328 Ultrashort Sources I: Fundamentals
In the case of passive mode-locking, the ratio of beam waists in the
absorber–amplifier is important, because this is the parameter that deter-
mines the relative saturation of the gain and absorber.
• a small spot size in the amplifier may be desirable for efficient pumping and
heat removal.
• a round spot is desirable in the passive mode-locking element (for instance,
the amplifier rod in the case of a Ti:sapphire laser, or the saturable absorber
jet in a dye laser) to have the most uniform possible wavefront across the
beam, because it is a region of the cavity which contributes to the phase
modulation.
Mirrors or lenses can be used to create beam waists in a laser cavity. Because
of the requirement of minimum losses and dispersion, one will generally choose
reflective optics over lenses.
5.5.2. Astigmatism and Its Compensation
It is not always an easy task to create a waist of minimal size with off-axis
reflective optics. Indeed, let us consider the typical focusing geometry sketched
in Figure 5.17. The smaller the focal spot in A, the larger the diameter w of the
incident beam on the mirrors, hence the larger the clearance angle θ required to
have the focal point fall outside of the incident beam cross section. However,
the astigmatism caused by a large angle of incidence θ will make it impossible
to obtain the desired small focal spot with a cylindrically symmetric Gaussian
beam incident from the left.
Because a tight focusing is required in the nonlinear elements of a mode-locked
laser, there is clearly a need for minimizing or reducing the astigmatism. There are
A
B
2
Figure 5.17 Off-axis focusing of a Gaussian beam leading to astigmatism. In the plane of the
figure the focal distance of the mirror is (R/2) cos θ, where R is the radius of curvature of the mirror.
The first focalization is therefore a line perpendicular to the plane of the figure originating from A.
In the orthogonal plane the focal distance of the mirror is (R/2)/ cos θ. There will therefore be a focal
line in the plane of the figure at B.
Cavities 329
some exceptions. A large astigmatism may sometimes be desirable in the gain
medium. This is the case when it is desirable to take maximum advantage of a
self-lensing effect. Another example of such a need is to match the elongated
shape of the gain region of semiconductor lasers.
Let us choose as transverse coordinate y for the plane of incidence (the plane
of the figure in Fig. 5.17), and x for the orthogonal direction. The locations of
the two focal lines corresponding respectively to the plane of the figure and to
the orthogonal plane are:
f
y
= f cos θ
f
x
=
f
cos θ
, (5.106)
where θ is the angle of incidence and f = R/2 is the focal length of the mirror
(see for example Kogelnik et al. [37]).
Other elements in a cavity, such as Brewster plates, also have astigmatic
properties that can limit the performance of the system. The gain medium of a
Ti:sapphire laser or of a dye laser is generally a plane parallel element put at
Brewster’s angle. Kogelnik et al. [37] have shown under which condition the
astigmatism can be compensated by such elements.
To analyze the astigmatism of such elements let us consider the propagation
of a Gaussian beam through a plate of thickness d and refractive index n put at
the Brewster angle directly next to the beam waist (size w
0
)atz = 0, as sketched
in Figure 5.18.
On entering the medium, the beam waist takes the values:
w
0x
= w
0
w
0y
= w
0
cos θ
r
cos θ
B
= w
0
sin θ
B
cos θ
B
= nw
0
, (5.107)
where θ
B
= arctan n is the Brewster angle, and θ
r
= 90
◦
− θ
B
is the angle of
refraction. To a thickness d, there corresponds a propagation distance
χ =
d
cos θ
r
= d
√
1 + n
2
n
. (5.108)
Applying the propagation law (1.182) for the beam waist across the thickness d
yields:
w
x
= w
0
;
<
<
=
1 +
λ
χ
nπw
2
0
2
= w
0
;
<
<
=
1 +
λ
πw
2
0
d
√
1 + n
2
n
2
2
(5.109)
Get Ultrashort Laser Pulse Phenomena, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
