Ref. p. 40] |
1.1 Fundamentals of the semiclassical laser theory |
17 |
||||
Steady-state equations |
∂∆n |
|||||
The temporal variations of the radiation field are slow |
= 0 |
∂P A0 |
= 0 |
|||
∂t |
∂t |
|||||
compared with T1. |
||||||
Adiabatic equations |
∂∆n |
|||||
no transient e ects of the atom, T2 T1. |
= 0 |
∂P A0 |
= 0 |
|||
∂t |
∂t |
|||||
Coherent equations |
∂∆n |
|||||
The width τ of the interacting pulses is short compared |
= 0 |
∂P A0 |
= 0 |
|||
∂t |
∂t |
|||||
with T1, T2; (1.1.45a), (1.1.45b) can be applied. |
||||||
In steady state inversion density ∆n0, polarization P A0, and intensity J of the field are constant in time, but may depend on the spatial coordinates.
The stationary solutions of (1.1.48a), (1.1.48b) are obtained immediately:
∆n = |
∆n0 |
(inversion density, homogeneously broadened), |
|||||||||||||||
1 + (J/Js) f (ω) |
|||||||||||||||||
A |
k0 |
∆ωA/2 |
|||||||||||||||
χ |
= |
nrσ |
ω − ωA |
+ i |
∆n (susceptibility), |
||||||||||||
P A0 = ε0χAE0 |
(polarization) |
||||||||||||||||
with |
|||||||||||||||||
J = |
1 |
ε0c0nr|E0|2 |
(intensity of the field), |
||||||||||||||
2 |
|||||||||||||||||
Js = |
ωA |
(saturation intensity of the two-level transition), |
|||||||||||||||
2σ0T1 |
|||||||||||||||||
σ = σ0f (ω, ωA) |
(frequency-dependent cross section of the transition), |
||||||||||||||||
σ0 = |
| µA|2 ωAT2 |
(cross section in resonance), |
|||||||||||||||
ε0c0nr |
|||||||||||||||||
fL(ω, ωA) = |
(∆ωA/2)2 |
(spectral line shape, Lorentzian), |
|||||||||||||||
(ωA − ω)2 |
+ (∆ωA/2)2 |
||||||||||||||||
∆ωA = 2/T2 (line width of the transition),
(1.1.49)
(1.1.50)
(1.1.51)
(1.1.52)
(1.1.53)
(1.1.54)
(1.1.55)
(1.1.56)
(1.1.57)
Landolt-B¨ornstein
New Series VIII/1A1
18 |
1.1.4 Steady-state solutions |
[Ref. p. 40 |
|||||||||||||
gh(ω, ωA) = ∆ n σ = |
∆ n0σ0f (ω, ωA) |
(gain coe cient, homogeneously |
(1.1.58a) |
||||||||||||
1 + (J/Js) f (ω, ωA) |
|||||||||||||||
broadened), |
|||||||||||||||
g |
inh |
(ω, ω |
R |
) = |
∆ n0σ0 |
h(ω, ω |
R |
) |
∆ωA |
(gain coe cient, inhomogeneously |
(1.1.58b) |
||||
1 + J/Js |
2 |
broadened, see Sect. 1.1.6.3). |
|||||||||||||
In Table 1.1.3 some numbers of relevant laser transitions are compiled, in Table 1.1.4 some typical values of the small-signal gain coe cient in resonance are given. The susceptibility strongly depends on the frequency as shown in Fig. 1.1.6. According to (1.1.26) the real part of χA produces an additional refractive index, and the imaginary part absorption or amplification:
A |
r − |
k0 ∆ωA/2 |
|||||||
Re χ |
= n2 |
1 = |
nrσ |
ω − ωA |
∆n , |
||||
Im χA = −nrαk0 = |
nrσ |
||||||||
∆n . |
|||||||||
k0 |
|||||||||
The steady-state propagation of the electric field is obtained from (1.1.48c):
dz |
− 2 |
2 |
∆ ωA |
0 |
|||||||
dE0 |
= |
α |
+ |
σ∆ n |
+ iσ∆ n |
ω − ωA |
E |
, |
|||
where ∆ n is a function of the field or the intensity.
(1.1.59a)
(1.1.59b)
(1.1.60)
Table 1.1.3. Examples of resonance wavelength λ0, resonance cross section σ0, upper-level lifetime T1 and saturation intensity Js. The simple relation (1.1.53) for the saturation intensity holds for two-level systems only and is not applicable in general [01Men].
λ0 |
σ0 |
T1 |
Js |
|||
[µm] |
[m2] |
[s] |
[W/m2] |
|||
Amplifiers |
10−20 |
10−5 |
× 105 |
|||
CO2-gas (1300 Pa) |
10.6 |
2 |
||||
Neodymium-ion in glass |
1.06 |
4 × 10−24 |
3 |
× 10−4 |
8 . . . 12 × 107 |
|
Neodymium-ion in YAG |
1.06 |
5 × 10−23 |
2 |
× 10−4 |
2 |
× 107 |
Chromium-ion in Al2O3 |
0.69 |
2 × 10−24 |
3 |
× 10−3 |
2.4 × 107 |
|
(ruby, T = 300 K) |
3 × 10−17 |
10−8 |
5.3 × 105 |
|||
Neon (25 Pa) |
0.63 |
|||||
Rhodamine 6G in ethanol |
0.57 |
4 × 10−20 |
5 |
× 10−9 |
109 |
|
Absorbers |
8 × 10−22 |
× 10−4 |
2.5 × 105 |
|||
SF6 |
10.6 |
4 |
||||
KODAK dye 9860 |
1.06 |
4 × 10−20 |
10−11 |
5.6 × 1011 |
||
KODAK dye 9740 |
1.06 |
6 × 10−20 |
10−11 |
4 |
× 1011 |
|
Cryptocyanine-dye |
0.7 |
5 × 10−20 |
5 |
× 10−10 |
2 |
× 1010 |
in methanol
Table 1.1.4. Typical values of the small-signal gain coe cient g0 = ∆n0σ0 in resonance. The exact values depend on pumping, doping, and other parameters of operation [01Men].
System |
λ0 [nm] |
g0 [m−1] |
He/Ne laser |
632.8 |
0.1 |
Nd-doped glass |
1060 |
5 |
Nd-doped YAG |
1060 |
50 |
GaAs-diode |
880 |
4 × 103 |
Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 40] |
1.1 Fundamentals of the semiclassical laser theory |
19 |
,Pχ$ JDLQ
∆ω$
5H χ$ SKDVH VKLIW
ω $ |
Fig. 1.1.6. Real and imaginary part of |
||
)UHTXHQF\ω |
|||
the susceptibility vs. frequency. |
|||
The solutions for low intensities are discussed. Low means that the intensity J is small compared with the characteristic parameter Js of the system (see Table 1.1.3).
At low intensities J Js, the inversion density is not a ected by the intensity,
∆n = ∆n0 ,
and (1.1.60) can be integrated. Together with (1.1.23), the complete field is obtained:
E(z) = E0(0) exp[i(ωt − ntk0z) − |
1 |
(α − ∆n0σ)z] |
(1.1.61) |
||||||||
2 |
|||||||||||
with a total refractive index nt |
|||||||||||
t |
r |
nrk0 |
∆ωA |
||||||||
n |
= n |
1 + |
σ∆n0 |
ω − ωA |
. |
(1.1.62) |
|||||
The active atoms of the two-level system cause an additional phase shift or refractive index and an additional absorption or amplification, depending on the sign of ∆n0. The small-signal gain factor according to (1.1.30)/(1.1.50) is:
G0 = exp[σ(ω)∆n0z] . |
(1.1.63) |
Amplification, G0 > 1, requires inversion ∆n0 > 0. The complex amplitude transmission factor A is defined as the ratio of the monochromatic field amplitudes and can be written:
A = E0(0) |
= exp i |
2 |
(ω − ωA) + i ∆ωA/2 |
z . |
(1.1.64) |
||
E0(z) |
σ0∆n0 |
∆ωA/2 |
|||||
It depends on the frequency of the field, which means dispersion. Time-dependent fields and especially short pulses are distorted by the amplifying system, pulse broadening and chirping occur.
The steady-state solutions are discussed for intensities which saturate the inversion, see Fig. 1.1.7.
The inversion now depends on the intensity. For the propagation of the intensity, (1.1.48c) gives in steady state
Landolt-B¨ornstein
New Series VIII/1A1
20 |
1.1.5 Adiabatic equations |
[Ref. p. 40 |
|
dJ |
= (g(J ) − α) J , |
(1.1.65) |
|
dz |
where g(J ) is the saturated gain coe cient of (1.1.58a), (1.1.58b). For a homogeneously broadened transition and without losses (α = 0) this equation can be can be integrated and provides a transcendental relation for the gain factor G:
G0 |
= exp |
J (0) |
f (ω) (G − 1) |
(1.1.66) |
|
G |
Js |
||||
with G0 the small-signal gain factor of (1.1.62) and G the ratio of output/input intensities
G = J (z)/J (0) .
For inhomogeneously broadened transitions a more complicated relation is obtained [81Ver].
* |
||||||||
- -6 |
Fig. 1.1.7. Saturation of the gain factor G for a homogeneously and inhomogeneously broadened transition. 1: G0 = 1, 2: G0 = 4, 3: G0 = 6.
1.1.5 Adiabatic equations
If the polarization is in equilibrium with the applied field, without transient oscillations of the electronic system, the interaction is called adiabatic.
The field is replaced by the intensity, most spectral e ects are neglected and the rate equations are obtained. They represent an energy balance.
T2 is the time constant, which characterizes the transient behavior of the polarization. In most cases (see Table 1.1.6) T2 is much smaller than T1, and the transient oscillations of the electrons can be neglected. In (1.1.48a) the polarization is replaced by its steady-state value (1.1.50)/(1.1.51) and the rate equations are obtained. They have to be completed by the time-dependent pump term, here labeled as ∆ n0. It depends on the specific pump scheme (see Sect. 1.1.5.3). The rate equations are widely used in laser design to evaluate output power, spiking behavior and Q-switching dynamics. The spontaneous emission contributes to the intensity of the interacting field, but only with a very
Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 40] |
1.1 Fundamentals of the semiclassical laser theory |
21 |
small amount and is neglected here. Nevertheless it is important, because the laser is started by spontaneous emission and in the lower limit it determines the laser band width (Chap. 5.1).
With these approximations the field equations (1.1.48a)/(1.1.48b)/(1.1.48c) for the interaction with a monochromatic field reduce to one equation for the inversion density and a transport equation for the intensity:
∂∆n |
= |
− |
J f (ω) |
∆n |
− |
(∆n − ∆n0) |
, |
(1.1.67) |
∂t |
JsT1 |
T1 |
||||||
∂ |
+ |
1 ∂ |
J = (∆n σ0f (ω)) J |
(1.1.68) |
||||
∂z |
c ∂t |
|||||||
(rate equations for a homogeneously broadened two-level system and a plane monochromatic wave)
with
J (z, t): local intensity,
Js: saturation intensity, depends on the level system (2,3, or 4 levels), see Sects. 1.1.4.1/1.1.5.3, ∆n(z, t): local inversion density.
So far the interaction with a monochromatic field of intensity J (ω) was discussed. Now the intensity is replaced by the spectral energy density ρω of black-body radiation, providing the Einstein coe cients of spontaneous and induced emission.
Einstein published in 1917 [17Ein] his famous work on the quantum theory of radiation, where for the first time induced emission was introduced, the cornerstone of laser physics. He discussed the two-level system in equilibrium with thermal radiation of spectral energy density ρω (energy per volume and spectral range d ω). The density is given by Planck’s law [61Mor]:
ρω = |
ω3 |
1 |
VAs2 |
(1.1.69) |
|||
π2c3 |
exp [ ω/κT ] − 1 |
m3 |
|||||
with
κ = 1.38 × 10−23 VAs/K: Boltzmann’s constant.
In thermal equilibrium the levels |ϕ1 , |ϕ2 are populated according to Boltzmann’s law [61Mor]:
n2 |
= exp [− ωA/κT ] . |
(1.1.70) |
n1 |
These two fundamental laws can only be fulfilled, if induced emission is introduced, and Einstein postulated the following equation in steady state for the interaction of thermal radiation with a two-level system:
B12 ρω n1 = B21 ρω n2 |
+ A21 n2 |
(1.1.71) |
(absorption = induced emission + spontaneous emission)
with
B12, B21, A21: Einstein coe cients of induced and spontaneous emission.
The transition of atoms from the lower level to the upper level by absorption of radiation must be balanced by induced emission and spontaneous emission from the upper level. This equation was
Landolt-B¨ornstein
New Series VIII/1A1