1. Introduction

Asymmetric control of light by introducing different physical responses of counter-propagating signals is of great significance on asymmetric imaging devices [1,2] and nonreciprocal directional optical filters [3]. In general, excluding the magnetic-field-induced breaking of time-reversal symmetry [4,5] and spatiotemporal modulation of system permittivity [6–8], the nonlinear material [9–14]is the third resort to realize asymmetric control of light. Nonlinear metasurfaces [15–21] and nonlinear photonic crystals (NPCs) [22–27] are two promising media for highly efficient nonlinear wavefront shaping. However, asymmetric control of light at new frequency has only been realized with nonlinear metasurfaces [1,2,28–32] and the use of NPCs for asymmetric optical transmission, to the best of our knowledge, is still an untouched area of research.

NPCs are microstructures characterized by a spatially modulated second-order nonlinear coefficient (χ⁽²⁾) [33–35] that have been extensively used for the generation and control of coherent light at new frequencies. 3D NPCs are mainly fabricated by femtosecond laser direct writing to realize χ⁽²⁾ spatial modulation by either its selectively erasure [36,37] or inversion [38,39], with which the second-harmonic (SH) amplitude or SH phase can be modulated respectively. Two traditional χ⁽²⁾ pattern encoding methods are nonlinear computer-generated holography (NCGH) and nonlinear volume holography (NVH) [40–42], respectively. With both methods, SH wavefront shaping in NPCs has been experimentally demonstrated, i.e. Hermite-Gaussian beam [43–45], vortex beam [46–49], Airy beam [50,51], and holographic imaging [52,53]. However, holograms used to generate a complex intensity profile, for example, an image of Einstein, features very fine ferroelectric domains, which are quite challenging to fabricate.

Recently, detour phase theory [54,55] has been introduced into SH holographic imaging with laser-induced χ⁽²⁾ erasure NPCs. For example, the quasi-phase-matching (QPM)-division multiplexing SH holography in 3D NPCs has been realized [26]. In 2021, the so-called 2D nonlinear detour phase holography was proposed to realize SH holographic imaging with laser-induced χ⁽²⁾ inversion NPCs [27]. However, the current detour phase holograms show limitations in the following two aspects. First, the SH twin-image problem due to the non-collinear QPM leads to reduced efficiency. Second, as of today, none of the reported encoding methods can achieve asymmetric SH wavefront shaping, which seriously limits the development of asymmetric imaging devices and nonreciprocal filter devices.

In this work, we conceptually extend, for the first time, to the best of our knowledge, the 2D nonlinear detour phase holography [27] into 3D space to build the nonreciprocal 3D nonlinear detour phase hologram for achieving SH twin-image elimination and asymmetric SH wavefront shaping. SH twin-image elimination is due to the collinear QPM scheme, and the asymmetric SH wavefront shaping is attributed to the nonreciprocal property of the 3D nonlinear detour phase hologram. The nonreciprocity mentioned here lies in the dependence of the SH phase on the propagation direction of the excitation beam in a 3D nonlinear detour phase hologram. Considering the superiority of the laser-induced χ⁽²⁾ inversion approach over the laser χ⁽²⁾ erasing method in the nonlinear efficiency [56], we choose the former approach to design a 3D detour phase hologram in an as-grown ferroelectric Sr_0.61Ba_0.39Nb₂O₆ (SBN61) crystal. With the proposed 3D nonlinear detour phase hologram, SH beam steering with a steering angle independent of the period and in agreement with the generalized laws of refraction [57] is demonstrated. The nonreciprocity of the designed 3D hologram is verified by the opposite beam steering angles for the forward and backward propagation regimes. Moreover, the asymmetric SH orbital angular momentum (OAM) beam generation and the dual-channel optical device for beam steering and OAM beam generation are implemented with the proposed nonreciprocal 3D hologram for the first time. This work opens a new avenue for the design of nonreciprocal nonlinear optical devices.

2. Theory of 3D nonlinear detour phase algorithm

The general concept of 3D nonlinear detour phase holography for frequency conversion and asymmetric SH wavefront shaping under a plane wave excitation is shown in Fig. 1. 3D nonlinear detour phase hologram as shown in Fig. 1(a) (Left) is composed of a set of 3D basic units characterized by layered antiparallel distributed ferroelectric domains showing inverted sign of χ⁽²⁾. Structure of the (n, m)-th sampling 3D basic unit is shown in Fig. 1(a) (Right). Unlike its 2D counterpart [27], the arrangement of inverted ferroelectric domains in a 3D basic unit is along the propagation direction (y-axis). We denote the SH phase generated by the (n,m)-th base unit as φ_nm. To quantitatively describe the relationship between the SH phase (φ_nm) and the distance that the inverted domain displaced from the center of the 3D basic unit along the y-axis, we define the following parameters. Specifically, the thickness of the 3D basic unit and inverted domain are σy and 0.5σy; the distance between the center of inverted domain and the center of 3D basic unit is P_nm·δy; the length and width of the 3D basic unit are (σx, σz) which are determined by the fabrication resolution, and for convenience, we set σx =σz = 2µm in this theoretical concept demonstration part. In this configuration of an e-polarized fundamental frequency (FF) excitation, only the ee-e interaction is available to reconstruct the nonlinear wavefront via the maximum nonlinear coefficient of d₃₃ in the sample. The thickness of the 3D basic unit (σy) is determined by the collinear-type QPM scheme (2k_1e+ G = k_2e, where G = 2π/σy) in the 3D nonlinear detour phase hologram, as shown in Fig. 1(b). Here, G is the distribution of reciprocal lattice vectors (RLVs) of the 3D hologram, k_1e and k_2e are the wave vectors of the e-polarized FF and SH beams. To simplify the encoding process, we split each 3D basic unit into eight (N = 8) layered antiparallel domain structures, then it will give eight typical 3D basic units whose P_nm values as -1/2, -3/8, -2/8, -1/8, 0, 1/8, 2/8, and 3/8 as shown in Fig. 1(c) (up). We conducted the numerical simulation of the SHG when the FF plane wave passed through these eight 3D basic units, using the split-step fast Fourier transform-based beam propagation method [58]. The numerically calculated constant SH amplitude and eight discrete SH phases (φ_nm = -π, -3π/4, -π/2, -π/4, 0, π/4, π/2, 3π/4) are shown in Fig. 1(c) (middle and bottom). This numerical simulation demonstrates that like its 2D counterpart, within the 3D nonlinear detour phase encoding method, the relationship between φ_nm and P_nm can also be expressed as φ_nm= 2π·P_nm. However, unlike the 2D counterpart, within the 3D nonlinear detour phase encoding method the arrangement of inverted ferroelectric domains in a 3D basic unit is along the propagation direction (y-axis) to yield an SH phase satisfying φ_nm= 2π·P_nm. This proposed encoding method yields opposite P_nm values for forward and backward transmission, which further leads to opposite φ_nm values. It is equivalent to a 3D hologram that yields SH phases with opposite signs (±φ) for the forward and backward propagation regimes. In this work, the opposite SH phase generated by forward and backward transmission is the theoretical basis for realizing the nonreciprocity of the 3D nonlinear detour phase hologram.

 

Fig. 1. The general concept of 3D nonlinear detour phase holography for frequency conversion and asymmetric SH wavefront shaping under a plane wave excitation. (a) (Left) Schematic illustrations of the structure of 3D nonlinear detour phase hologram and its nonreciprocity by the opposite SH phases (±φ) for the forward and backward propagation regimes. (right) Structure of the (n, m)-th sampling 3D basic unit, which is composed of inverted positive (white) and negative (gray) domains. (b) Illustration of the collinear-type QPM scheme. (c) Numerical demonstration of the nonlinear properties of the eight 3D basic units. (Up) The structures of the eight 3D basic units. (Middle and Bottom) The numerically calculated amplitude and phase of SHG after being transmitted through the corresponding units.

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3. Results and discussion

In this section, we use the above-mentioned 3D nonlinear detour phase holography to build nonreciprocal 3D nonlinear detour phase hologram and realize asymmetric SH wavefront shaping for the counter-propagating FF excitations. To verify the nonreciprocity of the designed 3D hologram, SH beam steering angles for the forward and backward propagation regimes were studied. Specifically, three holograms for realizing the SH beam steering at theoretical emission angles of 2.79°, 8.41°, and 14.11° under the forward-propagated FF excitation at 1.56 µm wavelength were designed. These theoretical emission angles were calculated with the general law of SH refraction for NPCs, and more details of the general law of SH refraction can be seen in Section S1 (Supplement 1).

The schematic diagram illustrating the nonreciprocity of the 3D nonlinear detour phase hologram is shown in Fig. 2. The target SH phase distribution with a phase gradient of π/8 rad·µm^-1 for realizing +2.79 degrees’ beam steering is shown in Fig. 2(a). The 3D nonlinear detour phase hologram as shown in Fig. 2(b) was obtained by placing the eight 3D basic units at the corresponding grids according to the SH phase distribution. The thickness of the 3D hologram along the propagation y-axis is σy|_eee= 2π/(k_2e-2k_1e)= 16µm. The inserts show the projection of the eight layers domain structure on the xz plane with dimensions of 160 × 160 µm². Numerically calculated far-field SH intensity distribution shows the steering angle of +2.8° for the forward transmission match the theoretically designed value. In addition, the steering angle of -2.8° for the backward propagation indicates that the phase gradient (-π/8 rad·µm^-1) introduced by the 3D hologram during backward propagation is opposite to that of the forward propagation situation, which verifies the nonreciprocity of the designed 3D hologram. In Figs. 2(d) and 2(e), we illustrate SH beam steering at angles of ±8.41° and ±14.11°. Figure 2(d) (Left) describes the target SH phase distribution with a phase gradient of 3π/8 rad·µm^-1 for +8.41 degrees’ beam steering. The designed 3D hologram is displayed in Figure S2 (Middle) (Supplement 1), with which the numerically calculated SH intensity distribution in the far field is shown in Fig. 2(d) (Right). The steering angles of ±8.4° for the forward and backward transmission match the theoretically designed values. Figure 2(e) (Left) shows the target SH phase distribution with a phase gradient of 5π/8 rad·µm^-1 for +14.11 degrees’ beam steering. The designed 3D hologram is displayed in Figure S2 (Bottom) (Supplement 1), with which the numerically calculated SH intensity distribution in the far field is shown in Fig. 2(e) (Right). The steering angles of ±14.1° for the forward and backward transmission match the theoretically designed values. The comparison of these three 3D holograms for achieving ±2.79, ± 8.41, and ±14.11 degrees’ deflection is shown in Figure S2 (Supplement 1).

 

Fig. 2. Schematic diagram illustrating the nonreciprocity of the 3D nonlinear detour phase hologram. (a) The target SH phase distribution for +2.79 degrees’ beam steering. (b) 3D nonlinear detour phase hologram. The inserts show the projection of the eight layers domain structure on the xz plane. (c) Numerically calculated far-field SH intensity distributions show the steering angles of ±2.8° for the forward and backward transmission situations. This opposite beam steering angle for the forward and backward propagation regimes demonstrates the nonreciprocity of the designed 3D hologram. (d) Demonstration of the SH beam steering at angles of ±8.41°. (Left) The target SH phase distribution for +8.41 degrees’ beam steering. (Right) Numerically calculated far-field SH intensity distributions show the steering angles of ±8.4° for the forward and backward transmission situations. (e) Demonstration of the SH beam steering at angles of ±14.11°. (Left) The target SH phase distribution for +14.11 degrees’ beam steering. (Right) Numerically calculated far-field SH intensity distributions show the steering angles of ±14.11° for the forward and backward transmission situations.

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After verifying the nonreciprocity of the 3D nonlinear detour phase hologram, in the following part, we will use the nonreciprocal 3D hologram to realize asymmetric SH wavefront shaping for the counter-propagating FF excitations. The asymmetric SH OAM beams generated with a 3D nonlinear detour phase hologram are depicted in Fig. 3. As shown in Fig. 3(a), the phase plate used to construct the 3D hologram has a topological charge of 2. The eight layers domain structure of the 3D hologram is shown in Fig. 3(b). Each layer of the domain structure has a size of 160 × 160 µm² at the xz plane and a thickness of σy/8 = 2µm along the y-axis. Demonstrations of the directional formation of asymmetric OAM beams under different FF excitations are shown in Fig. 3(c). The phases of the FF OAM excitation beams with topological charges of -2, -1, 0, 1, and 2 are shown in Fig. 3(c) (Column 1). The numerically calculated SHG under forward FF excitation is shown in Fig. 3(c) (Columns 2-4). Specifically, the numerically calculated SH amplitude and SH phase in the near field is shown in Fig. 3(c) (Column 2-3), where the constant SH amplitude and modulated SH phases with topological charges of -2, 0, 2, 4, and 6 can be seen. The OAM conservation formula under forward FF excitation can be written as: LC_SHG = 2·LC_FF + LC_HOLO, where LC_SHG, LC_FF, and LC_HOLO are the topological charges of the SHG, FF beam, and the 3D hologram, respectively; The corresponding SH intensity distribution in the far field is shown in Fig. 3(c) (Column 4). The numerically calculated SHG under backward FF excitation is shown in Fig. 3(c) (Columns 5-7). Specifically, the numerically calculated SH amplitude and SH phase in the near field is shown in Fig. 3(c) (Column 5-6), where the constant SH amplitude and modulated SH phases with topological charges of -6, -4, -2, 0, and 2 can be seen; The OAM conservation formula under backward FF excitation can be written as: LC_SHG = 2·LC_FF - LC_HOLO; The corresponding SH intensity distribution in the far field is shown in Fig. 3(c) (Column 7). The results of Fig. 3 demonstrate that in general the OAM conservation formula within the nonreciprocal 3D hologram can be written as: LC_SHG = 2·LC_FF ± LC_HOLO, where the plus sign ‘+’ corresponds to the forward excitation and the minus sign ‘-’ corresponds to the backward excitation. This clearly demonstrates the nonreciprocal character of the device.

 

Fig. 3. The asymmetric SH OAM beams generated with the designed nonreciprocal 3D nonlinear detour phase hologram. (a) Phase plate used to construct the 3D hologram. (b) The eight layers domain structure of the 3D hologram. (c) Demonstrations of the directional formation of asymmetric OAM beams under different FF excitations. (Column 1) Phases of the FF OAM excitation beams with topological charges of -2, -1, 0, 1, and 2. (Columns 2-4) Numerically calculated SHG under forward excitation. (Columns 2-3) The near-field SH amplitude and SH phase. (Column 4) The far-field SH intensity distribution. (Columns 5-7) Numerically calculated SHG under backward excitation. (Columns 5-6) The near-field SH amplitude and SH phase. (Column 7) The far-field SH intensity distribution.

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The implementation of dual-channel SH beam steering and SH OAM beam generation by using the designed nonreciprocal 3D nonlinear detour phase hologram are demonstrated in Fig. 4. The phase plate used to generate a vortex beam with a topological charge of 2 at a steering angle of 2.79° is shown in Fig. 4(a). Figure 4(b) is the corresponding eight layers domain structure of the 3D hologram calculated by the phase plate in Fig. 4(a) and the 3D nonlinear detour phase encoding algorithm. Each layer of the domain structure has a size of 160 × 160 µm² at the xz plane and a thickness of σy/8 = 2µm along the y-axis. The directional formation of SH beam steering and SH OAM beam with continuously varied angles under different FF excitation is demonstrated in Fig. 4(c). The phases of the FF excitation beams consisting of the vortex phase component with a topological charge of -1 and gradient phase component for steering angles at 0°, 1°, 2°, and 3° are shown in Fig. 4(c) (Left). Under forward excitation, the numerically calculated SH intensity distribution with beam steering angles at 2.8°, 4.8°, 6.8°, and 8.8° are shown in Fig. 4(c) (Middle), which have good agreement with the theoretical steering angles at 2.79°, 4.79°, 6.79°, and 8.79°. Under backward excitation, the numerically calculated SH OAM beams with steering angles at -2.8°, -0.8°, 1.2°, and 3.2° are shown in Fig. 4(c) (Right), which have good agreement with the theoretical steering angles at -2.79°, -0.79°, 1.21°, and 3.21°. According to the previously mentioned OAM conservation formula under backward FF excitation, LC_SHG = 2·LC_FF - LC_HOLO, theoretically the topological charge of the SH OAM beam is -4. Furthermore, this topological charge value of the SH OAM beam is also numerically verified by the SH phase in the near field. The details are demonstrated in Figure S3 (Supplement 1).

 

Fig. 4. Demonstration of the dual-channel SH beam steering and SH OAM beam generation by using the designed nonreciprocal 3D nonlinear detour phase hologram. (a) Phase plate used to construct the 3D hologram. (b) The eight layers domain structure of 3D hologram. (c) Demonstration of the directional formation of SH beam steering and SH OAM beam under different FF excitations. (Left) The phases of the FF excitation beams consist of the vortex phase component and the gradient phase component. (Middle) Under forward excitation, the numerically calculated SH intensity distributions with beam steering angles at 2.8°, 4.8°, 6.8°, and 8.8°. (Right) Under backward excitation, the numerically calculated SH OAM beam with steering angles at -2.8°, -0.8°, 1.2°, and 3.2°.

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Having demonstrated the theoretical concept of the nonreciprocal nonlinear photonic crystals, in the following, we would like to address the SH twin-image elimination (Section S2, Supplement 1) and the practical feasibility of this nonreciprocal device. As previously mentioned in Fig. 1, the 3D base unit that constitutes the 3D hologram consists of layered modulated χ⁽²⁾ regions and the number of layers (N) of each 3D basic unit is equal to the number of discrete phases, φ_nm. The number of discrete phases (N) is depending on the complexity of the target nonlinear wavefront and/or the practical fabrication accuracy of domain engineering. Although we used eight discrete phases (N = 8) in the theoretical conceptual part, in general, four discrete phases (N = 4) are sufficient to generate a complex SH wavefront [27]. It can be understood from the collinear QPM scheme of the 3D hologram, the key technique for fabricating the 3D hologram is to control the thickness of the layered χ⁽²⁾ region, whose theoretical value is defined as t = 2π/(k₂-2k₁)/N. Where N is the number of layers of each 3D basic unit, k₁ and k₂ are the wave vectors of the FF and SH beams. When considering the 3D hologram located on a SBN61 crystal [59], the ee-e interaction and oo-e interaction are available to reconstruct the nonlinear wavefront via the nonlinear coefficient of d₃₃ and d₃₁, respectively. The required thickness of layered domain structures under the interaction of ee-e and oo-e is depicted in Figure S6 (Supplement 1). Taking N = 4 for example, the required thickness of layered domain structures would be t|_ooe =2π/(k_2e-2k_1o)/4 = 16 µm under o-polarized FF excitation at 2.5 µm wavelength, which is three times larger than the required thickness (t’|_eee =2π/(k’_2e-2k’_1e)/4 = 4 µm) under e-polarized FF excitation at 1.56 µm wavelength. It indicates the required thickness per layer is highly dependent on the wavelength and/or polarization state of the FF excitation. To better demonstrate the practical feasibility of implementation of our design, let us also consider Lithium Niobate (LiNbO₃) [60]as another potential candidate, where 3D laser-induced domain engineering has been recently experimentally demonstrated [22]. Demonstration of the 3D nonlinear detour phase hologram located on an x-cut LiNbO₃ crystal and the required thickness of layered domain structures under the interaction of ee-e and oo-e are depicted in Figure S5 (Supplement 1). In this case, we take N = 8 for example, then the required thickness of layered domain structures would be t|_eee =2π/(k_2e-2k_1e)/8 = 4.4 µm under e-polarized FF excitation at 2.5 µm wavelength. Moreover, when considering the o-polarized FF excitation, the required thickness of layered domain structures can be even larger. Specifically, the required thickness of layered domain structures would be t’’|_ooe =2π/(k’’_2e-2k’’_1o)/8→∞ for the FF wavelength at 1.074 µm due to the phase matching condition (k’’_2e= 2k’’_1o), and the required thickness of layered domain structures would be t’’’|_ooe= 2π/(k’’’_2e-2k’’’_1o)/8 = 5.0 µm for the FF wavelength at 1.0 µm. According to the state-of-the-art ferroelectric domain engineering with LiNbO₃ crystal [22], the 3D domain structure fabricated in an x-cut LiNbO₃ crystal can have periods of 3µm(x) by 2µm(z) by 3µm(y). This fabrication accuracy is better than the theoretical value of the required thickness for layered domain structures making the laser domain engineering in LiNbO₃ a promising candidate for the implementation of our design.

4. Conclusion

In conclusion, we employed the concept of 3D nonlinear detour phase holography to build the nonreciprocal 3D nonlinear detour phase hologram for achieving SH twin-image elimination and asymmetric SH wavefront shaping. The 3D basic units composing a 3D hologram yield opposite SH phases for the forward and backward transmission, which is the theoretical basis for realizing the nonreciprocal 3D hologram and asymmetric SH wavefront shaping. With the proposed 3D nonlinear detour phase hologram, we presented the first demonstration of SH beam steering with a steering angle in agreement with the generalized laws of refraction. The nonreciprocity of the designed 3D hologram is verified by the opposite beam steering angles for the forward and backward propagation regimes. Moreover, the asymmetric SH OAM beam generation and the dual-channel optical device for beam steering and OAM beam generation are also demonstrated with the designed nonreciprocal 3D hologram. This study provides a way to achieve twin-image elimination and asymmetric wavefront shaping at new frequencies. This approach not only can be easily extended to asymmetric 3D nonlinear holographic imaging but also promises new possibilities for the design of nonreciprocal nonlinear optical devices. We believe that the fabrication of the proposed nonreciprocal nonlinear photonic crystal is practically feasible with the currently available technique of laser-induced χ⁽²⁾ inversion or/and erasure.