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

n0: dipole density (m−3),

µ: expectation value of the dipole moment (Asm).

In this section the induced dipole moment will be evaluated quantum-mechanically, which requires some simplifications. It is not the intention to discuss in detail the mathematics, but only to summarize briefly the main results of interest for laser technology and to emphasize the approximations and the range of validity. A consistent presentation of the interaction light–matter, starting from first principles, is given in many textbooks [61Mes, 68Sch, 77Coh, 95Man].

From the infinite number of energy levels of an electronic system only two, E1 and E2, are taken into account for the interaction [75All, 89Yar, 69Are], see Fig. 1.1.5. This is a reasonable approach if the field is nearly resonant with the transition from E1 to E2. In this case the other levels of the system will not or only very weakly interact with the field.

It applies

|ωA − ω| ∆ωA

with

ωA: resonance frequency of the transition, ∆ωA: bandwidth of the transition,

ω: frequency of the radiation field,

= 1.0546 × 10−34 Ws2: Planck’s constant.

1.1.3.2 The dipole approximation

The oscillating electric field E deforms the electron cloud of the two-level system and generates a complicated, oscillating charge distribution. A first order approximation is an oscillating dipole. The interaction of this dipole with a monochromatic wave is evaluated quantum-mechanically.

1.1.3.2.1 Inversion density and polarization

The interaction of an electromagnetic field with a two-level system was first investigated by Bloch [46Blo] and extensively discussed by Allen and Eberly [75All]. It is characterized by its dipole moment and the population densities in the two levels:

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n1, n2 : density of states (atoms, molecules) in the lower/upper level, ∆n = n2 − n1 : inversion density,

n0 = n1 + n2 : total density, const.

The following assumptions are made:

–Non-relativistic interaction. The velocity of the electrons is small compared with the velocity of light. This does not hold for inner-shell electrons, hot plasmas and free-electron lasers.

–The wavelength of the light is large compared with the diameter of the atoms/molecules. It

means that in the domain of the atomic wave function the electromagnetic field is locally constant. Bohr’s radius with rB = 5.3 × 10−5 µm is a typical atomic dimension. The wavelength in the visible range of the spectrum is about 0.5 µm, thus this condition is fulfilled in the visible and UV-part of the spectrum. It is called the dipole approximation [97Scu].

–The permanent dipole moments of the two-level system µ11 = µ22 are zero. Even if larger molecules have a permanent dipole moment, their response to the high-frequency field is small. Only for very strong fields are the permanent dipole moments of importance (see Part 4 on nonlinear optics). A dipole moment exists only for the transition from level 1 to 2 and vice versa. Non-degenerated levels are assumed with µ = µ12 = µ21.

The two-level system is completely described by its state vector |ϕ , which in general is timedependent:

with |ϕ1 , |ϕ2 the eigenfunctions and E1, E2 the energy eigenstates. The eigenfunctions are normalized, orthogonal and depend on the position vector r:

The state vector has to fulfill the time-dependent Schr¨odinger equation:

with H0 the Hamilton operator of the undisturbed system (Hint = 0) and Hint the interaction energy. For the undisturbed system holds [89Yar]:

which follows directly from (1.1.35) by replacing |ϕ by |ϕi exp (−iEit/ ). The parameters of interest, the inversion density ∆n = n2 − n1 and the macroscopic polarization

are determined by the coe cients c1, c2. The probability of the system to be in the lower/upper state is given by |c1|2 , |c2|2 , respectively, which requires:

The number of atoms in the lower/upper level is then given by:

n1 = n0 |c1|2 , n2 = n0 |c2|2 , n1 + n2 = n0

and hence the inversion density :

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The expectation value of the dipole moment µ = −e ϕrϕ is obtained from (1.1.33). Using the afore mentioned assumptions:

µ11 = −e ϕ1rϕ1 = 0 , µ22 = −e ϕ2rϕ2 = 0

one obtains finally for the polarization from (1.1.33), (1.1.34), (1.1.38)

with µ12 , µ21 the dipole moment of the transition E1 ↔ E2 and vice versa. For non-degenerated transitions one has µ12 = µ21 = µA. In the following only µA will be used, which is a characteristic parameter of the specific transition:

1.1.3.2.2 The interaction with a monochromatic field

The interaction Hamiltonian for a non-quantized real field Ereal corresponds to the classical energy of an electric dipole in an electric field. It reads [97Scu]:

Substitution of (1.1.42) into (1.1.35), using the orthogonality (1.1.34) and (1.1.41) provides two

The time dependence of inversion density and polarization is obtained from (1.1.39), (1.1.40) by di erentiating and applying (1.1.43). After some simple mathematics the following two equations

(1.1.44b) terms with the frequency 2ω appear, which are neglected. This approach is called the rotating-wave approximation [97Scu, 72Cou]. The above equations simplify to

(rotating-wave approximation) with

µA: electric dipole moment of the transition, ω: frequency of the interacting field,

ωA: resonance frequency of the two-level system,= 1.0546 × 10−34 Ws2: Planck’s constant.

Some typical values of dipole moments are given in Table 1.1.2.

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Table 1.1.2. Typical values of dipole moments [01Men].

1.1.3.3 The Maxwell–Bloch equations

The idealized rotating-wave approximation is adapted to the real situation and combined with the SVE wave equation. Incoherent perturbations by the environment are taken into account.

So far the interaction of the two-level system with the electromagnetic field is purely coherent, no perturbations by external influences on the system are considered. Stochastic processes will modify the interaction considerably. Here only a very basic description is presented. A detailed analysis of these statistical processes is given in [70Hak, 97Scu].

1.1.3.3.1 Decay time T1 of the upper level (energy relaxation)

Three incoherent processes reduce or increase the upper-level population and have to be considered in (1.1.45a), (1.1.45b):

–spontaneous emission,

–interaction with the host material (collisions, lattice vibrations),

–increase of the population by pumping (light, electron collisions, or other processes).

1.1.3.3.1.1 Spontaneous emission

The two-level system is coupled to the modes of the optical resonator or to the free-space modes. Spontaneous emission into these modes reduces the upper-level population. Moreover, by each spontaneous emission process the phase relation between the field and the two-level eigenfunction is destroyed. If the dimensions of the resonator are large compared with the wavelength, the decay is given by ∂n2/∂t = −n2/Tsp , with A21 = 1/Tsp , the Einstein coe cient of spontaneous emission. If the resonator dimensions are comparable with the wavelength, spontaneous emission is strongly influenced by the resonator geometry, it can be enhanced or reduced (see Chap. 8.1).

1.1.3.3.1.2 Interaction with the host material

This interaction reduces the population density. Energy is transferred to the host material and converted into heat. A simple approach for this decay is again an exponential ansatz ∂n2/∂t = −n2/TH . This decay time together with the spontaneous decay time delivers a resulting decay T1 of the upper-level population, also called energy relaxation time or longitudinal relaxation time.

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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.

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 :

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:

(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]:

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:

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