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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 E i and E j in
thermal equilibrium with a thermal radiation field with energy
density u(ν) at temperature T
- the transition frequency ν ij in the individual atom is
- the probability of an atom in state i to absorb a photon is
proportional to u(ν ij ) (related to the number of photons at that
frequency) and to the probability B ij (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 N i is the number of atoms in state i
- an atom in state j has a probability A ji (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 B ji to decay into state i by
emitting a photon at frequency ν ij stimulated by the presence of other
photons at that frequency the number of which is proportional to u(ν ij )
- this process is called stimulated emission
- thus the total number N ji 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 N i or excited state N j 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 A ji , B ji , B ij 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 limited only by optics
brightness
- intensities
- large photon flux
in pulsed operation
- short pulses
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The LASER
1. active LASER medium (gas, dye, solid)
2. pumping (light, electrical energy)
3. 100% reflecting mirror
4. coupling mirror (< 1% transmission)
5. 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 (B ij ) and emission (B ji ) 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 Al 2 O 3 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 Aleksandr M. Prokhorov
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Laser-Pointer
• housing
• electronics
• LASER
and optics
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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
• CO 2 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 c V : 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 = N 0 = 6.022 10 23 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 ν max
- 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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