked Questions
How do I calculate Malus's law by hand without a calculator?
First convert the angle to degrees if needed, then take the cosine, square it, and multiply by the incident intensity: \(I = I_0 \cos^2\theta\). For common angles you can skip trigonometry entirely: \(\cos^2(0°)=1\), \(\cos^2(30°)=3/4\), \(\cos^2(45°)=1/2\), \(\cos^2(60°)=1/4\), and \(\cos^2(90°)=0\). So 100 W/m² at 60° gives \(100 \times 0.25 = 25\) W/m².
Why is my result wrong when I enter the angle in radians?
The most common mistake is mixing angle units. If your calculator or software is in radian mode but you type 30 (meaning degrees), you compute \(\cos^2(30 \text{ rad})\) instead of \(\cos^2(30°)\), which gives about 0.75 % instead of 75 %. Always check which unit you selected: \(30° = \pi/6 \approx 0.5236\) radians, and \(90° = \pi/2 \approx 1.5708\) radians.
What happens if unpolarized light passes through two polarizers?
For unpolarized light of intensity \(I_0\), the first polarizer always transmits exactly half: \(I_1 = I_0/2\), regardless of its orientation. Then apply Malus's law to the now-polarized light with the second polarizer: \(I_2 = (I_0/2)\cos^2\theta\). For example, unpolarized 100 W/m² through two polarizers at 45° to each other yields \(50 \times 0.5 = 25\) W/m².
Can light pass through three polarizers even if the first and last are crossed?
Yes — a classic physics puzzle. With the first and third polarizers at 90°, inserting a middle polarizer at 45° to the first transmits \(I = I_0 \cos^2(45°)\cos^2(45°) = I_0/4\), even though two crossed polarizers alone block everything. This happens because each polarizer projects the field onto its own axis, re-orienting the polarization. It demonstrates that polarization is not simply a filter that removes rays.
Why doesn't Malus's law apply to circularly or elliptically polarized light?
Malus's law assumes the incident light is linearly polarized, meaning the electric field oscillates along one fixed direction. Circular or elliptical polarization has a rotating field, so the transmitted intensity through a linear analyzer stays constant (or nearly constant) as the analyzer rotates. Circularly polarized light of intensity \(I_0\) always transmits \(I_0/2\) through a linear polarizer, just like unpolarized light.
How do I convert between W/m², mW/cm², and W/cm²?
Since 1 m² = 10,000 cm², you multiply by 10 to go from mW/cm² to W/m²: 1 mW/cm² = 10 W/m². Similarly, 1 W/cm² = 10,000 W/m² (1 × 10⁴). To go the other way, divide by the same factors: 1 W/m² = 0.1 mW/cm² = 0.0001 W/cm². This calculator converts automatically and displays the result in all three units so you can cross-check.
Does Malus's law apply to real polarizers like Polaroid filters?
Only approximately. Real polarizers have imperfections: they absorb 10–20 % of the aligned component (so even at θ = 0° less than 100 % passes) and leak a small amount at θ = 90° instead of transmitting nothing. Polarizing beam splitters do better because they redirect rather than absorb the blocked component. For precision work, multiply the ideal result by the polarizer's measured transmission coefficients for the parallel and perpendicular components.