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

scalar version of the SVE-approximation is su cient. It reads in rectangular/cylindrical coordinates

This is the fundamental equation in paraxial di raction optics. It gives the Fresnel-integral and the eigenmodes of free propagation (Gauss-Hermite/Gauss-Laguerre polynomials, see Chaps. 3.1 and 8.1). Equations (1.1.24a)/(1.1.24b)/(1.1.24c) hold for a homogeneous medium, but can be extended to quadratic index media [86Sie].

1.1.2.3 Propagation in doped media

The active medium of a laser amplifier consists of a host material, doped with the active atoms (molecules). Host and doping interact di erently with the laser radiation.

A plane wave without transverse structure interacts with active atoms or molecules and induces a polarization P A. In most cases the active atoms are embedded in a host medium (glass, crystal, liquid, gas), which is also polarized by the field, generating an additional polarization P H. The total polarization is:

The response of the host medium is in most cases very fast (10−12 . . . 10−14 s), no transient behavior occurs and nonlinear e ects are assumed to be small. Then the host polarization is proportional to the applied field:

P H = ε0χHE .

χH is the complex susceptibility of the host material and is related to the refractive index nr and the loss coe cient α according to (1.1.17)/(1.1.20) [99Ber]:

The imaginary part of χH is called extinction coe cient. Some values of refractive indices nr and absorption coe cients α are given in Table 1.1.1. For the polarization of the active atoms one has

where χA depends on the field and has to be evaluated quantum-mechanically. Neglecting first and second order derivations of P A0 and second order derivations of E0, the SVE-approximation for the interaction is obtained, assuming a plane wave without transverse structure:

(SVE-approximation for the amplitude of a plane wave in an active medium)

with c = c0/nr the phase velocity of the wave in the host medium. The above equation describes the amplification/attenuation of cw-fields and pulsed radiation by an active medium. It provides also the widely used rate-equation approach, as will be shown in Sect. 1.1.5.1. It fails for fields

Landolt-B¨ornstein

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with amplitudes varying very rapidly in time or space (fs-pulses). If the intensity J (1.1.19) and the susceptibility of the active medium (1.1.27) are introduced, (1.1.28) reduces to:

The active atoms enhance or reduce the losses of the medium, depending on the sign of the imaginary part Im χA of the susceptibility, which is a function of the intensity. In steady state and for constant χA, which holds for low intensities, (1.1.29) can be integrated and delivers for the intensity

J (z) = J (0) exp −α + k0 Im (χA) z . nr

The amplifying factor is called the small-signal gain factor G0 of the medium and the exponent the small-signal gain coe cient g0:

Some typical values of g0 are compiled in Table 1.1.4.

1.1.3 Interaction with two-level systems

Most quantum systems as atoms or molecules have an infinite number of energy levels. To demonstrate the essential features of light–matter interaction, a simplified model with only two levels is presented.

1.1.3.1 The two-level system

The relevant parameters are the energy di erence ∆E of the two levels, the inversion ∆n, the dipole moment µ, and the polarization P A.

The two-level system can be part of an atom, ion, molecule, or something more complicated. A monochromatic electric field E of frequency ω in the SVE-approximation according to (1.1.23) acts via the Coulomb force on the bound electrons of the active medium. In linear systems (parabolic potential) the negative electrons will oscillate sinusoidally, whereas the heavy positive nucleus remains more or less at rest. An oscillating dipole is induced with a dipole moment µ(t), which is given by

with

e: electron charge,

x: displacement of the electron.

The dipole moment per volume is the macroscopic polarization P A of the active medium. As all single dipoles are aligned by the electric field, the resulting polarization reads:

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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:

Landolt-B¨ornstein

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