16 1.1.3 Interaction with two-level systems [Ref. p. 40

1.1.3.3.1.3 Pumping process

The dynamics of upper-level excitation depend on the special pumping scheme and are discussed in Sect. 1.1.5.3 and in Vol. VIII/1B, “Solid-state laser systems”. In any case the pump produces in steady state and without a coherent field (E0 = 0) an inversion density ∆n0.

These three processes are included into (1.1.45a) by the term:
	∂∆n	=	−	∆n − ∆n0	(1.1.46)
	∂t			T1

with

T1: the resulting time constant.

1.1.3.3.2 Decay time T2 of the polarization (entropy relaxation)

An external field E induces dipoles, which generate the macroscopic polarization P A. If the external field is switched o , the polarization will disappear for several reasons:

The energy of the two-level system decays with T1, which means that the polarization disappears at least with the same time constant.

Due to incoherent interaction with the host material (collisions), the single dipoles are disoriented in their direction or dephased. The resulting polarization becomes zero, although the single dipole still exists. This process can be much faster than T1 (see Table 1.1.6) and is characterized by a time constant T2. This decay strongly depends on the interaction process. The simplest approach is :

∂P A0	= −	P A0	,	(1.1.47)
∂t		T2

and (1.1.45b) has to be completed by (1.1.47). T2 is called the transverse relaxation time, the entropy time constant or the dephasing time. Finally, the two-level equations together with the SVE-approximation, (1.1.28), of the wave equation read:

∂∆n

=

−

i

(E P

A0 −

E

P

)

−

∆n − ∆n0

,

(1.1.48a)

2

∂t

0

0

A0

T1

∂P A0

= − i δ +

1

P A0

+ i

µA

µAE0

∆n , δ = ω − ωA ,

(1.1.48b)

∂t

T2

∂

+

1

∂

+

α

E0 = −i

k0

(1.1.48c)

P A0

∂z

c

∂t

2

2ε0nr

(Maxwell–Bloch equations).

They describe the propagation of radiation in two-level systems and are called Maxwell–Bloch equations. Equation (1.1.48c) holds, if the transition frequency ωA for all two-level atoms is the same (homogeneous system). In inhomogeneous systems (see Sect. 1.1.6.3, Fig. 1.1.13) di erent groups of atoms exist with center frequencies ωA of each group and a center frequency ωR of the ensemble. Therefore (1.1.48c) has to be replaced by [81Ver]:

∂		1			∂			k
	+						E0 = −i	0		h(ωA, ωR)P A0(E0, ωA)dωA .	(1.1.48d)
∂z			c		∂t			2ε0nr

h(ω, ωA) is the spectral density of atoms with the transition frequency ωA according to (1.1.92)/ (1.1.93). For the solution of these equations, three di erent regimes are distinguished:

Landolt-B¨ornstein

New Series VIII/1A1