Efficient two dimensional nonlinear photonic crystals for cascaded third
harmonic generation, which complies with fabrication constraints.
R. T. Bratfalean and N. G. R. Broderick
Optoelectronics Research Centre, University of Southampton, Southampton, SO17 1BJ, UK.
Phone: +44 (0)23 80594523, Fax: +44 (0)23 80593142, email: rtb@orc.soton.ac.uk
Two-dimensional nonlinear photonic crystals (2D NPC), with their increased design flexibility in relation to 1D NPCs,
are particularly attractive for cascaded two-step processes, such as 3-rd harmonic generation (THG), where two quasiphase-matching (QPM) conditions need to be satisfied simultaneously. The efficiency of these processes is influenced
by the shape and size of the poled domain of the direct lattice unit cell and this has been studied by A. Norton et al. [1].
They show that the best solution is found in an n-1 dimensional subset of poling functions, where n is the number of
independent Fourier coefficients of the nonlinear process. The search criterion can be the initial growth of the nonlinear
output [1] or, a much stronger one [2], which numerically integrates Maxwell equations to find the best efficiency in the
shortest length of crystal. This theory of optimal poling is important because it indicates the ultimate fundamental limit
of the optimisation problem but cannot ensure that the resulting pattern complies with the usual fabrication constraints.
In our paper we present an optimisation of the generic THG pattern, based on the initial growth of the 3-rd
harmonic, in lithium niobate (LN), which takes into account several practical constraints. First we require a collinear 1st step of the process as this increases the overall efficiency by maximising the overlap of the waves generating the 3-rd
harmonic. We also require a polygonal poled domain with 60 or 120 degrees angles, consistent with the crystallographic
symmetry of LN, and with a size of at least 5 microns along any direction. As the poled domain is connex it is important
to choose the RLV in the 1-st step of the process, G1, to be of the lowest order, namely G1 = G(1,0), as this also
maximises the efficiency. In the case of parallelogram shaped domains the figure of merits (FOMs) for various m and n
indexes of G2 = G(m,n), are shown in table 1, with generally lower FOM values for higher orders of G2. The largest
FOM value is for G2 = G(0,1) but the spacing between poled domains, for this pattern, is too small, less than 1 micron,
to be safely fabricated. The careful examination of all the other patterns yields that corresponding to G2 = G(3,1) to be
the best choice. This is shown in figure 1 and has a FOM value of 0.114, just 30% less than the fundamental limit [1],
which is 0.164 = (4/pi^2)^2. The relatively large FOM value can be explained by the low angle between G1 and G2; in
the limit of a zero angle, the whole cascaded process would be collinear, meaning that it essentially takes place in a 1D
NPC. Both patterns for G2 = G(0,1) and G2 = G(3,1) have large aspect ratio (AR) for the parallelogram shaped poled
domain. This indicates that, for equilateral triangles and regular hexagons, which have a fixed AR, the figures of merits
are likely to be lower and this was confirmed by our optimisation code. This also means that the remaining polygons to
be considered, compliant with the crystallographic symmetry of LN, which can equally have a variable AR, will need to
have similar AR values for their best patterns, of the same m and n indexes. This will result in a similar amount and
distribution of the poled area within the direct lattice unit cell and, therefore, in similar FOM values, as those obtained
for the parallelogram case.
In conclusion, we presented a numerical optimisation of the THG cascaded scheme in a 2D NPC, where the 1st step of the process is collinear and which takes into account the usual fabrication constraints. The figure of merit we
found is at 70% of that resulting from the theory of optimal poling [1]. Future work will extend our optimisation code to
4-th harmonic generation patterns. We are also planning to fabricate such crystals and present experimental
observations in the future.
[1] A. Norton et al. Optics Letter 28 (3): 188-190 (2003);
[2] A. Norton et al. Optics Express 11 (9): 1008-1014 (2003).
m\n
1
2
3
4
0
0.122
0.090
0.066
0.052
1
0.909
0.086
0.064
0.050
2
0.9352
0.072
0.053
0.042
3
0.114
0.067
0.048
0.037
4
0.085
0.062
0.045
0.035
Table 1. Figures of merit for THG patterns in LN at 140 deg.
Celsius corresponding to various m and n indexes of G2.
Figure 1. THG pattern for G2 = G(3,1) and G1 = G(1,0).
Domain’s sides are 11 and 65.3 microns, respectively.