Theoretical investigation of three-dimensional quasi-phase-matching photonic structures

We present a full theoretical analysis of quasi-phase-matching (QPM) in three-dimensional (3D) periodic structures and point up optimum nonlinear structures, which promote the best nonlinear conversion efficiencies and are close to real structures. The QPM properties of 14 Bravais lattices are investigated as a function of motifs (orthorhombic and spherical) and of modulation types (“+/–” and “+/0”). This full 3D QPM theory allows us to produce all results of one- and two-dimensional QPM structures by choosing appropriate lattice periodicity and motif. The optimization of nonlinear conversion efficiencies in 3D QPM is obtained by analyzing four particular structures (simple cubic, body-centered cubic, face-centered cubic, and diamond cubic lattices) with different filling factors and motifs. In particular, 3D structures, which are very close to those realized in practice, are proposed and simulated, creating a guide for fabrication of real optimum QPM structures.

Figure 1

Convolution of a simple cubic lattice with a triangular motif to model 3D periodic QPM structure.

Figure 2

Illustration of a triclinic unit cell, with different parameters defined in the text.

Figure 3

(a) 1D-like structure with parameters $X/a=0.5, Y/b=1$, and $Z/c=1$; (b) 2D-like structure with parameters $X/a=0.5, Y/b=0.5$, and $Z/c=1$; (c) 3D structure with parameters $X/a=0.5, Y/b=0.5$, and $Z/c=0.5$.

Figure 4

Optimum configurations of a 3D QPM structure with orthorhombic lattice, spherical motif, and the shortest side $a$ depending on QPM order: (a) $\left(1,0,0\right)$ order for collinear QPM means that fundamental light propagates along the shortest side of the lattice, (b) $\left(0,1,0\right)$ order, and (c) $\left(0,0,1\right)$ order both have propagation of fundamental light perpendicular to the shortest side.

Figure 5

Schematic comparison of QPM structures with (a) rectangular and circular motif in 2D and (b) orthorhombic and spherical motif in 3D.

Figure 6

Extension of 2D structure to 3D space using cylindrical and spherical motifs, with different motif/period ratios.

Figure 7

Four 3D structures with fixed ratio $R/a=\sqrt{3}/8$: (a) SC, (b) BCC, (c) FCC, and (d) DC. The tables present Fourier coefficients of first orders for each corresponding structure. Normalized efficiency is calculated for the “+/–” structure.

Figure 8

Normalized efficiency of SC lattice for the three first orders $\left(1,0,0\right), \left(1,1,0\right)$, and $\left(1,0,0\right)$ as a function of ratio between the size of cubic motif, $X$, and the the lattice periodicity, $a$. Normalized efficiency is calculated for the “+/–” structure.

Figure 9

Normalized efficiency of SC lattice for three first orders $\left(1,0,0\right), \left(1,1,0\right)$, and $\left(1,0,0\right)$ as a function of ratio between the spherical motif diameter, $D$, and the lattice periodicity, $a$. Normalized efficiency is calculated for the “+/–” structure.

Figure 10

Normalized efficiency for $\left(1,0,0\right)$ order of a SC lattice with a cubic motif. The solid and dashed lines represent efficiency curves of 3D QPM structures made by solid and inverse (hollow) motifs, respectively. The insert shows the evolution from full material to null material (hollow motif) and from null to full material (block motif).

Figure 11

(a) Spherical motifs with “circular-cut” connections. $a$ is lattice period, $R$ is radius of sphere located in the center of unit cell, and $r$ is radius of circular cut. (b) Surface of normalized efficiency of corresponding 3D QPM structure as a function of $r/a\le 0.1$ and $0.3\le R/a\le 0.5$. Normalized efficiency is calculated for $\left(1,0,0\right)$ order for “+/0” QPM structure.