Stimulated Emission of Radiation
- stimulated emission is referring to the emission of radiation (a photon) from one quantum
system at its transition frequency induced by the presence of other photons at that frequency
- introduced by Einstein in 1917 to derive Planck's blackbody radiation law
- is the basis for the operation of a LASER (Light Amplification by Stimulated Emission of
Radiation)
Consider an assembly of N atoms with energy levels Ei and Ej in
thermal equilibrium with a thermal radiation field with energy
density u(ν) at temperature T
- the transition frequency νij in the individual atom is
- the probability of an atom in state i to absorb a photon is
proportional to u(νij) (related to the number of photons at that
frequency) and to the probability Bij (Einstein B coefficient) of
the atom to absorb a photon at that frequency
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- thus, the total number of atoms per second that absorb a photon is
where Ni is the number of atoms in state i
- an atom in state j has a probability Aji (Einstein A coefficient) to
spontaneously decay into state i by emitting a photon at frequency ν in a
process called spontaneous emission
- also, an atom in state j has a probability Bji to decay into state i by
emitting a photon at frequency νij stimulated by the presence of other
photons at that frequency the number of which is proportional to u(νij)
- this process is called stimulated emission
- thus the total number Nji of atoms per second that make a transition into
the lower state is given by
- in thermal equilibrium the number of atoms that make a transition to lower energy
must be identical to the number of atoms making a transition to higher energy
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Transitions in a collection of atoms
interacting with a radiation field
- in thermal equilibrium
- solve for the energy density of the radiation field u(ν)
- with the number of atoms in the ground Ni or excited state Nj is given by classical M.-B.
statistics
- thus
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- hence the energy density of a radiation field in thermal equilibrium is given by
- this is Planck's formula for blackbody radiation for
- for identical upwards and downwards transition rates
- and with the ratio of spontaneous emission to
induced emission rate given by
note:
- stimulated emission does occur and is consistent with the form of the
blackbody spectrum
- the probability for stimulated absorption is equal to the probability
stimulated emission
- the likelihood of spontaneous emission relative to stimulated emission
increases rapidly with frequency
- knowing one of the parameters Aji , Bji , Bij we can determine the other
ones
- spontaneous emission is 'stimulated' by vacuum fluctuations (QED)
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Light Amplification by Stimulated Emission of Radiation (LASER)
LASER
Light Amplification by Stimulated Emission of Radiation
properties:
• monochromatic
• coherent
• directed
• bright
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Properties of LASER light
monochromatic
- relative line width
coherence
- temporal coherence
- spatial coherence
brightness
limited only by optics
- intensities
in pulsed operation
- large photon flux
- short pulses
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The LASER
1.
2.
3.
4.
5.
active LASER medium (gas, dye, solid)
pumping (light, electrical energy)
100% reflecting mirror
coupling mirror (< 1% transmission)
LASER beam
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The LASER Principle
- generation of a large number of excitations in a medium
by (stimulated) absorption
- a long lived metastable state to achieve population inversion
- stimulated emission of radiation
- storage of radiation field in a resonant cavity with
partially transmitting mirror
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Generation of Population Inversion
- excitation by optical pumping in a three level system
- transition (possibly non-radiative) to a metastable state
- stimulated emission
- at least a three level system is required to achieve
population inversion as the symmetry between the
probability for absorption (Bij) and emission (Bji) results
at best in equal population of the two states such that
emission cannot be the dominant process (i.e. there would
be no amplification in number of photons)
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Examples of Lasers
Ruby Laser: Cr+ ions in Al2O3 crystal (sapphire)
- optical pumping using flash light
- Helium-Neon Laser: gas mixture (90 % He, 10 % Ne) where He is excited electrically in gas
discharge and energy is transferred in collision to Ne
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Nobel Prize in Physics 1964
for fundamental work in the field of quantum electronics,
which has led to the construction of oscillators and
amplifiers based on the maser-laser principle
Charles H. Townes
Nicolay G. Basov
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Laser-Pointer
•
housing
•
electronics
•
LASER
and optics
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Aleksandr M. Prokhorov
CD Player
•
compact disc
•
•
the mechanism
portable CD player
(older modell)
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Laser-Printer
•
resolution up to 20 µm or 50 points per mm (= 1270 dpi)
•
colors with multiple stages
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Material Processing
•
CO2 gas LASER
•
up to 20 kW power
www.trumpf.com
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Lasers in Medicine
•
eye surgery
•
dermatology etc.
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Specific Heat of Solids
- dependence of thermal energy stored in a solid on temperature
- molar specific heat at constant volume cV: energy that needs to be added to 1 mol of a solid
to increase its temperature by 1 Kelvin
- thermal energy is stored in solids in the vibrations of its constituents (atoms, ions or
molecules)
- in a simple model the atoms can vibrate along 3 independent (x, y, z) directions that each
can be considered as an independent harmonic oscillator
- the equipartition theorem tells us the average energy stored in each degree of freedom, thus
the total energy of a solid containing 1 mol = N0 = 6.022 1023 of atoms is given by
- R = 8.31 J/mol is the ideal gas constant (pV = nRT)
- specific heat (Dulong Petit law)
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Dulong petit law fails for light elements (e.g.
carbon) and at low temperatures, see plot
- problem can be solved using the Bose-Einstein
distribution function for the probability of an
harmonic oscillator to have energy hν
- average energy for a harmonic oscillator of frequency ν
- total energy per mol
- thus Einstein's specific heat formula is
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- the high temperature limit of Einstein's formula
- therefore
- at high temperatures the thermal energy is much larger than the typical level separation of
the harmonic oscillators in the solid and the specific heat is close to the classical prediction
- at low temperatures the thermal energy is comparable to the level separation and quantum
statistics start to play a role
- for lighter elements with smaller masses the oscillator frequencies are higher and thus
quantum effects are more important even at higher temperatures
- the zero point motion does not play a role for the specific heat as it is only a constant offset
in energy
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Debeye theory
- Debeye regarded a solid as a continuous elastic medium with elastic standing waves
instead of considering the atoms as individual oscillators the way Einstein did
- these standing waves can have frequencies ν up to a maximum value νmax
- they are like transverse and longitudinal elastic waves that propagate at the speed of
sound in the solid
- their energies are quantized as in a harmonic oscillator
- each quantum of the elastic wave is called a phonon
- Debeye assumed that 3N different modes exist per mol in a solid
- the resulting formula describes the specific heat of the solid better than Einstein's
formula
- to be discussed in detail in solid state physiscs
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