Classical Oscillator model and electric dipole transitions ...

Classical Oscillator model and electric dipole transitions in real

atoms

Tomas Kristijonas Uždavinys

0

Laser Physics SK3410 2015-02-26

Somewhere in Alba Nova

1

Based on “Lasers” by A. E. Siegman, 1986

Outline

1 2

3

What is CEO?

2

3 CEO – chief executive officer

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Electronic models for real atoms Classical electron oscillator model

CEO - Classical electron oscillator

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Mathematics (Siegman book p. 81)

Short CEO description

Restoring force

External electrical field

Resonance frequency

6

Random iliustration

But wait! We are in quantum world!

Real atoms

What is resonance frequency?

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8

No references- random google search pictures.

Almost real life illustration

9

Siegman book p. 83

Energy damping #1

Ideal life

Real life

10

Siegman book p. 82-83

Energy damping #2

1 No electric field

2

3 exact resonance frequency

Errata – displacement is in real part

4

11

Siegman book p. 83

Emission of electromagnetic radiation

Only in movies…

12


Real life- aka non “radiative” mechanisms

Collisions with other atoms Lattice vibrations

Errata – absolute values

Internal charge cloud oscillation within atom

13

Siegman book p. 84

Radiative decay rates

Parameter Value

e 1.602e-19 (C)

ε 8.85e-12 (F/m)

m 9.11e-31 (kg)

c 3e8 (m/s) 3x1014

1x1015

2x1015

3x1015

4x1015

105

106

107

108

1000.0 230.8 130.4 90.9 69.8

Wavelength (nm)

De

ca

y r

ate

(s

-1)

Frequency (Hz)

CEO

14 “UV vision”

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Microscopic dipole moments

microscopic dipole

moment

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Macroscopic electric polarization

Medium density (1e12 to 1e22) in cm^3

2 waves 500 nm ( 2*(5e-5 cm) )^3 = 1 e-12 cm^3 With density (1e18) we get 1e6 CEO’s

Macroscopic dipole moment

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Siegman book p. 87

Classical world illustration

+

-

+

-

Surface

Free space

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Coherent dipole oscillations #1

x electron chage

+ microscopic

dipole

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Coherent dipole oscillations #2

Unfortunately this is fantasy world

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Random collisions

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Siegman book p. 93

Dephasing mechanisms

Brownian motion (atoms, or ions, or molecules). Even if coalition is elastic, no energy is transferred, electronic oscillation phase will be changed.

Solid state lasers, the quantum energy-level spacing and ω are affected by neighboring atoms. Thermal vibrations of lattice, will modulate them.

In materials there atoms are sufficiently dense, oscillating dipole may spread out to, and be felt by, other neighboring laser atoms. Even week coupling tends to randomize overall response. (1e-13 sec)

22


The dephasing time

Errata

Macroscopic polarization initial state

N(t)- dipoles that have not suffered at least one coalition

Suppose that collisions occur at

a random rate

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Siegman book p. 95

Decay of number of uncollided dipoles

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Slightly tricky part

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Siegman book p. 100

Mental exercise

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

+

+

-

-

t = 0

-

t = T2

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Siegman book p. 106

The Lorentzian Line Shape

Normalized frequency shift:

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Siegman book p. 108

A plot

Imaginary part

Real part

FWHM

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Random picture from somewhere

Finally atomic levels

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Siegman book p. 110

Lets convert it to real atomic transitions

Ref– page 13

𝜔 = 2𝜋𝑓 = 2𝜋𝑐

𝜆

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Population difference

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Quantum susceptibility equation

Replace CEO with actual radiative decay rate

Classical Quantum

Spontaneous emission

is proportion to

Frequency variations

32

Siegman book p. 112

Quantum polarization equation

CEO

Quantum

33


Substitutions #1

1 Transition frequency

2 Atomic population difference

3 Radiative decay time

4 Transition linewidth

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Substitutions #2

5 Transition lineshape

6 Tensor properties

7 Polarization properties

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Substitutions #3

8 Degeneracy effects

9 Inhomogeneous broadening

ELECTRIC-DIPOLE TRANSITIONS IN REAL ATOMS

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Energy decay in real atoms

“Rate equation”

“Decay rate”

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Fluorescent Lifetime Measurements

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Without authors permission

Example #1

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Some unpublished data

Temporally-resolved fluorescence measurements

•Pulsed pump

[HeCd laser + pockels cell]

•Time-gated photon counting

Can resolve fluorescence lifetime

Fluorescence lifetime ~ 1-10 ms

[depending on waveguide]

42


Radiative and non radiative decay. Again… Nj

Ni

Total decay rate

γnr γrad

Einstein A coefficients

43


Oscillator strength

“General” rule

Examples:

44

Siegman book p. 84

Radiative decay rates

Parameter Value

e 1.602e-19 (C)

ε 8.85e-12 (F/m)

m 9.11e-31 (kg)

c 3e8 (m/s) 3x1014

1x1015

2x1015

3x1015

4x1015

105

106

107

108

1000.0 230.8 130.4 90.9 69.8

Wavelength (nm)

De

ca

y r

ate

(s

-1)

Frequency (Hz)

CEO

Duplicate of slide 13

45

Siegman book p. 123

Level degeneracy #1

46

Siegman book p. 155

Level degeneracy #2

47


Nd:YAG transitions

48


Line-broadening mechanisms in real atoms

CEO Real atoms

Pressure broadening in gases

Homogeneous linewidth dependence:

49

Siegman book p. 129

Pressure-broadening coefficients

Relationship between partial pressure and density of each species in gas mixture.

50


Phonon broadening

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Zeeman-split atomic transitions

52

Siegman book p. 139

Electron density distribution in atoms

• The electron density is proportional to the product of the wavefunction and its complex conjugate.

• Since the electron density distribution does not change with time, the atom does not radiate in these stationary states.

53

Siegman book p. 138

Mathematical interpretation of the picture

x2 Errata

54

Siegman book p. 140

Linear dipole

Laser radiation that is linearly polarized in the z-direction couples very efficiently with superposition states with the same MJ value.

55

Siegman book p. 140

Circular dipole

Laser radiation that is circularly polarized in the x-y plane couples very efficiently with superposition states with MJ value that differ by +1 or -1.

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Siegman book p. 141

Linear and circular micro-summary

Atom whose charge distribution can oscillate only in a certain direction on a given transition will obviously respond

only to applied fields that have the same direction or sense of polarization

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Tensor susceptibilities

Linear

Circular

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The “factor of three”

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Siegman book p. 158

Inhomogeneous line broadening

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Convolution of inhomogeneous and homogeneous profiles

Different atoms will have different kinetic velocities

through space

Atoms at different sites in a crystal may see slightly

different local surroundings, or different

local crystal structures, because of defects,

dislocations, or lattice impurities

Gasses

Solids

61

Siegman book p. 160

Example – Doppler broadening

Average Doppler shift

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Doppler lineshape

Gaussian distribution in gasses

Errata

64 Number 5

65 Number 4

66 Number 3

67 Number 2

68 Number 1

Insert text here

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Classical Oscillator model and electric dipole transitions ...

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