Ref. p. 40]						1.1 Fundamentals of the semiclassical laser theory	37
Jpeak =	ω		g0			2
						(peak intensity) ,	(1.1.117)
	2σ0T2			α
Tπ = 2 τ = 2 T2				α		(pulse duration) .	(1.1.118)
				g0

The pulse propagates approximately with c, depletes at each position the upper level, and converts this energy via the broadband losses α into heat. The saturated gain just compensates the losses. The pulse is only stable for α > 0 and g0 > 0.

So far solutions of the steady-state SVE-equation were presented, assuming resonance and a homogeneously broadened two-level system. O -resonance interaction and inhomogeneously broadened systems are much more complicated and are discussed in detail in the literature [74Sar, 69Ics, 72Cou]. Moreover, the stability of the pulses with respect to small perturbations was not yet mentioned. It is controlled by the area theorem [67McC, 74Sar].

1.1.7.3.2 Superradiance

The spontaneous emission was neglected in the coherent interaction. An initial state, R = (0, 0, µ ∆ n), complete inversion, without external field F would be stable according to the interaction equations (1.1.103). But due to spontaneous emission and amplified spontaneous emission, the R-vector will be pushed a bit out of equilibrium and decay into the stable position R = (0, 0, −µ ∆ n). This phenomenon is called superradiance and discussed in detail in Chap. 6.2.

1.1.8 Notations

Symbol	Unit	Meaning
A21	s−1	Einstein coe cient of spontaneous emission
B	Vs/m2	magnetic induction
B12, B21	m3/VAs3	Einstein coe cient of induced emission
C	As/m2	component of the Feynman vector R
c0	m/s	vacuum velocity of a plane wave
c	m/s	phase velocity of light in a medium
c1,2	–	coe cients of the eigenvector
D	As/m2	electric displacement
E	V/m	electric field
E0	V/m	electric-field amplitude
E1,2	VAs	energy eigenstates of the two-level system
Ein	VAs	amplifier input energy
Eout	VAs	amplifier output energy
ES	VAs/m2	amplifier saturation energy density
f (ω, ωA)	–	line shape factor
G	–	gain factor
G0	–	small-signal gain factor
g	m−1	gain coe cient
g0	m−1	small-signal gain coe cient
g1,2	–	degeneracies of lower/upper laser level

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38	1.1.8 Notations	[Ref. p. 40

gh

ginh

H

H0

H0

Hint

h(ω, ωA) j

J

J

J +, J −

Js, Js4

k k k0

n nr n0

n1,2

PA,real

PA

PA0

PH

R

R

R1,2

r S

T1

T2

T2

T2

Tsp

Tπ , T2 π V

v Z Z0

α

χA χe χH χm

δ

∆n

∆tr

∆ωA

∆ωC

m−1	gain coe cient of a homogeneously broadened
m−1	transition
	gain coe cient of an inhomogeneously broadened
	transition
A/m	magnetic field
A/m	magnetic-field amplitude
VAs	Hamilton operator of the undisturbed transition
VAs	Hamilton operator of interaction
s	line shape factor
A/m2	current density
Vs/m2	magnetic polarization
VA/m2	intensity
VA/m2	intensity inside the resonator
VA/m2	saturation intensity of 2-, 3- and 4-level system
m−1	wave number
m−1	wave vector inside the medium
m−1	wave vector in vacuum
m	geometrical length of the active medium
–	complex refractive index
–	real refractive index
m−3	density of active atoms
m−3	density of lower/upper population
As/m2	real polarization of the active atoms
As/m2	complex polarization of the active atoms
As/m2	amplitude of the complex polarization
As/m2	complex polarization of the host material
As/m2	Feynman vector
–	= √		, average mirror reflectivity
		R1 R2
–	reflectivity of mirror 1, 2
m	position vector
VA/m2	Poynting vector
s	upper-laser-level life time
s	dephasing time due to homogeneous broadening
s	dephasing time due to inhomogeneous broadening
s	resulting dephasing time
s	spontaneous decay time
s	pulse duration of π-, 2 π-pulses
–	resonator loss factor per transit
m/s	pulse peak velocity
V/A	impedance
V/A	vacuum impedance
m−1	absorption coe cient
–	susceptibility of the active atoms
–	electric susceptibility
–	susceptibility of the host material
–	magnetic susceptibility
s−1	detuning
m−3	inversion density
m−2	transverse delta-operator
s−1	line width of homogeneous broadening
s−1	line width of collision broadening

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Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	39

∆ωR

∆ωS

∆ωL,inh, ∆ ωL,h

ε

ε0

|ϕ |ϕ1,2

κ

λ0

Λ

µ

µ0

µ12, µ21

µA

θ

ρω σ(ω) σe σ0

τ

ω

ωA

ωR

s−1 s−1 s−1

–

8.8542 × 10−12 As/Vm

–

–

1.38 × 10−23 VAs2/K

m s−1

–

4 π × 10−7 Vs/Am Asm

Asm

–

VAs2/m3 m2

A/Vm m2

s s−1 s−1

s−1

line width of inhomogeneous broadening line width of saturation broadening

lasing bandwidth of inhomogeneous/homogeneous transitions

permittivity electric constant

state vector of the two-level system eigenfunctions of the two-level system Boltzmann’s constant

vacuum wavelength Rabi frequency permeability magnetic constant

= µA, dipole moment of the two-level transition dipole moment of the two-level transition

beam divergence, slope of the Feynman vector spectral energy density (per dω)

cross section of the two-level system electric conductivity

cross section of the two-level system in resonance pulse width

frequency of the radiation field

resonance frequency of the homogeneously broadened transition

resonance frequency of the inhomogeneously broadened transition

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40	References for 1.1

References for 1.1

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