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Final Answers
© 2000-2015   Gérard P. Michon, Ph.D.

Masers & Lasers
( Laser Effect  &  Gaussian Beams )

The least part of Light, I call a Ray of Light.
Isaac Newton (1643-1727)   Opticks, 1704.

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Related Links (Outside this Site)

Invention of the Maser and Laser   (American Physical Society, 2005)

Optical pumping   |   Alfred Kastler (1902-1984; Nobel 1966)
Beam divergence   |   Gaussian beam

Make a laser pointer burn!  (WickedLasers).

From Stimulated Emission to Laser Beams

(2012-07-09)   Stimulated Emission of Radiation  (Einstein, 1916)
Stimulated emission is crucial to the equilibrium of blackbody radiation.

In 1916, Albert Einstein turned his attention away from General Relativity  to investigate the interaction of radiation with matter.  Although he was certainly guided by  Planck's law for blackbody radiation  (which Planck had formulated in 1900)  Einstein's most brillant discovery was that simple thermodynamical considerations imply the existence of what's called  stimulated emission of radiation  (and, incidentally, also impose the general form of  Planck's law ).

A bound electron interacts  stochastically  with photons in one of three ways :
Absorption ( Spontaneous )
 Absorption  Spontaneous Emission  Stimulated Emission
Rate = B12 I (T) Rate = A21 = A Rate = B21 I (T)

Both  absorption  and  stimulated emission  are  induced  transitions, whose rates are proportional to the intensity  I (T)  of the surrounding radiation  (a  density of energy  expressed in pascals (Pa) or joules per cubic meter).  Being exact time-reversal of each other, they must occur at the same rate:

B12   =   B21   =   B

The coefficients  A  and  B  are properties of the atom and, thus, do not depend on the temperature  T  of the surrounding  photon gas.  When thermal equilibrium is achieved at a certain temperature  T,  the numbers of atoms in both states  (which may depend on T)  remain constant.  So, the total transition rates from either energy level to the other are equal:

N1 [ B12 I (T) ]   =   N2 [ A + B21 I (T) ]

Solving for  I,  this yields:     I (T)   =   ( A / B ) / [ N1 / N2  - 1 ]

On the other hand, the ratios of the populations of the two energy levels is an exponential function of the ratio of the energy difference to the thermal energy  (kT)  according to Boltzmann's statistics:

N2 / N1   =   exp [ -( E2- E1 ) / kT ]

Einstein's equation:

A / B   =   8p hn3 dn / c3

In this,  dn  stands for the width of the atomic transistion spectrum which is very narrow compared to the whole blackbody spectrum and is thus adequately represented by a delta distribution  (instead of properly convoluting the blackbody spectrum with the so-called  atomic lineshape function ).

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Absorption (electromagnetic radiation)   |   Spontaneous emission   |   Stimulated emission
Detailed balance   |   Boltzmann distribution   |   Planck's law   |   Einstein coefficients   |   Atomic spectral line

(2012-07-17)   Bose-Einstein Statistics   (1924)
The reason  why  lasers work...
   Satyandra N. Bose (1925)
Satyandra N. Bose

In a large enough cavity, the number of modes per unit of volume per unit of frequency interval is:

8p n2 / c3

Each electromagnetic mode of a cavity corresponds to a possible quantum state for a photon.  At thermal equilibrium, the occupation number per quantum state is:

exp ( hn / kT ) - 1

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 still working on this one...

Bose-Einstein statistics   |   Satyendra Nath Bose (1894-1974)
Test of Bose-Einstein statistics for photons (animation)

(2012-07-15)   Population Inversion
Producing an abnormal abundance in the  more energetic  states.

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 still working on this one...

Population inversion (animation)

(2012-07-09)   Ammonia Maser  (Charles H. Townes, 1954)
"Microwave Amplification by Stimulated Emission of Radiation"

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 still working on this one...

Maser   |   Hydrogen Maser   |   Charles H. Townes (1915)

(2012-01-15)   LASER Cavity  (between two mirrors)
"Light Amplification by Stimulated Emission of Radiation"

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 still working on this one...

How Lasers Work (in theory).  "Minute Physics"
How Lasers Work (in practice).  "Smarter Every Day" #33

(2012-01-15)   Gaussian Beam
The shape of an  ideal  laser beam.

zo   =   p n wo2  /  l

½ q   =   l  /  ( p n wo )

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 still working on this one...

Propagation of Gaussian Beams:  Part I  Part II  Part III  (Oklahoma State University)

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