Chen C.J. Physics of Solar Energy

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42  Nature of Solar Radiation

equations explaining electromagnetic phenomena, now known as Maxwell’s equations.

Based on those equations, he predicted the existence of electromagnetic waves, propa-

gating in free space with a speed that equals exactly the speed of light, which was then

veriﬁed experimentally by Heinrich Hertz. Maxwell’s bold postulation that light is an

electromagnetic wave has since become one of the cornerstones of physics.

2.1.1 Maxwell’s Equations

In vacuum, or free space, Maxwell’s equations are

∇·E =

, (2.1)

∇·B = 0, (2.2)

∇×E = −

∂B

∂t

, (2.3)

∇×B = ε

∂E

∂t

+ μ

J. (2.4)

Electric current cannot exist in free space. For linear, uniform, isotropic materials, the

current density J is determined by the electric ﬁeld intensity E through Ohm’s law,

J = σE. (2.5)

The names, meanings, and units of the physical quantities in these equations are

listed in Table 2.1. For example, the electric constant has an intuitive meaning as

follows. A capacitor made of two parallel conducting plates with area A and distance d

has a capacitance C = ε

A/d in farads. Similarly, the electric constant has an intuitive

meaning as follows. An inductor made of a long solenoid of N loops with cross-sectional

area A and length l has an inductance L = μ

A/l in henrys.

Table 2.1: Quantities in Maxwell’s Equations

Symbol Name Unit Meaning or Value

E Electric ﬁeld intensity V/m

B Magnetic ﬁeld intensity T (tesla) N/A·m

ρ Electric charge density C/m

J Electric current density A/m

Electric constant F/m 8.85 × 10

−12

F/m

(permittivity of free space)

Magnetic constant H/m 4π × 10

−7

H/m

(permeability of free space)

σ Conductivity (Ω · m)

−1

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2.1 Light as Electromagnetic Waves  43

2.1.2 Vector Potential

To treat the electromagnetic ﬁeld in space, a convenient method is to use the vector

potential. From Eq. 2.2, it is possible to construct a vector ﬁeld A which satisﬁes

B = ∇×A. (2.6)

Then, Eq. 2.2 is automatically satisﬁed. Substituting Eq. 2.6 into Eq. 2.3, one obtains

∇×E = −

∂

∂t

∇×B. (2.7)

For any function φ(r), ∇×[∇φ(r)] = 0, it is possible to set up the vector potential A

such that

E = −

∂A

∂t

−∇φ, (2.8)

where φ is the electrostatic potential arising from the charges. The choice of the vector

potential is not unique. By adding a gradient of an arbitrary function to it, values of

the electric ﬁeld and magnetic ﬁeld do not change. This is called the gauge invariance

of the vector potential. It is possible to deﬁne a vector potential which satisﬁes the

condition

∇·A =0. (2.9)

Equation 2.9 is called the Coulomb gauge, which is the most convenient gauge to treat

nonrelativistic problems of an atomic system and an independent electromagnetic wave.

In fact, using Eq. 2.9 and the ﬁrst Maxwell equation Eq. 2.1, one obtains

∇

φ = −

, (2.10)

which means that the scalar potential is generated by the static charges only. It is

thus convenient for treating the problems of interactions between the radiation ﬁeld

and atomic systems. For details of the gauge problem, see, for example, The Quantum

Theory of Radiation by Walter Heitler [37].

2.1.3 Electromagnetic Waves

In this section, we study the electromagnetic waves in free space, that is, where the

electric charge ρ and current J are zero. Substituting Eqs 2.6 and 2.8 into Eq. 2.4, we

have

∇×∇×A + ε

∂

∂t

=0. (2.11)

Using the identity

∇×∇×A ≡∇(∇·A) −∇

A (2.12)

and Eq. 2.9, Eq. 2.11 becomes

∇

A − ε

∂

∂t

=0. (2.13)

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44  Nature of Solar Radiation

Introducing

c =

√

, (2.14)

Eq. 2.13 becomes

∇

A −

∂

∂t

= 0, (2.15)

which is a wave equation with velocity c. Because of Eqs. 2.6 and 2.8, the electric ﬁeld

intensity and the magnetic ﬁeld intensity also satisfy the same wave equation,

∇

E −

∂

∂t

= 0 (2.16)

and

∇

B −

∂

∂t

=0, (2.17)

Accordingtovaluesofε

and μ

coming from electromagnetic measurements in 1860s,

the velocity of electromagnetic waves should be 3.1 × 10

m/s. On the other hand,

experimental values of the speed of light at that time were 2.98 ×10

–3.15 × 10

m/s.

The diﬀerence was within experimental error. Maxwell proposed thusly [58]:

The agreement of the results seems to show that light and magnetism are

aﬀections of the same substance, and that light is an electromagnetic dis-

turbance propagated through the ﬁeld according to electromagnetic laws.

Maxwell’s theory of electromagnetic waves was experimentally veriﬁed by Heinrich

Hertz in 1865. From recent electrical measurements, one ﬁnds 1/

√

=2.998 ×

m/s, which is exactly the speed of light in a vacuum, c.

2.1.4 Plane Waves

An electromagnetic wave with circular frequency ω in space is deﬁned as

A(x, y, z , t)=A(x , y, z ) e

−iωt

. (2.18)

To study the properties of electromagnetic waves, we consider the case that the

wave propagates in one direction, say z. In this case, the ﬁeld intensities only depend

on z. Equation 2.15 becomes

A = 0. (2.19)

The general solution is

A = A

i(k

z −ωt)

. (2.20)

where A

is a constant, and the z-component of the wavevector k

is deﬁned as

. (2.21)

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2.1 Light as Electromagnetic Waves  45

2.1.5 Polarization of Light

Although in general the vector potential could have x, y, z-components, because of Eq.

2.9, the z-component of the vector potential must be zero,

∇·A =

∂A

∂x

∂A

∂y

∂A

∂z

= ik

=0. (2.22)

This means that A

must be a constant over the entire space. Because we are interested

in electromagnetic waves, or the variation of electromagnetic ﬁelds, we can simply

set A

=0. Thewavesaretransverse. In other words, the intensity vectors are

perpendicular to the direction of propagation.

The direction of the vector potential could be either x or y or a linear combination

of x-andy-components. For the x-component of A,wehave

= A

i(k

z−ωt)

=0,A

=0. (2.23)

The electric ﬁeld intensity, according to Eq. 2.8, is

= iωA

i(k

z−ωt)

=0,E

=0. (2.24)

And the magnetic ﬁeld intensity, according to Eq. 2.6, is

=0,B

= ik

i(k

z−ωt)

=0. (2.25)

Therefore, the only nonvanishing components of the electric ﬁeld intensity and the

magnetic ﬁeld intensity are E

and B

. According to Eq. 2.21, they are in phase and

proportional,

= cB

. (2.26)

Figure 2.2 Electromagnetic wave. The electromagnetic wave is transverse, where the intensity

vectors E and B are perpendicular to the direction of propagation. The electric ﬁeld intensity E is

perpendicular to the magnetic ﬁeld intensity B. The energy ﬂux vector S = μ

−1

E × B is formed from

E and B by a right-hand rule.

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46  Nature of Solar Radiation

In summary, according to the electromagnetic theory of light, the electric ﬁeld inten-

sity vector is perpendicular to the direction of the propagation of light. The magnetic

ﬁeld intensity is perpendicular to both the direction of the electric ﬁeld intensity vector

and the direction of the propagation of light, and its magnitude is proportional to the

electric ﬁeld intensity. See Fig. 2.2.

2.1.6 Motion of an Electron in Electric and Magnetic Fields

In this section, the interaction of the radiation ﬁeld—electric and magnetic ﬁelds vary-

ing with time—with the electrons is studied using classical mechanics as a preparation

for a quantum-mechanical treatment.

The standard method of developing the quantum mechanics of a dynamic system

is ﬁrst to cast the classical equation of motion into Hamiltonian format. The Hamilto-

nian H(p, r) of a dynamic system is a function of its coordinate r and corresponding

momentum p, representing the total energy. For example, for an electron with charge

q movinginanelectricﬁeldwithpotentialφ(r), the Hamiltonian is

H =

+ qφ(r). (2.27)

The equations of motion in the Hamiltonian format are a pair of ﬁrst-order ordinary

diﬀerential equations:

˙p

= −

∂H

∂x

, (2.28)

˙x =

∂H

∂p

. (2.29)

There are similar equations for y and z. Using Eqs 2.29 and 2.27, the expression of

momentum is found to be identical to the usual deﬁnition,

p = m

v = m

r, (2.30)

where a dot means taking a derivative with respect to time t. Applying Eqs. 2.28 and

2.29 to Eq. 2.27, one ﬁnds

r = −q∇φ(r)=qE, (2.31)

which is Newton’s equation of motion, where E is electric ﬁeld intensity.

For the motion of an electron in both electric and magnetic ﬁelds, the standard

method is to insert the vector potential A into the expression of momentum by simply

substituting p with p − qA in the Hamiltonian,

H =

(p − qA)

+ qφ(r), (2.32)

H =



− qA

)

+(p

− qA

)

+(p

− qA

)



+ qφ(x, y, z). (2.33)

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2.2 Optics of Thin Films  47

Applying Eq. 2.29 to Eq. 2.32, one obtains

= m

˙x + qA

(2.34)

and so on. In vector form, it is

p = m

r + qA, (2.35)

which is the deﬁnition of momentum in a magnetic ﬁeld. Applying Eq. 2.28 to Eq. 2.32

and using Eq. 2.35, for the x-component, yield

= q



∂A

∂x

˙x +

∂A

∂x

˙y +

∂A

∂x

˙z



− q

∂φ

∂x

. (2.36)

The familiar Newton equation of motion, similar to Eq. 2.31, can be obtained from

Eqs. 2.33 and 2.36. The x-component is given as

¨x =

− q

. (2.37)

Note that

∂A

∂t

∂A

∂x

˙x +

∂A

∂y

˙y +

∂A

∂z

˙z, (2.38)

for example, for the x-component, one obtains,

¨x = q



˙y



∂A

∂x

−

∂A

∂y



+˙z



∂A

∂x

−

∂A

∂z



−

∂A

∂t

−

∂φ

∂x



. (2.39)

Using Eqs. 2.6 and 2.8, in vector form, the equation of motion is

r = qE + q

r × (∇×A)=qE + q

r × B. (2.40)

which is Newton’s equation of motion including the magnetic force. Therefore, the

correctness of the Hamiltonian, Eq. 2.32, is veriﬁed. We will use the Hamiltonian in

the quantum-mechanical treatment of the interaction of radiation with atomic systems.

2.2 Optics of Thin Films

Maxwell’s theory of light plays a critical role in the understanding of selective absorp-

tion ﬁlms for solar thermal applications and antireﬂection ﬁlms in photovoltaics. The

general theory with an arbitrary incident angle is rather complicated. However, for ap-

plications related to solar energy, it suﬃces to study the case of normal incidence, which

demonstrates most of the related physics. First, let us extend Maxwell’s equations to

dielectrics.

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48  Nature of Solar Radiation

2.2.1 Relative Dielectric Constant and Refractive Index

Maxwell’s equations, Eqs. 2.1–2.4 are used for the case of a vacuum. To describe elec-

tromagnetic phenomena in a nonmagnetic medium, the electric constant ε

is replaced

by the electric constant of the medium, ε. Maxwell’s equations are

∇·E =

, (2.41)

∇·B = 0, (2.42)

∇×E = −

∂B

∂t

, (2.43)

∇×B = εμ

∂E

∂t

+ μ

J. (2.44)

Following the procedures in Section 2.1.1, we found the wave equations for the electric

ﬁeld intensity and the magnetic ﬁeld intensity:

∇

E −

∂

∂t

=0, (2.45)

∇

B −

∂

∂t

=0, (2.46)

where the velocity v is given as

v =

√

εμ

. (2.47)

Table 2.2: Dielectric Constant and Refractive Index of Selected Materials

Material Wavelength ε

Silicon 1.39 μm 12.2 3.49

Germanium 2.1 μm 16.8 4.10

TiO

2.0 μm 5.76 2.4

SiO

Visible 2.40 1.55

Window glass Visible 2.40 1.55

ZnS Visible 5.43 2.33

CeO

Visible 3.81 1.953

CaF

Visible 2.06 1.435

MgF

Visible 1.91 1.383

Source: American Institute of Physics Handbook,

3rd Ed, McGraw-Hill, New York, 1982.

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2.2 Optics of Thin Films  49

Comparing with Eq. 2.14, the relation of v with c is



. (2.48)

Deﬁning the relative dielectric constant of the medium as

≡

, (2.49)

the ratio of the speed of light in a vacuum and the speed of light in the medium, deﬁned

as the refractive index n,is

n ≡

√

. (2.50)

In general, the relative dielectric constant and the refractive index depend on the

frequency or wavelength of the electromagnetic wave. For application in solar energy

devices, the most relevant case is solar radiation in the visible or infrared. Table 2.2

shows the relative dielectric constant and refractive index of several materials often

used in solar energy devices.

For electromagnetic waves propagating in the z direction with wavevector k and

electric ﬁeld intensity in x, similar to Eqs. 2.24–2.26, the nonzero components are

= E

i(kz−ωt)

(2.51)

i(kz−ωt)

. (2.52)

The wavevector k is given as

k =

ωn

. (2.53)

And, according to Eq. 2.50, the electric and magnetic ﬁelds are in phase and propor-

tional,

. (2.54)

2.2.2 Energy Balance and Poynting Vector

Let us study the energy balance in an electromagnetic ﬁeld by considering a unit volume

with relatively uniform ﬁelds. If the current density is J and the electric ﬁeld intensity

is E, the ohmic energy loss per unit time per unit volume is J ·E. Using Eq. 2.44, the

expression of energy loss becomes

J · E = −

E · (∇×B)+εE ·

∂E

∂t

. (2.55)

Using the mathematical identity

E · (∇×B)=−∇· (E × B)+B ·(∇×E), (2.56)

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50  Nature of Solar Radiation

Eq. 2.55 becomes

J · E = ∇·



E × B



B · (∇×E) − εE ·

∂E

∂t

. (2.57)

Using Eq. 2.43, Eq. 2.57 becomes

J · E = −∇ ·



E × B



−

∂

∂t



2μ



. (2.58)

The right-hand side of Eq. 2.58 has a straightforward explanation. The energy density

of the electromagnetic ﬁelds is

W =

2μ

, (2.59)

and the power density of the electromagnetic ﬁeld per unit area is

S =

E × B. (2.60)

The vector S is called the Poynting vector after its discoverer.

For an electromagnetic wave, according to Eq. 2.54, cB

= nE

. The magnitude of

the Poynting vector along the direction of propagation is

. (2.61)

2.2.3 Fresnel Formulas

Consider two media of refractive indices n

and n

with an interface at z =0,asshown

in Fig. 2.3. The incident light is moving in the z direction with wavevector k

ωn

. (2.62)

The ﬁeld intensities of the incident light are

= Ie

i(k

z−ωt)

, (2.63)

i(k

z−ωt)

, (2.64)

where I is a constant characterizing the intensity of incident light.

For transmitted light, the wavevector is determined by the refractive index of

medium 2,

ωn

. (2.65)

The ﬁeld intensities of the transmitted light are

= Te

i(k

z−ωt)

, (2.66)

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2.2 Optics of Thin Films  51

Figure 2.3 Derivation of Fresnel for-

mulas. Two media with indices of refrac-

tion n

and n

share an interface at z =0.

The incident light has a wavevector k

The wavevector of transmitted light is k

The wavevector of reﬂected light is identi-

cal to that of incident light but with oppo-

site sign. By applying Maxwell’s equations

at the interface, the relations between the

three components of light can be derived.

i(k

z−ωt)

, (2.67)

The constant T characterizing the intensity of the transmitted light is to be determined

by the boundary conditions required by Maxwell’s equations.

For reﬂected light, because it is in the same medium as the incident light, the

absolute value of the wavevector is identical to that of the incident light. However, the

direction of z is reversed. By using the same notation k

, the ﬁeld intensities of the

reﬂected light are

= Re

i(−k

z−ωt)

, (2.68)

= −

i(−k

z−ωt)

. (2.69)

Notice the negative sign of the magnetic ﬁeld intensity B

. Again, the constant R

characterizes the intensity of the reﬂected light.

On the interface, z = 0, following Eqs. 2.1 and 2.2, both electric ﬁeld intensity and

magnetic ﬁeld intensity should be continuous. In other words,

+ E

= E

, (2.70)

+ B

= B

. (2.71)

Using Eqs. 2.63–2.71, we ﬁnd

I − R = T, (2.72)

(I + R)=n

T. (2.73)

The solutions of Eqs.2.72 and 2.73 are

R =

− n

+ n

I, (2.74)

T =

+ n

I. (2.75)

Equations 2.74 and 2.75 are the Fresnel formulas for the case of normal incidence. Ob-

viously, if n

= n

, there is no reﬂected light, and 100% of incident light is transmitted

through the interface.