10	1.1.2 The electromagnetic field	[Ref. p. 40

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

∂2

∂2

∂

E0

+

−

2ik0

= 0 ,

(1.1.24b)

∂x2

∂y2

∂z

∂2

1 ∂

1 ∂2

∂

E0

+

+

− 2ik0

= 0 .

(1.1.24c)

∂r2

r

∂r

r2

∂ϕ2

∂z

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:

P = P A + P H = (P A0 + P H0) exp[i(ωt − nrk0z)] .

(1.1.25)

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]:

χH = (nr2 − 1) − i	nrα	, α k0 .	(1.1.26)
	k0

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

P A = ε0χA(E0)E ,

(1.1.27)

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:

∂

+

1

∂

+

α

E0 = −i

k0

(E0) ,

div E = 0

(1.1.28)

P A0

∂z

c

∂t

2

2ε0nr

(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

New Series VIII/1A1