As laser light propagates through a nonlinear medium, the associated electric field induces polarization within the material. This nonlinear polarization, acting as a driving source, can generate electromagnetic waves at various frequencies other than the incident pump light frequency. Typically, the induced polarization $P$ consists of both linear and nonlinear components. The second-order nonlinear susceptibility ${\chi }^{(2)}$ governs phenomena such as second-harmonic generation (SHG), sum-frequency generation (SFG), and difference-frequency generation (DFG). SHG, one of the most extensively studied parametric processes in nonlinear optics, is often limited by phase mismatch due to material dispersion in nonlinear crystals. To overcome this, two widely used phase-matching techniques are birefringence phase matching (BPM)^[1,2] and quasi-phase matching (QPM)^[3,4], with the latter offering a more versatile and flexible solution to high-efficiency nonlinear frequency conversion.

QPM has been most extensively and popularly demonstrated in periodically poled lithium niobate (PPLN) and other thin plate nonlinear crystals (which cannot be very thick due to technological limitations). QPM is commonly employed for the collinear propagation scheme of the fundamental wave (FW) and second-harmonic waves (SHWs) along the modulation direction (say, $x$ axis) of ${\chi }^{(2)}$ [namely, ${\chi }^{(2)}(x)$] to facilitate the longest coupling length and, thus, the highest conversion efficiency^[5–9]. On the other hand, when FW propagates along the $z$ axis [perpendicular to the modulation direction of ${\chi }^{(2)}(x)$] and shines upon the PPLN thin plate, fruitful nonlinear diffraction phenomena and processes, such as nonlinear Bragg diffraction (NBD)^[10–13], nonlinear Cherenkov diffraction (NCR)^[14–23], and nonlinear Raman–Nath diffraction (NRND)^[24–33], can be generated and observed. In these situations, FW and SHW interact in a non-collinear scheme with respect to the modulation direction of the ${\chi }^{(2)}$ nonlinear microstructure within the nonlinear grating^[34,35]. Due to the more relaxed phase-matching conditions in the transverse direction along the reciprocal lattice vector (RLVs), NRND has garnered significant interest since its first experimental observation in 2008^[24]. NRND produces multiple diffraction orders, with the SHG spot emerging at various diffraction angles, analogous to the multi-order diffraction observed in linear gratings.

In nonlinear optics, the interaction between light and a medium can be manipulated by adjusting optical parameters such as the incident angle^[12,21,23], the beam parameter^[11,36], and the internal structure of the material^[26,31,33]. Previous studies primarily focus on normal or oblique incidence within the vertical plane that is perpendicular to the optical axis of the nonlinear grating, which is categorized as the in-plane normal incidence configuration. However, the case where the incident beam is distributed along a conical surface incorporating variations in both the polar angle $\alpha$ and the azimuthal angle $\phi$, which is categorized as the off-plane conical incidence configuration, remains largely unexplored.

It is well known that, in classical linear optics, when light shines upon a grating in the normal configuration where the incident plane of light is perpendicular to the optical axis of grating, ordinary Bragg diffraction will occur. However, when light is incident in the general conical configuration where the incident plane of light is oblique to the optical axis of the grating, the Bragg diffraction becomes much more complicated^[37]. This phenomenon is known as “conical diffraction,” where diffracted light rays distribute along a conical surface with the incident light direction as its axis, creating distinctive ring-shaped or arc-shaped patterns when projected onto a screen perpendicular to the optical axis^[38,39]. Extending this concept to nonlinear optics, it is then interesting to ask a question: what happens to the nonlinear diffraction of a laser beam by a nonlinear grating if one goes from normal incidence configuration to conical incidence configuration? Would the nonlinear diffraction pattern exhibit similar conical characteristics as observed in linear optics?

To investigate this unexplored aspect of nonlinear optical interactions and answer the above fundamental question of nonlinear diffraction, in this work, we experimentally and theoretically study general conical nonlinear diffraction phenomena and physics. We shine a Ti:sapphire femtosecond pulse laser beam (with a central wavelength at 800 nm) upon a PPLN thin plate nonlinear grating and systematically investigate the nonlinear diffraction pattern of the SHG laser beam in various conical configurations characterized by both the polar angle $\alpha$ and azimuthal angle $\phi$. We aim to explore the spatial distribution characteristics of NRND under conical incidence conditions by directing a focused laser beam to be obliquely incident along the generatrix of a cone onto the PPLN nonlinear crystal. These experimental and theoretical studies enrich the fundamental understanding of conical nonlinear optical interactions and diffraction by nonlinear microstructures.

In traditional linear diffraction experiments with gratings, the simplest situation involves the observation of the diffraction phenomena against normally incident light. This is because, under normal incidence, the diffraction angles are solely dependent on the grating period and the wavelength of the incident light, without being influenced by the incidence angle. When light is incident at a specific oblique angle, the situation becomes a little bit more complex. Yet, when light is incident at the off-plane conical configuration, things become even more complicated. Figures 1(a) and 1(c) depict the diffraction phenomena of linear dielectric gratings and PPLN nonlinear gratings under two-dimensional (2D) oblique incidence, while Figs. 1(b) and 1(d) illustrate the diffraction phenomena under conical incidence for both linear dielectric gratings and PPLN nonlinear gratings. This more complex geometrical configuration introduces unique diffraction patterns, further highlighting the distinct optical properties of PPLN compared to traditional gratings.

We define two coordinate systems: the intrinsic crystal coordinate system ($x–y–z$) and the laboratory coordinate system ($X^{\prime }–Y^{\prime }–Z^{\prime }$). As shown in Fig. 1(a), the grating can be analyzed along the $x$ and $y$ directions, with the grating lines periodically arranged along the $x$ axis and the slits extending along the $y$ axis. The incident light beam strikes the crystal at an oblique angle to the $z$ axis within the $xoz$ plane, leading to variations in the distance between adjacent diffraction spots, which can be explained using the grating equation $m\lambda =d(\sin \alpha +\sin \beta )$^[37], where $m$ is the diffraction order, which can take values of $0,\pm 1,\pm 2$, and so on. $\lambda$ is the wavelength of the incident light, $\alpha$ is the incidence angle, $\beta$ is the diffraction angle, and $d$ is the grating period. The relative angle between the light beam and the grating determines the angular distribution of the diffracted light, with the diffraction patterns for each order distributed perpendicular to the grating lines. This figure can serve as a reference standard for subsequent conical incidence [Fig. 1(b)] and other incidence conditions.

Figure 1(b) shows the diffraction pattern when the incident light beam lies on the surface of a cone and enters the grating along its generatrix. Unlike Fig. 1(a), where the light enters in a fixed normal plane (the $xoz$ plane), here the beam enters from a conical surface, with the incident direction distributed along the generatrix of the cone. Conical incidence implies that the incident beam is no longer confined to a single direction but is distributed across a 3D conical surface. This geometrical configuration results in a more complex diffraction phenomenon, with the diffraction angles not only distributed in a single plane but extending in multiple directions. This can also be described using the grating equation $m\lambda =d(\sin \alpha \cos {\phi }_1+\sin \beta \cos {\phi }_2)$, where ${\phi }_1$ and ${\phi }_2$ represent the azimuthal angles of the incident and diffracted beams, respectively^[37]. While the periodic structure of the grating still plays a role, the multidirectional nature of the incident light causes the diffraction pattern to differ significantly from that in Fig. 1(a). Compared to Figs. 1(a) and 1(b), through conical incidence, one greatly expands the angular distribution of the incident beam and increases the diffraction angles along the $y$ axis, and the diffraction pattern takes on a more complex arc-shaped distribution. This lays the foundation for subsequent nonlinear diffraction analyses of the femtosecond laser beam upon the PPLN crystal nonlinear grating in the conical incident configuration, as shown in Figs. 1(d) and 1(e).

In the experiment, the PPLN sample we used is a $z$-cut thin plate crystal, having dimensions of $20(x)\mathrm{mm}\times 5(y)\mathrm{mm}\times 1(z)\mathrm{mm}$ and a fixed poling period $\mathrm{\Lambda }$ of 6.97 µm. In the crystal coordinate system, the origin is located at the center of the $xoy$ plane of the crystal, where the $x$ axis is aligned with the modulation direction of the effective nonlinear coefficient ${\chi }^{(2)}(G_m)$ with a series of auxiliary RLVs^[40], the $y$ axis is aligned along the domain wall extension, and the $z$ axis is aligned with the optical axis of the $z$-cut LN crystal, which is a uniaxial birefringence crystal. Similar to Fig. 1(a), the incident light beam is obliquely incident on the PPLN within the $xoz$ plane. While the periodic structure of the PPLN resembles that of the grating, it is a nonlinear optical material that generates SHG diffraction through nonlinear optical effects. Its nonlinear characteristics complicate the diffraction behavior. The nonlinear nature of PPLN leads to distinct diffraction phenomena, especially at higher energies and various incidence angles, potentially resulting in intricate diffraction patterns. Figs. 1(a) and 1(c) illustrate that while both gratings and the PPLN exhibit diffraction under similar incidence conditions, the diffraction beams produced by PPLN are a result of the crystal’s frequency-doubling effect, and the diffraction pattern is denser than that of the grating. Unlike the oblique incidence in the $xoz$ plane shown in Fig. 1(c), in Fig. 1(d), the beam also enters from a conical surface, similar to the scenario in Fig. 1(b), but the diffraction occurs in PPLN. The nonlinear optical effects in PPLN result in more pronounced curvature of the diffraction patterns compared to those from the grating. Although both structures exhibit periodicity, the nonlinear characteristics of PPLN can lead to greater wavelength dependence in the diffraction patterns.

Due to the difficulties associated with rotating the beam and to ensure that the domain direction of the crystal (along the $x$ axis) remains parallel to the optical platform, we opted to rotate the crystal instead. This led to the experimental setup shown in Fig. 1(e). The laboratory coordinate system ($X^{\prime }–Y^{\prime }–Z^{\prime }$) is fixed, where the $Z^{\prime }$ axis is parallel to the normal incidence direction of the horizontal-propagating pump laser beam, the $Y^{\prime }$ axis is perpendicular to the optical platform, and the $X^{\prime }$ axis is aligned with the initial domain wall modulation direction when the crystal is placed vertical to the optical platform.

The input pump beam is a Ti:sapphire femtosecond laser with a central wavelength of 800 nm, a pulse duration of 50 fs, a repetition rate of 1 kHz, and a pulse energy up to 4 mJ. A Glan prism is placed in the optical path to ensure that the outgoing beam is horizontally polarized (i.e., extraordinary wave against the LN crystal), with its electric field polarized parallel to the modulation direction of the poled domains, namely, the $x$ axis. The laser beam is focused using a lens with a focal length of 400 mm, and the crystal is positioned just before the focus to avoid any excessive power density that could damage the crystal. At this location, the laser beam forms a 3 mm diameter spot on the crystal surface. The crystal is mounted in a crystal holder [as illustrated in the inset of Fig. 1(e)] that is embedded in a horizontal rotational adjustment stage; besides, the crystal holder can be rotated vertically. The collective rotational action of the horizontal stage and the crystal holder can generate any value of polar angle $\alpha$ and azimuthal angle ${\phi }_1$.

When a laser beam is perpendicularly incident on the $xoy$ surface of the PPLN crystal, the extraordinary wave always remains parallel to the principal plane $xoz$. However, when the crystal is rotated with the beam incident at a specific angle, the extraordinary wave is no longer necessarily parallel to the principal plane $xoz$. Under these conditions, four distinct types of SHG can occur during the laser propagating through the PPLN. The first type involves the interaction of two extraordinary waves producing an ordinary SHW [$ee\text{-}o$]; the second type results from the interaction of two extraordinary waves generating an extraordinary SHW [$ee\text{-}e$]. The third type arises from the combination of an extraordinary and an ordinary wave producing an ordinary SHW [$eo\text{-}o$], while the fourth type involves an extraordinary and an ordinary wave generating an extraordinary SHW [$eo\text{-}e$]. The tensor elements involved in these processes include $d_{22}$, $d_{31}$, and $d_{33}$^{[13,15,17,18,40]}. The input power is set to 0.82 W, which is well below the damage threshold of the PPLN crystal. A low-pass filter below 500 nm is placed behind the PPLN crystal sample to filter out the residual fundamental signal light, thereby ensuring the precise measurement of parameters related to the NRND effect and the accuracy of the NRND experimental results. Since the SHG of different orders has different polarization states, a polarizer is placed after the filter to measure the power of light in both vertical and horizontal polarization directions. A flat screen is positioned 38 cm behind the crystal to receive the output NRND light spots, which are recorded using a Nikon D7200 camera.

Figure 2(a) illustrates the schematic of the beam path of the incident FW and the generated SHW. We suppose the incident beam enters the PPLN nonlinear grating from air at a certain conic angle of $(\alpha ,{\phi }_1)$, where $\alpha$ is the incident polar angle with respect to the $z$ axis and ${\phi }_1$ is the azimuthal angle with respect to the $x$ axis, both measured in the air side. It undergoes refraction within the crystal due to the interfacial refraction law of light, resulting in a refraction polar and azimuthal angle $(\theta ,{\phi }_1)$, which will totally determine the transmitted FW wave transport features together with the FW wave vector $k_1$ [denoted in the red plane in Fig. 2(a)] within the LN crystal. When the interaction between FW and PPLN crystal occurs, an SHW is generated with a wave number $k_2$ and propagates in a distinct plane [shown in blue plane in Fig. 2(a)], differing from the FW’s red plane. We define the SHG angular coordinates as $(\beta ,{\phi }_2)$, where $\beta$ is the angle between the SHW and the optical axis, and ${\phi }_2$ is the azimuthal angle in the $xoy$ plane relative to the $x$ axis. Both the polar and azimuthal angles of the SHW differ from those of the FW. Similarly, following Snell’s law, when the SHW propagates to the rear surface of the crystal and exits into the air, the propagation can be characterized by the refraction angle of $(\gamma ,{\phi }_2)$, where $\gamma$ is the angle between the external SHW and the optical axis of the crystal.

To provide a clearer understanding of this conical nonlinear diffraction process with the PPLN slab, we construct and analyze the 3D physical diagram of NRND QPM conditions, as presented in Fig. 2(b). Obviously, the general vector phase-matching condition, $2{\overrightarrow{\mathit{k}}}_1+G_m+\mathrm{\Delta }k={\overrightarrow{\mathit{k}}}_2$, holds under various incident conditions. We readily find that the phase matching components along the $x$ axis, $y$ axis, and $z$ axis within the PPLN thin plate satisfy the geometric configuration defined by $$\{\begin{array}{l}{k}_{2x}=2k_1\sin \theta \cos {\phi }_1-k_2({\lambda }_2)\sin \beta \cos {\phi }_2=G_m\\ m=0,\pm 1,\pm 2,\dots \\ k_{2y}=k_2({\lambda }_2)\sin \beta \sin {\phi }_2=2k_1\sin \theta \sin {\phi }_1\\ k_{2z}=k_2({\lambda }_2)\cos \beta =2k_1\cos \theta +\mathrm{\Delta }k\end{array},$$(1)where $k_2=2\pi n_2/{\lambda }_2$ represents the SHW signal’s wave vector, $G_m=2\pi m/\mathrm{\Lambda }$ is the RLV, $m$ is the NRND order, and $\mathrm{\Lambda }$ is the poling period of the PPLN crystal (6.97 µm). $\mathrm{\Delta }k$ represents the phase mismatch along the $z$ axis. By solving the phase-matching equations, we can determine the SHW beam’s angle $\beta$ with respect to the $z$ axis, as well as the projection angle ${\phi }_2$ in the $xoy$ plane, as $$\beta =\arcsin \left[\sqrt{{\left(\frac{\sin \alpha }{n_2}\right)}^2-\frac{2\sin \alpha m{\lambda }_2}{\mathrm{\Lambda }{n}_2^2}\cos {\phi }_1+{\left(\frac{m{\lambda }_2}{n_2\mathrm{\Lambda }}\right)}^2}\right],$$(2)$${\phi }_2=\arctan \left(\frac{2\sin \alpha \sin {\phi }_1\mathrm{\Lambda }}{2\sin \alpha \cos {\phi }_1\mathrm{\Lambda }-m{\lambda }_1}\right).$$(3)

Using Snell’s law described by $\sin \gamma =n_2(\beta ,{\phi }_2)\sin \beta$, we find that the external polar angle of SHW in the air is $$\gamma =\arcsin \left[\sqrt{{(\sin \alpha )}^2-\frac{2m{\lambda }_2\sin \alpha }{\mathrm{\Lambda }}\cos {\phi }_1+{\left(\frac{m{\lambda }_2}{\mathrm{\Lambda }}\right)}^2}\right].$$(4)

Equations (3) and (4) clearly reveal that changes in the incident angle $\alpha$ and azimuthal angle ${\phi }_1$ of the FW beam, the wavelength $\lambda$ of the interacting waves, the diffraction order $m$, and the poling period $\mathrm{\Lambda }$ of PPLN would result in corresponding changes in the exit angles $\gamma$ and ${\phi }_2$ of SHW. Notably, $\gamma$, which is the angle between the SHW beam in air and the crystal’s $z$ axis, is independent of the refractive index $n_2$ of the SHW beam within the crystal. This indicates that the emission angle of the SHW beam is unaffected by both its polarization state and refractive index.

Figures 3(a)–3(d) illustrate the geometric diagram of the PPLN crystal rotation and the generated NRND spots observed in the experiment, where the pump beam consistently propagates along the $Z^{\prime }$ axis in the laboratory frame. In this experiment, the laser pulse energy was set at 0.82 W, with a duration of 50 fs, and the beam diameter incident on the front surface of the crystal was approximately 3 mm. The diffraction screen was positioned 38 cm behind the PPLN crystal, allowing for the clear observation of the NRND diffraction pattern generated by the crystal on the screen. These parameters ensured the reproducibility and consistency of the experimental results, providing a reliable basis for the analysis of the observed diffraction patterns.

Figures 3(e)–3(i) present the experimental results under varying incident angles. From top to bottom, the incident angles are $\alpha =0°,{\phi }_1=0°$; $\alpha =-10°,{\phi }_1=0°$; $\alpha =-11.169°,{\phi }_1=-26.302°$; $\alpha =-14.106°,{\phi }_1=-44.561°$; and $\alpha =-17.964°,{\phi }_1=-55.734°$. Here, negative angles are defined as follows: when viewed from the negative $z$ axis, the polar angle $\alpha$ is defined as the negative angle formed by the incident beam rotating from the $yoz$ plane toward the positive $x$ axis. Similarly, the azimuthal angle ${\phi }_1$ is characterized as the negative angle formed by the incident beam rotating from the $xoz$ plane toward the negative $y$ axis. Consequently, when both $\alpha$ and ${\phi }_1$ are negative, the beam resides in the eighth quadrant of the crystal coordinate system.

Figure 3(a) shows the case of normal beam incidence, corresponding to the experimental result in Fig. 3(e), where $\alpha =0°,{\phi }_1=0°$. The observed and recorded diffraction spots correspond to seven nonlinear diffraction orders on the left and six on the right of the central spot, defining negative orders on the left and positive orders on the right. Overall, the emitted diffraction spots remain on the same horizontal coordinate line (parallel to the $x$ axis) in this case. Moving from Figs. 3(a) and 3(b), the PPLN crystal is rotated about the $y$ axis by angle $\alpha$, thus introducing polar angle $\alpha$. Here, a counterclockwise rotation of 10° around the $y$ axis results in the beam rotating clockwise relative to the $yoz$ plane, giving a result of $\alpha =-10°$, as reflected in Fig. 3(f). On this condition, the NRND spots still remain on the same horizontal coordinate line but with a slightly larger diffraction angle. Subsequently, from Figs. 3(b) and 3(c), the crystal is rotated about the $x$ axis by angle ${\phi }_1$, introducing azimuthal angle ${\phi }_1$. A 5° counterclockwise rotation around the $x$ axis causes the beam to rotate clockwise relative to the $xoz$ plane. Using the angle transformation formula, this results in $\alpha =-11.169°,{\phi }_1=-26.302°$. At these stages, both the azimuthal angle and the half-cone angle have changed, which can be calculated from formulas $\alpha =-\arccos [\cos (y_{\text{rotate}})\cos (x_{\text{rotate}})]$ and ${\phi }_1=-\arctan [\cot (y_{\text{rotate}})\sin (x_{\text{rotate}})]$, where $x_{\text{rotate}}$ and $y_{\text{rotate}}$ represent the rotation angles with respect to the $x$ and $y$ axes. The corresponding experimental result in Fig. 3(g) shows that the diffraction spots no longer align horizontally and begin to deviate away in the vertical direction (namely, the $z$ axis).

As the crystal continues to rotate around the $x$ axis [from Figs. 3(c) and 3(d)], the absolute values of both the azimuthal angle ${\phi }_1$ and the polar angle $\alpha$ increase. This indicates that the cone angle of the beam’s conical surface is widening, as observed in the experimental results in Figs. 3(h) and 3(i). The diffraction pattern takes on an irregular, arc-shaped arrangement, and as the cone angle increases, the curvature of the arc formed by the diffraction spots also increases. The distribution of the diffraction orders is asymmetric relative to the central diffraction spot, with the diffraction spots on the right shifting toward the positive $Y^{\prime }$ axis. As the diffraction order increases, the distance between the higher-order spots and the zeroth-order spot in the $Y^{\prime }$ direction grows. In contrast, the diffraction spots on the left deviate less dramatically than those on the right. The asymmetry can be well explained by Eqs. (3) and (4). As derived from Eqs. (3) and (4), when the pump laser beam is incident at an angle, the exit angles $\gamma$ and azimuthal angles ${\phi }_2$ for each diffraction order differ, and the principal planes of the diffraction orders are no longer the same. Furthermore, due to differences in the diffraction orders of the emitted beam, the spots show deviations in the $Y^{\prime }$ direction, thus forming a conical arc-like arrangement of diffraction patterns.

To better visualize and quantify the distribution characteristics of NRND spots across different incident angles, the diffraction spots at different orders are connected by green dashed lines, as illustrated in Figs. 4(a1)–4(b4), again clearly revealing an increasing conical curvature from top panel to bottom panel. Experimentally, the NRND diffraction angles ${\sigma }_m=\arctan (d/{Z^{\prime }}_{C-S})$ are determined by measuring the distance between the crystal and the detection screen, defined as ${Z^{\prime }}_{C-S}$ [as indicated in Fig. 3(a)], along with the separation $d$ between the diffraction spots of each order and the zeroth-order spot on the screen. This angle can be theoretically calculated using the formula $\cos {\sigma }_m=({\overrightarrow{\mathit{k}}}_{20}·{\overrightarrow{\mathit{k}}}_{2m})/(|k_{20}||k_{2m}|)$, and we find $${\sigma }_m=\arccos (\sin \alpha \cos {\phi }_1\sin \gamma \cos {\phi }_2+\sin \alpha \sin {\phi }_1\sin \gamma \sin {\phi }_2+\cos \alpha \cos \gamma ).$$(5)

Here, one can find that the deviation is influenced by both the incident FW orientation angles of $(\alpha ,{\phi }_1)$ and the output SHW orientation angles of $(\gamma ,{\phi }_2)$. A comparison of the experimental and theoretical diffraction angles is shown in Fig. 5. The data show a strong consistency between the experimental measurements and theoretical predictions. The changes in the incident angles $\alpha$ and ${\phi }_1$ lead to slight differences in the output diffraction angles. Notably, the angular distribution exhibits a non-symmetric pattern between the positive and negative diffraction orders. This asymmetry points to potential new insights into the interaction between incident light and the crystal structure under such a special conical pump configuration.

Further, we introduce a deviation angle to describe the degree of the arc-shaped arrangement of the diffraction spots for various incident angles, which is defined as the angle between each diffraction order and the zeroth-order output beam, named ${\psi }_m=\arctan (\mathrm{\Delta }Y/{Z^{\prime }}_{C-S})$ Experimentally, the deviation angle ${\psi }_m$ is determined by measuring the distance between the crystal and the detection screen ${Z^{\prime }}_{C-S}$, as well as the vertical displacement $\mathrm{\Delta }Y$ between the diffraction spots of each order and the horizontal line passing through the zeroth-order spot on the screen. These parameters are clearly illustrated in the inset of Fig. 4(b₄). To facilitate a more intuitive observation and explanation of this diffraction angle deviation phenomenon, we have made a comparison between theoretical predictions and experimental measurements and show the result in Fig. 6.

In the absence of the azimuthal angle, namely ${\phi }_1=0$, all diffraction spots align on the same horizontal straight line, resulting in a deviation angle being exactly 0°. However, when the azimuthal angle ${\phi }_1$ is present, we can examine the deviation angles numerically, and the results are shown in Figs. 6(b)–6(d). Here, we define the zeroth-order horizontal line as 0°, with the diffraction spots located below this line yielding negative deviation values, and those above this line yielding positive values.

Figures 6(c) and 6(d) all reveal that, as the diffraction order increases, the deviation angle on the right side of the central spot gradually increases, revealing increasingly significant cone-angle radiation characteristics. In contrast, the deviation angle on the left side exhibits a more complex behavior. As shown in Fig. 6(d), during the variation from the zeroth-order to the negative fifth-order, the deviation angle ${\psi }_m$ initially increases and then decreases along the $-Y^{\prime }$ direction. This result visually indicates that the NRND effect does not exhibit a linear conical influence on the radiated directions of different diffraction orders under specific cone angle incidence conditions. Theoretical and experimental analyses reveal minor discrepancies, which may stem from insufficient precision in measuring the size and position of the diffraction spots.

Finally, we present the diffraction power of each diffraction SHG beam at different incident angles using a line graph. A suitable low-pass filter is selected and placed before the power meter to eliminate any residual pump light energy, ensuring that the power meter probe remains perpendicular to each diffracted beam. Due to the phase mismatch in the longitudinal direction ($z$ direction), the conversion efficiency of this process is relatively low, with only a small fraction of the FW pump light being converted to the NRND SHG beam. The measured power levels are illustrated in Fig. 7. As illustrated in Figs. 7(a)–7(d), for every incident angle, the zeroth-order diffraction exhibits relatively higher power, while higher-order diffractions show progressively weaker power, following a decreasing trend consistent with typical energy distribution in diffraction processes. Despite variations in the incident angle, the nW-level powers can still be accurately measured, and the variations in power for the same diffraction order are not significant. Further analysis of the energy distribution across various diffraction orders reveals that, although power variations between the orders are minimal, distinct energy differences still exist between the diffraction orders. Although the overall diffraction power is not particularly high, it nonetheless effectively reflects the nonlinear optical properties of the PPLN crystal.

In summary, we have realized conical NRND in a single PPLN nonlinear grating shined by a high-peak-power Ti:sapphire femtosecond pump laser with different incident polar angels and azimuthal angles. To comprehensively validate the diffraction characteristics under this general conical incident configuration of the pump laser, we investigate the angular distribution of SHG across different diffraction orders including both the diffraction and deviation angles. We further measure the NRND power and polarization angles for different diffraction orders in PPLN crystals. The experimental findings reveal that, as the conical incident angle increases, the zeroth-order beam remains collinear with the incident beam, while the diffraction spots at different spots located on the both sides of the central diffraction spot are followed by a curved trajectory, with the curvature of the trajectory becoming more pronounced at larger conical incident angles.

In the context of conical NRND, transverse phase matching exhibits significant angular tuning flexibility due to the introduction of the physical parameters of polar angles and azimuthal angles. This novel approach of conical incidence provides profound insights into the physical mechanisms underlying NRND in nonlinear optical microstructures, offering enhanced control over ultrashort pulse laser interactions in nonlinear media. The conical tuning technique greatly increases the degrees of freedom for NRND TPM control in nonlinear crystals. We anticipate that this method would lead to advanced techniques for ultrashort pulse characterization and high-precision distortion measurement of domain structures and beam collimation diagnostics, thereby providing promising and powerful tools for nonlinear optical systems and metrology.