Wave Optics:
Wave nature of light, including interference, diffraction, and polarization.
Double-slit experiment and Young's modulus.
Fresnel and Fraunhofer diffraction.
Quantum Mechanics:
Wave-particle duality and the uncertainty principle.
Schrödinger equation and its solutions.
Quantum s...
Optics is the science that describes the properties of light and its interaction with matter. It is
categorized into three different types- Ray optics, Wave optics and Quantum optics. Ray optics or
geometrical optics considers light as rays and explains the phenomena like reflection, refraction
and dispersion. Wave optics or Physical optics considers light as wave and explains phenomena of
interference, diffraction and polarization while the Quantum optics assumes light as tiny packets
of energy called Photons and is used to explain Photoelectric Effect, Davisson and Germer
experiment and Compton Effect etc. The applications of optics led to design of mirrors, lenses,
prisms, telescopes, microscopes, gratings, polarizers, spectrometers, spectrographs, photo-
multipliers tubes, CRO and CRT’s, Lasers, Optical Fibers, CDs and DVDs. Nowadays a new
branch of optics called Photonics combines electronics and optical aspects and is used in
communications, medical diagnosis, optical computing etc.
1.2. CONCEPT OF WAVEFRONT
A wavelet is a point of disturbance due to the propagation of light and wavefront is the
locus ofpoints(wavelets)having the same phase of oscillations.
It is also defined as a surface on which the wave disturbance is in same phase at all the
points. The direction of propagation of a wave at a point is always perpendicular to the
wavefront through that point.
Depending on source, the shape of the wavefront may be circular, spherical, cylindrical
or planar. The best example of wavefront produced naturally is when one drops a small
pebble in calm pool of water.
The waves spread out at the point of impact on the surface of water.
Such wavefronts are circular in shape as the waves on the surface of water are two
dimensional in nature.
A point source of light in homogeneous medium or a source of sound, spreads waves in
all directions uniformly and are therefore three dimensional or spherical in nature.
The wavefronts will be then a family of concentric spheres.
A linear source produces cylindrical wavefronts.
At a distance far away from the source, both the spherical and cylindrical wavefronts can
be treated as plane wavefronts.
All these types of wavefronts are shown in fig.1.1.
1.3 SUPERPOSITION PRINCIPLE AND SUPERPOSITION OF WAVES
1.3.1 SUPERPOSITION PRINCIPLE
The Superposition principle states that: When two or more waves arrive at a point in a
medium simultaneously, the resultant displacement at that point is the algebraic sum of
their individual displacements.
After the superposition, the wave trains travel as if they have not interfered at all. Each
wave train retains its individual characteristics. They pass through each other without
being disturbed.
To understand this, as shown in fig.1.2, consider two waves travelling in opposite
directions pass through a point in a medium.
Let the separate displacement of the particle by individual wave be y1 and y2.
If the two waves are incident with the same phase [fig.1.2(a)], then the resultant
displacement at the point,
y = y1 + y2. ----------- (1.01)
After superposition they continue to move in opposite directions.
On the other hand, if these two waves are incident on the particle in opposite phase [fig.
1.2(b)] then the resultant displacement,
y = y1 ~ y2. ----------- (1.02)
Fig.1.2: Superposition of two waves a) in phase b) out of phase
1.4. COHERENT SOURCES
Two sources of light waves are said to be coherent when they emit light waves of same
amplitude, same frequency and have constant phase difference between them.
Therefore, the two sources must emit radiation of same wavelength.
2
, Engineering Physics
1.4.1 RELATION BETWEEN PATH DIFFERENCE AND PHASE DIFFERENCE
If the path difference between the two waves is λ, the corresponding phase difference is 2𝜋.
Suppose for a path difference 𝑥, the phase difference is δ.
2𝜋𝑥 2𝜋
Then the phase difference δ is given by δ = = × (path difference)
λ λ
1.5 PHENOMENON OF INTERFERENCE OF LIGHT
1.5.1. INTERFERENCE:
The phenomenon of interference of light is based on superposition principle.
DEFINITION: Interference is defined as the redistribution of light intensity (or energy)
due to the superposition of light waves coming from two or more coherent sources.
Therefore, when two or more waves having same amplitude and same frequency,
travelling along the same direction in the region of same medium and having constant or
zero phase difference, combine together, the energies of these waves are added up and
then redistributed in such a way that some areas are dark and some are bright. This
phenomenon is called Interference.
For example, the bright colours in feathers of peacock and hummingbird are due to
interference phenomenon. The structure of feathers split and recombine visible light so
that interference occurs for certain colours.
1.5.2. CONSTRUCTIVE INTERFERENCE
When two waves of same wavelength and of same phase superimpose on each other,
constructive interference occurs.
The resultant displacement of the particle is given by y y1 y 2 and the resultant
amplitude is two times the amplitude of the initial wave, while the intensity increases by 4
times (Intensity α Amplitude2) as shown in Fig.1.3, provided the amplitude of both the
waves are same.
Fig.1.3:Constructive interference
Conditions for constructive interference are:
Phase difference = 0, 2π, 4π… .= 2nπ, where, n=0,1,2,3,4…. . -----------(1.03)
Path difference = nλ ----------- (1.04)
Constructive interference leads to appearance of bright fringes on screen.
1.5.3 DESTRUCTIVE INTERFERENCE
When two waves of same wavelength which are out of phase superimpose on each other,
destructive interference occurs.
The resultant displacement of the particle is given by y y1 y 2 .
Thus, the resultant amplitude and hence the intensity will be almost zero as shown in
Fig.1.4, provided the amplitude of both the waves are nearly same.
3
, Engineering Physics
Fig.1.4: Destructive interference
Conditions for destructive interference are:
Phase difference = π, 3π, 5π….= (2n+1)π, where, n=0,1,2,3,4…..----------- (1.05)
Path difference = (2n+1) λ/2 ----------- (1.07)
Destructive interference leads to the appearance of dark fringes on screen.
1.5.4 METHODS FOR OBTAINING INTERFERENCE PATTERN
The methods for obtaining interference pattern are based on the method to obtain coherent
sources. The two methods are:
1. Division of Wavefront:
The wavefront generated from a source of light is divided into two parts.
These two divided parts of the same wavefront travel unequal distances and
recombine to produce interference pattern.
For example: Young’s double slit, Fresnel’s Biprism and Lloyd’s mirror.
2. Division of Amplitude:
The amplitude of a beam of light is divided into two parts by partial reflection
and refraction.
These divided parts after travelling by different paths recombine to produce
interference pattern.
For example: In Newton’s Rings, Wedge shaped film and antireflection coatings.
QUE: What is meant by interference of light.
QUE: What are the necessary conditions on the path difference and phase difference
between two waves that interfere (a) constructively(b) destructively.
1.6. YOUNG’S DOUBLE SLIT EXPERIMENT- (Division of wavefront)
Thomas Young in 1802 demonstrated the experiment on the interference of light using the
method of division of wavefront.
He made a pinhole S in a cardboard and allowed sunlight to pass through it producing a
single wavefront.
This light was then allowed to fall upon another cardboard having two pinholes S 1 and S2
very close to each other.
These two pinholes were at equal distance from the source S and act as two coherent
sources. The single wavefront from source S gets divided into two wavefronts passing
through S1 and S2.
The emergent light was received on a plane screen placed at some distance. The
experimental arrangement is shown in fig.1.5.
At a given point on the screen the wavefronts from the two holes had different phases.
These wavefronts interfered to give a pattern of bright and dark alternative lines.
The variation of intensity on the screen demonstrated that the interference taking place
between the light wavefronts reaching the screen from the two pinholes.
The pattern of bright and dark areas is sharply defined only if light of a single wavelength
is used.
Young's original experiments were performed with white light.
This experiment successfully proved the existence of the wave nature of light.
4
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