Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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B.4 Light and its wave properties 363

understand light to be electromagnetic radiation in the range from ultraviolet to infrared

waves, with corresponding wavelengths from 0.01 µm to 100 µm, although these bound-

aries are not exact, whereas the light visible to human beings has much narrower bound-

aries and lies in the interval of wavelengths from 0.35 µmto0.67 µm. It is quite obvious

that all of the properties of electromagnetic waves, such as interference, polarization,

diffraction, and dispersion, apply to light. Let us brieﬂy discuss interference and the last

two phenomena with a speciﬁc focus on light.

B.4.1 Interference

During the simultaneous propagation of several waves in a medium, overlapping of the

waves occurs. From the linearity of Maxwell’s equations with respect to the ﬁeld vectors

E, D, B,andH, and the lack of dependence of the material constants , µ,andσ on

these ﬁelds, it follows that the wave processes described by Maxwell’s equations satisfy

the superposition principle. This means that the oscillations of the ﬁeld vectors at each

point are the vector sum of oscillations of the ﬁeld vectors of the individual waves, which

propagate separately. Media for which the superposition principle is valid are called

linear media. A linear medium represents an idealized model. We can use this model

for the description of waves propagating through a real medium only if the amplitudes

of propagating waves, E

, are much smaller than the internal electric ﬁeld, E

int

, acting

on the valence electrons of the constituent atoms. An estimate of the magnitude of this

ﬁeld is

int

∼

∼ 10

−1

, (B.147)

where a is the radius of the valence electron’s orbit (∼10

−10

m). Such magnitudes

are much larger than most wave ﬁelds. For example, the ﬁeld intensity in the beam

of a continuous-wave laser with power of about 1 W and beam cross-sectional area

1mm

attains magnitudes of E

∼ 10

−1

, which is many orders of magnitude less

than the internal ﬁeld intensity of Eq. (B.147). Thus, the medium for such a relatively

powerful laser beam is linear. Vacuum is always a linear medium for electromagnetic

ﬁelds of any intensity. Its dielectric and magnetic permittivities, 

vac

and µ

vac

,depend

neither on the magnitude of the ﬁeld nor on its frequency: 

vac

= µ

vac

= 1, and the

conductivity σ = 0.

Let us consider the superposition of two plane harmonic waves of the same polar-

ization, propagating in a non-absorbing medium along two arbitrary directions. At any

given point the magnitudes E

and H

of these two waves undergo harmonic oscilla-

tions. The vectors of the electric and magnetic ﬁelds of these oscillations at a point,

r,are

= e

cos(ω

t − 

) =



i(ω

t−

)

+ e

−i(ω

t−

)



, (B.148)

= e

γ A

cos(ω

t − 

) =

γ A



i(ω

t−

)

+ e

−i(ω

t−

)



, (B.149)

364 Appendix B. Electromagnetic ﬁelds and waves

where j = 1, 2; 

= k

· r;andγ =





/(µµ

). The wavenumber, k

,ofeachwave

is related to the frequency, ω

, and wavelength, λ

,by

2π

, j = 1, 2, (B.150)

where the speed of both waves is deﬁned by Eq. (B.55)(v = c/

√

µ). The sums of the

electric and magnetic ﬁelds of both waves will depend only on time, t, and will change

from point to point, i.e.,

E(r, t) = E

(r, t) +E

(r, t), (B.151)

H(r, t) = H

(r, t) +H

(r, t). (B.152)

Let us calculate the total intensity I =





. Using the deﬁnitions (B.74)and(B.69), we

obtain

I =







+ A

+ 2 A



cos

[

(ω

− ω

)t − (

− 

)

]



, (B.153)

where the period, T , of total oscillations is deﬁned in terms of the periods of individual

oscillations, T

,as

T =

2π

− ω

− T

. (B.154)

If the frequencies, ω

and ω

, of the added waves are different, then the integral in

Eq. (B.153) becomes equal to zero, and the intensity of the superposition of the two wave

ﬁelds is equal to the sum of the intensities of the individual waves. If the two individual

oscillations are not correlated with each other and the total phase difference of oscillations

of added wave ﬁelds

 = (ω

− ω

)t + (

− 

) (B.155)

changes with time, then such oscillations are called incoherent. If the phase difference

does not change with time (this occurs when the frequencies of the two waves are

the same), then such oscillations (and, correspondingly, waves) are called coherent.In

the case of incoherent waves their superposition does not produce a so-called interference

pattern, i.e., an increase of the intensity at some points and a decrease at other points.

The intensity of the total ﬁeld at a particular point for coherent waves (ω

= ω

), taking

into account Eq. (B.153), is given by the expression

I(r, t) =



+ A

+ 2A

cos(

− 

)



. (B.156)

Since the intensities of the individual waves are equal to

, (B.157)

B.4 Light and its wave properties 365

Eq. (B.156) can be rewritten as

I = I

+ I

+ 2



cos , (B.158)

where we introduced the phase difference of the two waves,

 = 

− 

. (B.159)

The last term in Eq. (B.158),

int

= 2



cos , (B.160)

is called the interference term. The intensity in the interference pattern greatly depends

on the magnitude of the interference term.

At the points of space where cos >0 the total intensity is I > I

+ I

.Atthe

points where cos <0 the total intensity I < I

+ I

. As a result of the superposition

of coherent waves the intensity of waves is redistributed in space – at some points the

intensity increases and at others it decreases. This phenomenon is called the interference

of waves. At the points of space where  = 2mπ and cos  = 1 the intensity is maximal

(these are the so-called points of constructive interference),

max







, (B.161)

and at the points where  = (2m − 1)π and cos  =−1, the intensity is minimal (these

are the so-called points of destructive interference),

min





−





. (B.162)

The interference effect becomes especially apparent when the amplitudes of the interfering

waves are equal to each other, I

= I

. In this case, at the points with maximal intensity

max

= 4I , and at the points with minimal intensity I

min

= 0. For incoherent waves in

this case the intensity is the same everywhere, and, according to Eq. (B.158), is given by

I = 2I

Interference is a natural phenomenon in which wave properties become strikingly

apparent. Interference is characteristic for waves of any nature and it is relatively easily

observed for acoustic (sound) waves or for waves on the surface of water. Interference

of electromagnetic waves, in particular light waves, can be observed only under certain

conditions. Conventional sources (i.e., not laser sources) do not radiate monochromatic

light waves. Non-laser sources of light have a wide spectrum of radiation. Light waves

that originate from such sources are not coherent and do not create a stable interference

pattern. The simplest way to obtain two coherent waves is the following. A wave from

one source (with spectral width λ and mean wavelength λ) is divided into two waves,

which then are brought together at a screen. The two waves formed after the division can

be considered as waves coming from two individual points or linear sources.

366 Appendix B. Electromagnetic ﬁelds and waves

Screen

m = 0

Distant linear light source

(a)

(b)

Wavelets

Distant linear light source

m = 0

−1

Figure B.9 Young’s

double-slit experiment. (a)

White stripes at the

screen correspond to

maxima of intensity and

black stripes to minima.

(b) Plane waves from the

distant linear source (e.g.,

the Sun) are transformed

into concentric wavelets,

which originate from the

slits S

,andS

There are many other ways to bring about such division of one wave into two. In

particular, it can be achieved by using two narrow slits, S

and S

, separated by a small

distance, d, which are illuminated by a distant linear source (e.g., the Sun). This method is

called Young’s scheme. The wavelets coming from the slits overlap, establishing a system

of alternating maxima and minima at the screen (see Fig. B.9). Let us introduce the so-

called path-length difference, L, of two waves propagating from corresponding slits to the

point SP at the screen:

L = r

−r

. (B.163)

If the path-length difference, L, contains an integer number of wavelengths (in the medium

being considered)

L =

mλ

√

µ

= mλ, (B.164)

then the oscillations of the ﬁeld vectors of these waves at the given point are in phase,

which results in interference maxima. Note that, if the path-length difference, L,ofthese

waves propagating from the source to the point of the screen SP is not greater than a

characteristic length l

coh

= λ

/λ, which is called the coherence length, then the random

changes of the amplitude and phase of the ﬁelds of the two waves happen coherently, and

a stable interference pattern is established. The points where the path-length difference is

equal to an integer multiple of half-wavelengths correspond to the interference minima.

The formation of a standing wave by two monochromatic waves with the same fre-

quencies and amplitudes, which are propagating in opposite directions, is an important

particular case of interference. For a wave propagating in the positive direction of the

y-axis the ﬁeld intensities of the electric, E

, and magnetic, H

, ﬁelds are deﬁned by the

B.4 Light and its wave properties 367

expressions

= e

cos(ωt − ky), (B.165)

= e

cos(ωt − ky). (B.166)

The equations for the wave propagating in the opposite direction can be written as

−

= e

cos(ωt + ky + ϕ), (B.167)

−

=−e

cos(ωt + ky + ϕ), (B.168)

where ϕ is the phase shift between the two waves traveling in opposite directions. In

writing the expression for H

−

it is necessary to take into account that the vectors E

−

,andk

−

form a right-handed system. Therefore, the polarization of the ﬁeld H

−

must

be directed opposite to the x -axis. As a result of overlapping of ﬁelds with opposite

directions we obtain a wave whose electric and magnetic ﬁelds are given by the following

expressions:

E = E

+ E

−

= 2e

cos



ky +



cos



ωt +



, (B.169)

H = H

+ H

−

= 2e

sin



ky +



sin



ωt +



. (B.170)

Each of these equations represents a so-called standing wave, because for such a wave,

in contrast to the case of a traveling wave, there is no displacement of the wavefront in

space. Equations (B.169)and(B.170) describe the oscillations of the ﬁeld intensities with

the frequency ω and amplitudes, A

and A

, for electric and magnetic ﬁelds that change

from point to point according to the relations

= 2E



cos



ky +





, (B.171)

= 2H



sin



ky +





. (B.172)

For each ﬁeld separately we can indicate the points where the amplitude is maximal (anti-

nodes) or equal to zero (nodes). For the electric ﬁeld the anti-nodes are located at the

points with coordinates

anti-node

mπ − ϕ/2

, (B.173)

and the nodes are located at the points with coordinates

node



m +



π −ϕ/2

, (B.174)

where m is an arbitrary integer number. For the magnetic ﬁeld all anti-nodes and nodes

are shifted along the y-coordinate by y = π/(2k). Thus, at the points where the electric

368 Appendix B. Electromagnetic ﬁelds and waves

ﬁeld has anti-nodes, the magnetic ﬁeld has nodes, and vice versa. The distance between

adjacent nodes and anti-nodes for each of the ﬁelds is equal to the quarter wavelength

of the traveling wave: y

n+1,n

= π/k = λ/4. Between two adjacent nodes of the ﬁeld it

oscillates in phase. After crossing the node the phase of the ﬁeld oscillations changes by

π, i.e., the symmetric ﬁeld oscillations on the two sides of the node are in anti-phase.

The magnitude of the phase difference, ϕ, of the waves propagating in opposite directions

depends on the conditions of their formation. Control of this magnitude allows us to shift

all the nodes and anti-nodes along the space coordinate.

There is a shift of a quarter of the period between the oscillations of electric and

magnetic ﬁelds not only in space but also in time. If at some instant in time at all points

|E| attains its maximum and H is equal to zero, then after a time t = T/4 at all points

E = 0and|H |attains its maximum. Thus, during the oscillations in the standing wave the

electric ﬁeld transforms into the magnetic ﬁeld and vice versa. The standing wave does not

transfer energy because for each of the ﬁelds the energy is concentrated between the nodes

of the ﬁeld. At the same time, after each quarter of a period, the energy of a standing wave

transforms from electric to magnetic energy, and vice versa. The amplitudes of the ﬁelds

satisfy Eq. (B.72), i.e.,

√



√

µµ

. Such a process of energy transformation in

a standing wave is analogous to the processes in harmonic oscillators, whereby the energy

is transformed from one type into another. For a spring–mass system and the simple

pendulum, the energy of oscillation is transformed from kinetic to potential energy. Thus,

a standing wave can be considered as a wave harmonic oscillator from the point of view

of energy.

B.4.2 Diffraction

We understand diffraction of light to be the deviation of light waves from direct propagation

in the presence of obstacles in their path, wave bending, and penetration into the region

of geometrical shadow. To observe the effect of diffraction, various types of obstacles

that limit the propagation of a part of the wavefront can be placed in the path of light

waves. If we put a screen behind the obstacle, then under certain conditions a diffraction

picture in the form of a pattern of intensity maxima and minima of the light ﬁeld appears

on the screen. The interference of the inﬁnite number of coherent spherical wave pulses

(wavelets) emitted by every point of the wavefront is the physical origin of diffraction.

One of the most important examples of electromagnetic wave diffraction is the diffrac-

tion of X-rays on crystalline lattices, by means of which the geometries of many crystalline

structures are investigated. For observation of X-ray diffraction it is necessary to have

the period of the diffraction grating comparable to the wavelength of the X-rays. Natural

diffraction gratings for such radiation are crystalline lattices with period about 0.1−1 nm.

If a monochromatic plane wave is incident on a crystalline lattice with basis vectors a

, a

and a

, at angles α

, β

,andγ

with respect to the crystallographic axes, then the directions

of the diffraction maxima, deﬁned by the angles α, β,andγ , satisfy the so-called Laue

equations:

(cos α − cos α

) = m

λ,

(cos β −cos β

) = m

λ, (B.175)

(cos γ − cos γ

) = m

λ,

B.4 Light and its wave properties 369

L/2

Sites of crystalline lattice

Parallel planes

L/2

Figure B.10 Diffraction of

X-rays from crystalline

planes.

where m

1,2,3

are integer numbers that deﬁne the order of the interference maximum.

The conditions (B.175) can be formulated in a different way. The diffraction in a

three-dimensional crystalline structure can be considered as a result of mirror reﬂections

from a system of parallel planes, on which the sites of the crystalline lattice are located

(see Fig. B.10). The wavelets reﬂected from the different atomic planes are coherent and

may interfere with each other. The path-length difference of two waves that are mirror

reﬂected from neighboring parallel planes, as in Fig. B.10, is equal to

L = 2d sin θ, (B.176)

where d is the distance between two adjacent planes and θ is the angle between the

incident ray and the plane. The directions for which the diffraction maxima are developed

are deﬁned by Bragg’s law:

2d sin θ

= mλ, (B.177)

where m = 1, 2, 3,...We can plot many sets of crystalline planes with different distances

between two adjacent planes. Each set of crystalline planes gives its own diffraction

maxima in the directions deﬁned by Eq. (B.177). However, the diffraction maxima that

occur near the planes with the highest density of atoms have the largest magnitudes of

intensity.

B.4.3 Dispersion

The dependence of the propagation speed, v, of a monochromatic light wave in a medium

on its frequency, ω, is called dispersion. Since the speed of the electromagnetic wave in a

medium, according to Eq. (B.55), is deﬁned by the expression

v(ω) =



µ(ω)(ω)

, (B.178)

the dispersion is actually deﬁned by the dependence of the medium’s material parameters,

µ and , on the frequency, ω. All substances are to some extent dispersive, but the

frequency dependence of the dielectric constant, , and magnetic permeability, µ,for

370 Appendix B. Electromagnetic ﬁelds and waves

x(t)

(a)

(b)

q q

Figure B.11 (a) The

polarization of an atom.

(b) The dipole moment d.

most media becomes apparent only in relatively narrow regions of the spectrum. For

the dielectric constant, , such a dispersive region usually lies in the visible and infrared

regions. In many weakly magnetic materials the magnetic permeability practically does not

depend on frequency and is close to unity throughout the entire frequency range. Only in

strongly magnetic materials (such as ferromagnetic materials and ferrites) is the frequency

dependence, µ(ω), substantial in the ultra-high-frequency region of the spectrum (ω ∼

−10

−1

). Therefore, let us concentrate on the analysis of the frequency dependence

of the dielectric constant, (ω), which can be considered qualitatively in the framework

of classical concepts of the behavior of materials in electromagnetic ﬁelds.

Let us consider an isotropic dielectric medium, which consists of non-interacting

atoms. Let us suppose that the atoms of the medium do not have a dipole moment in the

absence of an external electric ﬁeld, i.e., the centers of positive charges of nuclei coincide

with the centers of negative charges of electrons. In the presence of an electric ﬁeld

the centers of valence electrons’ orbits shift with respect to the nucleus, atoms acquire a

dipole moment oriented along the direction of the electric ﬁeld, and the medium in general

acquires polarization (see Fig. B.11(a)). The dipole moment, d, is deﬁned as the product of

two physical magnitudes: the charge q =|q

−

|=q

and the distance, L, between charges

−

and q

. The dipole moment, d, is a vector, which is directed from the negative charge

−

to the positive charge q

(see Fig. B.11(b)):

d = qL . (B.179)

Considering this, let us take into account that in the electric ﬁeld of the electromagnetic

wave the center of the valence electron’s orbit, coupled with the atom, undergoes an

oscillatory motion. The motion of the electron can be described by Newton’s second law:

=−kx −b

− eE(t). (B.180)

B.4 Light and its wave properties 371

Here, the ﬁrst term on the right-hand side of Eq. (B.180) deﬁnes the restoring force exerted

on the electron by an atom, whereas the second term is the friction force. The third term

describes the periodic driving force of the wave’s electric ﬁeld, E(t):

E(t) = E

iωt

= E

iωt

. (B.181)

The steady-state solution of Eq. (B.180) has the form

x(t) = e

x(t), (B.182)

with

x(t) = x

iωt

. (B.183)

Here we introduced the complex amplitude of the electron’s oscillations, x

− ω

+ 2iβω



− ω



+ 4β

(B.184)

and the standard parameters ω

= k/m

and 2β = b/m

. Finally, the dipole moment, d,

that the atom acquires is deﬁned as

d =−ex(t) =−ex(t)e

=−e

− ω

+ 2iβω



− ω



+ 4β

iωt

=−

− ω

+ 2iβω



− ω



+ 4β

E(t). (B.185)

Then, the dipole moment for unit volume of a medium, which is called the polarization

vector, P,isdeﬁnedas

P = nd =−n

− ω

+ 2iβω



− ω



+ 4β

E(t). (B.186)

Here, e is the charge of the electron and n is the number of atoms in the unit volume. The

electric ﬂux density, D, in the medium consists of the ﬂux density of the external electric

ﬁeld, 

E, and the internal ﬂux density, which is equal to the polarization vector, P:

D = 

E + P. (B.187)

For the so-called linear media the polarization vector, P, is linearly proportional to the

external electric ﬁeld, E:

P = κ

E, (B.188)

where we introduced the so-called electric susceptibility, κ, of the material. Thus,

D = 

E + κ

E = 

E, (B.189)

372 Appendix B. Electromagnetic ﬁelds and waves

Figure B.12 The

dependences of 



and 



on the frequency, ω.

where

 = 1 + κ. (B.190)

Finally, the polarization vector, P, can also be presented in the form

P = ( − 1)

E. (B.191)

By comparing Eqs. (B.186)and(B.191), we ﬁnd the complex dielectric constant, :

 = 1 −



− ω

+ 2iβω



− ω



+ 4β

. (B.192)

Now, we can ﬁnd the expressions for the real and imaginary parts of the complex dielectric

constant,  = 



− i





= 1 −



− ω



− ω



+ 4β

, (B.193)







2βω



− ω



+ 4β

. (B.194)

The phase velocity of the wave in the medium, v

= c/



µ



, is directly related to

the real part of the dielectric constant, 



. The damping of a wave in a medium, which

according to Eqs. (B.104)and(B.105) is proportional to 



, is related to the imaginary

part of the dielectric constant, 



Let us analyze the obtained frequency dependences of the magnitudes 



and 



,which

are shown in Fig. B.12. The entire frequency interval of change of 



and 



can be divided