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

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B.3 Reﬂection of a plane wave from an interface 353

On substituting these magnitudes into Eq. (B.69) and averaging over time, we obtain







(E + E

∗

) × (H + H

∗

)





E × H + E

∗

× H

∗

+ E

∗

× H + E ×H

∗



(B.87)

Taking into account that the dependence on time of the ﬁelds from Eq. (B.87) is propor-

tional to e

±iωt

, we obtain for two terms



E × H





∗

× H

∗



= 0. (B.88)

Let us prove that the ﬁrst term is equal to zero (for the second term the calculations are

similar):



E × H



= E

× H

−2ik·r



2iωt

dt =

× H

−2ik·r

2iω



2iωT

− 1



= 0,

(B.89)

since ω = 2π/T and

2iπ

= cos(2π ) +isin(2π) = 1. (B.90)

As a result, we obtain for the average density of ﬂux







∗

× H + E ×H

∗



. (B.91)

We have to leave only the real parts in the last equation. Let us show that the real parts

of both terms in Eq. (B.91) are the same. For this purpose, let us present the complex-

conjugate magnitudes we have,

∗

= E

−i(ωt−k·r)

∗

= H

−i(ωt−k·r)

(B.92)

The results of multiplication of the ﬁelds from Eqs. (B.86)and(B.92) show that both

products in Eq. (B.91) are real and that they are the same. Thus, the average density of

ﬂux can be written as follows:





Re(E ×H

∗

). (B.93)

B.3 Reﬂection of a plane wave from the interface

between two media

B.3.1 Plane waves in absorbing media

We have considered so far waves that are not damped while propagating. This is possible

only in vacuum or in a non-absorbing medium. All real media absorb waves to some extent.

354 Appendix B. Electromagnetic ﬁelds and waves

The magnitude of such absorption greatly depends on the frequency of the propagating

wave. For some regions of the electromagnetic spectrum this absorption can be so weak

that we can neglect it, but for other regions of the spectrum the absorption can be very

signiﬁcant. Let us consider the peculiarities of the propagation of plane waves in an

absorbing medium.

Absorption in the medium has to be taken into account if we are dealing with a medium

that has a conductivity, σ , that is not equal to zero. Since the current density, j,ofthe

free charges is related to the intensity of the electric ﬁeld, E, by Ohm’s law (in differential

form) as

j = σE, (B.94)

Maxwell’s equation (B.16) that contains j can be rewritten in the following form:

∇ × H = 

∂E

∂t

+ σ E. (B.95)

Using Maxwell’s equation (B.17),

∇ × E =−µµ

∂H

∂t

, (B.96)

we can obtain the wave equation for the case considered here (in the same way as we

obtained Eq. (B.52)):

∇

E =

µ

∂

∂t

+ µµ

∂E

∂t

. (B.97)

The corresponding wave equation can be obtained for the intensity of the magnetic

ﬁeld, H:

∇

H =

µ

∂

∂t

+ µµ

∂H

∂t

. (B.98)

These wave equations, in contrast to Eqs. (B.52)and(B.53), contain terms with the time

derivative in the ﬁrst order, which are responsible for the damping of the wave in a medium.

We will seek solutions of Eqs. (B.97)and(B.98) in the form of monochromatic plane

waves, which for convenience can be presented in the exponential form,

E(t, r) = E

i(ωt−k·r)

, (B.99)

H(t, r) = H

i(ωt−k·r)

, (B.100)

where E

and H

are the amplitudes of the electric and magnetic ﬁelds of the wave. This

form of presentation of wave equations is generally used in electromagnetism, where

the ﬁelds E and H are deﬁned as complex functions, while the real wave processes are

described by the real parts of Eqs. (B.99)and(B.100).

B.3 Reﬂection of a plane wave from an interface 355

By substitution of Eq. (B.99) into Eq. (B.97), we arrive at the equation



−



 − i





E = 0, (B.101)

from which we obtain the dispersion relation that connects the wavenumber, k, with the

wave frequency, ω, and the medium parameters:

= k



 − i



or k = k





 − i



. (B.102)

Here, we introduced the wavenumber, k

= ω/c, for the wave propagating in vacuum.

From the expressions obtained it follows that the wavenumber, k, for a wave propagating

in a conducting medium is a complex number, i.e.,

k = k



− ik



. (B.103)

The sign before the imaginary part of the wavenumber is chosen to be negative to make

the wave that propagates in the positive direction in an absorbing medium damped instead

of amplifying it. Let us substitute Eq. (B.103) into Eq. (B.102), and let us separate the real

and imaginary parts. As a result we obtain two equations from which to ﬁnd k



and k





)

− (k



)

= k

µ, (B.104)





= k

µσ



. (B.105)

By solving the system of (B.104)and(B.105), we obtain the following expressions for

the real, k



, and the complex, k



, parts of the complex wavenumber, k:



= k



µ





1 +



+ 1



, (B.106)



= k



µ





1 +



− 1



. (B.107)

The above relations for the real, k



, and imaginary, k



, parts of the wavenumber, k,are

written in the general form and are valid for weakly absorbing media as well as for strongly

absorbing media over a wide frequency range (from ω = 0toω = 10

Hz, where quantum

effects become signiﬁcant). Depending on the media and on the frequency ranges, the

magnitude of σ/(

ω) can be either large or small, which can be used for simpliﬁcation

of the above expressions.

356 Appendix B. Electromagnetic ﬁelds and waves

B.3.2 The skin effect

Let us assume that a plane wave, whose electric and magnetic ﬁelds are described by

Eqs. (B.99)and(B.100), is propagating in the y-direction. Then, the partial derivatives

with respect to x and z are equal to zero and in Eqs. (B.97)and(B.98) the operator ∇

simpliﬁed to the form

∇

∂

∂y

. (B.108)

Taking into account that the wavenumber of Eq. (B.103) is a complex variable, the

solutions (B.99)and(B.100) can be rewritten as

E(t, y) = E

−k



i(ωt−k



, (B.109)

H(t, y) = H

−k



i(ωt−k



, (B.110)

where E

and H

are the amplitudes of the wave ﬁelds in the plane y = 0. It follows from

Eqs. (B.109)and(B.110) that the amplitudes of wave ﬁelds decrease exponentially in the

direction of wave propagation (the y-direction):

E(y) = E

−k



, (B.111)

H(y) = H

−k



. (B.112)

The imaginary part of the wavenumber, k



, deﬁnes the rate of the decrease in amplitude

along the direction of wave propagation. Its reciprocal, δ, which is called the skin depth,

δ =



, (B.113)

deﬁnes the penetration depth of the ﬁeld in the absorbing medium. At the distance δ the

amplitude of the wave has decreased by a factor of e:

E(δ) = E

−k



= E

−1

. (B.114)

Figure B.5 shows the distribution of the electric ﬁeld of the wave in a medium with

damping for two time instants. If a wave with an amplitude of E

penetrates from vacuum

into a medium with non-zero conductivity, then it practically dampens at a depth of several

δ. Since the intensity of the wave I is proportional to E

, at the distance y = δ the intensity

of the wave will have decreased by a factor of I

/I = e

∼

7.4, and at the distance y = 2δ

the intensity will have decreased by a factor of I

/I = e

∼

55. We can conclude that the

energy of the transmitted wave in an absorbing medium is located in a layer of depth several

times δ. For media with high conductivity the depth of such a layer can be relatively small

(for example, for copper a wave of frequency f = ω/(2π) = 1 MHz has a penetration

depth of δ ≈ 0.07 mm). This is why the near-surface region of depth δ, where most of the

penetrated electromagnetic wave is concentrated, is called the skin layer, the magnitude δ

is called the skin depth, and the effect itself is called the skin effect. Let us consider two

B.3 Reﬂection of a plane wave from an interface 357

Figure B.5 The electric

ﬁeld of an

electromagnetic wave in

an absorbing medium at

two instants of time, t and

t + t. The magnitude of

the skin depth, δ,is

indicated.

limiting cases for Eqs. (B.106)and(B.107) for low and high conductivities.

(a) Low conductivity (or the high-frequency limit), σ  

ω. In this case

δ =



2









, (B.115)

i.e., the depth of wave penetration into the medium does not depend on its frequency.

With decreasing σ the wave ﬁeld penetrates further into the medium. In the case

of σ → 0 the medium becomes non-conducting and the penetration depth for the

electromagnetic ﬁeld tends to inﬁnity, δ →∞.

(b) High conductivity (or the low-frequency limit), σ  

ω.Inthiscasetheskin

depth, δ,isdeﬁnedas

δ = c



2

µσ ω



µσ ω

, (B.116)

i.e., with increasing conductivity, σ , and frequency, ω, the penetration depth

decreases. At ω → 0 the depth of penetration into the medium tends to inﬁnity,

δ →∞. This is a feature of static ﬁelds, both electric and magnetic.

B.3.3 Solution of the boundary problem

Real wave-guiding structures always have ﬁnite dimensions. Therefore, there is partial

reﬂection from boundary surfaces and transmission into the neighboring medium during

wave propagation in such structures. Depending on the type of material, the geometry

of the reﬂecting surfaces, the structure of the incident wave, and the angle of incidence,

the problems of ﬁnding ﬁelds of reﬂected and transmitted waves seem to be problems

of different levels of complexity. Let us consider the simplest problem of the reﬂection

and transmission of a monochromatic plane wave through a plane interface at normal

incidence.

358 Appendix B. Electromagnetic ﬁelds and waves

medium 2

medium 1

Figure B.6 The interface

between two semi-inﬁnite

homogeneous media:

medium 1 with 

, µ

,and

; and medium 2 with 

,andσ

Let the interface between two semi-inﬁnite homogeneous media coincide with the

plane y = 0 (see Fig. B.6). The medium located to the left of the interface (y < 0) is

characterized by the parameters 

, µ

,andσ

, whereas the medium located to the right

of the interface (y > 0) is characterized by the parameters 

, µ

,andσ

.Aplanewave

of frequency ω and wavevector k

= e

is normally incident (along the y-axis). The

reﬂected wave propagates along the negative y-axis, and the wave transmitted into the

second medium propagates in the positive direction of the y-axis (see Fig. B.6).

To ﬁnd the amplitude of the reﬂected and transmitted waves, let us solve the boundary

problem, which includes sewing ﬁelds at the interface using the boundary conditions

(B.26)–(B.29). In writing the expressions for the ﬁelds we will drop the time factor e

iωt

because it is present in each expression for the ﬁelds of each wave.

The ﬁeld of the incident wave can be written in the following form:

(y) = e

−ik

, (B.117)

(y) = e

−ik

, (B.118)

where e

and e

are the unit vectors of the corresponding coordinate axes, and wave-

number k

is deﬁned by Eqs. (B.106)and(B.107). The relation between the amplitudes

of these waves is given by the expression

= Z

, (B.119)

where the wave resistance Z

of the ﬁrst medium is





ω − iσ

. (B.120)

B.3 Reﬂection of a plane wave from an interface 359

The ﬁelds of the reﬂected wave propagating in the opposite direction have to be written

in the following form:

(y) = e

, (B.121)

(y) =−e

, (B.122)

where E

= Z

, and the negative sign appears in the expression for H

(y) because E

,andk

constitute a right-handed system.

For the wave transmitted across the interface we can write

(y) = e

−ik

, (B.123)

(y) = e

−ik

, (B.124)

where E

= Z

; Z

is deﬁned analogously to Z





ω − iσ

. (B.125)

At the interface, which does not have any charges and any currents along it, according

to Eqs. (B.26)–(B.29) the tangential components of the ﬁeld intensities of the electric and

magnetic ﬁelds must be continuous. On the left-hand side of the interface (y < 0) the ﬁeld

is presented as a sum of the incident and reﬂected waves, and on the right-hand side of

the interface (y > 0) it is given solely by the transmitted wave. So at the interface (y = 0)

we obtain the following system of equations:

(0) + E

(0) = E

(0), (B.126)

(0) + H

(0) = H

(0). (B.127)

Let us substitute into this system of equations the expressions for the ﬁelds of the incident,

reﬂected, and transmitted waves, and let us drop the common factors. As a result, we will

obtain the following system of equations for the amplitudes of the corresponding ﬁelds:

+ E

= E

, (B.128)

− H

= H

. (B.129)

On proceeding in the second equation with the amplitudes of the electric ﬁelds, we obtain

− E

. (B.130)

By solving this equation self-consistently with Eq. (B.128) we obtain

− Z

+ Z

, (B.131)

360 Appendix B. Electromagnetic ﬁelds and waves

+ Z

. (B.132)

From these relations we can ﬁnd the coefﬁcients of reﬂection, R, and transmission, T :

R =

− Z

+ Z

=|R|e

iϕ

, (B.133)

T =

+ Z

=|T |e

iϕ

. (B.134)

In the general case of absorbing media the coefﬁcients R and T are complex variables.

This means that the phases of reﬂected and transmitted waves do not coincide with the

phase of the incident wave. The corresponding phase shifts ϕ

and ϕ

can be found from

Eqs. (B.133)and(B.134).

B.3.4 Analysis of the reﬂection and transmission coefﬁcients

Let us consider several important special cases for several interfaces.

(1) Let the reﬂection happen from the interface of two transparent non-magnetic

dielectrics, for which µ

= µ

= 1andσ

= σ

= 0. For such media the impedances

are real:





, (B.135)





. (B.136)

Then, the coefﬁcients of reﬂection, R, and transmission, T , we obtain from

Eqs. (B.133)and(B.134),

R =

√



−

√



√



√



, (B.137)

T =

√



√



√



, (B.138)

have real values in this case. There is no phase shift between incident and transmitted

waves (i.e., ϕ

= 0) regardless of the magnitudes of 

and 

. The total electric ﬁeld

in the region y < 0 has the form

(y, t) = e

[

cos(ωt − k

y) + R cos(ωt + k

]

. (B.139)

The phase shift between the incident and the reﬂected wave depends on the relation

between 

and 

.If

<

, there is a phase shift equal to π (ϕ

= π ,seeFig.B.7)

B.3 Reﬂection of a plane wave from an interface 361

2 muidem1 muidem

(a)

(b)

Figure B.7 The incident,

, reﬂected, E

,and

transmitted, E

, waves for

media with (a) 

<

and

(b) 

>

between incident and reﬂected waves. When the wave is incident from the optically

more dense medium to the optically less dense medium (i.e., 

>

) there is no

phase shift between the incident and reﬂected waves (see Fig. B.7).

(2) Let the wave be incident from the non-absorbing medium (σ

= 0) to the interface

with an ideal conductor (σ

→∞). In this case Z

= 0, R =−1, and T = 0, i.e.,

the wave does not penetrate into the second medium and the wavenumber k

is real.

Therefore, a standing wave is formed in the ﬁrst medium, and, as we found from

Eq. (B.139)forR =−1,

(y, t) = 2e

sin(k

y)sin(ωt). (B.140)

The expressions for the magnetic ﬁeld can be obtained analogously:

(y, t) =

cos(k

y)cos(ωt). (B.141)

From this expression we see that the wave formed as a result of the overlapping of

incident and reﬂected waves with phase shift ϕ = π is a standing wave. The magnetic

ﬁeld of this wave has a phase shift equal to π/2 in space as well as in time relative to

the electric ﬁeld.

Example B.5. For real media the material parameters are functions of frequency. Find

the reﬂection coefﬁcient of an electromagnetic wave from a medium that has for a given

frequency the dielectric constant 

(ω) < 0andσ

(ω) = 0. The parameters of the medium

in which the wave is initially propagating are 

(ω) > 0andσ

(ω) = 0. Media 1 and 2

are non-magnetic media with µ

= µ

= 1.

362 Appendix B. Electromagnetic ﬁelds and waves

1010101010101010

, m

f, Hz

Visible range

Figure B.8 The visible

range of the

electromagnetic

spectrum.

Reasoning. According to Eq. (B.120) the expressions for the wave resistance of each of

the media have the forms





, (B.142)





= i





|

. (B.143)

On substituting these expressions into Eq. (B.133), which deﬁnes the reﬂection coefﬁcient,

we ﬁnd

R =

− Z

+ Z

√



−

√

|

√



√

|

√



+ i

√

|

√



− i

√

|

. (B.144)

The expression for the reﬂection coefﬁcient, R, can be rewritten in the form

R =|R|e

iϕ

, (B.145)

where

tan ϕ

√



|



+|

and |R|=1. (B.146)

Therefore, in the case considered here the modulus of the reﬂection coefﬁcient |R|=

1. The non-absorbing medium with 

< 0 completely reﬂects the wave, shifting its

phase by ϕ

. In this case it is said that we observe the effect of “metallic reﬂection.”

Media with (ω) < 0 in the corresponding frequency range are often used in devices

in which complete reﬂection of a wave with frequency ω is needed. This is important

when for any reason you cannot use metallic layers in the device because of their high

conductivity.

B.4 Light and its wave properties

The electromagnetic radiation that we observe in nature occupies a very wide range

of wavelengths: from 10

−15

m to hundreds of kilometers (see Fig. B.8). Nowadays we