2
Interaction of Electrons and Photons
This chapter provides the basis for the discussion in the following chapters by
summarizing the fundamental concepts and the quantum theory concerning
the interaction between electrons and photons in a form that is convenient for
theoretical analysis of semiconductor lasers [1–9]. First, quantization of
electromagnetic fields of optical waves is outlined, and the concept of a
photon is clarified. Quantum theory expressions for coherent states are also
given. Then the quantum theory of electron–photon interactions and the
general characteristics of optical transitions are explained. Fundamental
mathematical expressions for absorption, spontaneous emission, and
stimulated emission of photons are deduced, and the possibility of optical
wave amplification in population-inverted states is shown.
2.1 QUANTIZATION OF OPTICAL WAVES AND PHOTONS
2.1.1 Expression of Optical Waves by Mode Expansion
The electric field E and magnetic field H of optical waves, together with the
electric flux density D, magnetic flux density B, current density J, and charge
density , generally satisfy the Maxwell equations
J  E ¼
@B
@t
ð2:1aÞ
J  D ¼  ð2:1bÞ
J  H ¼
@D
@t
þ J ð2:1cÞ
J  B ¼ 0 ð2:1dÞ
The electromagnetic fields can be expressed using a vector potential A and
a scalar potential . For cases where there is no free charge in the medium
( ¼ 0; J ¼ 0), in particular, we can put  ¼ 0, and accordingly E and H

ÂÃ
ð2:3aÞ
a
m
ðtÞ¼a
m
expði!
m
tÞð2:3bÞ
ThenEcanbewrittenas
Eðr;tÞ¼
1
2
X
m
a
m
ðtÞE
m
ðrÞþa

m
ðtÞE

m
ðrÞ
ÂÃ
ð2:4aÞ
E
m

m
¼0,JEE
m
¼0ð2:5Þ
wherec¼1="
0

0
ðÞ
1=2
isthelightvelocityinvacuum.A
m
alsosatisfiesthe
sameHelmholtzwaveequationasE
m
E
m
andA
m
satisfyingthewave
equationgivenbyEq.(2.5)andboundaryconditionsconstituteamode,
andtheexpressionsinEqs.(2.3)and(2.4)arecalledmodeexpansions.The
conceptofthemodeexpansionisillustratedinFig.2.1.Notingthatthe
modes {E
m
(r)} form an orthogonal system, we normalize them so that
the energy stored in the medium of volume V satisfies
Z
V
"

ðrÞ¼E
m
expðik
m
E rÞ, k
m
jj
¼
n
r
!
m
c
, E
m
E k
m
¼ 0 ð2:7Þ
Each mode is a plane transverse wave propagating along the direction of a
wave vector k
m
. Since there exist many modes with different propagation
directions for a given frequency, we need the concept of mode density. As for
18 Chapter 2
Copyright © 2004 Marcel Dekker, Inc.
the space volume V, consider a cube of side length L much larger than the
optical wavelength. Then the periodic boundary condition requires that
the wave vector k (¼ k
m
) must be in the form of

in the k space, per unit volume in real space, is given by
dN ¼
1
ð2p=LÞ
3
dk
x
dk
y
dk
z
L
3
¼
1
2p

3
dk
x
dk
y
dk
z
¼
1
2p

3
k

, n
g
 n
r
þ
! dn
r
d!
ð2:10Þ
Arbitrary
electromagnetic
field in a space of
finite volume V
A(r, t)
A
m+1
(r) A
m+2
(r)A
m
(r)
a
m+1
(t) a
m+2
(t)a
m
(t)Expansion coefficients
Spatial mode functions
Figure 2.1 Schematic illustration of mode expansion of an optical wave in free

The energy H stored in the medium associated with the electric field E and
the magnetic field H of an optical wave can be given as follows, using the
mode expansion, orthonormal relation (Eq. (2.6)) and periodic boundary
condition:
H ¼
Z
V
1
2
n
r
n
g
"
0
E
2
þ
1
2

0
H
2
ÂÃ
dV
¼
X
m
hh!

!

1=2
a þ a

ðÞ ð2:13aÞ
p ¼
1
2i
2 hh!ðÞ
1=2
a  a

ðÞ ð2:13bÞ
Then the Hamiltonian is written as
H ¼
hh!
2
aa

þ a

aðÞ
¼
1
2
p
2
þ !
2

2
p
2
þ !
2
q
2
ÀÁ
¼ hh!ða
y
a þ
1
2
Þ
¼ hh! N þ
1
2
ÀÁ
ð2:16Þ
N ¼ a
y
a
Making the Heisenberg equation of motion from the above H yields
da
dt
¼
1
ihh
a, H½
¼i!a ð2:17Þ

1=2
jn  1ið2:19aÞ
a
y
jni¼ðn þ 1Þ
1=2
jn þ 1ið2:19bÞ
If the eigenvalue n is not an integer, from Eq. (2.19a) we expect the existence
of eigenstates of infinitively large negative n. Since such eigenstates are not
Interaction of Electrons and Photons 21
Copyright © 2004 Marcel Dekker, Inc.
natural, the eigenvalue n should be an integer. This means that eigenstates
for optical waves of a mode are discrete states of n ¼ 0, 1, 2, ... and, from
the relation between H and N (Eq. (2.16)), the energy is given by
E
n
¼ hh! n þ
1
2
ÀÁ
ðn ¼ 0, 1, 2, ...Þð2:20Þ
From Eq. (2.16), the eigenstates jni of N, are energy eigenstates that satisfy
Hjni¼E
n
jnið2:21Þ
and form an orthonormal complete system. As Eq. (2.20) shows, the
increase and decrease in energy of the optical wave of frequency ! are
limited to discrete changes with hh! as a unit. This implies that optical waves
have a quantum nature from an energy point of view, and therefore the unit
energy quantity hh! is called the photon. The operator N ¼ a

= hM (n + ), n integer
5.5 hM
4.5 hM
3.5 hM
2.5 hM
1.5 hM
0.5 hM
5>
4>
3>
2>
1>
0>
0
E
Classical optical wave
Continuous energy
Electromagnetic
sinusoidal wave
Complex amplitudes
a(t), a*(t)
Photon
Quantization
Amplitude operators a, a
†
Commutation relation
[a, a
†
] = 1
1

aji¼jið2:24Þ
The expectation values for amplitudes a and a
y
at time t ¼ 0 are hai¼
andha
y
i¼

, and those at time t are
haðtÞi ¼ hjaðtÞji¼ expði!tÞð2:25aÞ
ha
y
ðtÞi ¼ hja
y
ðtÞji¼

expðþi!tÞð2:25bÞ
The expectation value hEi of the electric field is given by substituting the
above equations for a, a
y
in Eq. (2.18b) and is sinusoidal. This is consistent
with the well-known observations of coherent electromagnetic waves such as
single-frequency radio waves and laser lights. The state ji is suitable for
representing such electromagnetic waves and is called the coherent state.
The fluctuations in the canonical variables q, p for a coherent state ji
are Áq ¼hÁq
2
i
1=2
¼ðhh=2!Þ

using Eqs (2.19a) and (2.24) and normalizing so as to have hji¼1:
c
n
¼hnji
¼hjni

¼fhjðn!Þ
1=2
a
yn
j0ig

¼ðn!Þ
1=2

n
h0ji
¼ðn!Þ
1=2

n
exp 
jj
2
2

ð2:27Þ
Therefore the probability of taking each eigenstate jni is given by
jc
n

a
y
m
a
m
þ
1
2
ÀÁ
ð2:29Þ
24 Chapter 2
Copyright © 2004 Marcel Dekker, Inc.