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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

−

(E P

A0 −

)

−

∆n − ∆n0

(1.1.48a)

∂t

∂P A0

= − i δ +

P A0

+ i

µA

µAE0

∆n , δ = ω − ωA ,

(1.1.48b)

∂t

∂

E0 = −i

(1.1.48c)

P A0

∂z

∂t

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

Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory				17

Steady-state equations		∂∆n
The temporal variations of the radiation ﬁeld are slow		∂∆n	= 0	∂P A0	= 0
The temporal variations of the radiation ﬁeld are slow		∂t	= 0	∂t	= 0
compared with T1.		∂t		∂t
compared with T1.
Adiabatic equations		∂∆n
no transient e ects of the atom, T2 T1.		∂∆n	= 0	∂P A0	= 0
no transient e ects of the atom, T2 T1.		∂t	= 0	∂t	= 0
Coherent equations		∂∆n
The width τ of the interacting pulses is short compared		∂∆n	= 0	∂P A0	= 0
The width τ of the interacting pulses is short compared		∂t	= 0	∂t	= 0
with T1, T2; (1.1.45a), (1.1.45b) can be applied.

1.1.4 Steady-state solutions

In steady state inversion density ∆n0, polarization P A0, and intensity J of the ﬁeld are constant in time, but may depend on the spatial coordinates.

1.1.4.1 Inversion density and polarization

The stationary solutions of (1.1.48a), (1.1.48b) are obtained immediately:

∆n =

∆n0

(inversion density, homogeneously broadened),

1 + (J/Js) f (ω)

∆ωA/2

nrσ

ω − ωA

+ i

∆n (susceptibility),

P A0 = ε0χAE0

(polarization)

with

J =

ε0c0nr|E0|2

(intensity of the ﬁeld),

Js =

ωA

(saturation intensity of the two-level transition),

2σ0T1

σ = σ0f (ω, ωA)

(frequency-dependent cross section of the transition),

σ0 =

| µA|2 ωAT2

(cross section in resonance),

ε0c0nr

fL(ω, ωA) =

(∆ωA/2)2

(spectral line shape, Lorentzian),

(ωA − ω)2

+ (∆ωA/2)2

∆ωA = 2/T2 (line width of the transition),

(1.1.49)

(1.1.50)

(1.1.51)

(1.1.52)

(1.1.53)

(1.1.54)

(1.1.55)

(1.1.56)

(1.1.57)

Landolt-B¨ornstein

New Series VIII/1A1

1.1.4 Steady-state solutions

[Ref. p. 40

gh(ω, ωA) = ∆ n σ =

∆ n0σ0f (ω, ωA)

(gain coe cient, homogeneously

(1.1.58a)

1 + (J/Js) f (ω, ωA)

broadened),

inh

(ω, ω

) =

∆ n0σ0

h(ω, ω

)

∆ωA

(gain coe cient, inhomogeneously

(1.1.58b)

1 + J/Js

broadened, see Sect. 1.1.6.3).

In Table 1.1.3 some numbers of relevant laser transitions are compiled, in Table 1.1.4 some typical values of the small-signal gain coe cient in resonance are given. The susceptibility strongly depends on the frequency as shown in Fig. 1.1.6. According to (1.1.26) the real part of χA produces an additional refractive index, and the imaginary part absorption or ampliﬁcation:

A	r −			k0 ∆ωA/2
Re χ	= n2	1 =	nrσ				ω − ωA	∆n ,

Im χA = −nrαk0 =				nrσ
						∆n .
					k0

The steady-state propagation of the electric ﬁeld is obtained from (1.1.48c):

dz		− 2		2			∆ ωA		0
dE0	=		α	+	σ∆ n	+ iσ∆ n	ω − ωA	E		,
	=			+		+ iσ∆ n		E		,

where ∆ n is a function of the ﬁeld or the intensity.

(1.1.59a)

(1.1.59b)

(1.1.60)

Table 1.1.3. Examples of resonance wavelength λ0, resonance cross section σ0, upper-level lifetime T1 and saturation intensity Js. The simple relation (1.1.53) for the saturation intensity holds for two-level systems only and is not applicable in general [01Men].

	λ0	σ0	T1		Js
	[µm]	[m2]	[s]		[W/m2]
Ampliﬁers		10−20	10−5			× 105
CO2-gas (1300 Pa)	10.6	10−20	10−5		2	× 105
Neodymium-ion in glass	1.06	4 × 10−24	3	× 10−4	8 . . . 12 × 107
Neodymium-ion in YAG	1.06	5 × 10−23	2	× 10−4	2	× 107
Chromium-ion in Al2O3	0.69	2 × 10−24	3	× 10−3	2.4 × 107
(ruby, T = 300 K)		3 × 10−17	10−8		5.3 × 105
Neon (25 Pa)	0.63	3 × 10−17	10−8		5.3 × 105
Rhodamine 6G in ethanol	0.57	4 × 10−20	5	× 10−9	109
Absorbers		8 × 10−22		× 10−4	2.5 × 105
SF6	10.6	8 × 10−22	4	× 10−4	2.5 × 105
KODAK dye 9860	1.06	4 × 10−20	10−11		5.6 × 1011
KODAK dye 9740	1.06	6 × 10−20	10−11		4	× 1011
Cryptocyanine-dye	0.7	5 × 10−20	5	× 10−10	2	× 1010

in methanol

Table 1.1.4. Typical values of the small-signal gain coe cient g0 = ∆n0σ0 in resonance. The exact values depend on pumping, doping, and other parameters of operation [01Men].

System	λ0 [nm]	g0 [m−1]
He/Ne laser	632.8	0.1
Nd-doped glass	1060	5
Nd-doped YAG	1060	50
GaAs-diode	880	4 × 103

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	19

,Pχ$ JDLQ

∆ω$

5H χ$ SKDVH VKLIW

ω $		Fig. 1.1.6. Real and imaginary part of
	)UHTXHQF\ω
		the susceptibility vs. frequency.

1.1.4.2 Small-signal solutions

The solutions for low intensities are discussed. Low means that the intensity J is small compared with the characteristic parameter Js of the system (see Table 1.1.3).

At low intensities J Js, the inversion density is not a ected by the intensity,

∆n = ∆n0 ,

and (1.1.60) can be integrated. Together with (1.1.23), the complete ﬁeld is obtained:

E(z) = E0(0) exp[i(ωt − ntk0z) −								1	(α − ∆n0σ)z]	(1.1.61)
E(z) = E0(0) exp[i(ωt − ntk0z) −								2	(α − ∆n0σ)z]	(1.1.61)
with a total refractive index nt
	t		r		nrk0	∆ωA
n		= n		1 +	σ∆n0	ω − ωA	.			(1.1.62)
n		= n		1 +			.			(1.1.62)

The active atoms of the two-level system cause an additional phase shift or refractive index and an additional absorption or ampliﬁcation, depending on the sign of ∆n0. The small-signal gain factor according to (1.1.30)/(1.1.50) is:

G0 = exp[σ(ω)∆n0z] .

(1.1.63)

Ampliﬁcation, G0 > 1, requires inversion ∆n0 > 0. The complex amplitude transmission factor A is deﬁned as the ratio of the monochromatic ﬁeld amplitudes and can be written:

A = E0(0)		= exp i	2	(ω − ωA) + i ∆ωA/2		z .	(1.1.64)
	E0(z)		σ0∆n0		∆ωA/2

It depends on the frequency of the ﬁeld, which means dispersion. Time-dependent ﬁelds and especially short pulses are distorted by the amplifying system, pulse broadening and chirping occur.