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Shortening a laser pulse at the focus of a lens
Yuelin LiAccelerator Systems DivisionArgonne National Laboratoryylli@aps.anl.gov
2CLEO 08, San Jose, May 4-9, 2008
Content Introduction
– Pulse lengthening at lens focus Fourier formulation of the problem
– Axial temporal history
– Pulse shortening due to GVDE and SPM
– Numerical simulation Experiment outlook Summary
3CLEO 08, San Jose, May 4-9, 2008
Content Introduction
– Pulse lengthening at lens focus Fourier formulation of the problem
– Axial temporal history
– Pulse shortening due to GVD
– Numerical simulation Experiment outlook Summary
4CLEO 08, San Jose, May 4-9, 2008
Pulse lengthening due to Group Velocity Delay
Z. Bor, Opt. Lett. 15, 119 (1989)
5CLEO 08, San Jose, May 4-9, 2008
Content Introduction
– Pulse lengthening at lens focus Fourier formulation of the problem
– Axial temporal profile
– Pulse shortening due to GVD, with SPM
– Numerical simulation Experiment outlook Summary
6CLEO 08, San Jose, May 4-9, 2008
Formula of the problem: Fourier optics
Full wave optics (Fresnel diffraction) adapted from Kempe et al. (JOSA B 9, 1158 (1992))
Group velocity dispersion and group velocity delay effect considered up to the second order
7CLEO 08, San Jose, May 4-9, 2008
On axis formula
),()(2)( rrdrAU
2
22
exp
)1(2)(exp
2exp),(
TTjk
fn
rkkjr
f
kjdjkr al
al
U: field in frequency domain representation at the focus f : is the focal length r: ray location Kl, ka: wave vectors in the lens and air,n: refractive index in the lens. A(): input filed (homogeneous) in the frequency domain. , Lens transfer function
r
rdratu 2)(
Inverse Fourier transform
a=a(t)=F-1A() and =(r,t)=F-1(r,)
Field at focus, on axis
.2
11,,
)1(2)(
2
22
d
nd
d
dn
fn
rdrT
df
8CLEO 08, San Jose, May 4-9, 2008
Pulse with no SPM
For a=a0exp(-2ln2t2/2), no SPM,
,)(tanexp)1(
)1(2ln2exp
1
12)( 1
22
2
4/120
jjTkt
rdratu
=4kT/2.
.2
11,,
)1(2)(
2
22
d
nd
d
dn
fn
rdrT
Chirping and broadening due to Group Velocity DIspersion (GVDI)
Relative delay between pulse slices due to Group Velocity DElay (GVDE)
GVDE and GVDI have both been studied.
9CLEO 08, San Jose, May 4-9, 2008
Max phase modulation, over lens of thickness d:
.2ln4expexp2ln2exp)(
2
2
2
2
0
Tkt
jTkt
rdratu
2
2
2
2
0 2ln4expexp2ln2exp),(t
jt
atra
Pulse with SPM
Assume smaller bandwidth and d2/dn2=0,
=4kT/2.
.2
11,,
)1(2)(
2
22
d
nd
d
dn
fn
rdrT
Delay due to GVDE
Case not been studied.
Tkttr ),(
Interference due to GVDE
dakn 202
10CLEO 08, San Jose, May 4-9, 2008
Pulse shortening at the focus
-2 -1 0 1 2
0.0
0.5
1.0
-2 -1 0 1 2
0
4
8
12
(a)
r (m
m)
(c)
|u|2 (
a.u
.)
t (ps)
(b)
-1
-0.5
0
0.5
1
(d)
t (ps)
Li and Crowell, Opt. Lett. 32, 92 (2007).
The real part of the complex field arriving at the focus as a function of time and radius for =T/d (a) and =1 (b), and the corresponding intensity of the integrated field (solid) and the input pulse (dashed) as a function of time for the same cases [(c) and (d)]. The FWHM of the pulse is shortened from 1 ps to 210 fs (c) and 200 fs (d), a reduction by a factor of 5. The calculation assumes an f=150 mm lens with R=12 mm and d=5 mm. The pulse wavelength is 0.249 nm with =15 at laser intensity of 5×1011 W/cm2.
11CLEO 08, San Jose, May 4-9, 2008
Parameter f/i
0 20 40 600.0
0.5
1.0
0 20 40 600.0
0.5
1.0
(rad)
0
0.2
0.4
0.6
0.8
1.0
R
NAfn
R2
)1(2
2
dakn 202
Max GVDE to pulse duration ratio
Max phase modulation
=2
=3
12CLEO 08, San Jose, May 4-9, 2008
Numerical simulation
-2 -1 0 1 2-0.6
-0.3
0.0
0.3
0.6
(a)
t (ps)
z (
mm
)
0
0.2
0.4
0.6
0.9
1.1
0
2
4
(b)
r (m
)
-2 -1 0 1 20
2
4
(c)
t (ps)
Li and Crowell, Opt. Lett. 32, 92 (2007).
On-axis laser pulse envelope as a function of the defocusing distance; the intensity as a function
of time and radius for a pulse without (b) and with the SPM effect (c).
13CLEO 08, San Jose, May 4-9, 2008
Content Introduction
– Pulse lengthening at lens focus Fourier formulation of the problem
– Axial temporal history
– Pulse shortening due to GVD
– Numerical simulation Experiment outlook Summary
14CLEO 08, San Jose, May 4-9, 2008
An envisaged experiment
C
AL
ZSL
SF
PP
D
ODL
15CLEO 08, San Jose, May 4-9, 2008
Summary
Pulse shortening demonstrated in simulation, up to 5 time shortening observed
Potential for few cycle pulse generation with simpler setup