Malus's law is the rule that tells you how much light gets through a linear polariser. If a beam is already linearly polarised with intensity I and the polariser's transmission axis sits at an angle θ to the beam's plane of polarisation, the intensity that emerges is I cos²θ. Two facts follow immediately and both surprise people the first time they meet them: the falloff is not linear, and a pair of polarisers turned to 90° blocks essentially everything while a third one inserted between them at 45° lets light through again.
This Malus law calculator from Arb Digital works through a stack of up to three polarisers one element at a time rather than collapsing the whole chain into a single angle, because that is where the interesting behaviour lives. It also separates the two things the first element does depending on what hits it — halving unpolarised light regardless of orientation, or applying the cosine-squared factor to light that already has a plane — and it lets you model a real polariser with a parallel transmittance below one and a crossed transmittance above zero instead of assuming perfection.
What This Malus Law Calculator Does
You give it an incident intensity, tell it whether the source is unpolarised or already linearly polarised, and enter the absolute axis angle of each polariser in the stack. The tool computes the intensity leaving each element, reports the final value in the result hero, and shows the overall transmission as a fraction, as a percentage and in decibels. The bar chart underneath makes the stage-by-stage loss visible, which is the part a single number hides.
Angles here are absolute orientations measured from the same arbitrary reference, not the differences between elements. That is deliberate. The physics depends only on differences, so a tool that asked for differences would work, but real benches are set up by reading each polariser's own scale. Entering 0°, 30° and 60° is how you would actually record the experiment, and the calculator does the subtraction for you and prints each crossing angle in the note.
The two transmittance boxes are the honest-polariser feature. An ideal polariser passes all of the component parallel to its axis and none of the perpendicular component, so k₁ = 1 and k₂ = 0. Real dichroic sheet absorbs some of the wanted component and leaks some of the unwanted one, so the transmitted intensity is I(k₁cos²θ + k₂sin²θ). Leave the defaults alone and you get textbook Malus; put the figures from your own component's datasheet in and you get a number you could actually compare with a photodiode reading.
How to Use It
- Enter the incident intensity in whatever unit you have. Watts per square metre, milliwatts per square centimetre, raw photodiode counts — the law is a ratio, so the output comes back in the same unit you put in. Only the transmission fraction and the decibel figure are unit-free.
- Set the source type correctly. Sunlight, a filament lamp, a white LED and most fluorescent tubes are unpolarised. A laser, the output of another polariser, and light reflected near Brewster's angle are polarised or partly so. Getting this wrong changes the answer by a factor of two.
- Type the axis angle of each polariser you are using. Fill one, two or all three boxes. A blank box means that element is not in the beam, which is different from setting it to zero degrees.
- Read the stage-by-stage bars, not just the headline. The bars show what fraction of the original intensity survives at each stage. In a three-element stack the middle bar is usually the one that explains the result.
- Use the target-transmission box to work backwards. Type the fraction you need and the note tells you the crossing angle that produces it, which is how variable attenuators built from polariser pairs are set.
The Formula and a Worked Example
For polarised light meeting an ideal polariser, I = I₀ cos²θ, where θ is the angle between the plane of polarisation and the transmission axis. For unpolarised light the average of cos²θ over all angles is one half, so the first element passes I₀/2 whatever its orientation, and the emerging beam is polarised along that element's axis. Every element after the first sees polarised light and follows the cosine-squared rule with respect to the element before it.
Take the default loaded above: 100 W/m² of unpolarised light through polarisers at 0°, 30° and 60°. The first element halves it to 50 W/m² and sets the plane to 0°. The second is 30° away, and cos²30° = 0.75, so 37.5 W/m² emerges polarised at 30°. The third is again 30° away, giving another factor of 0.75 and 28.125 W/m². Overall transmission is 28.125 %, which is an attenuation of 5.51 dB, and the plane of polarisation has been rotated by a total of 60°. The OpenStax University Physics section on polarization derives the same relation and works a similar example if you want the full argument rather than the result.
Why the Third Polariser Puts the Light Back
Cross two polarisers at 90° and the field goes dark. Slide a third one between them at 45° and light reappears, which looks like adding an obstacle to make a path clearer. It is the single most instructive demonstration in the whole subject and the arithmetic is short. The first element passes half. The middle element is 45° from it, so cos²45° = 0.5 and a quarter of the original survives, now polarised at 45°. The last element is 45° from the middle one, not 90° from the first, so it passes half again and one eighth of the original intensity emerges. Load the "Three at 45°" preset and you will see 12.5 %.
The resolution is that a polariser does not filter photons by inspecting a fixed label. It projects the field onto its own axis and re-emits what survives with a new plane of polarisation. The middle element genuinely rewrites the state of the beam, so the final element is no longer looking at anything perpendicular to itself. Remove the middle element and the perpendicular relationship is restored and the field goes dark again. This is a quantum measurement of a two-state system dressed up as an optics bench, and it is why the demonstration keeps appearing in introductory quantum courses as well as in optics.
The same reasoning generalises. With N polarisers spread evenly from 0° to 90°, each step is 90°/N and the total transmission after the first element is cos²(90°/N) raised to the power N − 1, which climbs towards one half as N grows. A large stack rotates polarisation through a right angle with almost no loss. Real stacks do not behave that well because every extra surface reflects and absorbs, which is exactly what the k₁ input is for.
Where the Law Stops Being Valid
Malus's law assumes the incident light is either fully unpolarised or fully linearly polarised, that the polariser is linear rather than circular, and that you only care about intensity. Each assumption fails somewhere useful.
Partially polarised light is the common real case. Skylight, glare off a wet road and light reflected from a painted surface all sit somewhere between the two extremes, and the honest model splits the beam into a polarised fraction and an unpolarised fraction and applies the right rule to each. This tool covers the two limits; for a beam with a degree of polarisation of, say, 0.4, run it twice and add the results in that ratio.
Circular and elliptical polarisation break the law outright, because there is no single plane for the angle to be measured from. Circularly polarised light passing through any linear polariser gives half the intensity at every orientation, which looks exactly like unpolarised light on an intensity meter. Telling the two apart needs a quarter-wave plate in front of the analyser, and this is why photographic circular polarisers behave differently from the linear kind in autofocus cameras.
Finally, the law says nothing about wavelength. Real polarisers have a working band, and outside it the transmittance figures change, sometimes dramatically. If you are working at a specific wavelength it is worth converting to photon energy with the photon energy calculator or checking the frequency relationship with the wavelength calculator before assuming a datasheet figure applies.
Polarisation by Reflection, and Why Sunglasses Work
Light does not have to pass through a sheet to become polarised. Reflection from a dielectric surface partly polarises the reflected beam parallel to the surface, and at one particular incidence angle — Brewster's angle, where the reflected and refracted rays are 90° apart — the reflected light is very nearly perfectly polarised. Because most troublesome glare comes off horizontal surfaces such as water, glass and tarmac, that reflected light is predominantly horizontally polarised, which is why polarised sunglasses have their axis vertical. Rotate a pair 90° while looking at a wet road and the glare returns.
Brewster's angle depends on the refractive indices of the two media through the arctangent of their ratio, which makes it a natural companion calculation to this one. The Snell's law calculator handles the refraction side of the same boundary, and the index of refraction calculator covers the material property itself. If the distinction between linear, circular and elliptical polarisation is new, the HyperPhysics page on the classification of polarisation sets the three states out clearly.
How This Differs From the Adjacent Arb Digital Tools
This page is about intensity surviving a polariser, and nothing else. The optics tools nearby answer different questions and it is worth being explicit about the boundaries so you land on the right one. The Snell's law calculator handles the direction a ray bends at a boundary but says nothing about how much energy is transmitted. The thin lens equation calculator and the mirror equation calculator locate images formed by refracting and reflecting surfaces respectively, treating light as rays with no polarisation state at all. The lumen to lux calculator converts between photometric quantities weighted by human vision, which is a different measure of brightness from the radiometric intensity used here. And the angle converter is there for the mundane but frequent job of turning radians or gradians into the degrees this tool expects.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Measuring the angle from the bench instead of from the previous element — only the difference between two transmission axes matters. Rotating the entire stack together changes nothing.
- Applying cos²θ to the first element when the source is unpolarised — natural light has no plane, so the first polariser always passes half and always defines the plane for everything downstream.
- Using cosθ rather than cos²θ — the field amplitude falls as cosθ, and intensity goes as the square of amplitude. Forgetting the square is the most common arithmetic slip in the whole topic.
- Assuming crossed polarisers give absolute darkness — a real pair leaks. That leakage is the k₂ term, and it sets the useful contrast limit of every LCD panel ever built.
- Applying the law to circularly polarised light — it gives half the intensity at every angle and is indistinguishable from unpolarised light on an intensity meter alone.
Related Free Tools From Arb Digital
Stay in optics with the Snell's law calculator for refraction at a boundary and the mirror equation calculator for curved reflecting surfaces. Move to the wave description with the wavelength calculator, or to the particle description with the photon energy calculator. For lighting work rather than physics, the lumen to lux calculator converts a source rating into an illuminance at a distance. Unit housekeeping is handled by the angle converter and the energy converter, and the full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
The intensity of linearly polarised light passing through a linear polariser equals the incident intensity multiplied by the square of the cosine of the angle between the light's plane of polarisation and the polariser's transmission axis.
Because unpolarised light contains every plane of polarisation equally, and the average value of cosine squared taken over all angles is exactly one half. The result does not depend on how the first polariser is oriented.
A polariser does not merely block light, it re-emits what survives with a new plane of polarisation. A middle element at forty-five degrees rewrites the beam so the final element is no longer perpendicular to it, and one eighth of the original intensity emerges.
For already-polarised light meeting an ideal polariser, forty-five degrees, because the cosine of forty-five degrees squared is one half. For unpolarised light the first element already costs you half, so fifty per cent overall is the best a single polariser can do.
Close to it, but not exactly. Real sheet absorbs part of the wanted component and leaks part of the unwanted one. Entering a parallel transmittance below one and a crossed transmittance above zero models both effects and reproduces the finite contrast a real pair achieves.
No. Circular polarisation has no single plane for the angle to be measured against, and it delivers half the incident intensity through a linear polariser at every orientation. Distinguishing it from unpolarised light requires a quarter-wave plate.
Glare reflected from horizontal surfaces such as water and road is predominantly horizontally polarised, so a vertical transmission axis rejects it. Rotating the lenses ninety degrees restores the glare, which is a quick way to check a pair is genuinely polarised.
This tool is provided for educational and study use. It applies the ideal and two-parameter polariser models as written and is not a substitute for measured component data or laser safety assessment.
