Focusing Ultrashort Pulses

1
Focusing Ultrashort Pulses
Chromatic aberration
Radially varying group delay
Achromatic lenses
Radially varying group-delay dispersion
2
Focusing Issues
Why focus an ultrashort pulse? Theyre high
power already focusing can yield ultrahigh
intensity. 1 mJ / (10 fs)
/ (3 mm)2 1018 W/cm2 But ultrashort pulses are
broadband, so a lens that focuses a single color
well wont necessarily focus an ultrashort pulse
well. Well need to keep the pulse short at the
focus, which wont be trivial. Spatio-temporal
distortions will occur.
3
Chromatic aberration distorts the pulse in space
and time.
An ultrashort pulse is broadband, and different
frequencies can focus at different points.
The lens refractive index is higher
for blue, so f is
smaller for
blue.
Parallel red and blue input rays
f(l)
Chromatic aberration distributes the focused
energy over a larger region than desired.
4
Chromatic aberration (contd)
Compute the change in focal length with
wavelength
The bandwidth, Dl, is
(for a Gaussian pulse)
And because
The lens-makers formula
Substituting
Simplifying
5
Chromatic aberration example
Fused silica lens focusing a 50-fs pulse from a
KrF laser
n 1.51 l dn/dl -0.17 l 248 nm pulse
length 50 fs f 30 mm desired focal spot
size radius, w0 0.6 µm
Is this a lot or a little? Compare with the
pulse Rayleigh range
So its a lot!
6
Example
Focusing a 6-fs 620-nm pulse with a 30-mm
focal-length singlet lens (n 1.51554, l0
dn/dl -0.03613, df/dl 2.102236)
7
Radially varying group delay also affects the
pulse focus.
For a lens, the phase delay at the focus is
independent of input radial position (r), so the
phase fronts are flat there.
L(r) Lens thickness vs. radial co-ordinate
Phase fronts
r
But the group velocity differs from the phase
velocity, so the intensity fronts (pulse
fronts) will not be flat.
8
Understanding the effects of radially varying
group delay on the focus.

• The group velocity is less than the phase
  velocity, so the more glass the later the pulse
  arrival time.
• Radially varying group delay lengthens and
  distorts the focus.

Longer pulse at the focus
9
Understanding the effects of radially varying
group delay on the focus.
The pulse fronts will become distorted and lag
behind the phase fronts, especially in the beam
center, where the lens is thickest
Interestingly, well be able to avoid this effect
and chromatic aberration in the same way!
10
Group vs. phase delay in a lens
The difference in propagation time between the
phase and intensity
where v??? is the phase velocity and vg is the
group velocity.
L(r) lens thickness vs. radial
co-ordinate, r
and where
r0
r
Lens axis
R1 Front-surface radius of curvature R2
Back-surface radius of curvature.
For this lens, R2 lt 0.
11
Group and phase delays in a lens (contd)
Expressions for the phase velocity, v???, and the
group velocity, vg
So
Substituting for L(r) and the inverse-velocity
difference
12
Practical formula/example focusing a UV pulse
Using lens-makers formula again
?
?
So the difference in group delay at the lens
edge, r0 , and on axis, 0, is
(The phase delays are equal and cancel out)
Example
Fused silica lens focusing a 50-fs pulse from a
KrF laser
n 1.51 l dn/dl -0.17 l 248 nm, pulse
length 50 fs f 30 mm, desired focal spot
size radius 0.6 µml input spot size radius 4
mm (required for 0.6 µm focus)
13
Group and phase delays in a lens (contd)

• Now rewrite Dt(r) using the lens-makers formula

?
Recalling the lens phase-minus-group time delay
Substituting for dn/dl 1/R1 1/R2 in the time
delay
This result relates the difference between the
group and phase delays to the chromaticity of the
lens.
And it says that an achromatic lens (for which f
is independent of l) has radially independent
group delay and hence flat pulse fronts!!! So an
achromatic lens solves two problems!!
14
Achromatic lenses solve two problems.
Combining two lenses in a doublet can create a
lens that is achromatic (to first order) and
that cancels out radially varying group delay.
15
Achromatic lenses solve two problems (contd)
The paths traveled through the lenses are
As a result, doublets have a phase-minus-group
delay difference with additional terms
Note that only the third term depends on the
radial co-ordinate, r, and this term is zero when
first-order achromaticity exists. Unfortunately,
the other terms have radially varying group delay
dispersion, which also distorts the pulse in time.
16
Radially varying group-delay dispersion also
broadens the pulse in time at the focus.

• Propagating through the lens also chirps the
  pulse due to group-velocity dispersion (GVD).
• And the magnitude of the chirp depends on radial
  position.

Without GVD
With GVD
Kempe, et al., JOSAB, 9, 1158 (1992)
17
Avoiding GVD in lenses (maybe)
You might think that a Fresnel lens, which has no
group velocity component to the delay, would
solve the problem...
But now the pulse fronts lead the phase fronts!
You cant win!
Z. Bor, Opt. Lett., 14, 119 (1989)
18
Focusing pulses numerical results
Solving the Fresnel Integral for the intensity
vs. position and time
19
How to focus an ultrashort pulse

• You cannot do it perfectly and easily. Options
• 1) Use an achromatic lens and pre-compensate for
  the average GVD.
• You cant really do this exactly.
• And some pulses have more bandwidth than any
  lens is designed for.
• 2) Use a curved mirror.
• On axis (Cassegrain design), the beam center is
  blocked.
• Off axis, astigmatism is a problem. So use an
  off-axis paraboloid, but it is very hard to
  align and manufacture.

20
Even a perfectly achromatic lens with no GVD may
not focus the way youd like
Different colors focus to different spot sizes
w0
f
w1
You might wish to begin with a beam with a
color-dependent spot size
21
Everything weve just said about focusing also
applies to collimating a diverging pulse.

• The beam divergence angle q depends on l
• q 2l/pw, where w beam spot size.
• So if w is independent of l, and l ranges from
  400 nm to 1600 nm, q varies by a factor of 4.
• To collimate such a beam, the lens focal length
  will have to depend on l.
• Such lenses do not yet exist. Worse, w
  typically wont be independent of l.