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Light Amplification by the Stimulated Emission of Radiation Source producing an intense, concentrated and highly parallel beam of coherent light and works on principle of Quantum theory of light

## Properties of laser

Intensity Laser beams are highly intense as a large number of photons are concentrated in a small region.

Coherence Perfectly coherent as the emitted light waves has the same phase with one another.

Directionality Travels in a single direction as the photons are traveling along the optical axis of the system.

## Basic concepts of Laser Action

Absorption: Atoms in the lower energy state absorb energy from the incident photon and moves to the higher energy state. Probability of absorption depends on:

• Number of atoms present in the lower energy state.
• Intensity of incident light

Spontaneous Emission: Atoms in higher energy state jumps to the lower energy state with emission of a photon at random (i.e. without influence from a photon) Probability of Spontaneous emission depends on:

• Number of atoms available in the excited state.

Stimulated Emission: Atoms in the excited state jumping to the lower state under the influence of another photon emit a photon of the same frequency as the incident photon. Probability of Stimulated emission depends on:

• Number of atoms available in the excited state
• Intensity of the incident light
The direction of propagation of energy, phase and state of polarization of energy of the emitted photon is exactly the same as those of the stimulating photon. Thus photons are coherent.

## Working principle of a laser

• Based on phenomenon of stimulated emission and spontaneous emission
• Active medium should have one metastable state besides excited state and ground state.
• The lifetime of atoms in excited state is 10^-8 sec but it is longer in metastable state.
• When atoms are excited with light of suitable wavelength, they jump from lower energy state to excited state by absorbing photons.
• But atoms can remain in excited state only for a small amount of time and they drop back by spontaneous emission.
• Many of them are trapped in the metastable state where its lifetime is greater and population inversion is obtained.
• After getting population inversion, a photon got from spontaneous emission is made to strike an atom of the metastable state.
• The excited atom of metastable state is stimulated to emit a photon of the same energy as that of the stimulating photon.
• The stimulating and stimulated photons yield a large number of coherent photons by repeated stimulated emissions as they pass through the atom.
• Hence light amplification occurs due to multiplication of photons all of which have same frequency, direction and phase.

Population Inversion An artificial situation that is established by generating a large number of atoms in the higher energy state than that of the lower energy state.

Pumping The phenomenon of achieving population inversion, i.e the process which raises the atoms from lower energy state to higher energy state in the active medium. Methods:

• Optical Pumping: a light source is used to supply luminous energy and create population inversion by optical photon
• Electrical Pumping: electrical discharge converts the gas medium into plasma which liberates electrons which, in turn, are accelerated by the strong electric fields present in the tube. These electrons, on collision with neutral gas atoms, makes some atoms jump to excited state.
• Chemical Pumping: an exothermic chemical reaction is used to produce energy.

Lasing The process which leads the emission of stimulated photons due to the transition of atoms from the metastable state to the ground state after achieving population inversion.

By Maxwell Boltzman statistics:
The number of atoms present in a particular energy state at any time:
N2 = N1e-ΔE/kT Where, ΔE = E2 – E1 = hυ
N2 & N1 => number of atoms present in excited state & ground state respectively
k & T => Boltzman’s constant and absolute temperature

Einstein’s Theory of Laser and relation between A and B coefficients Let N1 and N2 be the number of atoms present in ground state E1 and excited state E2 respectively of the active medium.

Energy required to raise atom from E1 to E2 = hυ where υ is frequency of radiation

But rate of absorption is proportional to N1 and the energy density u(υ) of incident light. Thus, the number of absorption per unit volume per unit time => Tab = B12N1 u(υ) Where, B12 = Einstein’s coefficient of absorption of radiation from lower energy state to higher energy state.

Atoms in excited state return to ground state by emission of energy by two process:

• Spontaneous emission
• Stimulated emission

Since spontaneous emission depends only on the number of atoms per unit volume present in excited state ( N2 ) Tsp = A21 N2                    where A21 = Einstein’s coefficient of spontaneous emission

N2 would relax in metastable state where stimulated emission occurs. Since stimulated emission is directly proportional to the number of atoms present in the excited state per unit volume ( N2 ) and energy density[u(υ)] of incident radiation: Tst = B21 N2 u(υ)    where B21 = Einstein’s coefficient  of stimulated emission

When thermal equilibrium is reached, rate of upward transmission = rate of downward transmission,
Tab =  Tsp + Tst
B12N1u(υ) = A21N2 + B21N2 u(υ)
(B12N1 - B21N2 )u(υ) = A21 N2
u(υ) =         N2 A21/ N2(B12 N1 / N2 – B21) = A21/( N1 / N2 B12 – B21)

N1 and N2 are related by Boltzman’s law as: N2 / N1 = e-(E2-E1)/kT = e- hv/kT

Thus inserting value, we get
u(υ) =  A21/ B12 (1/( e hv/kT - B21 / B12 ))
But according to planck’s law of radiation,
u(υ) = 8πhυ3/c3. 1/( e hv/kT – 1)
Thus comparing equations, we get:

B21/ B12 = 1          or, B21 = B12
And    A21 / B12  =  8πhυ3/c3
Thus   A21 / B21  =  8πhυ3/c3

Physical Significance:

• The probability of stimulated emission is numerically equal to probability of stimulated absorption. So stimulated emission is inverse process of absorption. Their rates are different because stimulated emission is proportional to number of atoms present in excited state while stimulated absorption is proportional to number of atoms present in ground state.
• The coefficient of stimulated emission (B21) is inversely proportional to the third power of frequency of radiation.
• The ratio of the rate of stimulated emission to the rate of spontaneous emission               R = B21 N2 u(υ)/ A21 N2 = 1/( ehv/kT – 1)
• The probability of stimulated emission is more compared to spontaneous emission in microwave region
• The probability of stimulated emission is negligible compared to spontaneous emission in visible region

Condition for higher probability of stimulated emission compared to that of simultaneous emission

From Einstein’s relation: A21 / B21 = 8πhυ3/c3

From Planck’s radiation law: u(υ) = 8πhυ3/c3 .1/( ehv/kT – 1)

Thus, A21/ B21 u(υ) = ehv/kT – 1 = R

1. In the microwave region, hυ<<kT, the number of stimulated emission will be higher compared to simultaneous emission
2. In the visible region of the spectrum, spontaneous emission will be predominant and stimulated emission negligible.

Condition for higher probability of stimulated emission compared to absorption radiation process

Ratio of stimulated transition to absorption transition from Einstein’s relation:
R’ = B21 u(υ) N2 / B12 u(υ) N1
R’ = N2/ N1

The number of atoms per unit volume in the ground state is very large compared to number of atoms per unit volume in excited state, i.e. N1 > N2

To make stimulated transition exceed the absorption transitions:

1. The state population inversion has to be achieved. For this reason, the lasing material is doped with certain impurities such that metastable atomic energy level is obtained.
2. The larger value of the ratio of stimulated transitions to spontaneous transitions is to be achieved by considering a metastable energy state as the higher energy level.

## Optical Resonator

The large number of excited atom produced due to population inversion and pumping emit photons spontaneously in various directions. These photons, in turn, strike atoms in metastable state and cause stimulated emission. The photons produced due to stimulated emission also travels in various directions. Since these photons cannot give coherent beam, the number of photon states must be restricted to obtain a coherent beam of laser.

This can be achieved by placing the active medium between a perfectly reflecting plane spherical mirror and a semi-transparent mirror (90% reflecting). The mirror system reflects most of the energy from the light incident on it back to the medium, thus acting as a positive feedback necessary to compensate the losses. A small amount of energy escapes through the semi-transparent mirror to form the laser. If enough population inversion takes place in the active medium, the light is amplified to a great extent since each passage of light across the medium after successive reflection causes gain in strength and multiple reflections occur. A steady intense laser beam emerges from the semi-transparent medium.

Frequency difference between two consecutive modes of vibration

The waves propagate along both directions due to successive reflections from the two mirrors. They interfere to form standing wave pattern.
L =>distance between two mirrors
λ =>wavelength of emergent light
φ =>phase change after reflection from both mirrors

Change in phase after one round trip: ρ = 2π/λ.2L + 2φ

For constructive interference, phase change must be integral multiple of 2π for standing waves to form
2π/λ.2L + 2φ = 2mπ
or,     ν = c/λ = mc/2L – φc/2πL

Standing waves are formed within the two mirrors giving us two nodes. Since the wavelength of laser light is much smaller than the length of the cavity, the number of half waves formed within the mirror is also very large. So the frequency difference between two consecutive modes of vibration:
Δν = c/2L

## Types of Laser

Four types:

1. solid state laser
2. gas lasers
3. liquid dye lasers
4. semiconductor lasers

## Ruby Laser

Solid state laser consisting of a pink ruby cylindrical rod whose ends are optically flat and parallel with a silvered and a partially silvered (50%) end. The rod is surrounded by a high intensity helical flash lamp filled with xenon gas which is intense enough to produce population inversion. Composition of Ruby: Crystalline aluminium oxide (Al2O3 or host crystal) doped with 0.05% of chromium atoms (activator atoms). Al3+ ions are replaced by Cr3+ ions in crystal lattice. Cr3+ ions impart red colour to the white Al2O3 crystal.

Working: Chromium atoms consist of a metastable state of lifetime ~3 X 10-3 sec. When a flash of light of wavelength 550nm falls upon the rod for a very short time (about a millisecond), the chromium ion, in the ground state, absorbs a photon and jumps to excited state E2. The excited ions drop to the metastable state E3 very soon as lifetime of ions in excited state is short. The transition is non radiative as the energy released is absorbed by the lattice in which it is absorbed and is dissipated as heat. But the number of atoms in metastable state goes on increasing as lifetime in metastable state is high and soon exceeds those in the ground state, thus bringing about population inversion.

After this state is achieved, one or two photons released due to spontaneous emission is sufficient to induce stimulated emission and light amplification will start. The transition from M ŕ G state radiates photons, which after repeated reflection from the mirrors of the laser cavity amplifies largely to an intense beam.

An intense, highly directional, coherent beam of red light (λ = 694.3 nm) emerges from the partially silvered end of the ruby rod as laser beam.

## The Helium Neon Gas laser

It is a gas laser consisting of a mixture of helium (He) and neon (Ne) in a ratio of about 10:1 inside a narrow long discharge tube at pressure of 1mm of mercury. The gas system is placed between a pair of plane mirrors or a pair of convex mirrors out of which one is perfectly reflecting while other is partially reflecting forming the resonating system. The distance between two mirrors is equal to an integral multiple of half wavelength of the laser light and supports standing wave pattern within the resonator system.

Working: Helium has three energy states. These are 3S, 2S and 1S where 3S and 2S are metastable states. When an electrical discharge passes through the gas mixture, the helium atoms are excited by the impacts of accelerated electrons in the discharge tube due to its lower mass. As a result, some of the helium atoms are raised to its metastable states 2S and 3S from its ground state. The energy of the two excited states 2S and 3S of Ne are slightly less than the energy of the two metastable states of He atoms. Thus, after the collision of the excited helium atoms with neon atoms, the neon atoms in the ground state are raised to its 3S and 2S excited states and helium returns to its ground state by exchanging energy.

The gas discharge process after sometime leads population inversion in these metastable Ne(3S) and Ne(2S) levels relative to its lower 3P and 2P states. After achieving population inversion, one of the two photons released due to spontaneous emission can trigger stimulated emission and produce three type of lasing actions(3S ŕ 3P, 3S ŕ 2P, 2S ŕ 2P). After that, the Ne atoms return to the lower laser levels 3P and 2P to the level 1S by spontaneous emission. From this level, Ne returns to ground state by collision with the walls of the tube. The cycle of events occur continuously as the discharge in the tube is maintained continuously. Thus it is known as continuous laser.

Uses of He-Ne Laser

1. in interferometry
2. in laser printing
4. in holography
5. for larger distance measurement, i.e. in laser modulation telemetry
6. in the target aiming device used in guns

The advantages of Gas laser over Solid state laser

• The light from He-Ne gas laser has high degree of monochromacity and directionality than that from solid state ruby laser. This happens due to imperfection in the crystal, thermal distortion and scattering.
• The solid state laser need cooling in time of operation while the gas lasers can operate continuously without any cooling.

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