Figure 1 THG in homogeneous silica sample. (a) Scheme for self-induced QPM. (b) Intensity of the pump at =12.4
0
. (c) Power of the
TH beam at the output vs. the angle, , normalized to the power in the “standard” case; L=lc, and d=0. (d) power spectra of TH beams at
angles that give rise to phase-matching. Note that each line-type corresponds to different angle, marked in (c). Intensity (e) and scheme
for the phase-matching (f) for THG at =16
0
. Intensity patterns of the TH beams at =19.7
0
(g) and =12.4
0
(h).
Phase-matching in isotropic and homogeneous materials via
Talbot effect
Oren Cohen, Tenio Popmintchev, Margaret M. Murnane, and Henry C. Kapteyn
JILA and Department of Physics, University of Colorado, Campus Box 440, Boulder, CO, 80309-0440
coheno@colorado.edu
Abstract: We suggest a method for obtaining phase-matching of harmonic generation processes
in homogeneous isotropic materials. The intensity of the pump beam is modulated along the
direction of propagation giving rise to QPM-like signal generation.
©2006 Optical Society of Americ
OCIS codes: (190.4410) Nonlinear optics, parametric processes, (190.4160) Multiharmonic generation
Phase matching is critical for the efficient implementation of any nonlinear parametric process, e.g.
harmonics generation. The most common approach for obtaining phase matching is through birefringence. However,
this technique is limited to anisotropic materials. In isotropic materials, on the other hand, efficient harmonics
generation is possible in inhomogeneous structures, e.g., waveguides [1] and quasi-phase matching (QPM) samples
[2]. QPM makes use of modulation of the generation process (e.g. by modulating the sign of the relevant nonlinear
coefficient or the intensity of the pump beam [3]), enabling to compensate for the intrinsic phase-mismatch by a
QPM wave vector.
Here, we suggest a technique for obtaining phase-matched frequency conversion in homogeneous and
isotropic materials (e.g., glass). The distinguishing feature in this method is that the intensity of the pump beam is
modulated through the spatial or temporal Talbot effect. In this case, harmonics are generated predominantly at
within confined regions where intensity is largest, giving rise to QPM-like effects. We specifically discuss phase-
matched processes in two different regimes. First, we consider third harmonic generation (THG) in fused silica
(SiO
2
), employing the spatial Talbot effect. Then, we discuss phase matching of high harmonic generation (HHG) in
plasmas, making use of the temporal Talbot effect.
For the THG example, we assume a non-depleted pump beam at
1
=1.5m in the form of
. . exp 2 exp 2 cos exp
2 2
1 1
c c t i d k i d x z ik E
P
where , 2
1 1 1
n k
1
2 c , c is the speed of
light in vacuum, and n
1
=1.4451 is the index of refraction at 1.5m. Such a beam can be obtained by imposing three
plane waves (Fig. 1a) with =sin
-1
(
1
/d). Upon propagation, the intensity pattern of such a pump beam, and
therefore also the nonlinear polarization, has period Z
T
2d
2
n
1
/n
2
where Z
T
is the so-called Talbot distance. For
example, Fig 1b shows the intensity for =12.4
0
. We consider third harmonic generation employing
(3)
in the non-
depleted pump approximation, and including diffraction. Consider a nonlinear medium of length L=1mm,
a633_1.pdf
CThEE4.pdf
© 2006 OSA/CLEO 2006