Malus Law Calculator

Malus Law Calculator that finds the intensity a polarizer passes using I = I0 cos²θ, solves for the angle or initial intensity, handles unpolarized light and runs polarizer stacks.

What the Malus Law Calculator Solves

The Malus Law Calculator works with light that passes through a polarizer. It finds the transmitted intensity, solves backward for the polarizer angle or the initial intensity, and follows light through a stack of polarizers set at any angles you list.

Each mode accepts already polarized light or unpolarized light sent through two polarizers, and an optional panel adds the losses of a real polarizer. Intensity can be entered in W/m², mW/m², W/cm², mW/cm², lux or foot-candles, and the answer comes back in the same unit, with the cos²θ curve, an angle table and the step-by-step math.

Malus’s Law Formula

Malus’s law says the intensity leaving an ideal polarizer equals the incoming intensity times the square of the cosine of the angle between them. The angle θ is measured between the light’s polarization direction and the polarizer’s transmission axis.

$$I = I_0 \cos^2\theta$$

Here I is the transmitted intensity and I₀ is the intensity of the polarized light arriving at the polarizer. The NCERT Class 12 Wave Optics chapter states it the same way in Section 10.7, with I₀ as the intensity of the polarized light after the first polaroid.

With the default inputs, 5 W/m² at 20° gives 5 × cos²(20°) = 5 × 0.883 = 4.415 W/m². The angle only matters through cos²θ, so 160°, 200° and 340° give the same 4.415 W/m², and card 4 of the Malus Law Calculator lists those matching angles for you.

Solving for the Angle or I₀

Rearranging the law gives the angle from two intensity readings, or the starting intensity from a reading and an angle. The Polarizer Angle and Initial Intensity modes of the Malus Law Calculator use these two forms.

$$\theta = \cos^{-1}\sqrt{\frac{I}{I_0}}$$

$$I_0 = \frac{I}{\cos^2\theta}$$

If 4 W/m² goes in and 1 W/m² comes out, the ratio is 0.25 and θ = cos⁻¹(0.5) = 60°. If a meter reads 2 W/m² behind a polarizer set at 60°, the light going in was 2 ÷ 0.25 = 8 W/m². At exactly 90° no light passes, so the tool asks for another angle rather than dividing by zero.

Unpolarized Light

Ordinary light from the sun or a bulb is unpolarized, so its polarization direction changes at random. The average of cos²θ over all directions is ½, which means the first polarizer passes half the light whatever its angle.

$$I = \frac{I_0}{2} \cos^2\theta$$

In the Malus Law Calculator, pick Unpolarized, 2 Polarizers and θ becomes the angle between the two polarizer axes. Unpolarized light of 5 W/m² through polarizers 20° apart gives 2.5 × 0.883 = 2.21 W/m², which is 44.2% of the light that started.

How to Derive Malus’s Law

Light is a transverse wave, and its electric field E₀ points along the polarization direction. A polarizer passes only the part of the field along its transmission axis, which is E₀ cos θ, and blocks the part at right angles to it.

Intensity is proportional to the square of the field amplitude. So the transmitted intensity scales with (E₀ cos θ)², or I₀ cos²θ, which is Malus’s law. Card 1 shows both numbers, the intensity ratio I/I₀ and the field ratio |cos θ|, so at 20° the field keeps 94.0% of its amplitude while the intensity keeps 88.3%.

$$\frac{I}{I_0} = \left(\frac{E_0 \cos\theta}{E_0}\right)^2 = \cos^2\theta$$

Stacks of Polarizers

Crossed polarizers at 0° and 90° block all light. Put a third polarizer at 45° between them and light gets through again, because each polarizer turns the light’s polarization part of the way and only loses cos² of that smaller step.

The Stack of Polarizers mode in the Malus Law Calculator takes a list of axes such as 0, 45, 90 and applies the law at each one. With polarized light along 0°, the output is 5 → 2.5 → 1.25 W/m², or one quarter of I₀. With unpolarized light the first polarizer halves it first, so the same stack passes one eighth.

Splitting a 90° turn into more equal steps lets through more light. The table shows the share of polarized light that survives a 90° turn made in equal steps, which is what card 2 reports for your own first and last axes.

For a single polarizer, card 2 gives the same idea in reverse. At 20° it shows that one extra polarizer at 10° would raise the output from 4.415 to 4.703 W/m², and two extra polarizers would raise it to 4.801 W/m². In stack mode, card 4 names the polarizer where the most light is lost and the turn that causes it.

Loss in OD, Decibels and Stops

Filter makers, lab optics and photographers each describe light loss on their own scale. Card 3 of the Malus Law Calculator converts the transmittance T = I/I₀ into all three, so a polarizer setting can be compared with a neutral density filter or a sensor’s exposure.

$$OD = -\log_{10} T$$

Attenuation in decibels is 10 × OD, and camera stops are −log₂ T. A polarizer at 45° passes half the light, which is 0.301 OD, 3.01 dB and exactly 1 stop. At 70° it passes 11.7%, which is 0.932 OD, 9.32 dB and about 3.1 stops. Crossed polarizers have no finite value on any of these scales.

Real Polarizers and Extinction Ratio

Malus’s law assumes an ideal polarizer that passes all light along its axis and none across it. Real polarizers lose some light on the pass axis and leak a little on the blocked axis, which is why crossed polarizers never read exactly zero in a lab.

The Real Polarizer Losses panel of the Malus Law Calculator takes two numbers from a polarizer’s data sheet. Pass-axis transmission k₁ is the share of light passed along the axis, and the extinction ratio ER is the ratio of maximum to minimum transmission, so the blocked-axis leak is k₂ = k₁ ÷ ER.

$$I = I_0\left(k_1 \cos^2\theta + k_2 \sin^2\theta\right)$$

That form applies to light that is already polarized. For unpolarized light through two identical real polarizers, the first one passes (k₁ + k₂)/2 of the light, and the pair follows a slightly different curve.

$$I = \frac{I_0}{2}\left[(k_1^2 + k_2^2)\cos^2\theta + 2 k_1 k_2 \sin^2\theta\right]$$

The RP Photonics Encyclopedia notes that simple sheet polarizers may reach only about 500:1, while high-quality Glan–Taylor prisms reach around a million to one. With 5 W/m² of polarized light, k₁ = 80% and ER = 500:1, the output at 20° falls from the ideal 4.415 to 3.533 W/m², and at 90° about 0.008 W/m² still leaks through.

For unpolarized light through two of those sheets, the open pair passes 1.600 W/m² instead of 2.5, and the crossed pair leaks 5 × 0.8 × 0.0016 = 0.0064 W/m². Card 4 then reports the turn to the minimum instead of a full block, and the reverse angle mode only accepts readings between that leak floor and the maximum.

Both fields start at ideal values, 100% and no extinction limit, so the calculator gives the textbook answer until you enter a data sheet’s figures. The curve, angle table, middle-polarizer card, stack mode and step list all switch to the real-polarizer model once you do.

Malus’s Law vs Brewster’s Law

Both laws deal with polarized light, but they answer different questions. Malus’s law gives how much polarized light a polarizer lets through at a given angle, while Brewster’s law gives the angle at which reflected light comes off a surface fully polarized.

$$\tan\theta_B = \frac{n_2}{n_1}$$

For light in air striking glass with n = 1.5, Brewster’s angle is 56.3°, and for water with n = 1.333 it is 53.1°. The Brewster’s Angle Calculator finds it for any pair of materials, and the Index of Refraction Calculator finds n when you only know the light speed or ray angles.

The two laws often meet in practice. Glare reflected near Brewster’s angle is strongly polarized, so a polarizer turned 90° to it removes most of that glare, exactly as Malus’s law predicts for crossed axes.

Malus Law in Class 12 Physics

Malus’s law appears in Chapter 10, Wave Optics, of the NCERT Class 12 Physics textbook. Section 10.7 on polarisation explains how two polaroids pass light and gives the law as Equation 10.18, I = I₀ cos²θ, without a separate treatment of Brewster’s law.

The CBSE Class 12 Physics syllabus for 2025–26 lists 14 chapters. Its Wave Optics content covers Huygens’ principle, interference, Young’s double slit and single-slit diffraction, but it does not list polarisation or Malus’s law, so check your own board’s current syllabus before you study it for an exam.

Polarisation means restricting the vibrations of a transverse light wave to one direction. Only transverse waves can be polarized, which is why polarization is also evidence that light is a transverse wave.

Who Gave Malus’s Law?

The law is named after Étienne-Louis Malus, a French military engineer and examiner at the École Polytechnique. According to the Florida State University optics timeline, he noticed in 1808 that sunlight reflected from the windows of the Luxembourg Palace changed as he turned an Iceland spar crystal in front of it.

Malus coined the term polarized for light made this way, and he received the Royal Society’s Rumford Medal while France and Britain were at war. The St Andrews MacTutor biography gives his life as 1775 to 1812 and dates the published discovery to 1809. His name is French and is said roughly as “ma-LÜS”, with the final s sounded.

How to Verify Malus’s Law

A classic lab test uses a light source, a fixed polarizer, a second polarizer on a rotating mount and a light meter or photodiode. Record the reading every 10° or 15° as you turn the analyzer from 0° to 180°, starting where the reading is highest.

Plot the readings against cos²θ. If the law holds, the points fall on a straight line with a slope set by I₀. Real polarizers absorb a little along their pass axis and leak a little when crossed, so the line sits below the ideal values and does not quite reach zero.

Enter your readings in the Malus Law Calculator’s Polarizer Angle mode to compare the measured angle with the dial, or in Transmitted Intensity mode to compare each reading with the predicted value. Add your polarizers’ transmission and extinction ratio, and the Intensity by Angle table gives the value a real setup should read at each standard angle.

Polarization in Everyday Life

Light can be linearly, circularly or elliptically polarized. Linear polarization keeps the field in one plane, which is the case Malus’s law describes, while circular and elliptical light have a field that rotates as the wave travels.

Polarized sunglasses use a vertical transmission axis to cut glare reflected from roads and water. Camera polarizing filters darken skies and remove reflections as you turn them, LCD screens place the liquid crystal between two polarizers, and many 3D cinema glasses use polarized lenses to send a different picture to each eye.

In electrostatics, polarization has a different meaning. It is the dipole moment per unit volume of a dielectric, with the SI unit C/m², and it arises through four mechanisms, namely electronic, ionic, orientational and space-charge polarization.

Can Humans See Polarised Light?

Most people can, faintly, through an effect called Haidinger’s brushes. Research reported in The Conversation describes it as a yellow bow-tie shape, set at right angles to the polarization, that fades within a few seconds.

In that study of 24 people, the average viewer needed light about 56% polarized to see it, and some could see it below 25%. The effect comes from inside the eye, so it shows which way light is polarized but not how strong it is, which is why measurement still needs a polarizer and a meter.

Malus Clause and Clawback

Malus also has a separate meaning in pay and finance that has nothing to do with light. The Bank of England’s PRA supervisory statement SS2/17 defines malus as reducing deferred variable pay before it vests.

Clawback goes further and makes a staff member return variable pay that has already been paid or vested. The physics law shares only the surname, so for optics questions the Malus Law Calculator and the Optics Calculators hub are the right place to work the numbers.