Quasi-Phase-Matching Based on Hilbert Fractal Superlattice Structure

Abstract

Objective The quasi-phase-matching theory was first proposed to compensate for the phase mismatching caused by different refractive indices of the fundamental and harmonic waves in nonlinear photonic crystals. The team from Nanjing University introduced the one-dimensional Fibonacci quasi-periodic domain structure into the optical superlattice. As the order of the quasi-periodic structure is weaker than that of the periodic structure, the quasi-periodic domain structure, especially that with the fractal structure, has richer Fourier components. Therefore, more extensive reciprocal vectors are in the reciprocal space. Recently, researchers have been exploring and designing more excellent optical superlattice structures to meet the increasing application requirements for superlattice materials. However, owing to its low Hausdorff dimension, the corresponding nonlinear frequency conversion efficiency is not attained in existing fractal superlattice structures, such as the Cantor set and Koch curve. In this paper, we propose the design and preparation of the Hilbert fractal superlattice structure, apply to the experiment of nonlinear optical high-order harmonic generation. Results from the novel fractal structure experimental methods and research findings can be helpful to the phase matching of multiple wavelengths and high-order harmonic output of quasi-continuous wavelengths. Methods Using the characteristics of a fractal, we illustrated that the fractal structure was obtained by broadening the fractal curve into a stripe structure. Further, we showed that the Hilbert fractal structure was a two-dimensional space-filling curve. Then, taking third-order Hilbert fractal structure as an example, we used an iterative method to deduce the design method of the n-order Hilbert fractal structure. Furthermore, the fractal dimension of the Hilbert fractal structure was obtained by calculating the Hausdorff dimension. Moreover, the derivation proves that the reciprocal space of the structure provides rich reciprocal vectors. Next, using the local layout analysis of the real space of the Hilbert fractal structure, we evaluated the influence of the increase of the fractal order on the reciprocal vector in the reciprocal space. Additionally, we obtained the reciprocal vectors of the Hilbert fractal structure from the distribution of the periodic structures reciprocal vector by proportional calculation. Using the momentum conservation and Sellmeier formula of lithium niobite crystals, we obtained the second harmonics generated by the Hilbert fractal superlattice structure. Results and Discussions A two-dimensional Hilbert fractal superlattice structure was designed. We performed the diffraction experiment for two-dimensional periodic superlattices. Furthermore, the Hilbert fractal structure and diffraction simulation were performed under the same conditions using lasers with wavelengths of 532 and 632 nm, respectively. From the diffraction pattern, the spots in the reciprocal space of the Hilbert fractal superlattice structure are equidistantly distributed on the horizontal and vertical axes. Results show that the brightness gradually decreases with an increase in the distance from the spot to the center. Also, the dark fringes in the Hilbert fractal diffraction experiment diagrams (Fig. 4) were owing to the change in the diffraction angle, making the ratio of the light intensity between this point and the center point attain its minimum. Thus, from the reciprocal vector distribution of the fractal structure, the Hilbert fractal superlattice structure can be achieved using collinear quasi-phase-matching for 543, 632, 696, and 900 nm wavelengths, respectively. The comparative analysis with H-shaped fractal (Fig. 5) proves that the Hilbert fractal structure has a stronger order and complexity, and can provide richer and denser reciprocal vectors. The normalized conversion frequency of Hilbert fractal structure, composed of several strip-shaped structures, was compared to that of the H-shaped fractal. Results show that the conversion efficiency can be further improved through optimizing the poled reversal duty ratio or increasing the incident fundamental frequency optical power. Conclusions In this paper, two-dimensional periodic and Hilbert fractal superlattices were manufactured, and their Fraunhofer diffraction experiments were performed using lasers with wavelengths of 532 and 633 nm, respectively. The experimental results agree well with simulation calculations. Owing to the stronger order and complexity of the Hilbert fractal structure, the distribution of reciprocal vectors is rich in reciprocal space. Furthermore, the second harmonics at wavelengths of 543, 696, and 900 nm were achieved in the lithium niobite of the Hilbert fractal superlattice structure using theoretical calculations. Additionally, owing to small reciprocal vectors, the second harmonic with a wavelength of 632 nm was generated after linear combinations, thereby obtaining the effective output in the near-infrared and mid-infrared bands. Therefore, Hilbert fractal nonlinear photonic crystals with rich reciprocal vectors are expected to be used in multiband frequency conversion, optical parametric oscillator optimization, broadband pulsed white laser design, realization of multi-period terahertz pulses, and nonlinear photonic crystal light field modulation.