3D nonlinear photonic crystals (NPCs) allow spatial modulation of the second-order nonlinear coefficient ${\chi }^{(2)}$ along arbitrary directions, which is a major advantage for the generation of light at new frequencies and enables new functions that otherwise cannot be obtained with one- or two-dimensional structures^[1-8]. For instance, the traditional 2D fork-shaped ${\chi }^{(2)}$ gratings are used for second harmonic vortex beam generation from a fundamental Gaussian wave^[9,10]. But this process is phase matched via nonlinear Raman–Nath diffractions, which is a non-collinear process, and the energy of the incident beam is divided into multi-order diffracted harmonics. Hence, the conversion efficiency of each vortex beam generation is not high. This issue can be solved by using 3D NPCs, for example, a periodic spiral structure shown in Fig. 1(a)^[11]. The 3D spiral structure is capable of generating a collinear second harmonic vortex beam, and the fully fulfilled phase matching condition ensures the growth of second harmonic energy with propagation distance. The resulting conversion efficiency is, hence, higher than those of 2D folk-shaped gratings. In addition, 3D NPCs also offer a unique opportunity to study optical analogs of subtle quantum mechanical phenomena, such as the topological Hall effect^[12].

To fabricate 3D NPCs, the ferroelectric domain engineering with femtosecond laser pulses is commonly used^[13-18]. In this process, the illumination of tightly focused femtosecond pulses creates an extremely high temperature gradient and consequently the appearance of a thermal-electric field to locally invert ferroelectric domains in the focal volume of the laser beam^[1]. The domain inversion leads to the reversal of the sign of the second-order nonlinear coefficient ${\chi }^{(2)}$. While the femtosecond laser writing is an accurate manufacturing technique itself, the precise fabrication and control of laser-induced ferroelectric domain structures remains a challenge. So far, the physical mechanism on the laser-induced domain inversion is not fully understood, and extra investigations are required.

It has been recently shown that those complex 3D domain structures can be simplified for the sake of easy fabrication in experiment, without losing the functions of original structures^[7]. As an example, the 3D periodic spiral structure mentioned above was simplified into a four-segment structure [Fig. 1(b)] for collinear second harmonic vortex beam generation from the fundamental Gaussian wave^[7]. Each segment contains a periodic grating, and the four segments are parallelly arranged, with their starting positions being shifted longitudinally to introduce phase delays of 0, $\pi /2$, $\pi$, and $3\pi /2$ among the generated second harmonics in these four segments. In this way, the second harmonic is generated with the major characters of an ideal Laguerre–Gaussian mode with topological charge $l_c=1$ being approximately remained. For comparison, the domain structures, transverse phase profiles, and far-field of the second harmonic beam obtained in the ideal and simplified cases are shown in Figs. 1(a)–1(f), respectively. The far-field second harmonic distributions are obtained by the Fourier transform of the corresponding domain structures in real space.

In this Letter, we focus on exploring more functions of the four-grating structure. Specifically, we experimentally study the second harmonic generation (SHG) with fundamental Gaussian beams of different wavelengths. We demonstrate that, besides the previously reported topological charge of $l_c=1$, the four-grating structure is also capable of generating higher order Hermite–Gaussian beams and Laguerre–Gaussian second harmonic with $l_c=-1$, depending on the wavelength of the incident Gaussian beam. This is because the phase differences among the second harmonics generated in the four gratings varying with the wavelength. Our study contributes to a deeper understanding of the nonlinear wave dynamics in 3D nonlinear photonic structures. It also opens new possibilities for nonlinear generation of structured light with versatile properties in a single piece of crystal.

The four-segment structure shown in Fig. 1(b) is designed to introduce $\pi /2$ phase difference between the second harmonic signals generated in each grating segment at the first-order quasi-phase matching (QPM) wavelength. In this case, the wave vectors of the fundamental wave and the second harmonic satisfy the phase matching condition $k_2-{2k}_1=G=2\pi /\mathrm{\Lambda }$, namely ($k_2-{2k}_1$)$\mathrm{\Lambda }=2\pi$, for the first-order SHG process, where $\mathrm{\Lambda }$ is the period of ${\chi }^{(2)}$ grating, and $G=2\pi /\mathrm{\Lambda }$ is the primary reciprocal lattice vector in Fourier space. To introduce $\pi /2$ phase delay, the starting position of each grating segment is shifted by a distance $d=\mathrm{\Lambda }/4$, as shown in Fig. 2(b). The successive phase shift of $\pi /2$ can be used to approximate the ideal vortex beam with topological charge $l_c=1$, whose phase continuously varies from 0 to $2\pi$ [Fig. 1(c)]. Consequently, the second harmonic exhibits a dark intensity center to form a donut-shaped beam in far field, as shown in Fig. 2(b).

When the fundamental wavelength is tuned to satisfy the second-order QPM condition, we have the relation ${(k_2-{2k}_1)}^{\prime }=2G=4\pi /\mathrm{\Lambda }$. Accordingly, the phase delay between the grating segment becomes $\mathrm{\Delta }{\phi }^{\prime }={(k_2-{2k}_1)}^{\prime }d=\pi$. As shown in Figs. 2(c) and 2(d), the second harmonic wave generated in this case is (1,1)-order Hermite–Gaussian mode, presenting four bright nobs to form two dark cross lines in between.

Similarly, when the fundamental wavelength satisfies the third-order QPM, the phase delays between the neighboring gratings are $3\pi /2$, such that the second harmonic forms a vortex beam again, but with topological charge of $l_c=-1$ [see Figs. 2(e) and 2(f)].

The 3D nonlinear photonic crystal with four grating segments is fabricated in an x-cut ${\mathrm{Ca}}_{0.28}{\mathrm{Ba}}_{0.72}{\mathrm{Nb}}_2{\mathrm{O}}_6$ (CBN) crystal using the near-infrared femtosecond laser poling technique. The sizes of the crystal are $5\mathrm{mm}\times 5\mathrm{mm}\times 0.5\mathrm{mm}$ in length, width, and thickness, respectively. The crystal was mounted on a translational stage that can be accurately positioned in three orthogonal directions. The infrared pulses (150 fs, 800 nm, Mira Coherent) were tightly focused using an objective lens ($\mathrm{NA}=0.65$, $50\times$, Olympus LCPLN50XIR). The beam was incident from the x-surface and scanned along the $z$-direction for 50 µm with a velocity of 0.02 mm/s to produce an inverted domain stack in sizes of $20\mathrm{\mu m}(x)\times 3\mathrm{\mu m}(y)\times 50\mathrm{\mu m}(z)$. Then the writing beam was blocked by an automatic shutter, and the sample was first translated back to the original position and then moved a small distance along $y$-direction to produce another inverted domain stack. The small distance was selected to be 1 µm to ensure the neighboring stacks merge into a thicker domain block. By repeating the above process for $N$ times, an inverted domain block in dimensions of $20\mathrm{\mu m}(x)\times N\mathrm{\mu m}(y)\times 50\mathrm{\mu m}(z)$ was created. To further enlarge the scale of the inverted domain block in the $x$-direction, the sample was transferred along $x$-axis for 20 µm, and the domain inversion steps above were repeated again to double the domain length in the $x$-direction. This large-scale single domain block forms the basic unit of the four-segment structure. The nonlinear grating was fabricated by producing such blocks periodically along the $y$-axis. The period of the gratings is 16 µm, which is selected to fulfill the first-order QPM SHG at a fundamental wavelength of 1400 nm. The width of each block (along the $y$-direction) was 8 µm to approach the optimal duty cycle of 50% and, hence, maximize the conversion efficiency for odd-order QPM interactions. The four-segment ${\chi }^{(2)}$ structure was constructed by fabricating four nonlinear gratings and shifting them along the $y$-direction by 4, 8, and 16 µm, respectively.

The four-segment domain structure was fabricated at a depth range of 500 to 600 µm below the $x$-surface of the crystal. The laser poling power was kept as 450 mW. This value was just 20 mW higher than the laser poling threshold to minimize the undesirable refractive index changes.

The fabricated structure of the four grating segments is shown in Fig. 3(a), which is imaged using the Čerenkov second harmonic microscopy^[19]. This microscopy works on the principle that the strength of the Čerenkov second harmonic is much larger when the scanning laser beam illuminates the ferroelectric domain walls, namely the interfaces between the antiparallel domains with opposite signs of the second-order nonlinear coefficient. In Fig. 3(b), the amplified view of the grating structures is given. The measured period of the gratings is 16 µm, corresponding to the first-order SHG at the fundamental wavelength of ${\lambda }_1=1.412\mathrm{\mu m}$, the second-order SHG at 1.141 µm, and the third-order SHG at 1.014 µm. The longitudinal displacement between the neighboring gratings is 4 µm, which is a quarter of the period and introduces the phase delays of $\pi /2$, $\pi$, and $3\pi /2$ for the first-, second-, and third-order QPM SHGs, respectively.

The SHG experiment was conducted with a laser source of the Coherent Chameleon OPO system. The pulse duration is 150 fs, the repetition rate is 80 MHz, and the wavelength is tunable between 1000 and 1600 nm. The beam was loosely focused into the nonlinear photonic structure by a lens (focal length of 50 mm), and the beam size was about 100 µm. The fundamental beam propagated along the $y$-axis of the sample and was linearly polarized along the $z$-axis to generate the second harmonic via the largest nonlinear coefficient d₃₃. The combination of a half-wave plate and a polarizer is used to adjust the incident power of the fundamental beam. A short-pass filter was placed behind the sample to block the fundamental beam while the outgoing spatially shaped second harmonic was directly recorded by a CCD. The power of the second harmonic wave was measured by a power meter (PM100D, Thorlabs).

The wavelength tuning responses of the SHG in the four-grating structures are measured to determine the resonant wavelengths for different orders of the QPM SHG. The result is shown in Figs. 4(a) and 4(b). It can be seen that the measured resonant wavelengths of 1.40, 1.14, and 1.01 µm, agree well with the predicted wavelengths of 1.412, 1.141, and 1.014 µm, respectively. The slight difference between the experimental and theoretical wavelengths may come from the imperfection of the produced ferroelectric domain structures. In addition, the dispersion equation used for the CBN crystal may also cause some errors^[20].

In Figs. 4(c)–4(e), the recorded second harmonic profiles are displayed for different QPM orders. As predicted, the first- and third-order second harmonics are both Laguerre–Gaussian modes, and the second-order SHG forms the (1,1)-order Hemet-Gaussian mode. The slight differences between these results and the theoretic predictions [shown in Figs. 2(b), 2(d), and 2(f)] are mainly caused by the imperfection of our fabricated samples. It is known that the random structure errors are unavoidable in any poling process. Such random deviations from the ideal structure deteriorate the beam quality of the second harmonic in experiment. In addition, the predictions were obtained using the refractive index data reported in Ref. [20], which may not be exactly the same as the real values of the crystal used in this work.

The conversion efficiencies of the SHG of different QPM orders were measured, and the results are shown in Fig. 5. The internal conversion efficiency excluding Fresnel losses at the crystal facets was 0.268%, 0.099%, and 0.002%, respectively. The second-order second harmonic is extremely low because the duty cycle of 50% leads to a very low Fourier coefficient for the even-order interactions. It is expected the harmonic conversions will be more efficient if the longer samples are used.

In summary, we have fabricated a 3D nonlinear photonic crystal with four segments of grating structures using the femtosecond laser poling technique. The SHG from the four-segment structure is investigated to show its exceptional performance on structured light generation at new frequencies. Depending on the wavelength of the fundamental beam, the second harmonics generated via different orders of QPM interaction vary from donut-like Laguerre–Gaussian modes to high-order Hemet–Gaussian beams and then Laguerre-Gaussian waves with opposite topological charges. These results are interpreted with the wavelength-dependent phase delays introduced by the four-grating structure. It is worth noting that the four-segment structure is a specific simplification of the ideal 3D spiral structure^[11] for the generation of second harmonic vortex beams with a topological charge of $l_c=1$. For realization of other topological charges, the 3D spiral structure needs to be simplified into structures other than the four-segment structure. The detailed principles and examples for the simplification process are presented in Ref. [7]. This study is useful for deeper understanding of nonlinear interactions in 3D nonlinear photonic crystals. It also offers new possibilities to generate multiple special beams at new frequencies in a single crystal.