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Abstract
The direction variation of the fundamental wave in the same nonlinear photonic crystal would cause different pattern of harmonics generation. In a 2D/3D crystal with dense reciprocal lattice vectors, there will be large numbers of conical harmonic beams evolving with direction change of the fundamental wave. By rearranging the Ewald sphere and superposing it into the Ewald shell, we have a hybrid Ewald construction. It becomes a simple but useful geometric method to comprehensively depict the distribution of these quasi-phase-matching second harmonics and their conical form evolution. It presents conical second harmonic beams by their related reciprocal lattice vectors and simplifies the beams’ distribution according to spatial arrangement of those reciprocal lattice vectors. It finds that the conical beams will create, annihilate, or get enhanced in specific order when fundamental waves change incident directions. We applied the method on a periodically poled 2D LiTaO3 crystal and all observed phenomena, meet the method’s predictions. In our experiment, we observed that the conical beams distorted along the optic axis of the sample due to anisotropy, which was generally overlooked by earlier researches. The eccentricities of their ring projections suggest a potential auxiliary approach for crystal dispersion measurement.
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1. Introduction
Conical radiation is a special nonlinear optical phenomenon generated when an ultrashort pulse laser is transmitted in a nonlinear photonic crystal (NPC) [1–3]. The ring pattern it produces has a certain symmetrical beauty, which often arouses research interest. Conical second harmonic generation (SHG) also have significant applications in optics and related fields of physics. For example, in the field of nonlinear optical imaging [4] and the generation of entangled photons [5]. In bulk dielectric crystalline materials, elastic scattered light is generated due to defects inside or on the surface of the crystal [6], and they can assist quasi-phase-matching (QPM) optical frequency conversion and conical SHG in one-dimension(1D) NPCs [7], two-dimension(2D) hexagonally poled LiTaO3, and LiNbO3 crystals [8,9].
As scattered fundamental wave (FW) involved in the frequency conversion, the previous rigorous collinear or non-collinear QPM condition is sometimes relaxed to a cone shaped SHG. The scattering feature of this QPM process provides a unique subcategory of broadband frequency conversion, alongside with partial phase-matching as in nonlinear Raman-Nath diffraction, or structure induced broadband QPMs, etc [10–12]. Reciprocal lattice vectors (RLVs) in NPCs may provide multiple phase-matchable options to form a series of such conical beams. With denser RLVs in reciprocal space of NPCs, whether 1D, 2D or even higher dimension, more simultaneous RLVs’ participation in scattering-assisted QPM, make the phenomena complicated. Early research focused on elastic scattering with low-order RLV participation for 2D structures [6], or only scattering with a fixed configuration of reciprocal space with the FW for 1D structures [7]. Mostly these cases centered their research around a static situation that the relative configuration of the incident laser and the crystal sample is fixed. This restricts full access to the details of the phenomena, especially evolution of the conical beams when the incident FW keeps changing orientation with respect to the sample and hamper its potential applications. When a dynamic process is introduced, NPC sample rotates, FW making diverse orientations with NPC causes conical beams evolve into lager/smaller shapes. With some orientations, new conical beams are created and old conical beams gone. Besides, during the course, higher-order effects with denser RLVs make conical radiations intersect each other more frequently and may confuse observers. In 2016, Huang et al. proposed the Ewald shell construction for a comprehensive investigation of this phenomena [13]. In their proposition, proper RLVs that could participate the QPM processes locate in a shell of the reciprocal space, and they can be used to represent its relative conical beam and simplify the understanding of this dynamic process. However, the method has shortcomings, such as not being able to explain the enhancement of harmonic intensity during the FW/sample rotation. Therefore, improvements must be made.
The Ewald construction inspired the Ewald shell, well known in X-ray scattering [14], also is widely used in optics, as in holography [15], photonic quasicrystals [16], light extraction from LEDs [17], or for structural colors [18]. Its early appearance in nonlinear optics is Ewald construction for SHG process in a nonlinear photonic crystal [19]. Proposed by V. Berger, the nonlinear Ewald construction is defined by the set of RLVs supporting collinear or non-collinear QPM in the SHG process, and that set of RLVs forms a sphere (Ewald sphere) with its radius as wave-vector ${\textbf{k}_2}$ of the SH. Readers may easily find the Ewald shell and Ewald sphere are different, as Ewald shell is a 3D construction (as in a space between two spheres) and Ewald sphere is a 2D construction (as a ball surface of radius k2). However, we found that with geometric rearrangement, they may combine to solve the missing link of SH intensity enhancement in conical beam dynamics.
In this paper, we propose a hybrid geometric method to enhance the former Ewald shell methods and apply it to our observation of the SH conical beams in a 2D LiTaO3 NPC, both statically and dynamically. It focusses the relative displacement of the RLV that participates the QPM process, and using its head trajectory in our vector diagram to assist the study of conical beam evolution. This newly improved method sets an order in the evolution process and reveals that the enhancement of QPM SHG during the dynamic process either happens before the annihilation of the conical beam relating to the same RLV or after the creation of a conical beam relating to the RLV. In our experiment, a series of ring-shaped SHG formed by the projection of conical beams were spotted. Our model well-explained the dynamics of the conical SH beams and SH intensity enhancement during the sample rotation process. The slight distortion observed in the conical beams due to birefringence in our experiments suggests a potential auxiliary approach for crystal dispersion measurement.
2. Methods
The sample used in our experiments is a z-cut congruent 2D NPC LiTaO3 with a rectangular lattice prepared by the electric field poling technique. Sample thickness being 0.05cm, its periods along the x and y directions are 12.21µm by 12.20µm with 0.5cm total length along both x/y directions. The FW is a 1064nm Nd: YAG laser (InnoLas SpitLight Compact 200) with 5ns pulse width, 10Hz repetition and beam waist of 0.5cm. A schematic diagram of 3D optical mounting is shown in Fig. 1. The FW passes through a combination of half-wave plate and Glan laser prism. This intensity-controlled z-polarized light rises a bit by a pair of beam benders to suit sample height. A plano-convex cylindrical lens with a focal length of 2cm directs this raised light into our sample. The lens is 3.6cm away from the entrance of the sample, it makes the FW focus at its focal plane, namely 1.6cm away from the sample. When the FW hits the entrance of the sample perpendicularly, it slightly diverges to 0.25cm along z-axis while keeping the 0.5cm beam waist unchanged along y-axis. So, only a small portion of the FW energy, that is, the center part of the beam, estimated to be about 25% of total FW energy goes into the sample. The sample is mounted atop a multi-axis stage and the focused FW perpendicularly goes onto its x-surface, while we partially shielded the sample to get a better observation of the SH phenomena. An imaging screen was placed at 8.5cm from output x-surface of the sample to collect the generated SH beams. The experiments were performed in a dark environment.
3. Discussions for conical SHs
In our experiments, we took photos with different FW input energy. We gradually increased the FW energy from 0mJ to 15mJ per pulse. The observed results are given in Fig. 2(a-d). At low FW energy, captured image shows a central horizontal SH line consisting most of random phase-matching SH [20] and some symmetrically distributed SH arcs. As the FW input power raised, more details reveal several discrete rings, with their top and bottom parts gradually appearing to fill the area and the horizontal line obtained from random phase-matching extended further to two sides. The thickness of the conical rings also gradually increased.
Fig. 2. The scattering-assisted conical QPM SHs with different incident FW powers(a-d). The FW is along the x-axis. The z-axis is the optical axis of the sample and parallel to the polarization of FW. (e) Elastic scattering QPM conical beams of two symmetrical RLVs with respect to FW.
Phase-matching condition for elastic scattering assisted conical SHG in 2D NPCs is
The SH conical beams are mostly e-rays [21]. Anisotropy of this polarization causes these SH cone deforms along the z-axis. As a result, the ring projection of the SH cone on the imaging screen is elliptical rather than circular, even if the RLV is parallel to the FW. Anisotropy demands effective refractive indexes of FW${n_{eff1}}$ and of SH ${n_{eff2}}$ being slightly different from principal values of e-rays when they don’t propagate vertically to the optic axis. This leads to size modification in ${k_1} = {n_{eff1}}{k_{10}}$ and ${k_2} = {n_{eff2}}{k_{20}}$, thereby reshaping the vector polygon for phase-matching and the SH cone outline. Here, the effective refractive index of FW is ${n_{eff1}} = {\{{{{({{{\cos }^2}\beta \cdot {n^2}_{e(1)} + {{\sin }^2}\beta \cdot {n^2}_{o(1)}} )} / {({{n^2}_{o(1)} \cdot {n^2}_{e(1)}} )}}} \}^{ - 1/2}}$ and of SH ${n_{eff2}} = {\{{{{({{{\sin }^2}\theta \cdot {n^2}_{e(2)} + {{\cos }^2}\theta \cdot {n^2}_{o(2)}} )} / {({{n^2}_{o(2)} \cdot {n^2}_{e(2)}} )}}} \}^{ - 1/2}}$, where $\theta$ is the angle between ${\textbf{k}_1}$ and ${\textbf{k}_2}$, and $\beta$ for deviation of ${\textbf{k}_1}^\prime$ from the optic axis.${n_0}$ and ${n_e}$ are principle refractive indices of o-rays and e-rays, respectively.
We analyze, as an instance, the case of the SH projection linked to RLV ${\textbf{G}_{2,0}}$ and calculate the eccentricity of this oval ring. Since the SH ring projection deforms along z-axis, the vector polygon of the conical SH defines the relation of wavevector components in xz-plane as
andOur measured result shows that the SH cone shrink a little bit along z-axis and the resulting ellipse projection has an eccentricity of 0.052, whereas its theoretical value is 0.087 by Abedin’s dispersion [22] and 0.044 by Bruner’s dispersion [23].
The eccentricity of conical radiation suggests birefringence feature of positive and negative uniaxial crystals. In our case, congruent LiTaO3 is a positive crystal and make the SH cone shrink along z-axis. However, with same RLV magnitude and incident FW orientation restrictions, but in a poled structure of negative uniaxial crystal, such as LiNbO3, the SH cone would stretch a bit along z-axis. The eccentricity of SH cone projections with these two situations of different crystals is shown in Fig. 3. Further calculations show that under different temperatures, these distortions remain their trends as shown in Fig. 3(b,c) with slight numeric differences. As QPM does not necessarily demand a poled crystal, the material’s nonlinear coefficient can be directly modulated by mechanical extrusion and other means [24,25],so as to achieve conical radiation without complex poling preparation and unleash the potential application of this conical QPM process.
Fig. 3. (a) Calculated eccentricity of conical radiation projection at different FW wavelengths for LT and LN (inset) with FW parallel to the RLV of ${\textbf{G}_{2,0}}$. Schematic projection of a deformed conical SH for (b) LT and (c) LN; Dotted circles represent conical SH on the isotropic assumption, while solid green ellipses are practical anisotropic situations. The marked points represent the eccentricity corresponding to an incident wavelength of 1064nm.
If we rotate the sample, and the rings pattern of the conical beams would change along. Figure 4 shows an evolution example of conical beam relating to RLV ${\textbf{G}_{2, - 3}}$. Figure 4(a) shows that ${\textbf{G}_{2, - 3}}$ aligns with FW ${\textbf{k}_1}$, the conical beam processes its largest spatial volume, while when the sample rotates and ${\textbf{G}_{2, - 3}}$ deviates away from ${\textbf{G}_{2,0}}$ further along as in Fig. 4(b-d), the conical beam keeps shrinking until it disappears in Fig. 4(e). The process could be explained by nonlinear Ewald shell model [13]. The Ewald shell model is constructed as follows (Fig. 5(a)). First, draw a series of concentric circles with their center at point S. Attach the tail of ${\textbf{k}_1}$ to point S and set the head of ${\textbf{k}_1}$ as the origin of the reciprocal space O. Subspace between two spheres of $|{{k_2} - {k_1}} |$ and $|{{k_2} + {k_1}} |$ constitutes a shell, where any RLV in this subspace may corresponds to the conical radiation one to one. The statement is based on Eq. (1) of the vector phase matching condition and is effective even if the crystal sample is rotating. So, we can use this principle to infer the evolution of each conical radiation. As ${\textbf{G}_{2, - 3}}$ rotates with the sample, the decreasing modulus of ${\textbf{k}_1} + {\textbf{G}_{2, - 3}}$ leads to a smaller half apex angle $\beta$ of the conical beam. When ${\textbf{G}_{2, - 3}}$ reaches onto the $|{{k_2} - {k_1}} |$ sphere, the conical beam of ${\textbf{G}_{2, - 3}}$ shrinks to one simple straight beam and about to disappear.
Fig. 4. Variation of the elastic scattering conical beam during sample rotation. (a-e) An experimental example: enhancement then annihilation process of the conical beam of RLV ${\textbf{G}_{2, - 3}}$; the small red arrows indicate the exact position of the corresponding conical beam. The small blue arrow suggests the SH intensity enhancement spot. (f) Analytical diagram of the sample rotation. $\theta$ is the rotation angle of the sample, ${\theta _1}$ is the angle of refraction when the fundamental wave enters the sample, $\theta - {\theta _1}$ is the rotation angle of the FW in the sample relative to ${\textbf{k}_{10}}$. ${\textbf{G}_{\textrm{m},\textrm{n}}}^\prime$ is the RLV ${\textbf{G}_{\textrm{m},\textrm{n}}}$ after rotation.
Fig. 5. Hybrid Ewald geometric method. (a) Original Ewald shell model. Reciprocal space origin (yellow dot). (b) Nonlinear Bragg diffraction, corresponding to the Ewald sphere. (c) Rearrangement of Ewald sphere in (b) with ${\textbf{k}_2}$ vectors removed. (d) Hybrid Ewald geometric method: we put the rearranged Ewald sphere and Ewald shell together, make their reciprocal spaces completely overlap as one, and remove all $\textbf{G}$/${\textbf{k}_1}$ arrows. All RLVs are defined according to their relative position to the origin of the reciprocal space O, and will not be explicitly shown in the plot. The vector $\textbf{G}$ undergoes rotation from phase-matchable configuration I to phase-unmatchable configuration IV. Configuration II is the RLV reaching onto the Ewald sphere to form the harmonic enhancement. Configuration III is the RLV reaching onto the |k2-k1| sphere, which will annihilate the conical beam.
Motions of other RLVs may cause their corresponding creation/annihilation process too, their results of conical beams are shown in the Table 1. The experimental results agree well with Ewald shell method. In a clockwise rotation 30°, conical beams of ${\textbf{G}_{1,5}}$, ${\textbf{G}_{1,4}}$, and ${\textbf{G}_{1,3}}$ were created in turn and the conical beam annihilations of ${\textbf{G}_{2, - 3}}$ and ${\textbf{G}_{2, - 2}}$ followed in order.
Table 1. Rotating angles to reach annihilation/creation configuration for observed conical beams (represented by RLVs).
A detailed geometry about the sample rotation is presented in Fig. 4(f) to help readers to duplicate the experiment. It reveals a common flippancy that when the sample and its reciprocal space rotate an angle of $\theta$ with the multi-axis stage while the FW incident direction fixed, the FW actually rotates an angle of $\theta - {\theta _1}$ inside the crystal, where ${\theta _1}$ is the refractive angle of the FW.
During this rotation of the sample, the scattering SH cones intersect with each other frequently (Visualization 1). When they do, sometimes the SH intensity increases sharply at specific locations along the horizontal line. Since the RLV of a 2D LiTaO3 sample with the same period can be denser than that of 1D crystal, it produces more conical beams, and the sharp increase in SH intensity as the rings intersect are more likely to occur. During the same 30° clockwise rotation process as mentioned earlier, conical beams of ${\textbf{G}_{1,6}}$, ${\textbf{G}_{1,5}}$, ${\textbf{G}_{1,4}}$, ${\textbf{G}_{1,3}}$, ${\textbf{G}_{2, - 3}}$, ${\textbf{G}_{2, - 2}}$, were enhanced in sequence. The corresponding photographs of these enhancements are in Fig. 6(a-e) and Fig. 4(d) and corresponding results also shown in the Table 2. The richness of the experimental phenomenon would appreciate a simple explanation.
Fig. 6. Experimental photo of SHG enhancement during sample rotation. Rotation angels are recorded in each panel. (a) Conical SH enhancement of RLV ${\textbf{G}_{1,6}}$, where it intersects with another conical beam of RLV ${\textbf{G}_{2,0}}$. (b-e) Conical SH enhancement of RLV ${\textbf{G}_{1,5}}$, ${\textbf{G}_{1,4}}$, ${\textbf{G}_{1,3}}$, ${\textbf{G}_{2, - 2}}$.
Table 2. Rotating angles to reach harmonic enhancement for some observed conical beams (represented by RLVs).
Our primary consideration is to make improvements to the nonlinear Ewald shell method as it’s already successful in explaining some crucial details of the phenomena while failing to provide insight about enhancement of these dynamic conical beams. More than that, we are inspired by nonlinear Bragg diffraction [26] which may cause single or multiple SH enhancement simultaneously and their SH intensity extremes demand the FW in phase-matching with collinear FW waves to secure their maximal interaction volume. The proper set of RLVs meet nonlinear Bragg diffractions that enhances SH constructs the nonlinear Ewald sphere (Fig. 5(b)), which is defined by the point set in the reciprocal space as
where ${\textrm{G}_{\textrm{m},0}}$ is the longitudinal component of ${\textbf{G}_{\textrm{m,n}}}$ that is parallel to the ${\textbf{k}_{10}}$, and ${\textrm{G}_{\textrm{0,n}}}$ is its transverse component perpendicular to ${\textbf{k}_{10}}$.We just superpose a rearranged Ewald sphere (Fig. 5(c)) of radius |k2| into the Ewald shell model with their reciprocal spaces coinciding to form a hybrid Ewald geometry (Fig. 5(d)). At the same time, we discard the $|{{k_2} + {k_1}} |$ sphere in the Ewald shell of Fig. 5(a) as RLVs near that region are extreme large and their QPM processes would be small. This simple method can get all things considered in the dynamics elastic scattering QPM, including the distribution, creation/annihilation, and enhancement of the conical beams.
As shown in Fig. 5(d), when the RLV and the sample are rotated, their configuration changes from I through II, III to IV. For configuration I, the RLV $\textbf{G}$ is phase-matched and can produce a corresponding conical beam. When the $\textbf{G}$ head of some specific conical beams reaches onto the Ewald sphere, that is, at configuration II, the nonlinear Bragg diffraction induces the horizontal part of the conical ring enhanced. Configuration III is another critical case, the conical beam of $\textbf{G}$ now degenerates to an SH spot. Beyond this spot (Configuration IV) will lead to complete annihilation of the conical beam. The relative order of configuration II (enhancement) and III (creation/annihilation) can be readily checked by comparing the data shown in Table 1 and 2. If one reverses the rotation, making the RLV $\textbf{G}$ go from configuration IV to configuration I, the annihilation at configuration III would change to be a creation process of the conical beam relating to this RLV $\textbf{G}$, and the conical beam keeps expanding with an enhancement at configuration II. Geometric and symmetry restrictions demand the creation/annihilation of conical SH beams and their enhancements take place only in the horizontal direction.
Sometimes, when two conical beams meet and nonlinear Bragg diffraction condition also satisfied, it appears that the intersection get the SH intensity to rise. A clear example is showed in Fig. 6(a), where conical beams of ${\textbf{G}_{1,6}}$ and ${\textbf{G}_{2,0}}$ intersect and get enhanced. However, when some conical rings intersect, harmonic enhancement points do not appear. For example, when the clockwise rotation goes to 11°, the SH rings corresponding to the RLVs of ${\textbf{G}_{1,5}}$ and ${\textbf{G}_{2, - 1}}$ clearly intersected, but no enhancement showed up (Fig. 7(a)). That is because, as Fig. 7(b) shows that neither ${\textbf{G}_{1,5}}$ nor ${\textbf{G}_{2, - 1}}$’s head is on the Ewald sphere.
Fig. 7. The intersection of the conical beams without explicit SH enhancement. (a) No explicit intensity enhancement in the intersection with the SH rings corresponding to the RLVs of ${\textbf{G}_{1,5}}$ and ${\textbf{G}_{2, - 1}}$. (b) The termina of RLV ${\textbf{G}_1}$ and ${\textbf{G}_2}$ of the two conical beams do not attach to the Ewald sphere and cannot form an enhanced SH in the SP direction.
4. Conclusion
In summary, we studied conical SHGs under assistance of the FW elastic scattering. Distortion of the conical SHGs observed in the experiment may provide an assistant method to measure birefringence of the nonlinear optical crystals. During this study, we introduced a rearrangement in the vector diagram of nonlinear Ewald construction and superposed it in the Ewald shell. The combined hybrid construction is used to solve the SH enhancement during SHG evolution in nonlinear photonic crystals with different FW orientation. This model sets order about all the QPM SHG processes relating to the same RLV when FW changes direction in the crystal, including the creation/annihilation or the enhancement of the SHGs. Although we didn’t put all details about other SHG possibilities in a nonlinear photonic crystal, readers may find that this model can also be used to portray the composition of SHGs relating to different RLVs at the same time, including nonlinear Raman-Nath diffraction as a partial phase-matching process. The method can also be easily extended to other parametric processes, as in sum frequency generation, third harmonic generation or to 3D nonlinear photonic crystal situations.
Funding
National Natural Science Foundation of China (12174193).
Acknowledgments
H. H. developed the concepts of this work, and R. C. P. did the experiment and detailed calculations, they wrote the article together. H. X. H. prepared the crystal sample. The authors would like to thank Prof. Lei Wang (王雷), Ji-Wei Hou (侯纪伟) and Mr. Kan-Yi Xu (徐侃奕) for help preparing the experiments. Ms. Xiao-Ran Li (李晓然) participated in some of our early discussions. Mr. Jing Li (李晶) also did some preliminary experimental tests for the work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. Huang, C. P. Huang, C. Zhang, et al., “Second-harmonic generation in a periodically poled congruent LiTaO3 sample with phase-tuned nonlinear Cherenkov radiation,” Appl. Phys. Lett. 100(2), 022905 (2012). [CrossRef]
2. L. H. Hong, B. Q. Chen, C. Y. Hu, et al., “Rainbow Cherenkov Second-Harmonic Radiation,” Phys. Rev. Appl. 18(4), 044063 (2022). [CrossRef]
3. H. J. Ren, X. W. Deng, Y. L. Zheng, et al., “Nonlinear Cherenkov Radiation in an Anomalous Dispersive Medium,” Phys. Rev. Lett. 108(22), 223901 (2012). [CrossRef]
4. C. Q. Li, G. S. Mitchell, and S. R. Cherry, “Cerenkov luminescence tomography for small-animal imaging,” Opt. Lett. 35(7), 1109–1111 (2010). [CrossRef]
5. H. Jin, P. Xu, X. W. Luo, et al., “Compact Engineering of Path-Entangled Sources from a Monolithic Quadratic Nonlinear Photonic Crystal,” Phys. Rev. Lett. 111(2), 023603 (2013). [CrossRef]
6. P. Xu, S. H. Ji, S. N. Zhu, et al., “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004). [CrossRef]
7. Z. D. Xie, G. Zhao, P. Xu, et al., “Study of optical elastic scattering in a quasiperiodically poled LiTaO3 crystal,” J. Appl. Phys. 101(5), 056104 (2007). [CrossRef]
8. L. Mateos, P. Molina, J. Galisteo, et al., “Simultaneous generation of second to fifth harmonic conical beams in a two dimensional nonlinear photonic crystal,” Opt. Express 20(28), 29940–29948 (2012). [CrossRef]
9. X. Y. Fang, D. Wei, Y. M. Wang, et al., “Conical third-harmonic generation in a hexagonally poled LiTaO3 crystal,” Appl. Phys. Lett. 110(11), 111105 (2017). [CrossRef]
10. Y. Z. Tang, T. T. Ding, C. Y. Lu, et al., “Broadband second-harmonic generation in an angle-cut lithium niobate-on-insulator waveguide by a temperature gradient,” Opt. Lett. 48(5), 1108–1111 (2023). [CrossRef]
11. L. H. Hong, B. Q. Chen, C. Y. Hu, et al., “Ultrabroadband nonlinear Raman-Nath diffraction against femtosecond pulse laser,” Photonics Res. 10(4), 905–912 (2022). [CrossRef]
12. N. M. Wang, H. Huang, and C. Zhang, “Broadband second harmonic generation in aperiodic nonlinear photonic crystals: 1D projection from 2D Vogel sunflower spiral array,” AIP Adv. 11(5), 055219 (2021). [CrossRef]
13. H. Huang, C. P. Huang, C. Zhang, et al., “From Ewald sphere to Ewald shell in nonlinear optics,” Sci. Rep. 6(1), 29365 (2016). [CrossRef]
14. V. K. Pecharsky and P. Y. Zavalij, Fundamentals of powder diffraction and structural characterization of materials, 2nd ed. (Springer, 2009), Chap. 7.
15. T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4(3), 188–193 (2010). [CrossRef]
16. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, et al., “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). [CrossRef]
17. C. F. Lai, J. Y. Chi, H. C. Kuo, et al., “Anisotropy of light extraction from GaN two-dimensional photonic crystals,” Opt. Express 16(10), 7285–7294 (2008). [CrossRef]
18. L. Maiwald, S. Lang, D. Jalas, et al., “Ewald sphere construction for structural colors,” Opt. Express 26(9), 11352–11365 (2018). [CrossRef]
19. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998). [CrossRef]
20. S. Stivala, A. C. Busacca, A. Pasquazi, et al., “Random quasi-phase-matched second-harmonic generation in periodically poled lithium tantalate,” Opt. Lett. 35(3), 363–365 (2010). [CrossRef]
21. T. L. Li, X. H. Zhao, Y. L. Zheng, et al., “Conical second harmonic generation in KDP crystal assisted by optical elastic scattering,” Opt. Express 23(18), 23827–23833 (2015). [CrossRef]
22. K. S. Abedin and H. Ito, “Temperature-dependent dispersion relation of ferroelectric lithium tantalate,” J. Appl. Phys. 80(11), 6561–6563 (1996). [CrossRef]
23. A. Bruner, D. Eger, M. Oron, et al., “Refractive index dispersion measurements of congruent and stoichiometric LiTaO3,” Proc. SPIE 4628, 66–73 (2002). [CrossRef]
24. H. Ishizuki and T. Taira, “Quasi phase-matched quartz for intense-laser pumped wavelength conversion,” Opt. Express 25(3), 2369–2376 (2017). [CrossRef]
25. S. Kurimura, M. Harada, K. Muramatsu, et al., “Quartz revisits nonlinear optics: twinned crystal for quasi-phase matching [Invited],” Opt. Mater. Express 1(7), 1367–1375 (2011). [CrossRef]
26. S. M. Saltiel, D. N. Neshev, R. Fischer, et al., “Generation of second-harmonic conical waves via nonlinear bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008). [CrossRef]
