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

spatial coordinates), isotropic (ε does not depend on the polarization of light), and linear (ε does not depend on the intensity of the field). The last assumption holds for low-intensity fields only.

The permittivity ε is a scalar and (1.1.11)/(1.1.12) reduces to the standard wave equation:

Simple solutions are the plane and the spherical waves.

1.1.2.2.1 The plane wave

The infinite, monochromatic wave with a plane phase front and constant amplitude reads:

H0 = [k0 × E0] . k0Z

It is a transversely polarized field with E H k0, as plotted in Fig. 1.1.3.

(

\

[

6 N

+

U

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8 1.1.2 The electromagnetic field [Ref. p. 40

Table 1.1.1. Values of refractive index nr and absorption coe cient α at wavelength λ0 [85Pal, 82Gra, 78Dri].

The intensity is the time average over one period T = 2π/ω and results in:

with

nr: real part of the refractive index,

α: absorption coe cient, in general the non-resonant broad-band absorption.

For a field propagating in z-direction (1.1.15)/(1.1.20) deliver an exponentially damped ampli-

Some numbers of nr, α are compiled in Table 1.1.1.

1.1.2.2.2 The spherical wave

One solution of the wave equation (1.1.13) in spherical coordinates is the quasi-spherical wave, generated by an oscillating dipole (Hertz’s dipole), see Fig. 1.1.4. The far field reads [99Jac]:

with µ the dipole moment and ϑ the angle between the dipole axis and beam propagation k0.

In the paraxial approach (ϑ π/2 , θ 1) the well-known spherical wave, useful for applying

=

Huygens’ principle, results:

where E is approximately parallel to the dipole axis.

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y

E

r, S, k0

ϑ

x

θ

1.1.2.2.3 The slowly varying envelope (SVE) approximation

In the Slowly Varying Envelope approximation (1.1.11) is solved approximately with the ansatz of a quasi-monochromatic, quasi-plane wave

The wave propagates mainly in z-direction and the amplitude is slowly varying with x, y, z, t, which means:

–slowly varying in time (quasi-monochromatic): ∂|E0|/∂t ω|E0|, or spectral bandwidth ∆ω ω,

–slowly varying in space (quasi-plane wave): ∂|E0|/∂z k0|E0|, which means low divergence of the beam ∆θ 1 (paraxial approach), and a smooth transverse profile,

–slowly varying polarization ∂|P 0|/∂t ω|P 0|,

–slowly varying electric susceptibility ∂|χe|/∂t ω|χe| and |grad χe| k0|χe|.

Then second order terms can be neglected and the SVE-approximations are obtained [84She, p. 47], [66War, 86Sie].

1.1.2.2.4 The SVE-approximation for di raction

Steady-state propagation in vacuum means ∂|E0|/∂t = 0 and P = 0. Equation (1.1.11) delivers with the ansatz (1.1.23) and neglecting ∂2E0/∂t2 the SVE-approximation used in di raction theory, also called the Schr¨odinger equation of optics:

The field in (1.1.24a) is a vector field, and the ∆-operator in cylinder coordinates is rather complicated, because the unit-vectors are no longer constant [99Jac], especially for non-uniform polarization in circular birefringent media [82Fer, 93Wit]. In most cases (except birefringence) the

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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

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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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