Laser Physics Chapter 4

7/28/2019 Laser Physics Chapter 4

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4

Electron Oscillator Model of Absorption

1. A. Siegman, Lasers (University Science Boks, Mill Valley, 1986), Chapters 3.

2. P. W. Milonni and J. H. Eberly (John Wiley and Sons, Inc., Hoboken, NJ,2010), Chapter 3.

3. A. Yariv, Quantum Electronics (John Wiley and Sons, Inc., Hoboken, NJ,1989), Chapter 8.

We present a simple classical model for the response of a medium subject to atime-dependent electromagnetic field. Under the influence of the field charges areperturbed from their equilibrium positions. For applied field strengths that are smallcompared to those produced by the nucleus, we can treat bound charges as simpleharmonic oscillators perturbed by the applied field. The charges will respond notto the macroscopic field but to the local field. For a dilute medium the distinctionbetween macroscopic and the local field is negligible. Since our main interest is inthe frequency response of the medium we will take this to be the case. For dense

media we can use Lorentz-Lorenz equation to express the microscopic field in termsof the macroscopic field.

4.1 Harmonically-bound Charged Oscillator

Consider now the response of a bound electron to the applied field. Let ro be themean position and r be the complex1 displacement of the electron from its meanposition. Then for small displacements, the equation of motion for the electron canbe written as

d2r

dt2+ 2o

dr

dt+ 2or = (e/m)

E(ro, t) +

dr

dtB(ro, t)

. (4.1)

This is the equation of a charged charged harmonic oscillator driven by an electro-magnetic field. Here 2o is the radiative decay rate (the rate at which the oscillatorloses energy due to radiation), o is the natural frequency (one of the transition fre-

quencies of the atom or molecule).E

andB

are the (complex) electric and magneticfields of the incident wave. We have assumed the displacement r from equilibrium

1The real displacement is given by the real part of r.

121


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122 Laser Physics

to be small compared with the wavelength of light (dipole approximation), the vari-ation of the field over dimensions of the order r can be ignored. This allows us touse fields evaluated at the equilibrium position ro of the electron in the equationof motion. Now the magnitudes of the electric and magnetic fields are related by|B| = |E|/c. This relation holds for a plane wave and for more general waves in

transparent media in the geometrical optics limit. Then a comparison of the electricand magnetic force terms in the equation of motion shows that for non-relativisticmotion of the electron the magnetic force is smaller by the factor | r|/c 1 com-pared to the electric force. For the non-relativistic electron motion considered herewe can therefore drop the magnetic term. The equation of motion for the electronthen reduces to

d2r

dt2+ 2o

dr

dt+ 2or =

e

mE(ro, t) (4.2)

To study the response of the oscillator to the field, we consider a monochromatic

driving field2, E(r, t) = E(r)eit. Recalling that the forced oscillations are at thedriving frequency, we write the steady-state displacement of the electron from itsequilibrium position as

r(t) = roeit (4.3)

where the steady-state amplitude ro is to be determined. Substituting this in Eq.(4.2) and solving for ro we find the steady-state electronic displacement is givenby

r

(t) = eE(ro, t)/m

2o 2 2io . (4.4)This field induced displacement of the electron from its equilibrium position willinduce a dipole moment er(t).

4.2 Induced Polarization and Susceptibility

Different electrons in an atom or molecule will have different natural frequenciesand damping. Therefore, for a fraction fj of electrons (

j fj = 1) with natural

frequency j and damping constant j , the displacement rj from the equilibriumposition will be

rj = eE(rj , t)/m2j 2 2ij

. (4.5)

Assuming that all the electron oscillators respond in identical fashion (homogeneousmedium) we can add the contribution from all such groups of electrons. By carrying

2If the field is not monochromatic, we can Fourier analyze the field and consider one of the Fourier

components.


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Electron Oscillator Model of Absorption 123

out a spatial averaging by introducing a density of electrons N (number of electronsper unit volume), we find the macroscopic polarization is given by

P(r, t) =Ne2

m

j

fj2j 2 2ij

E(r)eit o()E(r)eit . (4.6)

Here E(r) is the spatially averaged electric field. Using the definition of dielectricpermittivity () = o[1 + ()] we obtain

() = o

1 + Ne2

mo

j

fj2j 2 2ij

. (4.7)

In terms of the plasma dispersion frequency

p =

Ne2

mo (4.8)

we can write the real and imaginary parts of () as

() = o

1 + 2p

j

fj(2j 2)(2j 2)2 + 42j2

(4.9)

() = 2o2pj

fjj

(2j 2)2 + 42j2(4.10)

Note that () is an even and () is an odd function of.

4.2.1 Refractive index and attenuation index

For field frequencies comparable to the eigenfrequencies, the dielectric permittivity() is complex. In this range, it is convenient to introduce the complex refractiveindex n by writing

()/o = n() n() + i() , (4.11)where n and are the real and imaginary parts of (). The real part n is called

the refractive index and the imaginary part is called the attenuation index of themedium. Both are frequency dependent quantities. Equating the real and imaginaryparts from the two sides of Eq. (4.11), we can express n and in terms of the realand imaginary parts of() as

n() =

1

2o

() +

2() + 2()

(4.12)

() = 12o

() + 2() + 2() (4.13)

The intermediate frequency behavior will now be discussed in terms of the refractiveindex and the attenuation index.


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124 Laser Physics

4.2.2 Nonresonant behavior of refractive index

For field frequencies far removed from any of the resonance frequencies such thatthe inequality |j | j holds, the dielectric permittivity can be written as

() o 1 + 2pj

fj2j 2 + i2pj

2fjj(2j 2)2 () + i() (4.14)

On comparing the magnitudes of the real and imaginary parts of () in the non-resonant limit, we see that the imaginary part is small compared to the real part() () at least by the ratio / 1. This implies that in this frequencyrange dissipation is negligible. The medium is said to be transparent in this fre-quency range. It is important to bear in mind that transparency (or opacity) of amedium is frequency dependent phenomenon.

In the transparency frequency range, since the imaginary part of the permittivityis small compared to its real part ( ), Eqs. (4.12) and (4.13) lead to thefollowing expressions for the refractive index and the attenuation index

n2() =()o

= 1 + 2pj

fj2j 2

(4.15)

() =()

2on()=

2p

2n()

j

fjj

(2j 2)2(4.16)

For many common materials that are transparent at optical frequencies, the eigen-frequencies j lie in the ultraviolet or the infrared part of the spectrum. For suchmaterials the refractive index n is an increasing function of frequency at opticalfrequencies. This behavior of the refractive index is known as normal dispersion.The expression for the refractive index can be written in terms of the wavelength = 2c/ as

n2 1 = 2p

42c2

j

fj2j

2

2 2jj

bj2

2 2j. (4.17)

This formula is known as Sellmeirs dispersion formula. Its form is used to fit therefractive index of materials in their transparency range. For example, the followingformula gives the refractive index of fused silica at 20o C ( in microns) accurateto 3 105 in the wavelength range 0.2 2.1

n2 1 = 0.69616632

2 (0.0684043)2 +0.40794262

2 (0.1162414)2 +0.89747942

2 (9.896161)2 (4.18)

4.2.2.1 Resonant behavior of refractive index

When the field frequency lies within a few o

of an eigen frequency, say o

, wecan use the approximation 2o2 2(o) in the resonant term. For example,for atomic electrons with j 2 1014 Hz, j 2 109 Hz, even when differsfrom j by 10j, + j 2 to one part in 104. In all the other terms we can


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use the off-resonance approximation. By separating the non-resonant (background)and resonant contributions we can write the permittivity as

() = o 1 + 2p

j

fj

2

j 2

+2pfo

2

(o )

(o )2

+ 2

o

b +2pofo

2

(o )(o )2 + 2o

(4.19)

() = 2o2p

j

fjj

(2j 2)2+

2pofo

2

o

(o )2 + 2o

b +2pofo

2

o

(o )2 + 2o(4.20)

As noted earlier, for the non-resonant background terms, the inequality b b

4

o

nnb

b

2

o

+2

0+4

FIGURE 4.1

Beahvior ofn and as a function of frequency near a resonance for a homogeneous collection

of oscillators.

holds. On introducing a background refractive index nb b/o and attenuation


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126 Laser Physics

index b = b/2onb and writing

/o = n + i , we obtain

n nb +2pfo

4nb

(o )(o )2 + 2o

(4.21)

b +2pfo

4nb

o

(o )2 + 2o (4.22)

The resonant terms vary rapidly with frequency in the vicinity of the resonancefrequency o, whereas the background terms nb and b vary slowly. Dissipationcharacterized by is largest on resonance. A plot of n and in the neighborhoodof the resonance frequency is shown in Fig. 4.1. Note that near a resonance, ncan be a decreasing function of frequency. This behavior of n, in contrast to thenormalbehavior where n is an increasing function of frequency, is termed anomalous.Away from a resonance, n reverts to its normal behavior, viz, that it increases with

frequency.We have considered a simple model of dispersion for an isolated resonance. Clearly,

if resonances overlap, more complicated dispersion behavior can result. Other re-finements such as the modification of dispersion due to the thermal motion of atomsare possible and are discussed in the literature [See for example Lasers by A. Sieg-man (University Science Books, Sausalito, CA, 1986)]

4.3 Near-resonance susceptibility

The resonant atomic susceptibility can be written as

at =Nfoe2

mo

(2o 2) + i2o

(2o 2)2 + 422o

. (4.23a)

For driving field frequency close to o such that |o | o , and |o |

restricted to be less than, say, 10o, we can use the approximation

2

o 2

=(o + )(o ) 2(o ) and write the real and imaginary parts of thesusceptibility at =

at + i

at as

at =Nfoe2

mo

2o

(2o 2)2 + 422o

Nfoe

2

2mo

o

(o )2 + 2o

=Nfoe2

4mo

1

o/2

(o )2 + (o/2)2 Nfoe

2

4moS() , (4.23b)

at

=Nfoe2

mo (2o 2)

(2o 2)2 + 422o Nfoe2

2mo (o )

(o )2 + 2o =

Nfoe2

4mo

1

(o )(o )2 + (o/2)2

Nfoe

2

4mo

2(o )o

S() . (4.23c)


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The function S() is called the line shape function. For a homogeneous mediumconsidered here, it is a Lorentzian

S() =1

o/2

(o )2 + (o/2)2 . (4.23d)

with a width (FWHM) given by

o =o

. (4.24)

In terms of the FWHM o, we can write the Lorentzian in the standard form

S() =1

o/2

(o )2 + (o/2)2 . (4.25)

The expressions for the susceptibility can then be written as

at =Nfoe2

4moS() , (4.26)

at =2(o )o

at . (4.27)

4.4 Absorption by a Single Oscillator and Atomic Susceptibility

Time-averaged power absorbed per unit volume (rate of absorption of energy) byatomic oscillators from a monochromatic field is given by

dw

dt= E

dPatdt

=1

2Re [E (iPat)] . (4.28)

Note that this is the time averaged rate at which the field does work on atomicdipoles per unit volume. Now the induced polarization (dipole moment per unit

volume) is

Pat =Ne2E

4m

(0 ) + i(0/2)

(0 )2 + (0/2)2

= 0(

at + i

at)E, (4.29)

where we have used the resonance approximation | 0| 0 0, and ex-pressed 0 in terms of the FWHM 0 = 0/ of the Lorentzian. The averagepower absorbed by atomic oscillators per unit volume is then

dw

dt =

1

2ReE (i)0(

at + i

at)E

=

1

2|E|20nc

nc

at I

nc

at , (4.30)


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128 Laser Physics

where I is the intensity of the incident wave and S() is the Lorentzian line shapefunction.

If the wave is propagating in z-direction through a region containing a distributionof oscillators with density N (# per unit volume), its intensity will decrease as itloses power to the oscillators. The change in power of the wave in traversing a

cylinder of small cross-section A and length z will be

P = [I(z + z) I(z)]A . (4.31)

This must equal the negative of the power absorbed by the oscillators in the cylinderunder consideration

[I(z + z) I(z)]A = Az I(z) nc

at .

In the limit ofz

0 we find

dI(z)

dz=

nc

at

I(z) , (4.32)

where the quantity inside square brackets is called the absorption coefficient

() =

nc

at, (4.33)

which represents the fractional change in intensity per unit propagation distance andhas dimensions of inverse length. Thus the imaginary part of dielectric susceptibility

is related is related to absorption via the absorption coefficient.Using the classical expression for the radiative decay rate 2o = Arad in a medium

of refractive index n =/o,

2o Arad = 23

e2

4

2

mv3=

2

3

e2

4on22

m(c/n)3=

e2n2

6omc3, (4.34)

we can write the imaginary and real parts of the susceptibility as

at =NfoArad6c

3

4n3S() =

NfoArad33

16n2S() , (4.35a)

at =2(o )o

at . (4.35b)

Then the absorption coefficient can be written as

at =

ncat =

32

8n2AradNfoS() . (4.35c)

This relation between absorption coefficient and atomic susceptibility holds quan-tum mechanically as well if we interpret the resonance frequency

oas the transition

frequency for an allowed transition between two atomic levels of energy E2 and E1with E2E1 = ho, and replace Nfo by the population density difference N1N2,where N1 and N2 are, respectively, the number density of atoms in level E1 and E2.


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In our model of susceptibility we have assumed isotropic response of the mediumand the dielectric polarization to to be parallel to the electric field. In real atoms,the dielectric response, in general, has a tensorial character so that the inducedpolarization has the form

Pi = 0ijEj . (4.36)

Then the power absorbed per unit volume by the atoms is

dw

dt=

1

20

ijEi Ej (4.37)

The scalar product on the right hand side of this equation can be evaluated giventhe atomic response and the polarization of the field. The net result is that thefactor of 3 in the expression for the coefficient of absorption can be any numberfrom 0 to 3 depending on the polarization of the field and atomic response tensor

[Siegman Sec. 3.5]. If the field and the induced polarization line up, Eqs. (4.35)stand as written. On the other hand, if the fields are linearly polarized and theatomic response is in a random direction, the factor of 3 should be replaced by 1.In what follows, we will use these equations with 3 replaced by 1.

Let us now return to the power absorbed by atomic oscillators per unit volumefrom a monochromatic field of frequency

dw

dt= ()I (4.38)

If the incident field has many discrete frequencies we must sum over all such fre-quencies leading to

dw

dt=j

(j)Ij . (4.39)

For a continuous distribution of frequencies we obtain

dw

dt=

0

d()I() , (4.40)

where I() represents the power density per unit frequency interval (spectral powerdensity W/m2Hz) can be expressed in terms spectral energy density () (J/m3Hz)as

I() = ()c

n. (4.41)

Then power absorption per unit volume can be written as

dw

dt=

0d()() . (4.42)

When spectral density () is broad compared to the oscillator line function S()[see Fig. (4.2)] we say we are dealing with broadband radiation. In this case ()


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130 Laser Physics

()

S()

(a)

()

S()

(b)

FIGURE 4.2

Field spectral density and the oscillator absorption profile for (a) broad band and(b) narrow band fields.

may be assumed constant over the significant portion of the line shape functionS(). We can then evaluate the frequency integral approximately as

0

d()() (0)0

d() = (0)320AradN

8n2, (4.43)

where we have replaced the spectral density by its value at the peak 0 of theoscillator line function. Thus for broadband radiation the rate of power absorptionby an oscillator of frequency 0 can be written as

dw

dt =

320AradN

8n2 (0) . (4.44)

If the spectral density is a narrow function of frequency compared to the line functionS() [see Fig. (4.2)] and is centered at some frequency , we can treat S() asconstant over the spectral profile of the field and evaluate the integral as

0

dS()() S()0

d() = S()cIn

. (4.45)

Here we have replaced the oscillator line function by its value at the peak ofthe field spectral density. I is the total intensity [W/m2] in the narrow band of

frequencies contained in (). Thus the power absorption by the oscillator from anarrowband field is given by

dw

dt=

32AradN

8n2S()I (4.46)

Simple expressions, like the ones derived here, are not possible for arbitrary ()and S(). It is clear, however, from the two extreme cases considered here that therate of absorption has different frequency dependence in these limits and that it isimportant to explore shapes of absorption profile S() for different collections ofoscillators.


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4.5 Line Shape Functions

In our discussion of energy absorption by an oscillator we found that radiativeenergy loss ( at rate 2

0) by an accelerated charged oscillator leads to a broadening

of the absorption line. The shape of the absorption line is a lorentzian. The typeof broadening, which affects all oscillators in the same way, is an example of whatare called homogeneous broadening mechanisms. There are also mechanisms thataffect different oscillators differently and lead to inhomogeneous of absorption line.Homogeneous and inhomogeneous broadening lead to different line shapes as well.We will explore some of these line broadening mechanisms.

4.5.1 Homogeneous Line Broadening

4.5.1.1 Natural line broadening

Radiation reaction force on a radiating dipole leads to an energy damping rate 2o(=Arad) in the equation of motion

x + 2ox + 20x = 0 , (4.47)

where the classical energy damping rate 2o due to radiation by a charged oscillatorin a medium of refractive index n =

/0 is given by

2o = 23

e2

n20

40mc3

. (4.48)

This mechanism of line broadening, which effects all oscillators the same way, is anexample of homogeneous line broadening. For a collection of oscillators of naturalfrequency o this broadening leads to a lorentzian line shape

S() =1

rad/2

(o )2 + (rad/2)2 ,0

dS() = 1. (4.49)

The width (FWHM) of this line is given by

rad =o

. (4.50)

4.5.1.2 Collision Broadening

In a gas, collisions can cause an interruption of the phase of oscillations. Thesecollisions lead to a damping of oscillations when an averaging over the collection ofoscillators is carried out. Atoms suffering collisions have their phases randomizedand give no contribution to the average response of the collection. This leads to anexponential decay et/coll of the number of oscillators contributing to the coherentresponse of the collection, where coll is mean time between collisions or 1/coll isthe average rate of collision (collision frequency). This is also a homogeneous line


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132 Laser Physics

broadening mechanism and leads to a lorentzian line shape. The equation of motionfor the average oscillator has the form of Eq. with 20 replaced by

2o +2

coll. (4.51)

This leads to a linewidth

H

= rad + coll =o

+

1

coll. (4.52)

When we think of a collection of oscillators both the natural and collision broadeningmechanisms effect all the oscillators in the same way. For this reason natural andcollision broadening mechanisms are referred to as homogeneous line broadeningmechanisms.

A simple model of collision broadeningConsider a gas of identical atoms at temperature T , pressure P, and density ofatoms N. From ideal gas law these quantities are related by

P = NkB

T (4.53)

Assuming each atom to be a hard sphere of diameter d, the number of collision itsuffers in an interval t can be calculated as follows. Two atoms cannot come closerthan their center to center separation d. In time t each atom will sweep a volumed2vt where v is the average (relative) atomic speed. This volume will containd2vtN atoms. So each atom will suffer, on average, d2vtN collisions intime t if all the other atoms are at rest. This means the speed v in this calculationmust be the relative speed. It follows that the collision frequency is given by

d

M, v

FIGURE 4.3

Atomic centers cannot come closer thanatomic diameter d.

# of collisions per unit time collision frequency = 1coll

= d2vN (4.54)

According to the kinetic theory of gases, the rms speed of an atom of mass M isgiven by

1

2M v2rms =

3

2kB

T vrms =

3kB

T

M(4.55)


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To find the average relative speed v of colliding atoms, we replace M by the reducedmass (which for a gas of identical atoms will be M/2) in the equation for vrms. Thuswe have

v =

6k

BT

M. (4.56)

For the density N we use the ideal gas law (4.53). We then obtain for the collisionfrequency

1

coll= d2

6k

BT

M

P

kB

T= d2P

6

M kB

T. (4.57)

Then the contribution of collisions to line broadening will be

coll =1

= d2P

6

M kB

T. (4.58)

The diameter d can be estimated by noting that atomic radius is given by

r = r0n2 (4.59)

where n is the principal quantum number of the outermost occupied shell andr0 0.05 nm. By measuring d in nm, P in torr, and M in atomic mass units(amu), we can express collision frequency in a form more suitabel for making orderof magnitude estimates,

coll =

d(nm)1092

P(torr)

1.013

105

760 6

M(amu)1.66 1027 kB

T(K)

= 1018

6

1.66 1027 1.38 1023 1.013 105

760

[d(nm)]2P(torr)M(amu)T(K)

= 2.157 109 [d(nm)]2P(torr)

M(amu)T(K)Hz (4.60)

For pure N e with M(amu) 20, d = 2r0n2 = 20.054 = 0.40 nm at room tempT = 300K

coll = 4.46 106P(torr) Hz . (4.61)Since collision broadening is directly proportional to the pressure, it is also calledpressure broadening.

The simple model, treating the collection of oscillators as a gas of identical hardspheres, is not expected to give quantitative results for collision broadening but itdoes allow us to understand qualitative features of collision broadening. The exactcalculations taking into account different masses and more realistic inter-particlepotentials are, however, quite involved. The actual values ofcoll can be larger,by as much as an order of magnitude or more, than those predicted by Eq. (4.61).

H

= [2.6 107 + 1.44 108P(torr)] Hz (4.62)P. W. Smith, Journal of Applied Physics 37, 2089 (1966).


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134 Laser Physics

4.5.2 Inhomogeneous Line Broadening

Certain line broadening mechanisms distribute the resonance frequencies of theatoms over a frequency range centered at some frequency 0. Such mechanisms thataffect atoms differently are referred to as inhomogeneous broadening mechanisms.

An inhomogeneous broadening mechanism typical of gases is the Doppler effectdue to atomic motion. Consider an atom with transition frequency 0 (in a framewhere it is at rest) interacting with an electromagnetic wave of frequency (inthe laboratory). If vz is the velocity component of the molecule (in the laboratoryframe) in the direction of propagation of the electromagnetic wave (+z-direction),the frequency of the wave, as seen from the atom is

= (1 vz/c) . (4.63)Note that is higher than if the atom is moving in direction opposite to that of

the wave ( vz < 0) and smaller than if the atom is moving in the same directionas the wave. The condition for resonant absorption is satisfied when the apparentfrequency of the electromagnetic wave is equal to the transition frequency 0,

= (1 vz/c) = 0 . (4.64)If we rewrite this equation as

=0

1 vz/c , (4.65)

we can interpret this to mean (in the lab frame) that the electromagnetic wave is

in resonance with an atom of frequency

a =0

1 vz/c (4.66)

In other words, the resonance frequency of an atom moving with speed vz is

a =0

1 vz/c 0(1 + vz/c) . (4.67)

Since different atoms are affected differently (depends on vz) this mechanism clearly

belongs to the inhomogeneous category. To calculate the distribution of resonancefrequencies we recall that for a gas in thermal equilibrium at temperature T, thefraction of atom having a velocity component between vz and vz + dvz along thedirection of propagation of the em wave is given by the Maxwell-Botzmann distri-bution

f(vz) =

M

2kB

Te Mv

2z

2kBT , (4.68)

where M is the mass of the atom, kB

is Boltzmann constant, and T is the equilibriumtemperature of the gas. Note that the normalization integral and mean squared

velocity are given by

dvzf(vz) = 1 , v2z =

dvzv2zf(vz) =

kB

T

M. (4.69)


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It is easily checked that the mean velocity vanishes, since f(vz) is an even functionof velocity vz. From a = 0(1 + vz/c) we find

vz =c

0(a 0) . (4.70)

Then the distribution G(a) of resonant frequencies a is given by G(a)da =f(vz)dvz or

G(a) = f(vz)

dvzda = c0

M

2kB

Texp

a 0

0

2 M c22k

BT

. (4.71)

The function G(a) is normalized since

0

daG(a) =c

0 M

2kB

T0

da expa 00

2 M c2

2kB

T

,

=

M c2

2kB

T20

0

dx exp

x2 M c

2

2kB

T20

,

M c2

2kB

T20

dx exp

x2 M c

2

2kB

T20

= 1 . (4.72)

where, since M c2/kB

T

1, we have replaced the lower limit

0 by

with

negligible error. This is because the integrand is vanishingly small well beforex = 0. Thus the resonance frequency distribution for Doppler broadening is

G(a) =1

0

M c2

2kB

Texp

a 0

0

2 M c22k

BT

(4.73)

This curve is a gaussian whose maximum value occurs at a = 0

G(0) =

1

0M c2

2kBT (4.74)

The width of the distribution (FWHM) D

is determined by

G(0 +1

2

D) =

1

2G(0)

or1

4

D

0

2 M c22k

BT

= ln 2

Solving for the width D we find

D

= 0

kB

T8 l n 2

M c2= 0

kB

T8 l n 2

M c2(4.75)


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136 Laser Physics

This width is referred to as the Doppler width. Note that it depends on the restframe transition frequency 0. Sometimes the relative width

D

0=

kB

T8 l n 2

M c2(4.76)

is used, which is useful when the same species can exhibit absorption at severaltransition frequencies.

In terms of the FWHM D

we can rewrite the line shape function

G(a) =2

D

ln 2

exp

a 0

D

24 l n 2

(4.77)

The expression for Doppler width D

can be cast in a form that is more convenientfor estimating its numerical values

D

=1

0

kB

T8 l n 2

M=

1

0(nm) 109

1.38066 1023T(K) 8 l n 2M(amu) 1.659 1027

= 2.148 1011

1

0(nm)

T(K)

M(amu)

Hz

2.15 1011

1

0(nm)

T(K)

M(amu)

Hz (4.78)

Similarly, for the relative width we obtain

D

0=

1

c

kB

T8 l n 2

Mln 2 = 7.161 106

T(K)

M(amu) 7.16 106

T(K)

M(amu).

(4.79)For a He:Ne laser MNe 20 amu, T = 400 K, and 0 = 632.8 nm

D

= 2.15 1011 1632.8

400

20Hz = 1518 MHz 1.52GHz

For the 10.6 m line of CO2

, MCO2

44 amu, T = 400 K

D

= 2.15 1011 110.6 103

400

44Hz = 61 MHz

Doppler broadening is an example of inhomogeneous broadening. Atoms in acollection have different resonance frequencies because of their different velocities.Other examples of inhomogeneous broadening include broadening of active atomsby impurities or defects in a crystal that cause different local crystal fields. Thesein turn have the effect of shifting the resonance frequencies of active atoms slightlydifferently. The distribution of such shifts is very much like the Doppler distribution,and gives rise to an inhomogeneously broadened absorption line. This type ofbroadening is present in the Cr3+ line associated with the Ruby laser light, forexample.


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4.6 Voigt Line shape

In general both homogenous and inhomogeneous mechanisms contribute to line

broadening. In order to see the eff

ect of both inhomogeneous and homogeneous linebroadenings on the line shape function let us consider a gas of atoms of transitionfrequency 0 subjected to an em wave of frequency (all frequencies measured inthe laboratory frame). Atoms moving with velocity component vz (in the directionof the em wave) have a resonance frequency

a = 0(1 + vz/c) (4.80)

This group of atoms, a fraction daG(a) of the total number of atoms, is subjected

to collision and natural broadenings. Both of these are homogeneous broadeningmechanisms and lead to a lorentzian line shape

S(, a) =1

H

/2

(a )2 + (H/2)2(4.81)

where

H

= 0

+ coll . (4.82)

By adding the contribution of atoms from all frequency groups we obtain the lineshape function for the gas, we obtain

S() =

0

daG(a)S(, a)

=

0

da

H/2

(a )2 + (H/2)2

4 l n 2

2D

exp

(a )2 4 l n 2

2D

=

0du

H

ln 2

D3/2

exp

u2 4ln2

2D

(0 + u)2 + (H/2)2, u = a

0

13/2

H

2

ln 2

2D

dyey

2

(0)2

ln 2

D

+ y2

+

H

ln 2

D

2

Here in the last step we have extended the lower limit of integration to withnegligible error because 0/D 1 and the integrand is vanishingly small wellbefore the lower limit is reached. With a simple change of variables the line shapefunction can be written in a more compact form as

S() =2b2

3/2H

dyey

2

(x + y)2 + b2. (4.83)


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138 Laser Physics

y =(a 0)2

ln 2

D

(4.84a)

x =(0 )2

ln 2

D

(4.84b)

b =

H

ln 2D

(4.84c)

This line profile, which takes into account both homogeneous and inhomogeneousline broadenings, is known as Voigt profile. This function is encountered frequentlyin laser and atomic physics. Its values may be found tabulated in handbooks. Withdesk top computers it is a simple matter to evaluate this function numerically. Wecan obtain some intuition into the behavior of the line shape function by consideringsome limiting cases.

First let us remind ourselves that S() is the general line shape that takes intoaccount both homogeneous and inhomogeneous broadening mechanisms. In the in-tegral, when we look at the definitions ofx and b we realize that Doppler broadening

Dsets the scale for the variation of the integrand.

4.6.1 Large collisional broadening

At sufficiently large pressures, collisional broadening dominates [Fig. 4.4(a)]. Inthis case b 1 [

H

D] so that the lorentzian varies slowly with y so that we

can approximately evaluate the integral (x not too large) by replacing the lorentzian

by its value at the peak y = 0 of the gaussian

S() 2b2

3/2H

1

x2 + b2

dyey2

=1

H

/2

( 0)2 + (H/2)2(4.85)

exp[-y2]

(y+x)2+b2

y0

(a)

b2

x

exp[-y2]

(y+x)2+b2

y0

b2

x

(b)

FIGURE 4.4

The integrand of Eq.(4.83) consists of two factors: a gaussian centered at y = 0and a lorentzian centered at y = x. The figure shows these factors in the limitof (a) large collisional broadening b = 5 (

H

/D

6) and (b) large Dopplerbroadening b = 0.14 (

H/

D 1/6).


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4.6.2 Large inhomogeneous broadening

At low pressures in a gas inhomogeneous broadening due to Doppler effect domi-nates. In this case

H

Dso that b 1 [Fig. 4.4(b)]. Then the Gaussian

in the integrand varies slowly compared to the Lorentzian. Once again we can ap-proximately evaluate the integral by replacing the gaussian by its value at the peakof the Lorentzian (y = x)

S() 2b2

3/2H

ex2

dy1

(x + y)2 + b2=

2b2

3/2H

ex2

b=

2bex2

H

=

4 l n 2

2D

exp

( 0)2 4 l n 2

2D

(4.86)

and obtain pure Doppler broadened line shape.

4.6.3 Far Wing Limit

Finally, we consider the limit where the detuning 0 is large compared to both

Hand

D. This is the so called far wing limit.

exp[-y2

]

(y+x)2+b2

y0

b2|x|>>1

FIGURE 4.5

In the far wing limit |x| 1 only the tail of the Lorentzian contributes near thepeak of the gaussian.

In the far wing limit, the qualitative difference between the Lorentzian andthe Gaussian becomes important. The Gaussian falls to zero much more rapidlythan the Lorentzian. Thus the Gaussian is essentially zero near the peak of theLorentzian. On the other hand the Lorentzian tail still gives significant contributionnear the peak of the Gaussian. We can then evaluate the integral, approximately,


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140 Laser Physics

by replacing the Lorentzian by its tail at the peak (y = 0) of the Gaussian

S() 2b2

3/2H

1

x2

dyey2

=2b2

3/2H

1

x2

=

H

2

1

( 0)2(4.87)

Hence in the far wing the line behaves essentially as a homogeneously broadenedline even if the broadening is principally Doppler. This provides a justification fortreating off-resonance response of a dielectric medium essentially homogeneous.

Laser Physics Chapter 4

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Transcript of Laser Physics Chapter 4