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Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	29

given by the spontaneous life time Tsp, is reduced to the collision time τ . The Fourier transform of these shortened waves gives a Lorentzian line shape with the spectral width ∆ ωC = 2/τ or

√32 σC p

∆ ωC = (1.1.97)

π mA κT

with

σC : collision cross section of the atom, p : pressure of the gas.

The collision broadening is proportional to the gas pressure. Some numbers are given in Table 1.1.5.

1.1.6.2.4 Saturation broadening

A strong ﬁeld of intensity J , comparable with the saturation intensity Js, depletes the upper laser level. The gain is reduced according to (1.1.58a), (1.1.58b) and the gain proﬁle becomes ﬂatter and broader with the spectral width (see Fig. 1.1.13) [81Ver]:

∆ωS = ∆ωA 1 + J/Js .

1.1.6.3 Types of broadening

The interaction of the ﬁeld depends strongly on the type of broadening. Two idealized cases are the homogeneous and the inhomogeneous broadening [00Dav].

1.1.6.3.1 Homogeneous broadening

All transitions have the same resonance frequency ωA. The gain is saturated for all atoms in the same way as given by (1.1.58a) and shown in Fig. 1.1.13. Examples for this type of broadening are:

–spontaneous emission,

–collision broadening,

–saturation broadening,

–thermal broadening in crystals by interaction with the lattice vibrations.

Homogeneously	Inhomogeneously
∆ A	∆ A	∆ R

A	R

Frequency of the radiation field

Fig. 1.1.13. Saturation of homogeneously and inhomogeneously broadened systems by a radiation ﬁeld of frequency ω.

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30	1.1.6 Line shape and line broadening	[Ref. p. 40

1.1.6.3.2 Inhomogeneous broadening

Groups of atoms with spectral density h(ωR, ωA) and di erent frequencies ωA produce a resulting line proﬁle with center frequency ωR and width ∆ ωR as shown in Fig. 1.1.14. A strong monochromatic ﬁeld of frequency ω interacts mainly with the group ωA = ω and saturates this particular group. A dip appears in the proﬁle, which is called spectral hole-burning. Examples of inhomogeneous broadening are:

–Doppler broadening,

–Stark broadening in crystals due to statistical local crystalline ﬁelds.

h ( , R)

f ( , A)

∆ R

∆ A

A R

Frequency

Fig. 1.1.14. An inhomogeneously broadened proﬁle.

The resulting line proﬁle is a convolution of the individual group proﬁles and the broadening process, which results in complicated integrals. The saturation process for inhomogeneously broadened lines is quite di erent, as will be shown by a simple example. In this case (1.1.58a) holds only for one group of atoms with the spectral density h(ωA, ωR). Integration over all groups results in the total gain coe cient ginh:

	+∞
ginh(ω, ωR) =		f (ω, ωA)h(ωA, ωR)d ωA .	(1.1.98)

−∞

If the width ∆ ωA is much smaller than the total width ∆ωR, the function h(ωA, ωR) can be taken outside of the integral at ωA = ω. Assuming a Lorentzian proﬁle for the single group, (1.1.98) becomes:

ginh(ω) = ∆ n0σ0h(ω, ωR)

f (ω, ωA)

d ωA

1 + (J/Js) f (ω, ωA)

and can be integrated:

ginh(ω) =

∆ n0σ0

h(ω, ωR)

π∆ ωA

∆ n0 σ0

f (ω)

∆ ωA

(1.1.99)

∆ ωR

1 + J/Js

The gain saturates slower than in the case of homogeneous broadening, but the maximum gain is lower by the ratio of the line widths. Inhomogeneous gain proﬁles can also be caused by spatial hole burning in solid-state laser systems. The standing waves between the mirrors produce an inversion grating and holes in the spectral gain proﬁle [86Sie].

The spectral characteristics of lasers depend strongly on the type of broadening, see Fig. 1.1.15. In steady state the gain compensates losses and the gain proﬁle saturates to fulﬁll the condition GRV = 1. A homogeneously broadened gain proﬁle saturates till the steady-state condition is fulﬁlled for the central frequency. The bandwidth ∆ ωL,h is very small and depends on the thermal and

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Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	31

Total gain factor GRV

Homogeneous broadening		Inhomogeneous broadening
	∆ L,h		∆ L,inh

A		R

Frequency

Fig. 1.1.15. Spectrum of an inhomogeneously and homogeneously broadened laser transition in steady state. Total gain factor GRV vs. frequency of the radiation ﬁeld.

mechanical ﬂuctuations [00Dav]. In the case of solid-state lasers spatial hole burning will inﬂuence the spectral behavior and can produce even for homogeneous transitions multi-mode oscillation [63Tan, 66Men]. In the case of inhomogeneous broadening each spectral group of atoms saturates separately and many modes will oscillate, which produces a large lasing bandwidth ∆ ωL,inh. If single-mode operation is enforced by suitable frequency selecting elements, the left → right and the right → left traveling waves produce two symmetric holes, due to the Doppler e ect. This e ect can be used for frequency stabilization (Lamb dip [64Lam]).

1.1.6.4 Time constants

The line proﬁle of a real laser transition is in most cases a mixture of homogeneous and inhomogeneous proﬁles, depending on the temperature and the pressure. The following time constants are used in literature:

Tsp :	spontaneous life time,
T1 :	upper-laser-level life time (energy relaxation time, longitudinal relaxation time),
T2 :	Stochastic processes broaden the line homogeneously. The inverse of the line width is the
	dephasing time T2 .
T2 :	The line is broadened inhomogeneously. The inverse of this line width ∆ ωR is the de-
	phasing time T2 .
T2 :	For the resulting dephasing time (transverse relaxation time, entropy time constant),
	approximately holds (depends on the line proﬁles):
	1		1		1
			≈		+		.
		T22		T2 2		T2 2

Some examples of decay times are given in Table 1.1.6.

1.1.7 Coherent interaction

Radiation ﬁeld and two-level system are two coupled oscillators. Without stochastic perturbations the stored energy is permanently exchanged between these two systems.

If the interaction time of the radiation ﬁeld with the two-level system is small compared with all relaxation times, including the pump term, the stochastic processes can be neglected and

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32	1.1.7 Coherent interaction	[Ref. p. 40

Table 1.1.6. Spontaneous life time Tsp, upper-laser-level life time T1, transverse relaxation time T2, homogeneous relaxation time T2 and inhomogeneous relaxation time T2 [01I , 92Koe, 86Sie], [01Men, Chap. 6].

	Tsp [s]	T1 [s]	T2 [s]	T2 [s]	T2 [s]
Neon-atom (He/Ne-laser),	10−8	10−8	3 × 10−9	10−8	4 × 10−9
λ0 = 632.8 nm, He (p = 130 Pa),
Ne (p = 25 Pa)
Chromion-ion, λ0 = 694.3 nm,
R1-transition in ruby	3 × 10−3	3 × 10−3			2 × 10−7
T = 300 K	3 × 10−3	3 × 10−3	10−12	10−12	2 × 10−7
T = 4 K	4 × 10−3	4 × 10−3	2 × 10−7	3 × 10−3	2 × 10−7
SF6-molecule, λ0 = 10.5 µm,	10−3	10−3	6 × 10−9	7 × 10−6	6 × 10−9
p = 0.4 Pa
Rhodamin-molecule in ethanol,	5 × 10−9	5 × 10−9	10−12	10−12	–
singlet-transition, λ0 = 570.0 nm
Neodymium-ion in YAG-crystal,	5 × 10−4	2.3 × 10−4	7 × 10−12	–	–
λ0 = 1060 nm, T = 300 K

(1.1.45a)/(1.1.45b) hold. This kind of coherent interaction is of strong interest in nonlinear spectroscopy [84She, 86Sie, 71Lam, 72Cou], [01Men, Chap. 7] and conﬁrmed by many experiments. Examples of nonlinear coherent interaction are transient response of atoms, optical nutation, photon echoes, n π-pulses and quantum beats. Here only some very simple examples will be presented. A more detailed treatment is given in [95Man].

1.1.7.1 The Feynman representation of interaction

Feynman introduced a very elegant representation of interaction, which enables an easy-to-under- stand visualization.

A very compact description of the two-level interaction was given by Feynman [57Fey]. The real electric ﬁeld is

Ereal = 12 {E0 exp [i (ωt − kz)] + E0 exp [−i (ωt − kz)]} .

It generates a real polarization, (1.1.23), shifted in phase against the ﬁeld:

A,real

{

exp [i(ωt

−

kz)] + P exp [

−

i(ωt

−

kz)]

}

= C cos (ωt − kz) + S sin (ωt − kz)

(1.1.100)

with C, S real vectors:

(P A0 + P A0) , S

i (P A0 − P A0) .

C =

In the following an isotropic medium is assumed. Then µA, P A and E are parallel and can be treated as scalars. With these new real quantities the equations of interaction (1.1.45a), (1.1.45b) become:

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