Multiorder nonlinear diffraction
in frequency doubling processes
Solomon M. Saltiel,1,2,* Dragomir N. Neshev,1Wieslaw Krolikowski,1
Ady Arie,3Ole Bang,4and Yuri S. Kivshar1
1Nonlinear Physics Centre and Laser Physics Centre, Centre for Ultra-high Bandwidth Devices for Optical Systems
(CUDOS), Research School of Physics and Engineering, The Australian National University,
Canberra ACT 0200, Australia
2Department of Quantum Electronics, Faculty of Physics, Sofia University, BG-1164 Sofia, Bulgaria
3School of Electrical Engineering, Tel-Aviv University, Tel Aviv 69978, Israel
4Department of Photonics Engineering, DTU Fotonik, Technical University of Denmark,
DK-2800 Kongens Lyngby, Denmark
* Corresponding author: firstname.lastname@example.org
Received December 8, 2008; revised January 22, 2009; accepted January 30, 2009;
posted February 27, 2009 (Doc. ID 105073); published March 13, 2009
We analyze experimentally light scattering from ??2?nonlinear gratings and observe two types of second-
harmonic frequency-scattering processes. The first process is identified as Raman–Nath type nonlinear dif-
fraction that is explained by applying only transverse phase-matching conditions. The angular position of
this type of diffraction is defined by the ratio of the second-harmonic wavelength and the grating period. In
contrast, the second type of nonlinear scattering process is explained by the longitudinal phase matching
only, being insensitive to the nonlinear grating period. © 2009 Optical Society of America
OCIS codes: 190.4420, 190.2620, 050.1940.
Second-harmonic generation (SHG) is one of the most
studied parametric processes in nonlinear optics. Of-
ten, to achieve efficient SHG, this process requires
periodic modulation of the ??2?components, the so-
called quasi-phase-matching (QPM) . Usually,
QPM is realized by collinear propagation of the fun-
damental frequency (FF) and second-harmonic (SH)
beams along the QPM grating vector. However, when
the FF beam and the QPM grating vector are noncol-
linear, then the SH beam diffracts from the ??2?grat-
ing in a fashion similar to a linear beam diffraction
on an index grating.
In the linear optics it is well established that
propagation of a light beam in periodic refractive in-
dex structures leads either to Bragg diffraction when
the full vectorial phase-matching condition is satis-
fied or to Raman–Nath diffraction otherwise . In
the latter case since the full vectorial phase matching
is not fulfilled, a multiorder diffraction takes place,
similar to the case of light scattering by acoustic
Analogously, nonlinear Bragg diffraction is real-
ized at a specific angle between the beams , deter-
mined by the vectorial phase matching 2k1+Gn=k2,
where Gnis the grating vector and k1, k2are the
wave vectors of the FF and SH beams, respectively
[Fig. 1(a)]. If this vectorial condition is not satisfied,
multiorder nonlinear SH diffraction should be seen,
in an analog to linear Raman–Nath diffraction [Fig.
Nonlinear Bragg diffraction has been observed in a
number of structures, including naturally laminated
crystals [3–5], as well as artificially created nonlinear
one-dimensional [6–9] and annular  ??2?gratings.
On the other hand, to the best of our knowledge, the
nonlinear Raman–Nath diffraction has not been dis-
cussed so far, simply because QPM technology is
mainly used for photonics components with collinear
propagation. In fact, the first experimental indication
of this phenomena has been reported in our recent
work , where the first- and second-order nonlin-
ear diffraction in annular QPM structure was ob-
In this Letter we study the nonlinear diffraction of
beams from a one-dimensional nonlinear grating, re-
alized by periodic spatial alternation of the sign of
the relevant components of the ??2?tensor. We ob-
serve multiple-order nonlinear diffraction in the SHG
processes, which represents the Raman–Nath contri-
bution to the process. In addition, we analyze the dif-
fracted SH beams that are governed only by longitu-
classified as noninteger order nonlinear diffraction,
having no analog in linear optics.
nonlinear Bragg and (b) Raman–Nath diffraction. (c)
Schematic of the experiment. (d), (e) Experimentally ob-
served nonlinear diffraction patterns generated in one-
(Color online) Phase-matching conditions for (a)
OPTICS LETTERS / Vol. 34, No. 6 / March 15, 2009
0146-9592/09/060848-3/$15.00© 2009 Optical Society of America
In our experiments we use periodically poled
lithium niobate (PPLN) and stoichiometric lithium
tantalate (SLT) samples (typical thickness 0.5 mm)
in the arrangement shown in Fig. 1(c). The funda-
mental pump beam from a regenerative amplifier, de-
livering 10 ps, 1 mJ pulses at 1053 nm with repeti-
tion rate of 20 Hz, is directed at a small angle with
respect to the Z axis. The laser beam is focused with
a lens creating a spot size of ?400 ?m in the plane of
In Figs. 1(d) and 1(e) we show an example of the
SH diffraction patterns obtained in one-dimensional
??2?grating ??=14.6 ?m? in PPLN. The patterns con-
sist of two types of spots: (i) central diffraction spots,
grouped around the pump position, and (ii) periph-
eral diffraction spots, situated relatively far from the
pump at both sides of the diffraction array [top and
bottom pairs in Fig. 1(d)]. The two neighboring pe-
ripheral spots at each side of the array are orthogo-
nally polarized. The spots that are furthest away
from the pump are polarized in the x–y plane, the
other spots are polarized in the x–z plane. Since the
pump is ordinary polarized, we identify the interac-
tions that are responsible for these SH spots as
O2–O1O1and E2–O1O1, respectively. The PPLN ??2?
components that contribute to the O2–O1O1process
are d22and d21, while for the E2–O1O1interaction
the second-order polarization term involves addition-
ally both d32and d31components. Similar O2–O1O1
spots have been observed recently in periodically
poled KTP , however, in contrast to our experi-
ments, they were due to second-order nonlinearities
existing only in the domain walls.
The observed nonlinear diffraction patterns can be
explained by employing the phase-matching condi-
tions shown in Fig. 2. The general vectorial phase-
matching condition 2k1+Gn+?k=k2is valid for both
types of the SHG diffraction spots (the central and
peripheral ones). Figure 2 shows one of the phase-
matching triangles (OCD) that governs the first-
order SH diffraction patterns, appearing close to the
pump. It is constructed by the vectors k2,o, 2k1,o,
BD=?kTPM, and one of the grating vectors Go. The
angular direction of the SH diffraction spot is defined
only by the transverse phase-matching (TPM) condi-
tions: sin ?m=mGo/k2,o, for the mth diffraction order,
m=1,2,3.... The external angles then are
sin ?m= m?2/?,
m = 1,2, ... ,
where ?2is the SH wavelength. This is a generic con-
dition that holds for any periodically poled sample
and does not depend on its refractive index. Our mea-
surements of the first-order SH diffraction angles for
four (PPLN and SLT) samples with different grating
periods (7.5, 9.0, 14.6, 24.1 ?m) are shown in Fig.
3(a). The experimentally measured angles are in ex-
cellent agreement with the predictions of Eq. (1). We
note that Eq. (1) is known to describe normal inci-
dence Raman–Nath diffraction in linear optics.
For normal pump incidence, ?−m=−?+m. For the
pump incident at the angle ? with respect to the Z
axis toward the X axis (in the plane X–Z), again we
have ?−m???=−?+m??? but the angles are defined by
sin ?m???=?mGo/k2,o?cos ?, derived again using the
TPM condition only. Calculating the external angles
gives that ?m?????m??=0?. The dependence ?m???
has been verified experimentally and the result is
shown in Fig. 3(b).
In summary, so far we have demonstrated that the
observed central SH multiorder diffraction [Fig. 1(e)]
is a nonlinear optical analog of Raman–Nath diffrac-
tion. In this process, the directions of the SH dif-
fracted beams can be determined by applying the
TPM conditions only.
Next, we describe the phase-matching conditions
for the peripheral SH spots that appear far from the
pump. Experiments with four different samples show
that their angular positions, ?oand ?e[defined in
Figs. 1(d) and 2], do not depend on the poling period.
This is in strong contrast to the direct angular depen-
dence of the central diffraction spots on ?, as seen
from Eq. (1). The measured angles can be explained
only if we assume that the directions of these SH
beams are defined uniquely by the longitudinal
phase-matching (LPM) condition, seen as the tri-
angle OAB in Fig. 2, constructed by the vectors k2,e,
2k1,o, FA=?kLPM, and m grating vectors, BF=mGo.
Correspondingly, the phase-matching triangle for the
ordinary SH wave is OBG.
For normal incidence, the measured angles ?oand
?eare found to be in accordance with those derived on
the basis of LPM equations,
diffraction when pump is perpendicular to the QPM grat-
ing. The two semicircles correspond to the ordinary and ex-
traordinary wave vectors.
(Color online) Phase-matching diagrams for SH
TPM-SH in four samples with different grating periods. (b)
Increase of the first-order TPM-SH external angle with
the angle of FF beam incidence for a PPLN sample
(Color online) (a) External angle of first-order
March 15, 2009 / Vol. 34, No. 6 / OPTICS LETTERS
cos ?e= 2k1,o/k2,e??e?,
These equations are also known to define the emis-
sion of Cherenkov SH generation  and noncol-
linear SHG in one- and two-dimensional random
nonlinear gratings [13–15]. In the PPLN crystal no
?ne, so that ??o????e?. The angles measured for
O2–O1O1and E2–O1O1interactions are 40.3° and
38.4°, in good agreement with theory at 1.053 ?m,
giving values of 40.64° and 38.78°, respectively.
For the pump propagating at an angle ? with re-
spect to the Z axis [see inset in Fig. 4(b)], the phase
matching for the LPM diffraction spots is obtained
from Eq. (2) by replacing 2k1with 2k1cos ?. An in-
crease of the angle ? thus leads to an increase in both
??o? and ??e?. We should mention that the angular po-
sitions ?oand ?eare determined from the direction of
the Z axis, so that increasing the angle ? leads to a
reduction of the angle between the pump and one of
the diffraction spots. This is illustrated by an addi-
tional experiment where the SH pattern is recorded
as a function of the crystal rotation around the Y
axis. The recorded images are shown in Fig. 4(a).
With the increase of the angle ?ext, the phase-
matching angles for both interactions increase. As
seen from Fig. 4, the angle ?o,e,ext−?extdecreases ini-
tially with a growth of ?extbut, after reaching mini-
mum value for ?ext?40°, it starts to grow again. The
theoretical dependencies of the external angles ?o,ext
and ?e,exton ?ext[Fig. 4(b)] are in very good agree-
ment with the experimental results [Fig. 4(a)].
Although the period of the poled grating does not
appear in Eqs. (2), the LPM signals are not visible in
experiments with single-domain crystals. On the
other hand, if both TPM and LPM are satisfied si-
multaneously (the case of nonlinear Bragg diffraction
), the respective diffraction order is much more in-
tense than the neighboring SH diffraction spots.
cos ?o= 2k1,o/k2,o.
In conclusion, we have observed second-harmonic
patterns generated owing to multiorder nonlinear
diffraction. We have found that the angular positions
of the SH beams are defined only by the ratio of the
wavelength to the grating period as in Raman–Nath
diffraction in linear optics. The diffraction patterns
include additional spots defined only by the longitu-
dinal phase matching, not being observable without
the nonlinear grating. For this reason we believe
these peripheral SH spots can be identified as a non-
integer order nonlinear diffraction. As recently
shown , similar type SH radiation can be used for
single-shot reconstruction of femtosecond pulses. The
observed phenomena can also find application in non-
linear microscopy and nondestructive evaluation of
the domain structure in ferroelectric crystals.
This work was supported by the Australian Re-
search Council and Israeli Science Foundation. S.
Saltiel acknowledges the Nonlinear Physics Centre
for hospitality and support.
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spots versus ?ext, measured on a screen perpendicular to
the FF beam. (b) Theoretical prediction. Central curves,
TPM-SH; thick curves, LPM-SH. E2–O1O1, solid thick
curve; O2–O1O1, dashed thick curve. Inset, scheme of the
(Color online) (a) Positions of the SH diffraction
OPTICS LETTERS / Vol. 34, No. 6 / March 15, 2009