Malus Law: Formulation

It is difficult to distinguish who first discovered polarized light. Ancient people could notice a peculiar spot, looking at the sky in certain directions. Polarization has many quirks, manifests itself in different areas of life, and today it is the subject of mass research and application, the reason for everything is the law of Malus.

Discovery of polarized light

The Vikings might have used the polarization of the sky for navigation. Even if they did not, then they definitely discovered Iceland and the wonderful calcite stone. Icelandic spar (calcite) was known even in their times, it is the inhabitants of Iceland that it owes its name. The mineral was once used in navigation due to its unique optical properties. He played a major role in the modern discovery of polarization and continues to be the preferred material for the separation of the polarized components of light.

In 1669, the Danish mathematician from the University of Copenhagen, Erasmus Bartholinus, not only saw a double light, but also conducted some experiments by writing 60-page memoirs. This was the first scientific description of the polarization effect, and the author can be considered the discoverer of this amazing property of light.

Christian Huygens developed the pulse-wave theory of light, which he published in 1690 in his famous book Traite de la Lumiere. At the same time, Isaac Newton advanced the corpuscular theory of light in his book Opticks (1704). As a result, both were right and wrong, because light has a dual nature (wave and particle). Nevertheless, Huygens was closer to the modern view of the understanding of the process.

In 1801, Thomas Jung made the famous two-slit interference experiment. He proved that light behaves like waves, and the superposition of waves can lead to darkness (destructive interference). He used his theory to explain things like Newton's rings and the supernatural arcs of the rainbow. A breakthrough in science occurred a few years later, when Jung showed that polarization is due to the transverse wave nature of light.

The young Etienne Louis Malus lived in a turbulent era - during the French Revolution and the reign of terror. He participated with the army of Napoleon during the invasion of Egypt, as well as in Palestine and Syria, where he contracted the plague that killed him several years later. But he managed to make an important contribution to understanding polarization. Malus' law, which predicted the intensity of light transmitted through the polarizer, became one of the most popular in the 21st century when creating liquid crystal screens.

Sir David Brewster, a renowned science writer, studied optical physics, such as dichroism and absorption spectra, as well as more popular subjects, such as stereo photography. Brewster’s famous phrase is known: "Everything is transparent except glass."

He also made an invaluable contribution to the study of light:

• The law describing the "polarization angle."
• The invention of a kaleidoscope.

Brewster repeated the experiments of Malus for many precious stones and other materials, having discovered an anomaly of glass, and discovered the law - “Brewster’s corner”. According to him, "... when the beam is polarized, the reflected beam forms a right angle with the refracted beam."

The Polarization Law of Malus

Before speaking of polarization, one must first remember the light. Light is a wave, although sometimes it is a particle. But in any case, polarization makes sense if we imagine the light as a wave, as a line when it moves from the lamp to the eyes. Most light is a mixed mess of light waves that oscillate in all directions. This direction of oscillation is called the polarization of light. A polarizer is a device that cleans up this mess. It accepts everything that mixes light, and transmits only light that oscillates in one particular direction.

The formulation of the Malus Law is as follows: when completely flat polarized light falls on the analyzer, the intensity of the light transmitted by the analyzer is directly proportional to the square of the cosine of the angle between the transmission axes of the analyzer and the polarizer.

A transverse electromagnetic wave contains both an electric and a magnetic field, and the electric field in the light wave is perpendicular to the direction of propagation of the light wave. The direction of light vibration is the electrical vector E.

For an ordinary unpolarized beam, the electric vector continues to change its direction randomly, when the light passes through the polaroid, the resulting light is plane-polarized with its electric vector vibrating in a certain direction. The direction of the vector of the emerging beam depends on the orientation of the polaroid, and the plane of polarization is designed as a plane containing the E-vector and the light beam.

The figure below shows flat polarized light due to the vertical vector EI and horizontal vector EII.

Unpolarized light passes through Polaroid P 1, and then through Polaroid P 2, forming an angle θ with y ax-s. After the light traveling along the x direction passes through Polaroid P 1, the electric vector associated with the polarized light will vibrate only along the y axis.

Now, if we allow this polarized beam to again pass through polarized P 2, forming an angle θ with the y axis, then if E 0 is the amplitude of the incident electric field on P 2, then the amplitude of the wave emerging from P 2 will be equal to E 0 cosθ and, therefore, the intensity of the emerging beam will be according to the Malus Law (formula) I = I 0 cos 2 θ

where I 0 is the intensity of the beam emerging from P 2 when θ = 0 θ is the angle between the analyzer transmission planes and the polarizer.

Light intensity calculation example

Malus law: I 1 = I o cos 2 (q);

where q is the angle between the direction of polarization of light and the axis of transmission of the polarizer.

Unpolarized light with an intensity of I o = 16 W / m 2 is incident on a pair of polarizers. The first polarizer has a transmission axis aligned 50o from the vertical. The second polarizer has a transmission axis aligned at a distance of 20 ° from the vertical.

The verification of the Malus Law can be done by calculating what the light intensity is when it arises from the first polarizer:

4 W / m 2

16 cos 2 50o

8 W / m 2

12 W / m 2

Light is not polarized, therefore I 1 = 1/2 I o = 8 W / m 2.

Light intensity from the second polarizer:

I 2 = 4 W / m 2

I 2 = 8 cos 2 20 o

I 2 = 6 W / m 2

Then follows the Malus Law, the wording of which confirms that when the light leaves the first polarizer, it is linearly polarized at 50 °. The angle between this and the transmission axis of the second polarizer is 30 °. Hence:

I 2 = I 1 cos 2 30o = 8 * 3/4 ​​= 6 W / m 2.

Now the linear polarization of a light beam with an intensity of 16 W / m 2 falls on the same pair of polarizers. The direction of polarization of the incident light is 20o from the vertical.

The intensity of the light emerging from the first and second polarizers. Passing through each polarizer, the intensity decreases by 3/4 times. After exiting the first polarizer, the intensity is 16 * 3/4 ​​= 12 W / m 2 and decreases to 12 * 3/4 ​​= 9 W / m 2 after passing through the second.

The polarization of the Malus law says that for the rotation of light from one direction of polarization to another, the loss of intensity is reduced using more polarizers.

Suppose you need to rotate the direction of polarization by 90 ^o .

Calculation of the angle of reflection of Brewster

When light hits the surface, part of the light is reflected, and part penetrates (refracts). The relative amount of this reflection and refraction depends on the substances passing through the light, as well as on the angle at which the light hits the surface. There is an optimal angle, depending on the substances, which allows the light to refract (penetrate) as much as possible. This optimum angle is known as the angle of Scottish physicist David Brewster.

The calculation of the Brewster angle for ordinary polarized white light is performed according to the formula:

theta = arctan (n1 / n2),

where theta is the Brewster angle, and n1 and n2 are the refractive indices of two media.

To calculate the best angle for the maximum penetration of light through glass - from the table of refractive indices we find that the refractive index for air is 1.00 and the refractive index for glass is 1.50.

The Brewster angle will be arctan (1.50 / 1.00) = arctan (1.50) = 56 degrees (approximately).

Calculation of the best angle for light for maximum penetration into water. From the table of refractive indices it follows that the index for air is 1.00, and the refractive index for water is 1.33.

Brewster's angle will be arctan (1.33 / 1.00) = arctan (1.33) = 53 degrees (approximately).

Applying polarized light

A simple layman does not even realize how much polarizers are used in the world. The polarization of the light of the law of Malus surrounds us everywhere. For example, such popular things as Polaroid sunglasses, as well as the use of special polarizing filters for camera lenses. Various scientific instruments use polarized light emitted by lasers or by polarizing incandescent lamps and fluorescent sources.

Polarizers are sometimes used to illuminate rooms and stages to reduce glare and provide more even illumination and as glasses to give a visible sense of depth to three-dimensional films. Cross polarizers are even used in space suits to dramatically reduce the light from the sun falling into the eyes of an astronaut during sleep.

Secrets of optics in nature

Why blue sky, red sunset and white clouds? These questions have been known to everyone since childhood. The laws of Malus and Brewster clarify these natural effects. Our sky is really colorful, thanks to the sun. Its bright white light has all the colors of the rainbow embedded inside: red, orange, yellow, green, blue, indigo and purple. In certain conditions, a person meets either with a rainbow, or with sunset, or with a gray late evening. The sky is blue due to the "scattering" of sunlight. The color blue has a shorter wavelength and greater energy than other colors.

As a result, blue is selectively absorbed by air molecules, and then released again in all directions. Other colors are less scattered and therefore usually not visible. The midday sun is yellow, absorbing its blue color. During sunrise or sunset, sunlight penetrates at a low angle and must pass through the greater thickness of the atmosphere. As a result, the blue color is carefully scattered, so that most of it is completely absorbed by the air, other colors are lost and scattered, especially orange and red, creating a glorious horizon of colors.

The colors of sunlight are also responsible for all the shades that we like on Earth, whether it is green grass or the turquoise ocean. The surface of each object selects specific colors that it will reflect in order to distinguish itself. Clouds are often brilliant white because they are excellent reflectors or diffusers of any color. All returned colors are added to neutral white together. Some materials reflect all colors evenly, such as milk, chalk and sugar.

The value of polarization sensitivity in astronomy

For a long time studying the Malus law, the polarization effect in astronomy was ignored. The light of stars is almost completely polarized, and it can be used as a standard. The presence of polarized light in astronomy can tell us how light was created. In some supernovae, the emitted light is not unpolarized. Depending on the part of the star that is being contemplated, a different polarization can be seen.

This information about the polarization of light from different regions of the nebula may give researchers a clue about the location of the shaded star.

In other cases, the presence of polarized light can reveal information about the entire part of an invisible galaxy. Another use of polarization-sensitive measurements in astronomy is to detect the presence of magnetic fields. By studying the circular polarization of very specific colors of light emanating from the corona of the sun, scientists found out information about the strength of the magnetic field in these places.

Optical microscopy

A polarized light microscope is designed to observe and photograph samples that are visible due to their optically anisotropic nature. Anisotropic materials have optical properties that change with the direction of propagation of light passing through them. To accomplish this task, the microscope must be equipped with both a polarizer located in the path of light somewhere in front of the sample and an analyzer (second polarizer) placed in the optical path between the objective rear opening and the sight tubes or camera port.

The use of polarization in biomedicine

This trend, popular today, is based on the fact that in our bodies there are many compounds that are optically active, that is, they can rotate the polarization of the light passing through them. Various optically active compounds can rotate the polarization of light in different quantities and in different directions.

Some optically active chemicals are present in higher concentrations in the early stages of eye disease. Doctors can potentially use this knowledge to diagnose eye diseases in the future. One can imagine that the doctor shines with a polarized light source in the patient’s eye and measures the polarization of the light reflected from the retina. It is used as a non-invasive method for testing eye disease.

Present of the present - LCD screen

If you look closely at the LCD screen, you will notice that the image is a large array of colored squares located in a grid. The law of Malus found application in them, the physics of the process of which created the conditions when each square or pixel has its own color. This color is a combination of red, green, and blue light at each intensity. These primary colors can reproduce any color that the human eye can see, because our eyes are trichromatic.

In other words, they approximate specific wavelengths of light by analyzing the intensity of each of the three color channels.

Displays exploit this drawback by only displaying three wavelengths that selectively target each type of receptor. The liquid crystal phase exists in the ground state, in which the molecules are oriented in the layers, and each subsequent layer is slightly twisted to form a spiral pattern.

LCD display with seven segment LCD :

1. Positive electrode.
2. Negative electrode.
3. Polarizer 2.
4. Display.
5. Polarizer 1.
6. Liquid crystal.

Here the LCD is between two glass plates that are equipped with electrodes. LC transparent chemical compounds with "twisted molecules" called liquid crystals. The phenomenon of optical activity in some chemicals is due to their ability to rotate the plane of polarized light.

Stereopsis 3D Movies

Polarization allows the human brain to fake 3D by analyzing the differences between the two images. People do not see in three dimensions, our eyes can only see two-dimensional images. Nevertheless, our brain can understand how far objects are by analyzing the differences in what each eye sees. This process is known as Stereopsis.

Since our brains can only see pseudo-3D, film makers can use this process to create the illusion of three dimensions without resorting to holograms. All 3D movies work by delivering two photographs, one for each eye. By the 1950s, polarization had become the dominant method of image separation. Theaters began to have two projectors working simultaneously, with a linear polarizer above each lens.

For the current generation of 3D films, technology has switched to circular polarization, which takes care of the orientation problem. This technology is currently manufactured by RealD and makes up 90% of the 3D market. RealD has released a circular filter, which very quickly switches between polarization clockwise and counterclockwise, so only one projector is used instead of two.