Elements of Laser

Working Principle of Laser

Working Principle of Laser

Einstein’s assumptions & implications Thermodynamic equilibrium at arbitrary

temperature T exists between the radiation field and the atoms

The spectral density u(ν) of the radiation energy has the distribution characteristics of a blackbody at temperature T

The atom population densities Nl and Nu at energy levels El and Eu, respectively, are distributed according to the Boltzman distribution at that temperature

Population densities Nl and Nu are constant in time

Working Principle of Laser

The radiative process and assumptions above, it follows that the rate of change of atoms in level Eu is given by

uBNuBNANdt

dNlululuulu

u 0

The spectral energy density can be written as

1

18/3

33

kThec

hnu

The Boltzman distribution

kTh

l

ukTEE

l

u

l

u eg

ge

g

g

N

Nlu //

…..(1)

…..(2)

…..(3)

Working Principle of Laser

Solving eq.(1) in terms of u() and substituting Nu/Nl from eq. (3), we obtain

ul

kThlu

u

l

ul

ulullu

ul

BeBg

g

A

BNNB

Au

//

Compare it to eq. (2), gives

3

338

c

hn

B

A

ul

ul

lululu BgBg ……(5) ……(6)

……(5)

Working Principle of Laser

The importance of eqs. (5) and (6) cannot be underestimated. They tell us that:

(i) The fundamental Einstein’s coefficients Aul, Bul and Blu are all inter-related.

(ii) guBul = glBlu , i.e. stimulated emission and absorption are inverse processes. However note that the rate dNu/dt and dNl/dt differ depending on the population densities Nu and Nl. If Nu > Nl it leads to an increase in u(), an amplification. And if Nl > Nu it leads to a decrease in u(), an attenuation. For laser to operate, it is necessary that Nu be greater than Nl – a condition called population inversion.

(iii) Since Bul/Aul is proportional to the reciprocal of the cube of the frequency, the higher the frequency the smaller Bul becomes in comparison with Aul. Since Bul is related to stimulated emission and Aul is related to spontaneous emission, it would seem that lasers of short wavelength radiation would be more difficult to build and operate. Two important ideas for the successful operation of a laser

emerge from a review of Einstein’s study of the interaction of electromagnetic radiation with matter are, stimulated emission and population inversion.

Lasing condition

Population inversion Necessary condition for amplification. The case of the upper level being more populated than the lower

level. If stimulated emission rate exceeds absorption rate, net optical gain. The relationship for the intensity at a specific distance z into medium

at a frequency ν and width Δν can be expressed as,

Population inversion

Population inversion

If the value of the exponent is positive, the beam will increase in intensity and so amplification will occur.

If it is negative, the beam will decrease in intensity and absorption will occur.

The values of σul and z are always positive, thus amplification will occur only if

This condition is not normal under thermal equilibrium

ll

u Ng

gN u

Emission Broadening

Emission Broadening

Homogeneous Broadening Due to the isotropic collisions with other atoms, which also

causes non-radiative decay The processes lead to a Lorentzian distribution of emitting

frequencies All of the atoms in level u have an equal probability of

participating in the emission at any frequency ν of that emission shape – all atoms behave the same way

Emission Broadening

Homogeneous Broadening The process can decrease either the decay time u of

the atoms residing in the excited level u OR affect the linewidth – depending on collision intervals

Dephasing collisions interrupts the phase of radiating atoms without increasing their population decay rate

Emission Broadening

Inhomogeneous Do not affect the lifetime, but do affect the linewidth The processes include Amorphous Crystal broadening,

Doppler broadening and Isotope broadening Emission processes that lead to a Gaussian distribution of

emitting frequencies Specific portions of the population density Nu contribute to

different portions of the emission linewidth

Emission Broadening

Amorphous Crystal Broadening Glass materials have various small regions

oriented in slightly different directions Thus, each of the glass molecules can have

slightly different energy levels This leads to different radiating frequencies for different regions Since the emission line is composed of the sum of

all of the individual lines, this leads to a much broader emission spectrum

Emission Broadening

Doppler Broadening Due to random movements of radiating atoms in all

directions with a range of velocities This causes frequency shifts depending on the

directions of the movements The faster the atoms move on the average, the broader

the bandwidth A single photon might be able to stimulate one atom to

emit because that atom happened to be Doppler shifted to the photon’s frequency, but it might not be able to stimulate another atom because it had a different Doppler shift than the first

I.e. different atoms contribute to the gain at different frequencies within the laser bandwidth

Emission Broadening

Isotope Broadening Due to the presence of more than one isotropic

form of the species These different isotopes consist of atoms having

the same number of protons and electrons, but with different numbers of neutrons

Atoms with slightly different numbers of neutrons within their nuclei exhibit small differences in energy level values

The slightly different energy level values for different isotopes provide slightly different frequencies for the transitions