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

Ref. p. 134] 2.1 Ultrafast solid-state lasers 107
where we used the slowly-varying-envelope approximation (which is applicable for pulse durations
of more than 10 fs in the near infrared wavelength regime). Taking into account only the first-order
and second-order dispersion terms we then obtain:
[
à (z,ω) ≈ 1 − ik nΔ ′ ωz − i 1 ]
2 k′′ nΔ ω 2 z à (0,ω) . (2.1.39)
The linear term in Δ ω determines the propagation velocity of the pulse envelope (i.e. the group
velocity υ g ) and the quadratic term in Δ ω determines how the pulse envelope gets deformed due to
second-order dispersion. The influence of higher-order dispersion can be considered with more terms
in the expansion of k n (ω) (2.1.37). However, higher-order dispersion only becomes important for
ultrashort pulse generation with pulse durations below approximately 30 fs depending how much
material is inside the cavity. Normally we are only interested in the changes of the pulse envelope
and therefore it is useful to restrict our observation to a reference system that is moving with
the pulse envelope. In this reference system we only need to consider second- and higher-order
dispersion. In the time domain we then obtain for second-order dispersion:
]
A (z,t) ≈
[1+iD ∂2
∂t 2 A (0,t) , D ≡ 1 2 k′′ nz = 1 d 2 φ
2 d ω 2 , (2.1.40)
where D is the dispersion parameter which is half of the total group delay dispersion per cavity
round trip. Therefore we obtain for the change in the pulse envelope:
ΔA ≈ iD ∂2
A. (2.1.41)
∂t2 2.1.6.2.5 Self-phase modulation (SPM)
The Kerr effect introduces a space- and time-dependent refractive index:
n (r, t) =n + n 2 I (r, t) , (2.1.42)
where n is the linear refractive index, n 2 the nonlinear refractive index and I (r, t) the intensity
of the laser beam, typically a Gaussian beam profile. For laser host materials, n 2 is typically of
the order of 10 −16 cm 2 /W and does not change very much for different materials. For example,
for sapphire n 2 =3× 10 −16 cm 2 /W, fused quartz n 2 =2.46 × 10 −16 cm 2 /W, Schott glass LG-760
n 2 =2.9 × 10 −16 cm 2 /W, YAG n 2 =6.2 × 10 −16 cm 2 /W, and YLF n 2 =1.72 × 10 −16 cm 2 /W.
The nonlinear refractive index produces a nonlinear phase shift during pulse propagation:
φ (z,r,t) =−kn(r, t) z = −k [n + n 2 I (r, t)] z = −knz − δ L |A (r, t)| 2 , (2.1.43)
where δ L is the Self-Phase Modulation coefficient (SPM coefficient):
δ L ≡ kn 2 z/A L , (2.1.44)
where A L is the laser mode area inside the laser medium. Here we assume that the dominant SPM
inside the laser occurs in the gain medium. In this case, z is equal to twice the laser material
length. Of course the mode area can be also very small in other materials. In this case, we will
have to add up all the SPM contributions inside the laser resonator. The laser mode area A L is an
“averaged value” in case the mode is changing within the gain medium.
The electric field during propagation is changing due to SPM:
E (z,t) =e iφ E (0,t) ∝ e −iδL|A(t)|2 A (0,t)e iω0t−ikn(ω0)z . (2.1.45)
For δ L |A| 2 ≪ 1 we obtain:
Landolt-Börnstein
New Series VIII/1B1