2.1 Ultrafast solid-state lasers - ETH - the Keller Group

Ref. p. 134] 2.1 Ultrafast solid-state lasers 105
T R
∂A(T,t)
∂T
= ∑ i
ΔA i =0, (2.1.23)
where A is the pulse envelope, T R is the cavity round-trip time, T isthetimethatdevelopsona
time scale of the order of T R , t is the fast time of the order of the pulse duration, and ΔA i are the
changes of the pulse envelope due to different elements in the cavity (such as gain, loss modulator
or saturable absorber, dispersion etc.) (Table 2.1.10). Equation (2.1.23) basically means that at
steady state after one laser round trip the pulse envelope cannot change and all the small changes
due to the different elements in the cavity have to sum up to zero. Each element is modeled as a
linearized operator, which will be discussed in more detail below.
The pulse envelope is normalized such that |A (z,t)| 2 is the pulse power P (z,t):
E (z,t) ∝ A (z,t)e i[ω0t−kn(ω0)z] with |A (z,t)| 2 ≡ P (z,t) , (2.1.24)
where E (z,t) is the electric field, ω 0 the center frequency in radians/second of the pulse spectrum
and k n = nk with k =2π/λ thewavenumberwithλ the vacuum wavelength and n the refractive
index. Before we discuss the different mode-locking models we briefly discuss the linearized
operators for the differential equations.
2.1.6.2.1 Gain
A homogeneously broadened gain medium is described by a Lorentzian lineshape for which the
frequency-dependent gain coefficient g (ω) isgivenby
g
g (ω) = ( ) 2
≈ g
(1 − Δ )
ω2
ω − ω0
Ωg
2 for ((ω − ω 0 )/Ω g ) 2 ≪ 1 , (2.1.25)
1+
Ω g
where Δ ω = ω − ω 0 , g is the saturated gain coefficient for a cavity round trip, and Ω g is the Half
Width at Half Maximum (HWHM) of the gain bandwidth in radians/seconds. In the frequency
domain the pulse envelope after the gain medium is given by
à out (ω) =e g(ω) à in (ω) ≈ [1 + g (ω)] Ãin (ω) for g ≪ 1 , (2.1.26)
where à (ω) is the Fourier transformation of A (t). Equations (2.1.25) and (2.1.26) then give
[
à out (ω) = 1+g − g ] [
Ωg
2 Δ ω 2 Ã in (ω) ⇒ A out (t) = 1+g + g ∂ 2 ]
Ωg
2 ∂t 2 A in (t) , (2.1.27)
where we used the fact that a factor of Δ ω in the frequency domain produces a time derivative in
the time domain. For example for the electric field we obtain:
∂
∂t E (t) = ∂ ∂t
{ 1
2π
∫
}
Ẽ (ω)e iωt dω
= 1
2π
∫
Ẽ (ω)iω e iωt dω (2.1.28)
and
∂ 2
{
∂2 1
E (t) =
∂t2 ∂t 2 2π
∫
}
Ẽ (ω)e iωt dω = 1 ∫
2π
Ẽ (ω) [ −ω 2] e iωt dω, (2.1.29)
and similarly for the pulse envelope:
∂
∂t A (t) = ∂ { ∫
}
1
à (ω)e iΔωt dω = 1 ∫
∂t 2π
2π
à (ω)iΔω e iΔωt dω (2.1.30)
Landolt-Börnstein
New Series VIII/1B1