Spatial Four Wave Mixing in Nonlinear Periodic Structures

We present the first experimental study of spatial four-wave mixing in photonic lattices, demonstrating universal aspects of nonlinear processes in periodic media, such as engineered phase matching, Bloch-wave folding, and continuous control over the band at which the interaction products emerge.

Figure 1

Spatial FWM processes represented in the momentum space ($k_x$, $k_z$). Efficient FWM that satisfies the phase matching condition is presented as a straight line connecting $k_1$, $k_2$, and $k_4$. (a) Under the diffraction relations in homogeneous media, perfect phase matching is not possible. (b) The presence of a lattice enables perfect phase matching between input waves from Band 1 and Band 2, and a new wave emerging at Band 3. (c) Perfect phase matching between three FB waves from Band 1. The newly generated wave resides outside the 1st BZ, yet its propagation constant ($k_z$) belongs to Band 1. Consequently, the FWM interaction product “folds” back to the first BZ by one reciprocal lattice vector, and displays Band-1 characteristics (i.e., propagating at an angle smaller then $k_x=\pi /D$).

Figure 2

Observation of Bloch-wave folding via FWM in a lattice. The edges of the first Brillouin zone are marked by the yellow lines. (a) In the absence of a lattice, one of the FWM products lies at an angle larger than $\pi /D$ ($k_x=1.3\pi /D$; circled beam). (b) With the lattice “on,” that FWM product folds back by a reciprocal lattice vector into the first band and appears between $-\pi /D$ and $\pi /D$ ($k_x=-0.7\pi /D$; circled beam). (c) A plane wave launched at the same angle as the newly generate wave (at $k_x=1.3\pi /D$), results in a (linear) Bragg-reflected wave at $k_x=-0.7\pi /D$, at very low diffraction efficiency.

Figure 3

(a) 1D lattice in a 3D bulk material; the index of refraction is periodic in $x$ and uniform in $y$ and $z$. (b) Band structure of a 1D lattice in 3D space. The three lower bands (higher propagation constants) are presented.

Figure 4

Band control of the FWM product. The edges of the first Brillouin zone are marked by the yellow lines. The FWM products are marked by white circles. (a) For a small tilt angle in $y$, phase matching is satisfied for a Band-3 FB wave, hence the FWM product resides in the third band. (b) For a larger tilt angle in $y$, phase matching occurs for a Band-2 FB wave. (c) An even larger angular shift results in a FWM product residing in Band 1.