86 Femtosecond Optics
The example illustrates the differences between peak intensity reduction
at the focal point of a lens resulting from the difference between group and
phase velocity and effects of GVD in the lens material. The latter is strongly
chirp dependent, while the former is not. The spread of pulse front arrival times
in the focal plane is independent of the pulse duration and is directly related to
the spot size that will be achieved (the effect is larger for optical arrangements
with a large F-number). The relative broadening of the focus, ∝ T
/τ
p
, is how-
ever larger for shorter pulses. The GVD effect is pulse width dependent, and, in
typical materials, becomes significant only for pulse durations well below 100 fs
in the VIS and NIR spectral range.
2.4.2. Space–Time Distribution of the Pulse
Intensity at the Focus of a Lens
The geometrical optical discussion of the focusing of ultrashort light pulses
presented previously gives a satisfactory order of magnitude estimate for the
temporal broadening effects in the focal plane of a lens. We showed this type
of broadening to be associated with chromatic aberration. Frequently the exper-
imental situation requires an optimization not only with respect to the temporal
characteristics of the focused pulse, but also with respect to the achievable
spot size. To this aim we need to analyze the space–time distribution of the
pulse intensity in the focal region of a lens in more detail. The general pro-
cedure is to solve either the wave equation (1.67), or better the corresponding
diffraction integral,
8
which in Fresnel approximation was given in Eq. (1.187).
However, we cannot simply separate space and time dependence of the field
with a product ansatz (1.176) because we expect the chromaticity of the lens
to induce an interplay of both. Instead we will solve the diffraction integral for
each “monochromatic” Fourier component of the input field
˜
E
0
() which will
result in the field distribution in a plane (x, y, z) behind the lens,
˜
E(x, y, z, ).
The time-dependent field
˜
E(x, y, z, t) then is obtained through the inverse Fourier
transform of
˜
E(x, y, z, ) so that we have for the intensity distribution:
I(x, y, z, t) ∝|F
−1
{
˜
E(x, y, z, )}|
2
. (2.42)
The geometry of this diffraction problem is sketched in Figure 2.12. Assum-
ing plane waves of amplitude E
0
() = E
0
(x
, y
, z
= 0, ) at the lens input,
8
For large F-numbers the Fresnel approximation may no longer be valid, and the exact diffraction
integral including the vector properties of the field should be applied.
Focusing Elements 87
y
E
0
()
Plane of lens Observation plane
T
L
(x,y)
z=0
E(x,y,)
x
z
y
r
x
Figure 2.12 Diffraction geometry for focusing.
the diffraction integral to be solved reads, apart from normalization constants:
E(x, y, z, ) ∝
c
E
0
()T
L
(x
, y
)T
A
(x
, y
)e
−i
k
2z
6
(x
−x)
2
+(y
−y)
2
7
dx
dy
(2.43)
where T
L
and T
A
are the transmission function of the lens and the aperture
stop, respectively. The latter can be understood as the lens rim in the absence of
other beam limiting elements. The lens transmission function describes a radially
dependent phase delay which in case of a thin, spherical lens can be written:
T
L
(x
, y
) = exp
−i
c
6
nL(r
) + d
0
− L(r
)
7
(2.44)
with r
2
= x
2
+ y
2
and
L(r
) = d
0
−
r
2
2
1
R
1
−
1
R
2
= d
0
−
r
2
2(n −1) f
, (2.45)
where d
0
is the thickness in the lens center. Note that because of the dispersion
of the refractive index n, the focal length f becomes frequency dependent. For a
spherical opening of radius r
0
the aperture function T
A
is simply:
T
A
(r
) =
1 for x
2
+ y
2
= r
2
≤ r
2
0
0 otherwise
(2.46)
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