Page 338 - High Power Laser Handbook

306 So l i d - S t at e La s e r s Ultrafast Solid-State Lasers 307

where ϕ (w) is the material phase delay, L is the material length,
m
m
n(w) is the index of refraction, θ is the refracted angle inside the mate-
rial, and θ is incident angle.
i
The beam layout of the stretcher can be understood as follows:
The beam from the oscillator is directed onto the first diffraction grat-
ing and is then imaged via the focusing optics onto a second grating.
The beam is then retroreflected back through the grating pair, which
returns all the frequencies to a single spatial mode. It is critical that
the focusing optics are separated by 2f. If they are not, the output
beam will not return to the same spatial mode, leading to a condition
known as spatial chirp, which is a very undesirable frequency sweep
across the laser beam. When the gratings are placed at points other
than the object and image planes, the path lengths for the lower and
higher frequencies are different, which causes a temporally chirped
pulse to emerge from the stretcher. The degree of chirp depends on L
[see Eq. (12.3)], or the distance from the gratings to the image and
object planes. In the stretcher, the gratings are placed inside the image
and object planes, resulting in a shorter path length for the redder
wavelengths, and thus a positive chirp. Typically, a stretcher is aligned
to stretch a 15- to 20-fs pulse to a 150- to 400-ps pulse.
Dispersion is necessarily introduced whenever the beam passes
through any material; this is due to variations in refractive index over
the beam’s wavelength range [Eq. (12.5)]. Dispersion further posi-
tively chirps the pulse. However, higher-order terms of this disper-
sion are difficult to counteract when recompressing the pulse.
Therefore, curved mirrors, rather than lenses, are typically used as
the focusing optics in the stretcher design. After amplification, the
pulses will accumulate a certain amount of phase distortion, which is
defined as high-order phase terms that cannot be compensated for by
the stretcher and compressor. However, a slight mismatch in the inci-
dent angle and L of the compressor can compensate terms up to the
third order in the Taylor expansion of the total phase of the system
(stretcher, amplifier material, and compressor), given by

1
ϕ () = w ϕw ) + ( ϕw )( ww ) + ϕw )( ww ) (12.6)
−
2 2
( ′′
( ′
−
sys 0 0 0 2! 0 0
1
+ ϕ ′′′ w ( ww ) 3 + L
− )(
3! 0 0
where ϕ (w) is the total system phase delay and w is the pulse center
0
sys
frequency. The first two terms are constants related to the absolute
time delay of the pulses, and the ϕ" and ϕ"' terms are the group-
velocity dispersion (GVD) and third-order dispersion (TOD), respec-
tively. These terms, and their effect on the resulting output pulses,
will be discussed in Sec. 12.5.