Nonlinear wavefront shaping with optically induced three-dimensional nonlinear photonic crystals

Generation of coherent light with desirable amplitude and phase profiles throughout the optical spectrum is a key issue in optical technologies. Nonlinear wavefront shaping offers an exceptional way to achieve this goal by converting an incident light beam into the beam (or beams) of different frequency with spatially modulated amplitude and phase. The realization of such frequency conversion and shaping processes critically depends on the matching of phase velocities of interacting waves, for which nonlinear photonic crystals (NPCs) with spatially modulated quadratic nonlinearity have shown great potential. Here, we present the first experimental demonstration of nonlinear wavefront shaping with three-dimensional (3D) NPCs formed by ultrafast-light-induced ferroelectric domain inversion approach. Compared with those previously used low-dimensional structures, 3D NPCs provide all spatial degrees of freedom for the compensation of phase mismatch in nonlinear interactions and thereby constitute an unprecedented system for the generation and control of coherent light at new frequencies.

As far as I know, this is the first experimental report on nonlinear beam shaping with 3D modulated nonlinear photonic crystals. Since this demonstration opens very interesting new possibilities in nonlinear optics and in wavefront shaping, I recommend accepting the paper for publication in Nature Communications, pending revisions as detailed below: The structures that were fabricated are in fact a combination of separate two-dimensional structures, stacked inside the crystal along the Z axis. In each one of these separate structures, there is no dependence of the modulation pattern on the propagation coordinate. I believe that this design does not fully utilize the capability of modulating in the third dimension. Specifically, the current design is not phase matched in the propagation axis, and therefore exhibits low conversion efficiency (nano-Watts levels of second harmonic for hundreds of milli-Watts of the pump, as shown in Fig. 3c) into the shaped wavefront. I wonder if the authors can discuss or demonstrate more efficient schemes, based nonlinear volume holography, that require modulation along the Z axis? Can the authors refer to these possibilities and compare them to the designs that they chose? The authors write "To the best of our knowledge this is the first time to obtain dynamic nonlinear wavefront shaping with nonlinear photonic crystals in experiment." -Perhaps this is correct for 3D crystals, but for 2D crystals it is not correct. Dynamic shaping was already shown in the past in nonlinear photonic crystals and should be mentioned, see Trajtenberg-Mills et al, Optica 4, 153 (2017) Reviewer #2: Remarks to the Author: The authors demonstrate experimentally for the first time nonlinear beam shaping by quadratic nonlinear crystals with 3D engineered domain structures. They use a femtosecond laser writing technique, that was recently developed by them, to create the 3D structures in the materials. Specifically, experimental demonstrations include nonlinear emission from three layer or two layer structures. They show how different beams can be created simultaneously and how the depth difference of the different layers can be used to achieve dynamic control over the generated beams. I find that the paper is well written. It reports for the first time on a very important application of recently developed 3D laser writing technique of quadratic domain structures. I am certain that this demonstration will have a very strong impact on nonlinear optics community and on various laser applications. Therefore, I find it highly suitable for publication in Nature Communications Journal. I have only a few suggestions listed below for minor corrections that might improve the manuscript.
Suggestions for minor corrections: 1. Lines 62-63: "Furthermore, the m-th nonlinear diffracted order by a fork structure of the topological charge lc is a vortex beam with a "charge" lSH=mlc, representing 2πlSH azimuthal phase modulation." I think this should be explained better also for non experts, as it is a key concept for understanding the results. 3.Emphasize the difference in parameters for laser writing and for SHG experiments. 4.The specific demonstrations that were shown besides the "dynamic" one can be achieved also with 2D modulations of the three structures. add some more insight on benefit of the 3 layer approach or better yet its extension to true volume nonlinear shaping.
Reviewer #3: Remarks to the Author: In the paper entitled " Nonlineat wavefront shaping with optically-induced three-dimensional nonlinear photonic crystals" 1, Shan Liu et al demonstrate second harmonic generation in nonlinear photonic structures fabricated by using the femtosecond laser domain inversion technique, which has been developed by the same team. They perform frequency doubling in nonlinear grating comprising of fork, circular and one-dimensional nonlinear gratings fabricated in the same CBN crystal. The experiments demonstrated that one can change the nonlinear diffraction pattern by shifting the focal point of the fundamental beam along the propagation direction.
The presented results are interesting and I think the paper may deserve publishing in the Nature communication. However, my feeling is that prior to publication authors should address the following issues: 1. The paper essentially based on the Ref 29, in which authors have already reported the fabrication of the 3D NPCs via the femtosecond-laser domain inversion. That is the fabrication several NPCs in the same crystal reported in the current paper looks like an incremental improvement of the results obtained in Ref 29. Authors should clearly explain what is new in the current paper. 2. It is unclear from the text what was the conversion efficiency. Since the gratings are essentially transverse with respect to the fundamental beam, one may expect that the interaction length does not exceed several microns. It is definitely enough to register the SHG signal, however application potential of such a nonlinear optical device may be questionable.

Response to Reviewer #1
As far as I know, this is the first experimental report on nonlinear beam shaping with 3D modulated nonlinear photonic crystals. Since this demonstration opens very interesting new possibilities in nonlinear optics and in wavefront shaping, I recommend accepting the paper for publication in Nature Communications, pending revisions as detailed below:

Reviewer:
The structures that were fabricated are in fact a combination of separate two-dimensional structures, stacked inside the crystal along the Z axis. In each one of these separate structures, there is no dependence of the modulation pattern on the propagation coordinate. I believe that this design does not fully utilize the capability of modulating in the third dimension. Specifically, the current design is not phase matched in the propagation axis, and therefore exhibits low conversion efficiency (nano-Watts levels of second harmonic for hundreds of milli-Watts of the pump, as shown in Fig. 3c) into the shaped wavefront. I wonder if the authors can discuss or demonstrate more efficient schemes, based nonlinear volume holography, that require modulation along the Z axis? Can the authors refer to these possibilities and compare them to the designs that they chose?

Authors' Response
We appreciate the referee's insightful comments. What we present in this manuscript is the first demonstration of superiorities of 3D nonlinear photonic crystal (NPC) in nonlinear wavefront shaping and the proposed concept is quite straightforward. Stacking a number of 2D nonlinearity modulation patterns in the same crystal, along the propagation direction of fundamental beam (Z axis in our case), leads to emission of second harmonic waves having their wavefronts spatially shaped in arbitrary forms (e.g., vortices, Gaussian and conical beams in our experiment). While in this simple design the individual domain stacks play their roles independently, our 3D NPC still represents a significant advancement in nonlinear wavefront shaping showing new effects such as the dynamical wave shaping by tuning the focus position along the beam propagation direction, that cannot be realized in low dimensional structures. We agree with the referee that going over to more complex 3D structures will open up possibilities to observe entirely new effects provided by nonlinear holograms. The experimental realization of such spatially complex 3D structures is a subject of our ongoing efforts. However, it requires significant improvement and optimization of our femtosecond laser ferroelectric domain writing technique, which we hope to achieve in the near future. Hence, in this work we focused on demonstrating the application of simpler multilayer 3D structures. We commented on the formation of extended 3D nonlinearity structures, including those representing true nonlinear holograms, in the concluding parts of the revised version of the manuscript (Page 14,highlighted in yellow) Although the current design does not rely on a periodic nonlinearity modulation along the beam propagation direction (Z axis), the observed second harmonic generation processes can still be fully-phase matched at certain wavelengths, at which the angle of Čerenkov SH emission overlaps with the Raman-Nath nonlinear diffraction. In the fully phase matched regime higher conversion efficiency will be achievable by fabricating high quality, thicker domain structures. For example, with nonlinear gratings having period of 2 μm, the SHG is fully phase matched via 1 st order nonlinear diffraction at the fundamental wavelength of 1.76 μm. Since the longest wavelength available from our laser system was 1.6 μm, the results depicted in Fig.3 (c) were obtained at this particular wavelength. The relevant discussion is provided in the Method section (Emission angle of the second harmonic waves, highlighted in yellow, pages 17 and 18). Fig 3c -what was the pump beam size and what was the peak power? Also, for comparison, what would be the phase matched conversion efficiency in a crystal with similar length and pump beam.

Authors' Response
The pump beam size was slightly exceeding 70 μm in diameter, to ensure the illumination of the whole area of nonlinearity modulation (60x60 microns) and the used peak power was just below 22 kW. We add this information in the Method section (Nonlinear wavefront shaping experiment, page 16). Fig.3c is concerned, we estimated it (using experimental data for longitudinal phase mismatch Δk=0.12 μm -1 , and Eq. (6) in Ref [16]) to be roughly 3 times lower than that in the fully phase-matched case. This means that by going over to nonlinear Bragg regime and making structure longer one can substantially increase conversion efficiency.

Reviewer:
The authors write "To the best of our knowledge this is the first time to obtain dynamic nonlinear wavefront shaping with nonlinear photonic crystals in experiment." -Perhaps this is correct for 3D crystals, but for 2D crystals it is not correct. Dynamic shaping was already shown in the past in nonlinear photonic crystals and should be mentioned, see Trajtenberg-Mills et al, Optica 4, 153 (2017)