The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the best of our knowledge, example of nonlinear Talbot self-healing, i.e., the capability of creating defect-free images from faulty nonlinear optical structures. In particular, we employ the tightly focused femtosecond infrared optical pulses to fabricate

- Chinese Optics Letters
- Vol. 19, Issue 6, 060011 (2021)
Abstract
Keywords
1. Introduction
Nonlinear photonic crystals (NPCs), also known as periodically poled ferroelectric crystals, have been widely used to extend the wavelength range of coherent light sources[
Nonlinear Talbot self-imaging belongs to one of the most intriguing functionalities of NPCs. The original Talbot effect[
Figure 1.(a) SH diffraction from a nonlinear χ(2) grating. The grating has a periodic variation of the sign of the second-order nonlinear coefficient χ(2), which can generate SH waves with uniform amplitude but periodic phase difference of π. (b) Illustrating Talbot self-healing, where the initially missing point (#3) is restored in the first Talbot image plane.
One useful characteristic of the Talbot effect is its capability to produce defect-free images from imperfect structures, i.e., the property known as self-healing. As shown in Fig. 1(b), a defective grating with one missing element can be self-imaged without an apparent defect[
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In this Letter, we use the femtosecond laser poling technique[
2. Experiments
To fabricate the NPCs, we employ tightly focused femtosecond infrared optical pulses to invert ferroelectric domains in a
The setup of the nonlinear Talbot experiment is schematically shown in Fig. 2. A mode-locked Ti:sapphire femtosecond laser (Vitara Coherent) operating at a wavelength of 810 nm is used as the fundamental light source. The pulse width is 8 fs with a repetition rate of 80 MHz. The average power of the fundamental beam was controlled by the combination of a half-wave plate (HW1) and a polarizer (P). Then, another half-wave plate (HW2) was used to control the polarization state. Lens L1 (focal length
Figure 2.Experimental setup for nonlinear Talbot self-healing. HW, half-wave plate; P, polarizer; L, lens; S, nonlinear photonic sample; F, short-band-pass filter; G, Glan prism; CMOS, CMOS camera.
3. Results and Discussion
The experimental results of the nonlinear Talbot effect in the square lattice are shown in Fig. 3. As a reference, the nonlinear Čerenkov microscopic image[
Figure 3.(a) Čerenkov SH microscopic image of the fabricated NPCs with a square lattice. Several lattice points are missing on purpose to serve as the structural defects (marked by the yellow squares). (b) The ferroelectric domain structure imaged on the output surface of the NPC. (c) The SH self-image at the first nonlinear Talbot plane. The missing lattice points are all restored, indicating the nonlinear Talbot self-healing.
The experiment results of the hexagonally poled NPC are shown in Fig. 4. Nonlinear Talbot self-healing was observed, with the introduced structure defects restored in the first SH Talbot plane again. The SH Talbot distance depends on the symmetries of the periodic lattice. For a 2D hexagonal lattice, the nonlinear Talbot distance
Figure 4.(a) Čerenkov SH microscopic image of the hexagonally poled LiNbO3 NPCs. The designed defects are several missing lattice points located randomly throughout the sample (marked by the yellow hexagons). (b) The ferroelectric domain structure imaged on the output surface of the NPC. (c) The SH self-image at the first nonlinear Talbot plane, with the missing points being restored.
It is worth noting that nonlinear Talbot self-imaging and self-healing are strongly dependent on the periodicity of the sample. Structural defects can be restored only when they are not severe enough to destroy the structure periodicity. For example, an NPC with a sunflower seed pattern was also tested in experiments. The sunflower spiral has no periodicity in its structure [Figs. 5(a) and 5(b)]. Therefore, we did not observe the nonlinear Talbot self-healing with this structure. In experiment, the SH near-field diffraction from the sunflower pattern led to only blurred images, which got even worse with the increasing imaging distance, as shown in Figs. 5(c) and 5(d).
Figure 5.(a) Čerenkov SH microscopic image of the LiNbO3 NPC with sunflower seed pattern. (b) The ferroelectric domain structure imaged on the output surface of the crystal. (c), (d) The SH near-field diffraction patterns imaged at distances of 20 µm and 50 µm from the sunflower pattern.
4. Conclusion
In summary, we have employed tightly focused infrared femtosecond laser pulses to fabricate the
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