Conformal frequency conversion for arbitrary vectorial structured light

Vectorial structured light with spatially varying amplitude, phase, and polarization is reshaping many areas of modern optics, including nonlinear optics, as diverse parametric processes can be used to explore interactions between such complex vector fields, extending the frontiers of optics to new physical phenomena. However, the most basic nonlinear application, i.e., frequency conversion, still remains challenging for vectorial structured light since parametric processes are polarization dependent, leading to a change in the spatial topological structure of signals. In this work, to break this fundamental limit, we propose a novel conformal frequency conversion scheme that allows to maintain the full spatial structure of vectorial structured light in the conversion; and systematically examine its spatial polarization independence based on non-degenerate sum-frequency generation with type-0 phase matching. This proof-of-principle demonstration paves the way for a wide range of applications requiring conformal frequency conversion, and, particularly, to implement frequency interfaces with multimodal communication channels, high-dimensional quantum states, and polarization-resolved upconversion imaging.

Conformal frequency conversion for arbitrary vectorial structured light: supplemental document 1

. Detailed Theoretical Framework
Spatial Modes. -In the simulation and data analysis, we used LG modes, denoted by p LG  , and their superpositions to represent general spatial mode. The spatial complex amplitude of the LG mode in the cylindrical coordinates { , , } rz  , with the spatial indices of ( azimuthal) and p (radial), is given by [ In addition, similar to the relation between the helical LG modes and their parity counterparts, IG mode can also exist in the helical manner, denoted as  Nm IG , that carry net OAM, and the corresponding relation is given by [ LG LG LG LG LG LG

IG
LG LG LG LG LG LG from where we know that the average OAM carried by them are ±3ℏ, ±2.447ℏ, and ±3ℏ per photon, respectively.
Modal Transformation. -Here we derive the modal transformation of the general vector mode, shown in Eq. (1), in the Sagnac loop, i.e., 1 1 are not usually orthogonal to each other, unless In addition, note that the premise to achieve the conformal upconversion, i.e., , is using flattop beam. Here we show briefly why the commonly used Gauss pump would lead to the angular spread of spatial spectrum, or rather, radial-modal degeneration for LG modes. The spatial amplitude of an SFG (with   LG + , 4 1 LG + , and 4 2 LG + , i.e., a group data in Fig. 2(a), is shown below FIG. S1. Modal transformation of high-order LG modes during the SFG pump by Gauss-mode pump.
Super-Gauss Mode. -To realize the conformal frequency conversion, we need a perfect flattop beami.e., whose amplitude and phase are both spatially uniformas the pump. Crucially, the intensity-flattop beam obtained via phase-only modulation that we used in Refs. 5 and 6, cannot provide a flattop wavefront. To address this, we design a perfect flattop beam based on super-Gauss distribution [7], given by n SG u r r w  =− , its intensity distribution is shown in Fig. S2 (a), where we see that the degree of 'flattop' increases with the order n. Here we used the distribution with an order of n = 12 to define the super-Gauss mode at the beam waist, given by and its spatial complex amplitude upon propagation including lens transformation can be calculated using Collins integral [8]. To generate the super-Gauss mode at the focal region in the crystal, we used computed holography based on complex-amplitude modulation and following with a lens Fourier transformation, as shown in Fig. S2(b). The complex-amplitude mask loaded on the surface of SLM was designed by a Fourier integral, given by Beyond having a perfect flattop complex amplitude at the beam waist plane, the super-Gauss mode has also another important merit, i.e., maintaining a flattop wavefront over a short distance. For comparison, Fig. S3 shows beam propagations of the super-Gauss mode and the TEM00 mode in a 20 mm PPKTP crystal, which were used in the experiment. We see that the wavefront of super-Gauss mode, especially for the center region, keep flat through the crystal. Yet, for a Gauss beam, sphere wavefronts appear at the two ends of the crystal.

Additional Results
For the setup demonstrated in the main text, we also measured the efficiency of conversion in the depleted region by using pulse light, where the pulses were obtained via intensitymodulated 1560 nm wave and its SHG (780nm) wave. The quasi-continuous pulses have a 4ns duration with 100:1 duty factor. Before examine the conformal upconversion, the conversion efficiency between Gauss modes was characterized, as shown in Fig. S4, where the average power of signal was fixed at 1 mW. It was shown that, as illustrated in the Sec. A of the main text, i) both p  and q  are linearly proportional to the pump power before the signal-depleted region; and ii) p  for 780 nm signal are double of that for 1560 nm, while p  is identical for the two. Besides, it is worthy to note that the upper limit of q  is below 100% in the Gaussmode pumped SFG, even for a Gauss-mode signal. Then, we first consider the small signal in the depleted region. Specifically, a CV mode with 2 = at 780 nm was pumped by a flattop light at 1560 nm that can just well cover it, and the average power of the CV-mode signal was fixed at 1mW. The results in Fig. S5 show that, unlike the max q  was limited ~80% shown in Fig. S2, the 100% q  can be obtained experimentally by using only a 1.175 W pump with a flattop mode. The vectorial transverse structure was also well maintained during the upconversion. Finally, we consider cases in laser-based applications, that is, depleted SFG with a larger signal. Here, a group of vectorial IG modes based on IG44 with 1  = were played as signals, whose power was 200 mW at 1560 nm. According to the measured p  with respect to power of pump shown in Fig. S6(a), one can obtain a 200 mW structured laser at 520 nm with the same vector profiles by using a 513 mW pump.