Optical Sources, Detectors, and Systems

Chapter 2

Interaction of Radiation with Matter:

Absorption, Emission, and Lasers

In the preceding chapter we treated quantitatively the two major

sources of incoherent radiation, thermal emitters and diffuse scatterers.

Our treatment was very general, invoking only thermodynamic and sta-

tistical laws, and describing matter in terms of its absorptivity or reflec-

tivity. We now treat in much more detail the interaction of radiation

with matter, in particular, the absorption or emission of a photon, which

we again specify as an increment of energy, /z v, extracted or added to an

electromagnetic field of frequency, v. Einstein, in a classic (but not

classical) paper, (Einstein, 1917), placed an assembly of particles in

equilibrium with the Planck radiation field and deduced a fundamental

relationship for the induced and spontaneous transitions between the

discrete energy levels of the particles. His treatment proved that radi-

ation could induce or stimulate a downward transition, that is, one that

resulted in the

emission

of a photon of energy. From this theory, we de-

rive the requirements for net stimulated emission or amplification in a

medium. With an appropriate regeneration or feedback mechanism,

such as a Fabry-Perot resonator, we may then obtain

laser

action, or the

production of a single-frequency, single-plane-wave electromagnetic

field. We treat the Fabry-Perot and discuss the overall laser family after

a discussion of Fermi-Dirac or electron statistics, the latter topic being

essential for our treatment of semiconductor lasers in Chapter 3.

2.1 The Einstein A and B Coefficients and Stimulated

Emission

We consider the interaction of blackbody radiation with a simple

assemblage of particles each of which has two possible energy levels, L/^

and Uj' These particles might be, for example, ammonia molecules,

NH^, which were used in the first demonstration of stimulated emission,

the

maser,

a microwave device. As shown in Figure

2.1,

the particles

33

34

Chapter 2 Interaction of Radiation with Matter

U

U,

u.

N.

4

I

w

21

Ni

N

Figure 2.1 Transition rates and occupancy for states in equilibrium with blackbody field.

are distributed such that Nj are in the upper state and N^ in the lower

state.

The particles are contained in a blackbody chamber at temper-

ature T, and the occupancy probability is given by Eq. (1.1), the Boltz-

mann relation. Thus the ratio N2/N2 becomes

e'^^f^'^,

since energy

absorbing or emitting transitions between the two levels occur with a

change in electromagnetic field energy of (L/2 -

U-^)

=

hv. Defining the

transition

probabilities

per unit time as ]N^2^^^ ^2V ^^ "^'^Y write that

N^]Ni2

-

N2VJ21

since the total rates must be equal in thermal equi-

librium.

Now Einstein hypothesized that there were both spontaneous and

induced

or

stimulated

transitions. The spontaneous transitions occurred

from the upper to the lower state with a probability per unit time of A,

characteristic of the particle. In contrast the induced transition prob-

ability was assumed to be proportional to the electromagnetic spectral

energy density, u^, of Eq. (1.12), and transitions were induced in both

directions. It was this latter conjecture that was most surprising since

absorption had always been treated as the single process of excitation

from the lower to the higher energy state. As we will see, it is essential to

include the induced

downward

transitions to satisfy the rate equations,

which we write as

N2W22 = N2B22WV = N2W21 =N2(A + B2iUy) (2.1)