张利剑 lijian.zhang@nju.edu.

cn 025-89689217

Advanced Optics Lecture 9

Light & Matter

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 Solids: doped dielectric media
Ionic or covalent solids that are insulating and transparent in a particular
region of the spectrum are called transparent dielectric media. Such
materials often serve as hosts for active laser ions. Transition-metal and
lanthanide-metal ions are the most common dopants.

The extent to which the energy levels of the active laser ions are affected
by the host medium is determined principally by how well their
optically active electrons are shielded from neighboring lattice atoms.
The energy levels of transition-metal ions are substantially modi ed
by crystal- eld effects whereas those of lanthanide-metal (rare-earth)
ions are scarcely affected.
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 Solids: transition-metal dopant ions
Cr3+, Ti3+, Ni2+, Co2+, are popular transition-metal dopants ions for lasers

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 Solids: ruby and alexandrite
Ruby (Cr3+:Al2O3) is formed by replacing ~0.05% Al3+ by Cr3+ ions.
Alexandrite (Cr3+:BeAl2O4) is formed by doping a small amount of
chromium oxide (~0.1 %) into a chrysoberyl host (BeAl2O4). Both
materials has a refractive index that is close to that of ruby, n ≈1.74;
however chrysoberyl is biaxial whereas sapphire (Al2O3) is uniaxial.

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 Solids: ruby and alexandrite
Since the 3d electrons of the Cr3+ ions in both materials are exposed to neighboring
ions, the energy levels of these materials are determined in large part by the
surrounding crystal elds and therefore depend substantially on the host material.
The energy levels of the two materials are quite distinct even though they share
the same dopant. Alexandrite lases over a substantial range of wavelengths that is
not available in ruby.

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Solids: lanthanide-metal dopant ions
The lanthanides, comprising the series from 58Ce to 71Lu, reside in row
6 of the periodic table. Successive lanthanide elements are constructed
by adding electrons to the 4f subshell, which lies within the lled
5s25p6 and 6s2 subshells. The lanthanides usually exist as trivalent
cations; the con guration of their valence electrons takes the form 4fu,
with u varying from 1 (Ce3+) to 14 (Lu3+).
Nd3+, Er3+, and Yb3+ are particularly important dopants for laser
ampli ers and oscillators.
❖ Nd3+:glass and Er3+ :silica ber are widely used as laser ampli ers.
❖ Nd3+ :YAG, Nd3+:YVO4 , and Yb3+:silica ber often serve as laser
oscillators.
Among the other lanthanides, Pr3+ and Ho3+ are also extensively used
as active laser ions. 6
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Solids: lanthanide-metal dopant ions

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 Solids: lanthanide-metal dopant ions
The behavior of trivalent lanthanide ions in a dielectric host and in isolation is
rather similar. This results from the fact that the 4f electrons are well shielded
from external effects of the lattice by the lled 5s and 5p subshells. This is in
sharp contrast to the behavior of transition-metal ions. Rare-earth-ion energy
levels are essentially independent of the host material.

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End of energy levels
Occupation of energy levels
Boltzmann distribution
Consider a collection of
distinguishable objects, such as
atoms or molecules. Each atom is in
one of its allowed energy levels. If
the system is in thermal equilibrium
at temperature T

k: Boltzmann constant

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 Boltzmann distribution
Consider the Boltzmann distribution in the context of a large number of
atoms N. If Nm is the number of atoms occupying energy level Em,

If E2 > E1, in thermal equilibrium N2 < N1

Population inversion: a higher energy level has a greater average
population than the lower energy level.

If there is energy degeneracy

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 Fermi-Dirac distribution
Multielectron atoms and semiconductors are subject to the Pauli
exclusion principle. A state may then be occupied by at most one
electron; the number of electrons Nm in state m is either 0 or 1. The
probability of occupancy of a state of energy E is then described by the
Fermi-Dirac distribution (or Fermi function),
Ef : Fermi energy

f(E) is neither a probability density function

nor a probability distribution function.

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End of occupation of energy levels
 Interactions of photons with atoms
Consider the energy levels E1 and E2 of an atom placed in an optical
resonator of volume V that can sustain a number of electromagnetic
modes. We are particularly interested in the interaction between the
atom and the photons of a prescribed radiation mode of frequency ν ≈
ν0, where hν0 = E2 - E1.

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 Spontaneous emission
In a cavity of volume V, the probability density
(per second), or rate, for the spontaneous
transition into a particular mode, is

σ(ν): transition cross section

where θ is the angle between the dipole moment of the atom and the eld
direction of the mode.
If there are a large number N of such atoms

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 Absorption
Absorption can occur only when the mode contains
a photon.

The probability density for the absorption of a photon from a given

mode of frequency ν, in a cavity of volume V, is

However, if there are n photons in the mode, the probability density

that the atom absorbs one photon is

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 Stimulated emission
The presence of a photon in a mode of speci ed
frequency, propagation direction, and polarization
stimulates the emission of a duplicate ("clone")
photon with precisely the same characteristics.

If the mode originally carries n photons,

Pst = Pab = Wi

The overall probability density that the atom emits a photon into the mode

Spontaneous emission may be regarded as stimulated emission induced by the

zero-point uctuations.
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 The lineshape function
Transition strength
(oscillator strength)

g(⌫) = (⌫)/S Lineshape function

g(v) is centered about the resonance frequency ν0. Transitions are most
likely for photons of frequency ν ≈ ν0. The FWHM ∆v of the function g(v)
is known as the transition linewidth.

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 Spontaneous emission
Total spontaneous emission into all modes
Probability density for
spontaneous emission into a
speci c mode of frequency ν

Mode density in a 3D cavity (number of modes of

frequency ν per unit volume per unit bandwidth

Since modes at each frequency have an isotropic distribution of

directions, the averaged transition cross section

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 Spontaneous emission
Total spontaneous emission into all modes

The overall spontaneous-emission probability density

¯ (⌫) is sharply peaked around ν0

Psp is independent of the cavity volume V

tsp: spontaneous lifetime

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 Spontaneous emission
Relation between transition cross section and spontaneous lifetime

The transition cross section σ for stimulated emission into a particular mode
obey relations identical to those for spontaneous emission, except that the
effective spontaneous emission time is reduced due to the effect of θ.

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 Stimulated emission and absorption
Transition induced by monochromatic light
Consider the interaction of single-mode light with an atom when a stream
of photons impinges on it
Mean photon- ux density (photons/cm2-s)

Now try to determine Pst = Pab = Wi.

The number of photons n involved in the interaction process is

determined by constructing a volume in the form of a cylinder of base
area A, height c, and volume V = cA. The axis is parallel to the
propagation direction of light.

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 Stimulated emission and absorption
Transition induced by monochromatic light

Pst = Pab = Wi

σ(ν) is the effective cross-sectional area of the atom (cm2), and φ σ(ν) is the
photon ux "captured" by the atom for the purpose of absorption or
stimulated emission.
Whereas the spontaneous emission rate is enhanced by the many modes
into which an atom can decay, stimulated emission involves decay only
into modes that contain photons.
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 Stimulated emission and absorption
Transition induced by broadband light
Consider an atom in a cavity of volume V containing multimode
polychromatic light of spectral energy density ρ(ν) (energy per unit
bandwidth per unit volume) that is broadband in comparison with the
atomic linewidth. The overall probability

Since the radiation is broadband

where we ignore the difference

between S and S̄
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 Stimulated emission and absorption

Mean number of photons per mode

The probability density Wi is thus a factor of \bar{n} greater than that for
spontaneous emission, since each mode contains an average of \bar{n}
photons.

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 Einstein coefficients
Assuming that the atoms interacted with broadband radiation of spectral
energy density ρ(ν), under conditions of thermal equilibrium, Einstein
obtained the following expressions:

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Summary

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Summary

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To be continued

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Problems

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Problems

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