6.8) and thus obtain a relation describing the excitation energy of the system. This is

6.1 Classical radiation interaction 169

qEA

In

Eqs. (6.2)-(6.8)

and

(6.13),

the

eigenfrequency wo

has only to be replaced by the modified value

wo -coL.

Otherwise, the results can be taken over. With this theory, H. Lorentz was able to interpret the Zeeman effect, the shift and splitting of atomic resonance lines by external magnetic fields.

We finish by studying the effect of a transverse magnetic field on the motion of an electron.

For this purpose we take the vector product of Eq.

(6.9)

by

xx

(replacing p

x)

and obtain a new equation for the electronic angular momentum

L = mx x

X. Strictly speaking, this should be

mxx (X x B) = L x13 + mX x (x x B),

but in static fields the second term vanishes, and in alternating fields it is equivalent to a relativistic correction of first order,

(v/c)dxE,

and can be neglected. So:

—d

L +'yL = dxE + —LxB.

dt

It can be recognized from this equation that a circularly polarized electric light field as well as a transverse static field

(B 1 L)

can cause rotation of the electronic angular momentum.

The former case is usually called 'optical pumping' in spectroscopy

[40],

and the latter case occurs in the Hanle effect

[21].

6.1.2 Macroscopic polarization

The macroscopic polarization P(r,

t)

has already been introduced in section 2.1.2 in or- der to describe the propagation of electromagnetic waves in a dielectric medium. From the microscopic point of view, a sample consists of the microscopic dipole moments of atoms, molecules or lattice elements. The 'near field' of the microscopic particle does not play a role in the propagation of the radiation field, which is always a 'far field'.

If there are Nat atomic or other microscopic dipoles in a volume, the macroscopic polarization is obtained by averaging, P = Nat p/V. Here the volume V is chosen much larger than molecular length scales, e.g. dmoi < 5

A,

and the average volume of a single particle as well. If the microscopic polarization density p(r) is known, there is the more exact form:

P(r,

t) =

^Nat

f

p(r ^- r',

t) d3 r/ .

^(6.11)

V v

In our classic model the Fourier amplitudes of the polarization

p =

,F{P} and of the driving field

E

are linearly connected,

P(w) = coX(w)e(w), (6.12)

and the susceptibility x(w) =

V(w)-Fix"(w)

can be given using the results of Eq. (6.4),

x

r(6) _ Nat

3)0 ^26/

ii772- E 1 + (26N 2

"

( 6) 1-rat 3A3

¹

X V 47E2 1 + (26[7) 2

Since the temporal behaviour of the polarization is also characterized by transient processes, it usually depends on the field intensity also at earlier times. This becomes more apparent in the time-domain expression

(6.13)

P(r,

t) = co f x(t -

t')E(r,

t') de ,

^(6.14)

-00

v2E -

^{12 E — 1 (92 p.}

C2

a-t2

^E0C2

at

6.1 Classical radiation interaction

171

which requires

x(t — t') = 0

for (t ^— t') G

0

in order not to violate causality. Here as well the literal meaning of 'susceptibility' or 'after-effect' shows up. But for our pur- poses we assume that we are allowed to neglect relaxation processes occurring in solid materials within picoseconds or less, and therefore we can restrict our treatment to an instantaneous

interaction. 2

According to the convolution theorem of Fourier trans- formation, the relation is, however, much simpler in the frequency domain following Eq.

(6.12).

To be more exact, the 'dielectric function' (Eq.

(2.4)) co K(c.o) = co [1±x(w)]

and the susceptibility are second-rank tensors, e.g.

x, 3 = api lasi ,

and reflect the anisotropy of real materials. The magnetic polarization can mostly be neglected for optical phe- nomena (si r

1),

since the magnetic field

B

and the H field are identical except for a factor,

H = Bhio .

Only in an isotropic

(V • P = 0)

and, according to Eq.

(2.4),

linear medium does the wave equation take on a simple form. This is, however, an important and often realized special case where the polarization obviously drives the electric field:

(6.15)

Linear polarization and macroscopic refractive index

If the polarization depends linearly on the field intensity according to Eq.

(6.12),

then the modification of the wave velocity within the dielectric,

c2 c2 /k(w),

can be taken into account using the macroscopic refractive index

n(w)

(see eq.

(2.12)):

V2 E n2(w) a2

2

E 0.

(6.16)

C2

at

According to Eq.

(6.12)

we have

E ±Plco = [1+ x(w)je = n 2 (w)E

with

n2 (w) =

(w) =

1+ _X(w)*

Here the relation between the complex index of refraction

n = n' +

in" and the susceptibility

x

becomes simpler in a significant way, if, for example in optically thin (dilute) matter like a gas, the polarization is very low,

rx(c4.))1 < 1:

n'

o

1 + x'/2

and

n" x 11 /2,

or

N

A 3

i+ 2Sh

n 1 + (6.17)

— V 870 1 + (26/-0 2 '

Thus, by measuring the macroscopic refractive index, the microscopic properties of the dielectric requiring theoretical treatment by quantum mechanics can be deter- mined. Using

(N/V)A 3 /(87E2 ) > 0.1,

we can also estimate the density of particles

2The methods of femtosecond spectroscopy developed in the 1990s now also allow us to study such fast relaxation phenomena with excellent time resolution.

where we ultimately leave the limiting case of optically thin media. For optical wave- lengths (A 0.5 Jim), this transition occurs already at the relatively low density of

N IV =

2 x 10 14 cm-3 , which at room temperature for an ideal gas corresponds to a vacuum pressure of only 10 -2 mbar.

The solution for a planar wave according to Eq. (6.16) is then E(r,t) = E 0 e -1( w 1- n/k. r) e - nuk •r.

Propagation not only takes place with a modified phase velocity vo, =

c I n'

but also is exponentially damped according to Beer's law in the

z

direction with absorption coefficient a = 2n"kz,

/(z) = 1(0) e- 2n n k z z - (0) e-az. (6.18)

We have chosen n", x" > 0 for normal dielectrics according to Eq. (6.17); as we will see, in a 'laser medium' one can create n", x" < 0 as well, realizing amplification of an optical wave.

Let us briefly study the question of whether a single microscopic dipole can generate a refractive index, i.e. whether it could cause noticeable absorption or dispersion of an optical wave. For this consideration we again rewrite the absorption coefficient as

a

= 2nll k = N

3A 2 1

N aQ

(6 19)

V 2n 1 + (26/7) 2 V 1 + (26/7) 2 .

Therefore, the effect of a single atom is determined by a resonant cross-section of

cfQ = 3A 2 /27t at

6 = 0,

(6.20)

which is much larger than the atom itself. If we succeed in limiting a single atom to a volume with this wavelength as diameter (V A 3 ), then a laser beam focused on this volume will experience strong absorption. Such an experiment has in fact been carried out with a stored ion [110]. Dispersion is observed for nonzero detuning only, but for small values

6 =

+7/2 a single atom is predicted to cause a measurable phase shift

64) =

+1/(8n) as well.

Absorption and dispersion in optically thin media

Sometimes it is useful to consider directly the effect of polarization on the amplitude of an electromagnetic wave propagating in a dielectric medium. For this, we take the one-dimensional form of the wave equation (6.15),

E(z) e —i(ciit—kz) 1 82 P(z) ^{k z)}

) Co C2 8t 2

we fix the frequency w =

ck

for 1E(z, t)1 =

e(z) e - i(wt - kz)

and additionally we as- sume that the amplitude changes only slowly (on the scale of a wavelength) during

(

02

¹⁰²

Z

2 C2 8t 2

propagation. Thus:

z 2 «k 0z

6.1

Classical

radiation interaction 173 Then with 02/0z2

re

( z ) e zkz} ,_.,_, eikz

[2ikalaZ -

k 2 je (z), the wave equation is approxi- mately

[2ik

a

²

—aZ — k2 + —2 w:1

w e(z) = —

EpC2 P(z), which with k = 4.4 )Ic further simplifies to

a ik

e (z) = 2c0 P (z). (6.21)

Now we consider the electromagnetic wave with a real amplitude and phase, E(z) = A(z) e i'D( z ) , and calculate

dr ^(z)dA

^{. A2}

^A)

E(z) dz =

A + dz ï dz =

4k

de (z)

dA . A2 dil.

ik

*

2E0 P(z)e (z).

e* (z) dz = A dz i dz =

From this we can determine the change of the intensity

1(z)

= ^-21-ccoA 2 of an electro- magnetic wave while propagating within a polarized medium according to

-I(z) = —co

2 3m{e(z)P* (z)}

dz

and the phase shift according to

d w

cTz := 2I(z) Nele

The absorption coefficient a and the real part of the refractive index n' can be calcu- lated in an obvious way from

1 ^{dI (z)}

1(z)

^dz

1

d(z)

k dz

We naturally reproduce the results from the section on the linear refractive index, if we assume the linear relation according to Eq. (6.12). The form developed here also allows us to investigate nonlinear relations, and will be useful in the chapter on nonlinear optics (Section 12).

Dense dielectric media and near fields

Certainly, in a dilute, optically thin medium, we do not make a big mistake by ne- glecting the field additionally generated in the sample by polarization. But this is no longer the case in the liquid or solid states. In order to determine the 'local field' of the sample, we cut out a fictitious sphere with a diameter datom < dsph < A with

'frozen' polarization from the material (Fig. 6.4).

To determine the microscopic local field Eh), at the position of a particle, we de- compose it into various contributions, El., --= Eext +Esurf +ELor +Enear , which depend

a 1 , n — i

= w

2I(z)

=

21(z) c

Neg.

(z)P * (

z)} .

(6.22)

Fig. 6.4: Contributions to the lo- cal electric field in an optically dense medium. For a transverse wave the con- tribution of the surface vanishes in the case of normal incidence.

on the different geometries and structures of the sample and are in total called the 'depolarizing field' since they usually weaken the external field

Eext :

Esurf

The field of the surface charges generated by the surface charge density

psurf — n P(rsurf)•

It vanishes for a wave at normal incidence.

ELor

The field of the surface of a fictitious hollow sphere cut out from the volume (also known as the 'Lorentz

field').

For homogeneous po- larization, one finds EL,

= P/3c o .

Enear

The field of the electric charges within the sphere In the case of isotropic media, this contribution vanishes,

Enear =

O.

From

P = foXEloc = c o x(E +

Pk) ), we then obtain by insertion of

Ei0 = ELor = P/3

the macroscopic volume susceptibility

xv

of an isotropic and linear but dense material,

1

Pi _X

Xii 4.4)) = v

co Ei 1 —

x/3

From this by rearrangement can be obtained the

Clausius-Mossotti equation,

which describes the influence of the depolarizing field on the refractive index (density

Ar =