320 Ultrashort Sources I: Fundamentals
Integrating this equation over a propagation distance d yields for the output
pulse I
out
(t) in terms of the input I
in
(t)
I
out
(t) −I
in
(t) =−β
2
dI
in
(t)I
out
(t). (5.83)
If the pulse modification introduced by this element is small the change in pulse
intensity can be approximated:
I(t) =−β
2
dI
2
(t). (5.84)
For counter-propagating pulses of intensities I
1
and I
2
in an optically and
geometrically thin (d τ
p
c) absorber the induced change is
I
1
(t, d) =−β
2
d
I
2
1
(t,0)+ 2I
2
2
(t,0)
. (5.85)
This follows directly from integrating Eq. (3.166) using the approximations for
thin absorbers.
5.4.3. Self-Lensing
An intensity-dependent index of refraction results in spatial phase modulation,
because of the transverse variation of the intensity, as well as in temporal phase
modulation through the time-dependent intensity of the pulses. We will consider
here laser beams with an intensity profile that peaks on-axis. The radial inten-
sity distribution causes a variation in index resulting in a wavefront curvature.
Therefore, the nonlinear element can be adequately represented by a lens with an
intensity-dependent focal distance. Self-lensing can be caused either by the Kerr
effect (nonresonant nonlinearity), or by an off-resonance saturation (resonant
nonlinearity). The calculations presented in this section will take as an example
the Kerr nonlinearity. In Chapter 3, Eq. (3.168), we derived an expression for
the radial dependence of the phase in the vicinity of the beam center, assuming
a Gaussian beam profile
ϕ(r, t) =
2πd
λ
n
nl
(r, t) =−¯n
2
2π
λ
dI
0
(t)e
−(2r
2
/w
2
0
)
≈−¯n
2
2π
λ
dI
0
1 − 2
r
2
w
2
0
.
(5.86)
Here I
0
(t) is the intensity on axis (r = 0) and w
0
is the beam waist located at the
input of a thin sample of thickness d. This expression should be compared to the
Pulse Shaping in Intracavity Elements 321
phase factor that is introduced by a thin lens of focal length f , for example [34],
T(r) = exp
ik
r
2
2f
. (5.87)
Obviously the nonlinear element acts like a lens of focal length
f =
w
2
0
4¯n
2
dI
0
. (5.88)
Note that f = f (t) is controlled by the time dependence of the pulse envelope
I
0
(t). A similar expression applies to any nonlinear change in index that results
in a parabolic radial phase dependence ϕ(r) = Br
2
. The generalized expression
for the focal length is
f =
k
2B
. (5.89)
Another example is off-resonance interaction with a saturable absorber or
amplifier.
Let us now consider the transmission of a pulse with a Gaussian beam and
temporal profile,
I(r, t) =
ˆ
I exp
2(t/τ
G
)
2
exp
−2r
2
/w
2
0
,
through a sequence of a nonlinear lens element and an aperture of radius R a
distance z away. To explain the time dependence of the transmission analytically
we will make certain restrictive assumptions. One of these assumptions is that
the beam remains Gaussian after the nonlinear element. This requires to consider
the element as a thin lens of certain focal length f . Strictly speaking, the latter is
only true in the vicinity of the beam center. It will be obvious that similar effects
occur in the general case; its treatment, however, requires a numerical approach.
The waist of the incident Gaussian beam with Rayleigh range ρ
0
= πw
2
0
/λ is
placed at the nonlinear element (z = 0). After a lens of focal length f the waist
of the Gaussian beam develops as, see for example [34],
w
2
(z) = w
2
0
1 +
(z
m
− z)
2
ρ
2
0
(5.90)
322 Ultrashort Sources I: Fundamentals
where
w
2
0
= w
2
0
f
2
f
2
+ ρ
2
0
(5.91)
is the beam waist after the lens, which occurs at a distance
z = z
m
(f ) = f
ρ
2
0
f
2
+ ρ
2
0
. (5.92)
ρ
0
= πw
2
0
/λ is the Rayleigh range of the beam after the nonlinear element.
By way of Eq. (5.88) we can write the focal length of the nonlinear element
f (t) = f
0
exp
2
t
τ
G
2
, (5.93)
where f
0
= w
2
0
/(4d ¯n
2
ˆ
I). We will consider the behavior of the lens aperture
sequence in the vicinity of the pulse center, for which we can approximate
f (t) ≈ f
0
1 + 2(t/τ
G
)
2
. (5.94)
The aperture is placed in the plane of the beam waist produced by the pulse
center, that is, at z = z
m0
= z
m
(f
0
). The power transmitted through the aperture
is then
P
out
=
R
0
rdr
2π
0
dφ
w
2
0
w
2
(z
m0
)
ˆ
I exp
−2
r
w(z
m0
)
2
=
1 − e
−2R
2
/w
2
(z
m0
)
P
in
≈
2R
2
w
2
0
(z
m0
)
P
in
, (5.95)
where the input power P
in
=
ˆ
Iπw
2
0
/2, and R w(z
m0
) was assumed to derive the
last equation. Inserting Eqs. (5.90) through (5.92) with f (t) from Eq. (5.94) into
Eq. (5.95) yields for the time-dependent transmission through the lens–aperture
sequence
P
out
P
in
≈
2R
2
f
2
0
+ ρ
2
0
w
2
0
f
2
0
1 −
2ρ
2
0
f
2
0
+ ρ
2
0
at
2
, (5.96)
where, consistent with Eq. (5.94), we have kept expansion terms up to t
2
only.
The transmission is time dependent with the maximum at the pulse center.
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