Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf

∆ n > 0 is a state of non-equilibrium. To support this state, energy has to be pumped into the system. This pumping energy can be incoherent light, kinetic energy of electrons/ions, chemical energy or electric energy. The pumping schemes can become very complicated, and in most cases many energy levels are involved. To understand the principal process for the generation of inversion, two idealized pumping schemes will be discussed.

1.1.5.3.1 The three-level system

The simplified diagram of the three-level system is shown in Fig. 1.1.8. The level E3 is excited by absorption of light or by electron collisions, depending on the specific system. The decay from E3 to E2, the upper laser level, is very fast. Nearly all excited atoms are transferred into this level, which has a very long life time. If the pumping power is su ciently high to overcome the decay of level E2, atoms will be accumulated and finally n2 is larger than n1. The adiabatic rate equations give for the upper-level population without induced emission between the two levels (J = 0):

W is the pumping rate, the product of the cross-section σ13 and the specific pump parameters. T1 is the upper laser-level lifetime. This holds under the assumption that the population of level E3 is zero and that n1 + n2 = n0. Equation (1.1.81) reads with the inversion density ∆ n = n2 − n1:

and in steady state one obtains:

The relation between the inversion density and the pump rate is shown in Fig. 1.1.9. Inversion occurs for W T1 > 1. With increasing pump rate the inversion increases also and approaches finally one, all atoms are in the upper level. To obtain ∆ nsteady,3 > 0 requires at least 50 % of the active atoms to be pumped into the upper level, high pump rates are necessary and the e ciency is low. Equation (1.1.82) has to be completed by the coherent interaction term of (1.1.67). The complete rate equation for the three-level system with pump rate W , interacting with a monochromatic field of intensity J is given in (1.1.84). For the intensity (1.1.48c), (1.1.48d) hold, depending on the type of line-broadening (Sect. 1.1.6).

Fig. 1.1.8. The idealized threeand four-level system.

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The commonly used pump scheme, due to its high e ciency, is the four-level system as shown in Fig. 1.1.8. The two laser levels are E2 and E1, where the lower level E1 has a very short lifetime and its population n1 is nearly zero. This requires that the energy E1 − E0 is much larger than the thermal energy κT . The pump level E3 decays very rapidly to the upper laser level E2 and its population is again nearly zero. The inversion density now is ∆ n = n2 − n1 ≈ n2. Then the rate equation for the pump process reads:

with the steady-state solution (without coherent interaction):

Inversion is reached now at very small pump-power levels as shown in Fig. 1.1.9. The e ciency of such systems is much higher than of three-level systems. The complete rate equation for pumping and interaction with a field of intensity J is obtained by taking into account the corresponding term of (1.1.67). It has to be considered that n1 = 0, and therefore the saturation intensity is higher by a factor of 2.

(rate equation of a four-level system)

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1.1.5.4 Adiabatic pulse amplification

The amplification and shaping of light pulses by saturable two-level systems is presented.

The pulse is adiabatic if its width τ is small compared with T1 and large compared with T2. Then the variation of the upper-level population due to spontaneous emission and pump can be neglected and this term can be neglected. If such a pulse travels through an active medium of length , it depletes the upper level, is amplified and shaped as depicted in Fig. 1.1.10. The initial conditions at t = −∞ are:

The equations (1.1.67)/(1.1.68) can be solved for a loss-free-medium (α = 0) with a four-level system and yield for the output intensity [63Fra]:

with

G0 : small-signal gain factor, (1.1.63),

Es = Js,4T1 : saturation energy density,

Ein,out : input/output energy density.

Equations (1.1.88)–(1.1.90) also hold for saturable absorbers with G0 < 1. The pulse will be shaped in any case and the peak velocity will di er from the phaseand group velocities.

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1.1.5.5 Rate equations for steady-state laser oscillators

In the oscillator system, two counter-propagating traveling waves J +, J − appear, see Fig. 1.1.11, which are amplified by an intensityand z-dependent gain coe cient according to (1.1.58a), (1.1.58b):

For the two traveling waves the boundary conditions at the mirrors are:

J +(z = 0) = J −(z = 0)R1 ,

J −(z = ) = J +(z = )R2 .

The combination of (1.1.91a) and (1.1.91b) yields [81Ver]:

J +(z)J −(z) = const. ,

a useful relation for analytical solutions. The gain coe cient is saturated by both waves. In steady state (1.1.84)/(1.1.87) hold with J = J + + J −, depending on the level system and on the type of broadening. For homogeneous broadening a solution is given in [81Ver]. In general, numerical calculations are necessary. For optimization a diagram is o ered in [92Koe]. The intensity rate equations are very useful for laser design and optimization, but deliver no spectral e ects such as line width [58Sch, 74Sar, 95Man], mode competition [86Sie, 00Dav], mode hopping [86Sie, 64Lam, 74Sar], or intensity-dependent frequency shifts (Lamb dip) [64Lam]. Multimode oscillation can be described by rate equations with restrictions [64Sta, 63Tan, 93Sve].

1.1.6 Line shape and line broadening

Shape and width of the spectral response of the two-level system depend on the special stochastic perturbation processes, in detail discussed by [81Ver, 86Eas]. An easy-to-read introduction is given by [86Sie, 00Dav].

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