Nonlinear scattering in photonic crystals having dislocations with fractional topological character and multiple dislocations

The spectrum of the second harmonic signal generated in quadratic nonlinear photonic crystals, having different types of edge dislocations, was studied theoretically and experimentally. In the case of a dislocation with a fractional topological charge, we observed an asymmetric spectral conversion efficiency response, where the degree of asymmetry depends on the value of the fractional charge. Moreover, we have found that the conversion efficiency spectrum exhibits a periodic dependence on the topological charge value. In addition, nonlinear photonic crystals with multiple edge dislocations were studied. We show that for any number of dislocations characterized by even topological charge, the nonlinear spectral response will be identical to the response of the ideal, dislocation-free structure. This is a generalization of a previous observation that was made for crystals with a single even charge dislocation. Furthermore, for any number of dislocations with odd topological charge, two new peaks of maximal efficiency are observed in the second harmonic spectrum, in addition to a series of local efficiency peaks that are governed by the total number of dislocations. This is also a generalization of a previous observation that was made for a single dislocation case having an odd topological charge.

Figure 1

Microscope pictures of the different topological edge dislocations that were implemented in crystals where each one of them contained one dislocation positioned in the middle of the periodic structure relative to the crystal's edges. The bottom part of each figure presents, as an illustration of the designs, a magnification of the area where a dislocation was positioned in the structure. (a)–(c) Comparisons between the fabricated and the designed structures.

Figure 2

Microscope pictures of the different topological edge dislocations that were implemented in crystals that contained a pair of dislocations. An illustration of the designed structure is presented schematically at the bottom of the figure for one of the crystals which consists of a periodic structure, and the two dislocations in the structure are magnified. The left dislocation is located after one-third of the crystal's edge relative to the total length of the crystal and has a topological charge of $l=1$. The right dislocation is located after two-thirds of the edge and has a topological charge of $l=2$. (a)–(c) The three topological charge configurations that were examined experimentally. The fabricated dislocations in (c) can be compared with the illustration of the design (connected with the red dashed line).

Figure 3

The experimental setup. The red (dark gray) and green (light gray) colors of the beams illustrate the pump and SH beams, respectively.

Figure 4

SH conversion efficiency as a function of the input pump wavelength after the propagation through structures that include a single dislocation with a different value of the topological charge.

Figure 5

Comparison between the measured and the simulated spectral SH conversion efficiency curves for structures that contained a dislocation with a fractional topological charge of (a) $l=1/2$, (b) $l=3/4$, and (c) $l=3/2$.

Figure 6

Comparison between the measured and the simulated spectral SH conversion efficiency curves for structures that contained two dislocations with integer topological charges of (a) $l_1=2$ and $l_2=2$, (b) $l_1=1$ and $l_2=2$, and (c) $l_1=1$ and $l_2=1$. The insets in (a), (b), and (c) show simulations of the formation process of the SH intensity inside the crystals, during the propagation of a pump beam for selected wavelengths. The cases of cumulative construction, destruction, and recovery of the SH intensity inside the structures that include different dislocations are demonstrated.

Figure 7

Spectral SH conversion efficiency for structures that contain multiple dislocations with odd topological charges. The simulations were performed for different structures with a growing amount of dislocations, where the designed patterns contained dislocations with topological charges of $l=1$ and $l=3$ arranged alternately.