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11 Laser Physics 12

Laser Physics Laser Physics 1122..Interaction of light with matterInteraction of light with matter

MaákMaák PálPál

Atomic Physics DepartmentAtomic Physics Department

22 Laser Physics 12

Interactions in a volume V with a selected mode of frequency :spontaneous emission, absorption, stimulated emission

Probability densities of the processes ~ transition cross section [cm2]

Strength of interactions, lineshape function:

Total spontaneous emission into all modes:

Spontaneous lifetime:

Interaction with a photon beam of frequency travelling in a selected direction:

photon-flux density (photons / cm2 ·s)

S can be determined from measurement, g ( ?

Light-matter interactions (summary)

.VcnWPP,

Vcpp,

Vcp iieabieabsp

1s

.S

g,dS 1-2

0

scm1

28 sSPsp

gt

gS,t

S,t

Pspspsp

sp 881 22

.iWeffective interaction area

33 Laser Physics 12

Line-broadening mechanism

The frequency dependence of light-matter interactions is governed by the normalized lineshape function g . Materials can be classified into two basically different groups:

homogeneous

All atoms, molecules behave similarly in the light-matter interaction, they have the same individual lineshape function.

inhomogeneous

Group of atoms and molecules behave differently in the light-matter interaction, the whole system can be characterized by an average lineshape function.

The reality is always a mixture of the two properties!

Light-matter interactions

44 Laser Physics 12

Homogeneous broadening – lifetime or natural broadening

It is always present, the questions is how much dominant is?

Excited energy levels have finite lifetime. If level 2 is an excited level, its lifetime represents the inverse of the rate at which the population of that level decays to level 1 and to all other lower energy levels radiatively (tsp) or nonradiatively, therefore tsp.

The population of level 2 decays exponentially, therefore the amplitude of emitted electromagnetic field decays also exponentially

The spectral dependence can be calculated by the Fourier-transform of the exponentially decaying harmonic function.

., 01222/ 0 hEEeeE tjt

The energy decays with therefore the factor of ½!!

2

1

tsp

2


55 Laser Physics 12

Homogeneous broadening – lifetime or natural broadening (cont)

Time dependent field amplitude:

the Fourier-transform of f(t):

is the FWHM of the Lorentz-function. We could start from the uncertainty principle of Heisenberg:

.0ha,00ha,022

ttEteetE tjt

ggnormalizin22 FdtetfFtf tj

.g,,/

/go

22

12

2022

.12, EtE


66 Laser Physics 12


General case: both selected energy levels are excited levels, both have lifetime broadenings:

the broadening of the transfer is:

The reciprocal of the characteristic time is the sum of the reciprocal lifetimes and the broadening can be calculated:

,2

,2 1

12

2hEhE

,12

112 21

21hhEEE

.1121111

2121

MHz.161021

2110 88 ~s~

typnattyp


77 Laser Physics 12


Wavepacket emissions at random time and the Lorentz-function

,g Lorentzo2

.2

12

2

spo t

In ideal case 2 = tsp and 1/ 1=0 (lower level is stationary):

10-11-10-7 cm2 in 0.1 – 10 µm wavelength range.

.2

,2

1 2

ospt

( 0) has a typical order of magnitude of 10-20-10-11cm2

(small overlap)


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Homogeneous broadening – collision broadening

In gas and fluid can be important with increasing the density of particle (with increasing the pressure). All particle suffer from the same effect, therefore it is a homogeneous effect.

Collision will disturb the light emission process. Two different collisions:

inelastic – particle leaves the excited level lifetime of the excited level decreases similar to the previous case (lifetime broadening)

elastic, there is no transfer between energy levels, there is a disturbance in the mechanism of light emission random phase shift


d ~ 10-13 s

c 10-8 s

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Homogeneous broadening – collision broadening (cont.)

Harmonic function with random phase shift frequency spectra by Fourier transform again Lorentz-function.

If c is the average time between collisions, the normalized lineshape function:

c depends on the pressure, estimation with drastic simplifications:

„ideal” monatomic gas of radius r, hard spheres, we fix one particle, the others are moving toward the fix particle with .

2220 41

2

c

ccg

0max 2 ccg

cc

1

relavrv


1010 Laser Physics 12

relavrv


Suppose that the movement of particles toward the fixed particle takes place in a unit cross section cylinder of length :

The probability of collision is proportional with the area of the fixed particle:

The number of collisions in unit time is: , N is the atomic density.

The average collision time is: . If there is m mol gas in V volume

24rrelavrvNr 24

Nvr relavr

c 241

relavrv

fixed particleunit cross section

TkPN

VmNNTkVNTkNmTRmPV

B

ABBA

kN BA

)(




Collision broadening is proportional to the pressure, .

Rough guide suitable for estimation:

Different homogeneous broadening together

The sum of Lorentzian distributions is again a Lorentzian function:

Crystal field interaction is a ”non conventional” collision process, interaction with phonons in solids, but similarly homogeneous effect the lineshape function is Lorentzian.

PvrTk

relavr

Bc 24 kT

Pvr relavr

cc

241

Pc ~

.105~torrMHz

Pc

)21(21

121

ccnatLLL



,gg

Direction of observation

,000 cv

v

Temperature dependent velocity distribution in the gas

Light-matter interactionsInhomogeneous broadening –– Doppler-broadening

Different atoms have different lineshape functions or different center of frequency average lineshape function average with respect to the variable .

E.g. light emission of atoms movingwith velocity v, frequency shiftbecause of the Doppler-effect.


Inhomogeneous broadening –– Doppler-broadening (cont.)

p(v)dv is the probability that the velocity of an atom is in the interval of [v, v+dv] , the average lineshape function is:

In equilibrium the velocity distribution of atoms with mass M at temperature T is the Maxwell - Boltzmann distribution. The probability that the velocity component of the atom is in the range of [v, v+dv] in a given direction (e.g. in the direction of the resonator axis)

Gaussian-distribution.

FWHM:

.0 dvvpcvgg

dvkT

MvkT

Mdvpv 2exp

2

22/1

maximum at v=0

.2ln2 2/1

2/1 MkTv



Inhomogeneous broadening –– Doppler-broadening (cont.)

In case the homogeneous broadening its effect can be neglected:

where

the FWHM or Doppler broadening.

,2/10 c

vL

,2

exp2 2

22

max

2/1

dTk

MckT

Mcdgdvpo

o

B

imum

ov

o

.dcdv,cv,cv0

20

20

22

0

0

210

021

21

2002120

2021

2

2222

2222

1

/

symmetrytheofbecause

/D

/

//

MlnkT

c

,Mc

lnkT,kT

Mcexp

1



Different inhomogeneous broadenings together

Inhomogeneous distribution of doping materials in solids causes inhomogeneous broadening Gaussian distribution. The superposition of two different Gaussian distribution:

Homogeneous and inhomogeneous distributions together

Convolution of the Lorentz and the Gauss functions Voight-integral (can be numerically calculated).

Numerical examples - He-Ne laser

D=? T = 400 K, = 633 nm,

Collision broadening? Can be neglected at 3-4 torr pressure! Typical inhomogeneously broadened laser material.

.22

21 GGG

Js10636

JK10381106

g20

34

12323

.h

.k,MNe

GHz.51Hz1051 9 ..NeD



Numerical examples - ruby és Nd:YAG laser

T-dependent broadening,because the lattice vibrationincreases with T!Typical systems with homogeneous broadening.Inhomogeneous effect atT ~ 0 because of impurities.

/c0 ( 1 / , cm-1) data,

e.g. at 300 K L

Nd:YAG 120 GHzRuby 330 GHz

0100 200 300

4

8

12

16

20

Hõmérséklet ( K)o

Vonalszélesség[cm ]-1

Rubinm

mNd:YagNd:YAG


Linewidth

Temperature [K]


Typical broadenings of different laser materials


Type Gas Liquid Solid

natural 1 kHz - 10 MHz can be neglected can be neglected

collision 5 - 10 MHz/torr 9 THz -

crystal field interaction - - 300 GHz (300K)

Doppler 50 MHz - 1 GHz can be neglected -

local field - 15 THz 30 GHz - 15 THz

Homogeneous

Inhomogeneous


Problem

Possible decays of E2 and E1 in an atom, tsp = 5 ms, nr = 50 µs, 20 = 10 ps, 1 = 15 µs. Calculate 21 and nat of the transition! Is it an ideal choice to

use that transition for a laser transition?


2

1

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