Unit 8: Physical Optics and Gravitational Waves — Long Questions

Polarization
Definition: "Polarization is the process by which the electric and magnetic vibrations of light waves are restricted to a single plane of vibration. It is the property exhibited by transverse waves only such as light waves. It does not occur for longitudinal waves such as sound waves."

Explanation Interference and diffraction effects prove the wave nature of light. These phenomena, however, do not tell us whether the light waves are longitudinal or transverse. Polarization of light suggests that the light waves are transverse in character

In transverse mechanical waves, such as produced in a stretched string, the vibrations of the particles of the medium are perpendicular to the direction of propagation of the waves. The vibrations can be oriented along vertical, horizontal or any other direction (Fig.). In each of these cases, the transverse mechanical wave is said to be polarized. The plane of polarization is the plane containing the direction of vibration of the particles of the medium and the direction of propagation of the wave.

A light wave produced by oscillating charge consists of a periodic variation of the electric field vector E accompanied by the magnetic field vector B at right angle to each other Ordinary light has components of vibration in all possible planes. Such a light is unpolarized. On the other hand, if the vibrations are confined only in one plane, the light is said to be polarized. Unpolarized light is shown in Fig.

Examples of unpolarized light sources are sunlight, incandescent light bulbs, fluorescent light bulbs, light from a candle or fire.

Methods of Polarizing light
An unpolarized light can be made polarized by the following methods:
1 Passing light through a polarizing filter (e.g., Polaroid sheet).
2. Using a polarizing beam splitter.
3. Employing certain optical crystals or materials (e.g., calcite, quartz, etc.).

The most common method by which an unpolarized light can be polarized is by passing it through a polarizing filter, such as a polarizing beam splitter or a Polaroid sheet. When an unpolarized light passes through the polarized filter, only that electric field vector which is parallel to the axis of polarized filter can pass through it, while all other vectors are blocked. The resultant light then becomes polarized as shown in Fig.

Thus, in simple words, the process of transforming an unpolarized light in a polarized light is said to be polarization.

The orientation of the electric field vector E of light waves in a specific direction is the basis of Polarization.

Types of Polarization
Here are the basic types of polarization of light.

i. Linear Polarization
When the electric field vector oscillates in a single plane, light is said to be linearly polarized as shown in Fig. (a) Example is the light passing through a polarizing filter, like sunglasses.

ii. Circular Polarization
When the electric field rotates circularly, either clockwise (right-handed), or counterclockwise (left-handed), the light is said to be circularly polarized as shown in Fig. (b) Example is the light reflected off a CD (Compact Disc) or DVD (Digital Versatile Disc).

iii. Elliptical Polarization
A combination of linear and circular polarization, where the electric field vector traces an elliptical path is called elliptical polarization. In elliptical polarization as shown in Fig. (c), the two components of electric field E_x and E_y are not equal and/or they differ in phase by an arbitrary angle θ. Example is the light passing through a stress plate or a wave plate.

Production and Detection of Plane Polarized Light

The light emitted by an ordinary incandescent bulb (and also by the Sun) is unpolarized, because its (electrical) vibrations are randomly oriented in space as shown in Fig.

If unpolarized light is made incident on a sheet of Polaroid (polarizer), the transmitted light will be plane polarized. If a second sheet of Polaroid is placed in such a way that the axes of the polaroid, shown by straight lines drawn on them are parallel (Fig.), the light is transmitted through the second polaroid. If the second polaroid (analyzer) is slowly rotated about the beam of light, as axis of rotation, the light emerging out of the second polaroid gets dimmer and dimmer and disappears when the axes become mutually perpendicular as shown in Fig. The light reappears on further rotation and becomes brightest when the axes are again parallel to each other

This experiment proves that light waves are transverse waves. If the light waves were longitudinal, they would never disappear even if the two polaroids were mutually perpendicular.

Polarized Light by the Method of Reflection

In 1808, Malus discovered that polarized light is obtained when ordinary light is reflected (light bouncing back from a surface) by a plane sheet of glass. If the reflected light is viewed through a Polaroid which is slowly rotated about the line of vision, the light is practically extinguished at one position of the Polaroid. The most suitable angle of incidence i is about 57° for glass for which the reflected ray becomes plane polarized, as illustrated by ray diagram in Fig.

This proves that the light reflected by the glass is practically plane polarized. Light reflected from the surface of a table becomes darker when viewed through a rotated Polaroid, showing that it is partially plane polarized.

Brewster's Law

Statement "When light is incident on a surface at a certain angle (Brewster's angle), the reflected light is polarized in a plane perpendicular to the plane of incidence"

Explanation The particular angle of incidence on a transparent medium when the reflected light is almost plane polarized is called the polarizing angle. Let a beam of unpolarized light be made incident on the surface of medium 2 as shown in Fig. If the reflected beam of light is almost plane polarized, the reflected and refracted beams are at right angle to each other at the polarizing angle. i = θ_p Thus
θ_r + θ_t = 90°
θ_r = 90° - θ_p

or From Snell's law,
n₁ sin θ_p = n₂ sin θ_t

n₁ sin θ_p = n₂ sin θ_t (90° - θ_p)

n₂ sin θ_p = n₂ cos θ_p

sin θ_p / cos θ_p = n₂ / n₁

tan θ_p = n₂ / n₁

This equation is known as Brewster's law. In this equation, n₁ is refractive index of medium 1 and n₂ is refractive index of medium 2. If medium 1 is air, then equation becomes tan θ_p = n. Hence Brewster proved that the tangent of the angle of polarization is numerically equal to the refractive index of the medium 2 when medium 1 is air In Brewster's law, the angle "θ_p" for which the reflected ray and refracted ray makes an angle of 90° between them, is also called the Brewster angle θ_B, then tan θ_B = n holds.

Malus's Law

Malus's law states that the intensity I of plane polarized light after passing through an analyzer is directly proportional to the square of the cosine of the angle θ between the transmission axis of the analyzer and polarizer. That is:

Mathematically I ∝ cos² θ
I = I₀ cos² θ

Where I₀ = intensity of the incident polarized light. (after passing through a polarizer)

Explanation Actually, Malus's law gives a mathematical relation between the intensity of the light incident on the first Polaroid (i.e., polarizer) and the intensity of light obtained after passing it through the second Polaroid (i.e., analyzer). This is shown in Fig. (a) An analyzer is also a polarizer that is placed after a polarizer. Rotation of the analyzer affects the intensity of the polarized light. It is used to further reduce the intensity of light and also adjust it by adjusting the angle of the analyzer with respect to the polarizer.

Certain transparent crystalline materials, like tourmaline, calcite crystals, etc., are capable of confining vibrations of light waves in only one plane. Such materials are called polaroids which have high directionality in crystal structure. Light can also be polarized by natural phenomena like reflection, refraction and scattering.

If a piece of Polaroid is rotated in front of a polarized ray of light, it causes a variation in the intensity of the light that gets through. The reason that causes the variation of intensity is the angle between the initial polarizer and the axis of second polarizer

When the incident polarized light of amplitude A₀ strikes the analyzer at an angle θ, it is resolved into two components A₀ cosθ and A₀ sinθ as shown in Fig. (b) The component A₀ sinθ is absorbed in the analyzer. Since, only A₀ cosθ passes through the analyzer, the amplitude A of the transmitted light is therefore.

A = A₀ cos θ

Since intensity I is the square of the amplitude A it can be expressed as:
I = A²
or
I = A₀² cos² θ
I = I₀ cos² θ (. I₀ = A₀²)

Here I₀ is the intensity of the incident polarized light.

Extreme conditions of angle There are two extreme conditions of θ followed by the above equation given as.

i. If θ = 0°, then I = I₀. This means the intensity transmitted through the analyzer is equal to the initial light intensity that passes through the polarizer

ii. If θ = 90°, then I = 0. This means the light is extinguished completely, i.e., no light is allowed to pass through the analyzer.

Optical Activity

Optical activity is the ability of some substances to rotate the plane of polarization of light passing through them. The rotation is detected with a polarizer or analyzer as shown in in Fig.

Many crystals and solutions rotate the plane of polarization of light passing through them. Such substances are said to be optically active. Examples are quartz crystals, cinnabar (HgS), sugar water, insulin and collagen. The amount and direction of θ rotation depends on following factors:

i. The type of substance

ii. The concentration of the substance (the amount of a substance present in a given quantity of a mixture or solution).

iii. The distance the light travels through it, and

iv. The wavelength of light.

Optical activity occurs due to the asymmetric shape of molecules in the substance, such as being helical. A few mill meters thickness of such crystals will rotate the plane of polarization by many degrees. Certain organic substances, such as sugar and tartaric acid, show optical rotation when they are in a solution. This property of optically active substances can be used to determine their concentration in the solutions.

Importance of Polarization

i. Optics and Photonics
Polarization is essential for applications like polarized sunglasses, LCD (Liquid Crystal Display) screens, and optical communication systems.

ii. Imaging and Microscopy
Polarization enhances image quality, reducing glare (unwanted light that interferes with vision) and improving contrast, especially in microscopy and medical imaging.

iii. Medical Applications
Polarization is used in cancer diagnosis, tissue imaging, and laser surgery, leveraging its ability to distinguish between different tissue types.

iv. Astronomy
Polarization helps us to analyze cosmic phenomena, like the polarization of light from distant stars or the cosmic microwave background radiation.

v. Miscellaneous Field
Polarization has importance in miscellaneous fields such as optics and photonics, imaging and microscopy, biology and chemistry, communication system, etc.

Polaroid, also known as polarizing filters, have two main applications:

Two Main Applications of Polarization
i. Sky Photography
A camera which is used to photograph the clouds is fitted with a Polaroid. The light coming from the sky is polarized by the Polaroid.
In sky photography, polarizers are used to reduce the glare and haze which are produced by the scattering of light by small particles of molecules present in the atmosphere. Polarizers also enhance the contrast by clocking the excessive bright white light while allowing the other colours to pass through, thus creating a brighter detailed image. Thus, allows to improve the overall image quality.
ii. Stress Analysis of Materials
In materials science, polarizers are used to analyze the stress and strain on materials, such as plastics, metals, and glass.
When a material is stressed, its molecular structure changes, affecting the way, it interacts with light, and interference patterns on fringes are formed which in turn gives qualitative information about the material. By shining polarized light through the transparent material and analyzing the changes in the light's polarization, the researchers can:
• determine the material's stress patterns
• identify potential weaknesses or defects
• analyze the material's optical properties
iii. Photoelectricity
understand how a material will behave under different conditions. This technique is known as "photoelasticity" and is widely used in fields like engineering, materials science, and quality control.

Gravitational waves (GWs)
"A gravitational wave is a stretching and compressing of space and can be observed by measuring the change in length between two objects."

Gravitational waves (GWs) are actually:
"Ripples in the fabric of spacetime, produced by violent cosmic events, like colliding black holes or neutron stars that travel at the speed of light, carrying information about their source."

Examples The simplest example to understand GWs is given below:

If we throw a stone into a pond, the stone creates ripples on the water surface (spacetime). These ripples travel outward, carrying information about the stone (the cosmic event). They can be detected on the shore (by gravitational wave observatories), revealing the stone's presence and properties.

Prediction and Detection
Gravitational waves are a prediction of Einstein's theory of general relativity which is confirmed by observations and is opening a new window into the universe's most extreme phenomena.
According to Einstein's general theory of relativity, gravity is not a force, but a curvature of spacetime caused by massive bodies. Gravity is like a dent in a mattress. Heavy things wrap the space around them, and that is why we feel gravity.
Gravitational waves, as initially predicted by Albert Einstein in 1916, are ripples in spacetime that were first detected in 2015, but announced in 2016, the first observation of its kind: detection of gravitational waves, produced from two colliding neutron stars. In this type, there is a gradual increase in frequency and amplitude of GWs.
A Binary System (BS) in the context of gravitational waves refers to "a system consisting of two compact objects, such as black holes, neutron stars, or white dwarfs, which are orbiting each other and emitting gravitational waves."

Four Basic Types of GWs
There are four basic types of gravitational waves, each with different sources and characteristics:
i. Continuous GWs
When a single massive object spins with a constant rate, such as a neutron star, continuous GWs are produced with the same frequency and amplitude. White dwarf binary systems produce continuous GWs.
ii. Compact Binary Inspirational GWs
When a binary system, such as binary neutron stars, binary black holes, or a neutron star and black hole orbiting each other, compact inspirational GWs are produced.
iii. Burst GWs
These are produced by violent events like supernovae, gamma-ray bursts, or cosmic strings.
iv. Stochastic GWs
Stochastic GWs are weak, random signals of GWs which are produced by superposition of many weak gravitational wave sources, such as distant binary systems. These GWs are the most difficult to detect, e.g. gravitational waves of big bang.

Production of gravitational waves
Every physical object that accelerates, produces gravitational waves. Vehicles, airplanes, etc. are included in it. The masses and acceleration of objects on the Earth are too small to make gravitational waves big enough to be detected with our instruments.
As the binary systems (also termed as binaries) orbit each other, they emit gravitational waves, which can be detected by observatories like LIGO (Laser Interferometer Gravitational wave Observatory) which is situated in USA and Virgo, a large scale gravitational wave observatory in Cascina, Italy. The waves carry information about the system's mass, spin, and merger dynamics, offering insights into these extreme cosmic objects, and move with the speed of light.
The binary systems are significant sources of gravitational waves, and their mergers (collision- and union of two massive objects resulting more massive single object) are among the most intense cosmic events.
As the masses orbit and accelerate, their gravitational intensity fluctuates, generating waves that radiate outward in all directions. These waves are not bound by the binary system's gravity; instead, they travel freely through spacetime at the speed of light. The waves propagate through the universe, weakening in intensity as they distance themselves from the source.

Characteristics
The characteristics of GWs depend on the system's properties, such as:
• Masses of the objects
• Orbital period and frequency
• Eccentricity of the orbit: Eccentricity e is a measure of the amount by which an object deviates from a perfect circle.
e = 0 Circular orbit
e = 1 Parabolic trajectory
e > 1 Hyperbolic trajectory
0 < e < 1 Elliptical orbit

Spacetime Distortion / Tidal forces
Gravitational waves passing through a body with mass can cause the body to experience periodic stretching and compressing, also known as "spacetime distortion". This effect is known as "tidal forces" and is a result of the gravitational wave's oscillating nature. As the gravitational wave passes through the body, it causes the spacetime around the body to oscillate, leading to a periodic stretching and compressing of the body in the direction perpendicular to the wave's propagation. This effect is similar to how the tides on Earth are caused by the gravitational pull of the Moon and Sun.
The amount of stretching and compressing depends on the strength of the gravitational wave, as well as the mass and size of the body. This effect is an important prediction of Einstein's theory of general relativity.
Gravitational waves generated by far off celestial events, such as the merger of two black holes or neutron stars, pass through the Earth. However, the amplitude of these waves is extremely small, typically of the order of 10^-21 to 10^-22 metres. This means that the distortion caused by the gravitational wave is incredibly tiny, and requires extremely sensitive instruments to detect.
Despite their small amplitude, gravitational waves offer a unique window into the universe, allowing us to study strong-field gravity, test general relativity, and explore the universe in ways previously impossible.

Interferometer
An interferometer is an optical tool used in detecting gravitational waves. It is a very sensitive detection device that may use the interference of LASER (Light Amplification by Stimulated Emission of Radiation) beams. The basic Michelson's interferometer can be seen in Fig.
An interferometer that detects gravitational waves is a highly sensitive instrument that uses LASER light to measure tiny changes in distance between mirrors, caused by gravitational waves passing through the detectors. These interferometers are called LiGO (LASER interferometer Gravitational Wave Observatory).
The main differences between LiGO and conventional interferometers are:
(a) LiGO is 1000 times larger than conventional device, and
(b) LiGO uses LASER whereas conventional interferometer has normal light source.

Basic Components of GW interferometer
The basic components of a gravitational wave interferometer are:
i. Laser that produces a stable and high intensity beam of light.
ii. Power recycling mirror continually reflects LASER light that has already travelled through the instrument back into the interferometer and hence the term recycling is used.
iii. Beam splitter divides the LASER beam into two perpendicular beams.
iv. Mirrors reflect the beams, creating two perpendicular arms.
v. Fabry Perot cavity consists of two mirrors facing each other. The purpose of the cavity is to enhance the path length.
vi. Photodeteetors measure the returning beams, detecting tiny phase shifts (if any).
vii. Arm cavities enhance the LASER light, increasing sensitivity.

Working of Interferometer
A laser beam is split into two perpendicular beams, each travelling down two identical paths (arms) of the interferometer. The beams bounce off mirrors at the ends of each arm and return to the starting point, where they are recombined. If a gravitational wave passes through, it causes a tiny disturbance in the distance between the mirrors, resulting in a phase shift between the two beams.
When the beams recombine, they create an interference pattern, which is measured by a photodetector. The tiny phase shift caused by the gravitational wave alters the interference pattern, allowing the detector to sense the wave's pr

Virgo Detection
There is another facility for measuring gravitational waves. Similar to LIGO, this is called Virgo, which work under the European Gravitational Observatory (EGO) Cascina near Pisa, Italy. Virgo is also an interferometer with two arms of 3 km whereas LIGO has 4 km arms. The Virgo Observatory is named after the Virgo constellation, which is visible in the night sky during the months of March, April and May. The Virgo cluster is a group of about 1,500 galaxies about 50 mLYs (Million Light Years) away. Remember one Light Year (LY) is a distance which light travels in one year. The approximate value of 1LY = 9.5 billion km.
Virgo has been involved in detecting gravitational wave events, with the first detection in 2017.