Malus Law

Malus Law, also known as Malus Law of Polarization, is a fundamental principle in optics that describes how the intensity of polarized light changes as it passes through a polarizer. It is named after Étienne-Louis Malus, a French physicist who formulated the law in 1808.

In this article, we will discuss the concept of Malus Law which describes the intensity of change in the intensity of polarized light.

What is Malus Law?

According to Malus's rule, the square of the cosine of the angle formed between the polarizer's plane and the analyzer's transmission axes determines how much plane-polarized light changes in intensity as it passes through the device.

Malus's Law is significant in optics and helps in understanding how polarized sunglasses work, how stress patterns in transparent materials can be studied, and in various other applications where control of light polarization is required.

Who is Etienne-Louis Malus?

Étienne-Louis Malus was a mathematician, physicist, engineer, and commander from France. Malus was born in France's Paris. He served at the Institut d'Égypte's mathematics department and took part in Napoleon's invasion of Egypt.

Read More about Polarization of Light.

Malus Law Formula

Malus noticed that when the crystal spun, the intensity changed from highest to minimum. Consequently, he suggested that A = A cosθ must be the amplitude of the reflected beam. Malus calculated the intensity by square-rooting the amplitude relation i.e. I_o = A_o².

I(θ) = I_o cos² θ

Where

• I is the transmitted light intensity,
• I_o is the initial light intensity, and
• θ is the angle between the light's initial polarization direction and the axis of the polarizer.

This equation is known as Malus’s Law.

Derivation of Malus Law Formula

Assume the light source emits unpolarized light, which means the electric field vector oscillates randomly in all directions perpendicular to the direction of propagation.

When this light encounters a polarizing filter, the filter only allows the component of the electric field parallel to its polarization axis to pass through.

The electric field vector of the incoming light can be represented as = E₀​cos(ωt−kx) , where:

• E₀​ is the amplitude of the electric field,
• ω is the angular frequency of the light,
• k is the wave number,
• x is the position,
• t is the time, and
• is a unit vector in the direction of the electric field.

When the light hits the polarizer, the electric field vector can be decomposed into two components: parallel (E_∥​) and perpendicular (E_⊥​) to the axis of the polarizer. Since the perpendicular component is blocked, only E_∥​ passes through.

If θ is the angle between the light's initial polarization direction and the axis of the filter, the parallel component is given by E_∥​=E₀​cos(θ).

The intensity of light is proportional to the square of the amplitude of its electric field. If I₀​ is the intensity of the incident light, the intensity I of the transmitted light is proportional to E_∥²​.

Therefore, the intensity of the light after passing through the polarizer is I = I_o cos² θ.

Principles of Malus Law

According to Malus' law, the polarizer's angle affects how much-polarized light can flow through it.

• Only the portion of unpolarized light that is parallel to the polarization axis is transmitted via a polarizer. The square of the cosine of the angle formed between the polarization axis and the direction of the incident light determines the intensity of the light that is transmitted. We call this Malus' law.
• Malus' law may be mathematically represented as I = I0 cos²θ, where I is the transmitted light intensity, I0 is the incident light intensity, and θ is the angle formed between the polarization axis and the incident light direction.
• Malus' law is crucial to the study of polarization and has several uses, including the creation of polarising filters for LCD panels and cameras. It is also used in the science of optics to measure the birefringence of materials and to ascertain the polarization state of light.
• All things considered, Malus' law is a key idea in the study of polarization and has significant implications in many other domains.

Experimental for Malus Law

Malus Law Experiment. The goals of this experiment are to determine the connection between the intensity of light passed through the analyzer and the angle '′ between the polarizer and analyzer axes. The following equipment was used: The following are the experiment's requirements: A laser diode.

Setup and Equipment for Malus Law Experiments

To perform the experiment we can use the following setup and procedure for experiment.

1. Set up the laser, photodiode, polarizer and analyzer as shown in the picture to test Malus' law.
2. Make sure that the polarizer and analyzer are perpendicular to the laser beam and that the beam passes through point a. the "good" part of polarizers - look for minimum dispersion etc.
3. As the laser passes through the polarizer, the analyzer, and then the detector, make sure that the polarizer and the driving shafts of the analyzer are parallel. This is how you can work from 0o do that hold the polarizer still and rotate the analyzer until you detect the transmission maximum.
4. Note down the maximum current Imax and the angle as o.
5. Here, the emission axes of the polarizer and the analyzer are parallel.
6. Rotate the analyzer in 10o increments of o to obtain 'θ' from 0o to 360o . Take readings intensity in every corner.
7. The intensity of light beam that passes through polarizer and analyzer was measured by the light sensor.
8. The rotary motion sensor measures the angle that was obtained from rotating the second analyzer relative to the first polarizer.
9. In each case the current is noted and tabulated in
10. Plot a graph with the current 'Iexpt' on the Y-axis and the analyzer rotation angle on the X-axis. The graph clearly shows the cosineness of the curve, which confirms Malus' law.

Observations of Experiment

In above mentioned experiment we can observe the following things.

1. Smallest number of circular scales in polarizer and analyzer
2. First polaroid (polarizer) values ​​0₁
3. Lux meters (maximum) after I^st polaroid P (I₀)

Using these observations, we can calculate the value of required intensity.

Limitations of Malus Law

There are some limitations of this law in real life:

• Ideal Polarizers: Malus's Law assumes that the polarizer used is ideal, meaning it completely absorbs or blocks the perpendicular component of the electric field and only allows the parallel component to pass through without any loss. In reality, no polarizer is perfect, and there can be some attenuation of the transmitted light and minor transmission of the blocked component.

• Monochromatic and Coherent Light: The law is derived under the assumption that the light is monochromatic (single wavelength) and coherent. When dealing with polychromatic (multi-wavelength) or incoherent light, the behavior might differ, and the law may not hold precisely.

Read More,

• Dielectrics and Polarization
• Polarization by Scattering and Reflection
• Dual Nature of Light

Sample Questions on Malus Law

Question 1: What is the difference between unpolarized light and plane-polarized light?

Answer:

Ordinary (or unpolarized) light and plane polarized light vary in that the former oscillates in a single plane alone, while the latter has vibrations occurring inside it at random angles without any plane. Using an organic filter is another way to polarize light.

Question 2: Two consecutive polaroid (P1 and P2) are exposed to unpolarized light, and the polaroid P1 forms an angle of θ with the axis of P2. Determine the ultimate outgoing light's intensity?

Answer:

A light beam without polarization travels through two consecutive polaroid.

We are aware that the intensity of unpolarized light decreases to half as it travels through a Polaroid. In the event that it traverses any further polaroid, the intensity determined by Malus law .The angle θ in this case is between the polaroid's intensity variation axis and 0 to 2π. The curve is nothing (cos2 θ).

Question 3: How Is Light Intensity Measured?

Answer:

You must first measure the distance between the light source and the spot in order to determine the intensity of the light. Decide on the S.I. units for the distance.

For instance, the unit should be 0.56 m if the point of application is 56 cm from the light source.