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Physical Optics¶

Wave Front and Rays¶

Wave front is a locus of points which have same phase of vibrations. Following are the types of wave front:

Spherical wave front: Set of points, which determine the surface of sphere.
Cylindrical wave front: Set of points, which determine the surface of cylinder.
Plane wave front: A small part of spherical or cylindrical wave front at very large distance from source of light.
Point light sources produce spherical wave fronts.
When a point source is placed at focus of converging lens, plane wave fronts are obtained in laboratory.
Plane wave fronts reach from the Sun to the Earth, as the earth is far off from the sun.
The distance between two consecutive wave fronts is one wavelength.
A line normal to the wave front including the direction of motion is called a ray of light.

Info

A point source of light placed at principal focus of convex lens will produce a plane slave.

Huygen's Principle¶

Every point on wave front acts as a source of secondary spherical wavelets, which propagate in forward direction with speed of light.
Position of new wave front is tangent envelope to all of secondary wavelets.
Radius of hemisphere $=c\Delta t$
There is an infinite number of secondary wavelets present on wave front.
Light ray is associated with direction of flow of light energy.
In a homogeneous medium, the energy of wave is transmitted wave front remains spherical for long.

Interference of Light¶

Interference is the superposition or two light waves of same frequency and promoting in same medium along same direction very close to each other.
For constructive interference, light waves reach a point in phase their path difference $=n\lambda$
For destructive interference, light waves reach apoint in out of phase and their path difference $=(2n+1)\frac{\lambda}{2}$

Conditions for interference of light¶

Monochromatic (Having single wave length)
Coherence (Haying constant phase difference)
Same direction
Same medium
Very Close to each other.

There is no perfect monochromatic source, but by using filters it is possible to produce a source that gives light whose wavelength differ by $\pm5\times10^{-10}m$
If phase difference between two waves remains constant, then interference pattern will be stationary on screen otherwise it will change continuously.
For two ordinary sources, no interference pattern is obtained, because the phase changes rapidly and irregularly (that's why to get two coherent waves a single beam of light is split into two or more beams).
Optical path is equal to product of refractive index of medium and distance covered in air. $$ \text{Optical path} = nd $$ where $n$ is refractive index and $d$ is path in air.

Young's Double Slit Experiment¶

Path difference $=d\sin\theta$
For bright fringes: $$ d\sin\theta = m\lambda $$ 1^st bright fringe at $m=0$
For dark fringes: $$ d\sin\theta = (2m+1)\frac{\lambda}{2} $$ 1^st dark fringe at $m=0$
Position of mth bright fringe $$ y_m=\frac{m\lambda D}{d} $$
Position of mth dark fringe $$ y_m=(2m+1)\frac{\lambda D}{2d} $$
Wavelength from bright fringe $$ \lambda=y_m\frac{d}{m D} $$
Wavelength from dark fringe $$ \lambda = 2y_m\frac{D}{(2m+1)d} $$
Distance between centers of two consecutive dark fringes or bright fringes is called fringe width. $$ \text{Fringe Width}=\frac{\lambda D}{d} $$ applicable both for bright or dark fringes.

Interference in Different Types of Films¶

Thin film of refracting medium having thickness comparable to the wavelength of light rays e.g.
- Oil film on water
- Soap film
- Air film
When exposed to white light, thin film produces colorful pattern due to interference.
When exposed to monochromatic light, only bright and dark fringes are obtained

Types of thin Films¶

Uniform or Parallel Thin Films¶

Whose thickness is uniform. It gives straight interference pattern.

Wedge-shaped films¶

Whose thickness is zero at one end and then increases uniformly. Its interference pattern comprises a set of parallel fringes all parallel to the edge of wedge.
Wedge shaped films can be obtained using a spacer between the two slides of glass.
The thinnest part of wedge film shaped is always dark, due to additional path difference of $\lambda/2$, caused by phase reversal, at denser medium.

Newton's Rings¶

Newton's rings are practical study of interference in wedge shaped thin films.
When sodium light is incident on the plano convex lens system, light rays reflect from upper and, lower layers of the air present between lens and the glass plate. The sodium light source is almost monochromatic.
There is no phase change at the lens-air surface, because the wave is going from a higher to a lower refraction index medium. At the air-plate surface, however, there is a phase shift of $180^o$ with the reflection from a medium of higher refractive index.
Waves reflected from these two surfaces interfere, forming bright bands where the path length in air produces two waves in phase and dark bands where the waves are out of phase.
The centre of the pattern is black.
The fringes are circular as the locus of points of equal thickness of air is a circle
Conditions for interference are reversed for Newton's rings as:
- Path difference $= m\lambda$ (for dark ring)
- Path difference $=(2m +1)\frac{\lambda}{2}$ (for bright ring)
- This is due to phase reversal by 180° which is equivalent to an extra path difference of $\frac{\lambda}{2}$
Point of contact is always dark due to phase reversal at point of contact. Here actual physical path difference is zero.

Primary & Secondary Colors¶

Red, green and blue colors are called primary colors.
Complementary colors of white light are those two colors, whose combined effect is to produce white light on eye. They are:
- Red and blue - green
- Yellow and blue - violet
- Green and purple (mixture of red and blue)
If two primary colors of white light are mixed, we get complementary colors. e.g. red and green primary colors are mixed, we get yellow, which is complementary color of blue-violet.

Luminous, Non-Luminous & Incandescent¶

A luminous object is one that emits its own light e.g. sun
A non-luminous object is that which is visible by light it reflects. e.g. moon
Incandescent object is that which emits light due to heating, e.g. filament of electric bulb.

Michelson Interferometer¶

Michelson interferometers is an optical instrument used for following purposes
- Testing lenses, mirrors and prisms.
- Measurement of refractive indices.
- Thickness of thin plate through which light can pass.
Interferometers are based upon the Principle of division of wave front
Michelson's interferometer consists of following essential parts:
- Diffused source of monochromatic light.
- Beam splitter (semi-silvered glass plate).
- plane mirrors held perpendicular to each other, one is fixed and the other is movable
- Micrometer (it is attached to movables mirror)
- Compensator (glass plate equal in thickness to beams splitter and of the same material as that of beam splitter).
- Telescope (to observe interference fringes)
If mirror is moved by a distance of $\lambda/2$, with dark fringe in view, then fringe of same kind is observed because total path difference is $\lambda$.
If mirror is moved by distance of $\lambda/4$, then alternatively, dark and bright fringes can be observed because total path difference is $\lambda/2$.
If mirror is moved through distance $p$, and $m$ fringes pass before eye. $$ p=\frac{m\lambda}{2} $$
Interferometer can be used:
- To determine refractive index
- To test planes of glass slabs and lenses
- To determine wavelength of light

Diffraction of Light¶

Bending of light around sharp edges is called diffraction or the spreading of light waves into geometrical shadow of an obstacle and redistribution of light intensity resulting in dark and bright fringes is called diffraction of light.
The smaller is the size of diffracting object (obstacle), the higher the degree of diffraction is observed.

Interference	Diffraction
Superposition of few secondary wavelets is involved.	Superposition of large number of secondary wavelets is involved.
Interference fringes are equal in size.	Diffraction fringes wide near diffracting object and become small as one move away from it.
Interference fringes are equally spaced.	Diffraction fringes become narrow as distance from diffracting object increases.
Points of destructive interference are perfectly dark.	Points of minimum intensity are not perfectly dark.

Diffraction due to Narrow Slit¶

Diffraction due to a narrow slit has central maximum and alternating secondary minima and maxima on its both sides.
Condition for mth order minima on either side of center is given by $$ D\sin\theta=m\lambda $$ where $m = 1,2,3,...$ And $D$ is width of slit.

Diffraction Grating¶

Diffraction grating is a multi-slit arrangement of parallel and equally spaced slits. Suppose monochromatic light is directed at the grating parallel to its axis as shown. Let the distance between successive slits be $d$.
The diffraction pattern on the screen is the result of the combined effects of diffraction and interference. Each slit causes diffraction, and the diffracted beams in turn interfere with one another to produce the pattern. The path difference between waves from any two adjacent slits can be found by dropping a perpendicular line between the parallel waves. By geometry, this path difference is $d\sin\theta$. If the path difference equals one wavelength or some integral multiple of a wavelength, waves from all slits will be in phase and a bright line will be observed. Therefore, the condition for maxima in the interference pattern at the angle $\theta$ is $$ d\sin\theta=m\lambda $$ where $m = 0,1,2,3....$
Because $d$ is very small for diffraction grating, a beam of monochromatic light passing through a diffraction grating splits into very narrow maxima (bright fringes) at large angles 0.

Info

Practically a diffraction grating is a piece of glass with 400 5000 lines per cm.

Lines are opaque while separation between them is transparent, so space between two engraved lines behaves as slit.
Distance between two slits is called grating element. $d = 1/N$ where $N$ is the number of lines in one unit length.
Grating equation is given as $d\sin\theta = m\lambda$ where $m$ is called the order of diffraction pattern.
For white light, we see colored fringes.
Resolving power of grating is its ability to separate two wavelengths of light in given order of their spectrum. $$ \text{Resolving Power} = \frac{\lambda}{\Delta\lambda}= N \times m $$ where $N=$ number of lines ruled on grating, $m=$ order of diffraction $\Delta\lambda$ difference in two wavelengths to be resolved by the grating.

Diffraction of X-Rays by Crystals¶

Bragg's law is given as $$ 2d\sin\theta = m\lambda $$ Where $d$ is called lattice spacing and $\theta$ is called angle of diffraction.
Solid crystals behave as very good natural diffraction gratings and ultraviolet light is diffracted from layers of atoms.
Inter atomic layer distance is called lattice constant.
In 1913, Max Von Laue suggested that since atomic layers in solid $NaCl$ have layer separation of $10^{-10}m$, therefore, X-rays can be diffracted.
Max von laue pattern on film is in the form of dark spots/bright spots.
Analysis of the relative intensity of dark/bright spots gives rise to crystal structure of solid.

Note

Diffraction proves that wavelength of light is smaller than that of sound.

Polarization¶

Confinement of electric vectors of light into one plane is called polarization.
The material, which produces polarization, crystal is called polarizer. e.g. tourmaline
Polariod (polarizer) absorbs all magnetic vectors as well as randomly oriented electric vectors leaving only those electric vectors, which are in one plane.
Tourmaline crystal has internal molecular structure such that their interaction with incident light is to:
- Absorb all magnetic vectors
- Put (confine) all electric vector in one plane.
Polarization is possible only in e.m waves because their electric and magnetic vectors are perpendicular to each other as well as well to direction of propagation polarization Thus polarization has established that light is a transverse wave.
Analyzer is used to test plane polarization.
Plane determined by direction of propagation and polarized electric vectors of light is called plane of polarization.
Malus Law : $$ I = I_o\cos^2\theta $$
Different methods for producing plane polarized light are given below:
- Selective absorption technique (e.g. tourmaline crystal, calcite, crystal)
- Polarization by reflection.
- Polarization by scattering.

Intensity of Light Emerging from Polariod.¶

Intensity is given as $$ I = I_o\cos^2\theta $$
If the two polaroids have their transmission axes parallel to each other, i.e, $\theta' = \theta^o$ $$ I = I_o\cos^2\theta = I_o(\text{max}) $$
If the two polaroids are crossed, i.e., have their transmission axes perpendicular to each other, $\theta = 90^o$ $$ I = I_o\cos^2(90) = 0(min) $$

Uses of Polarized Light¶

Determination of concentration of optically active substance in a solution, eg sugar in blood & urine by using polarimeter in medical diagnostic labs.
Curtainless window.
To enhance effect of clouds & sky in photograph.
Headlights of vehicles to control the glare in night driving.