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Abstract
In this paper, we demonstrate a quasi-phase matched type-II waveguide fabricated by femtosecond laser direct writing in Z-cut MgO doped lithium niobate. Compared with the previous work that employs type-III configuration to realize the all-optical fabricated quasi-phase matched waveguide, our type-II waveguide has a simpler dual-line structure that can support TM-mode quasi-phase matched second harmonic generation which employs d33 naturally, thus the fabrication process can be significantly simplified. In experiment, an enhanced second-harmonic generation with a normalized conversion efficiency of 8.76 %W−1cm−2 is observed, corresponding to a nonlinear modulation depth of 0.89. The proposed design and fabrication method expand the application of femtosecond laser direct writing type-II waveguide in the field of monolithic integrated nonlinear devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Lithium niobate (LN) is an important material for optics and microwave photonics owning to its excellent electro-optic [1], acousto-optic [2], and nonlinear optic properties [3–5]. Thanks to the large second-order nonlinear susceptibility $\chi ^{(2)}$ and the capability of being poled [6], high-efficiency nonlinear frequency conversion is expected on LN through the quasi-phase matching (QPM) method [7], which usually requires periodical ferroelectric domain inversion (e.g. poling) of the LN crystal, corresponding to a periodical inversion from the $\chi ^{(2)}$ to $-\chi ^{(2)}$. Nowadays, numerous works on QPM enhanced nonlinear process on LN have been demonstrated [8–12], among which the most common and mature approach to engineering the domain structure is electric field poling [13]. However, this process is relatively complex and high-cost since requirement on prefabrication of the electrode and the precious control of poling. Besides, the fabricated domain structure is restricted to a 2D pattern, while the huge potential of 3D domain engineering is inapproachable under such configuration [14].
In the past decades, with the development of the ultrashort pulse technology, infrared femtosecond (fs) laser has emerged as a power tool to process materials [15–20]. As the fs laser is focused beneath the surface, ultra-high peak intensity in the focus volume can exceed the threshold of the multi-photon absorption, and then modify the localized morphology as well as the physical properties of the sample [21]. Compared with the standard nano-fabrication technology such as the electro beam or ultra violet lithography, fs laser micro-machining featuring lots of advantages such as the low-cost and 3D flexibility [22]. For the LN crystal, fs laser induced refractive index changes have been revealed in the previous works [23], based on which waveguides with different configurations have been investigated, including the single-line [23], dual-line [24] and multi-line [25,26] (named type-I, II, III, respectively). Meanwhile, such kind of modification also can reach the regime of nonlinearity [27–32]. By fs laser inverting or depleting $\chi ^{(2)}$, 3D nonlinear structure can be fabricated with the size from few microns [33–37] to hundreds manometers inside the LN crystal [38]. However, to completely inverse the domain by fs laser, fabrication parameters should be precisely controlled. In contrast, fs laser depleting $\chi ^{(2)}$ has attracted lots of interest since it is easily achieved while reserving the 3D flexibility. This phenomenon has been firstly used for volume QPM enhanced second harmonic generation (SHG)in a X-cut LN [27], and subsequently in Z-cut LN [33,39] where the light is confined in the fs laser fabricated type-III waveguide with the cladding structure. However, although the type-III waveguide has its advantage to support unpolarized light in the quasi-closed waveguide structure, the fabrication process is relatively complex and time-consumed, especially when internal structure like QPM grating is required. Besides, it also requires a specific fabrication sequence of writing tracks to avoid the defocusing of fs laser beam, and lots of fabrication parameters should be finely adjusted to ensure a proper cross section shape of the waveguide core as well as the QPM structure. While for the type-II waveguide, it has a simpler dual-line configuration that can naturally support the most efficient ee-e type SHG in LN, and its fabrication is more convenient and flexible for a less overlap between the dual-line and QPM structure, leading to a cost-efficient approach to the all-optics fabricated QPM waveguide on LN.
In this paper, a type-II waveguide with periodically modified nonlinear structure is fabricated inside Z-cut MgO-doped LN by fs laser direct writing. Compared with the previous work, the fabrication process is significantly simplified, while an enhanced SHG with a normalized conversion efficiency of 8.76 %$\textrm{W}^{-1}\textrm{cm}^{-2}$ is also observed, corresponding to a modulation depth of 0.89. Our work will provides the foundation of the further application for the fs laser fabricated type-II waveguide in the field of monolithic nonlinear devices.
2. Device fabrication
The QPM type-II waveguide is fabricated by a standard fs laser direct writing system, as the schematic diagram shows in the Fig. 1. Here an ytterbium-doped diode-pumped ultra-fast amplified laser serves as the laser source. A 40 $\times$ microscopic objective (NA = 0.65) is used to focus the fs laser beam into the sample which is mounted on a computer-controlled 3D motion stage, and the focus depth is about 60 $\mu\textrm{m}$ below the surface. To fabricated a desired pattern, the sample moves with the stage along a designed trace, while the optical beam shutter is commanded to open or close at each point. The fabrication procedure is controlled by the computer program, and simultaneously monitored by a CCD above the motion stage. In our experiment, the dimension of the Z-cut 5% MgO-doped LN sample is 10 $\textrm{mm}$ $\times$ 9 $\textrm{mm}$ $\times$ 1 $\textrm{mm}$. The pulse duration, central wavelength and repetition rate of the fs laser is 500 fs, 1030 nm, and 1 kHz, respectively. Two levels of pulse energy are employed, the lower one of 200 nJ is used for nonlinearity modulation while the higher one of 860 nJ is used for waveguide fabrication. The average velocity of the motion stage is 65 $\mu\textrm{m/s}$, and the whole fabrication time is about 1.5 hours.
Fig. 1. Femtosecond laser direct writing system. Inset: designed QPM type-II waveguide with modulated nonlinearity.
As the inset in Fig. 1 has shown, the total length of the waveguide is 9 $\textrm{mm}$, including 6-$\textrm{mm}$ QPM gratings. The distance between two tracks is 15 $\mu\textrm{m}$. The periodical structure inside the type-II waveguide is designed to fulfill the QPM condition of SHG with the fundamental wavelength at 1064 $\textrm{nm}$. For both the fundamental wave (FW) and second harmonic wave (SH) polarized along the Z-axis, thus the maximum nonlinear coefficient $d_{33}$ of LN can be employed, and the modulation period $\Lambda$ is designed to be 6.9 $\mu\textrm{m}$. Here we neglect the influence from the geometrical dispersion due to relatively large size of the type-II waveguide. Since the amplitude of $\chi ^{(2)}$ is periodically reduced, a modulation depth $M$ is defined to described the nonlinear coefficient, as we have displayed in the inset.
In our experiment, we first fabricated two identical type-II waveguides under the same condition, and then changed the pulse energy to inscribe the QPM structure inside the waveguide core of one of them. As the images shows in Fig. 2(a), only the left one processes a periodic structure for QPM, while in the right one the nonlinearity property is preserved. Different from the fs laser induced domain inversion which should be observed with assistance methods, here both the QPM structure and the waveguide can be clearly observed under the optical microscopic because they originate from the same reason of lattice damage. In Fig. 2(b), the cross section of the type-II waveguides is imaged by a microscopy from the end surface of the sample. Each fs laser induced track has a length of about 21 $\mu\textrm{m}$ and a maximum width of 2 $\mu\textrm{m}$. Such vertical enlargement of the focus point of the fs laser beam in the LN crystal is mainly attributed to the self-focusing effect [40] and the spherical aberration [41] originated from the large refractive index mismatch between the air and LN. The internal QPM structure can be observed when we move the focus spot beneath the surface, as the bottom image in Fig. 2(b) has shown. Here, the dual-line and the QPM structure are highlighted by the dashed blue line and dashed red rectangle, respectively, showing that the QPM structure is mainly located in the centre of the waveguide core.
Fig. 2. (a) Microscopical images of the fabricated waveguides from the top view. Inset: periodically modulated QPM structure. (b) Cross section images of the type-II waveguide (top) and internal QPM structure (bottom). Simulated stress field induced refractive index change for (c) ordinary light and (d) exordinary light.
According to the previous studies on the mechanism of fs laser induced refractive index change [21], both the ordinary index $n_{\textrm{o}}$ and the exordinary index $n_{\textrm{e}}$ in the fs laser irradiated tracks have a decrement due to the crystalline damage; while in the regime between two tracks, thermal expansion induced stressed field becomes the determined factor on the refractive index change [42]. Based on the cross section structure obtained in the experiment, the refractive index changes have been simulated in Fig. 2(c) and (d). As we can see that there is no obvious change of $n_{o}$ near the tracks, while an increment of about 2 $\times$ $10^{-3}$ of $n_{e}$ is found in the waveguide core. All of these analysis indicates that our type-II waveguide can support mainly the TM mode for light propagation, by which an eee-type SHG using the largest nonlinear coefficient $d_{\textrm{33}}$ can occur naturally.
3. Second harmonic generation
In this section, SHG in the type-II waveguides with and without QPM structure is experimentally investigated. The experimental setup is shown in Fig. 3. An optically pumped Nd:YAG laser with a pulse duration of 4 ns and a repetition rate up to 1 kHz is employed as the pump source of the FW. The polarization of the incident light is first adjusted to the Z-direction by a half-wave plate, and then focused into the end surface of the LN sample by an objective (5 $\times$, $\textrm{NA}$ = 0.1). A power meter (not appear in the diagram) is placed before this objective to measure the input FW power. The output FW and SH light is collected by another objective, and consequently divided by a dichroic mirror which has high-reflection in the infrared band and high-transmission in the visible band. As a result, the mode profile and the power can be measured separately for the FW and SH. The inset in the Fig. 3 shows the CCD captured intensity profiles of FW and SH, respectively. It can be seen that they all have a single mode property in the type-II waveguide, but the mode area of SH is much smaller than that of the FW. Origination of such difference will be discussed in the latter.
Experimental results are displayed in Fig. 4. In the QPM type-II waveguide, as the launched FW power is increased to 650 $\textrm{mW}$, a maximum output SH power of about 1 $\textrm{mW}$ is obtained in the output port. The inset in Fig. 4(a) shows the linear dependence of output FW power on the input FW power. Since the conversion efficiency is relatively low in our experiment, FW power in the waveguide is assumed to be undepleted by the nonlinear process, and an insertion loss of about 19.6 dB is estimated for the whole system. The directly measured conversion efficiency is 0.253$\%\textrm{W}^{-1}$. Considering external losses in the coupling system (see in the next section), a normalized conversion efficiency in the waveguide is estimated to be about 7.1 %$\textrm{W}^{-1}$, corresponding to a normalized conversion efficiency of 8.76 %$\textrm{W}^{-1}\textrm{cm}^{-2}$.
Fig. 4. (a) Experimentally measured output SH power versus the input FW power in the type-II QPM waveguide. Inset: the linear relation between the input and output FW power. (b) Comparison of the SHG in the type-II waveguides with and without QPM structure. Inset: simulated modal distributions and effective refractive indexes of high-order modes at FW and SH in the unmodified type-II waveguide.
In Fig. 4(b), the SH output is also measured in the type-II waveguide without nonlinearity modification. To exclude the impact from the difference in the waveguide loss, we adjust the input FW power in the normal waveguide to ensure the output power is maintained at the same level with that in the QPM waveguide. A relatively weak SH output is detected in the unmodified waveguide, and it may be explained by the modal phase matching between the weak-guided high-order TE mode at FW and TM mode at SH when input FW power is high enough and the initial adjustment of the polarization is inadequate, as the simulated result in the inset has indicated; while in the QPM waveguide, due to the modified internal structure and a higher propagation loss, only fundamental modes will be supported as Fig. 3 has shown, and QPM is the dominant issue for the nonlinear process. Here, QPM enhanced SH output power is about two times higher than that in a unmodified type-II waveguide, and the real contrast between the capabilities of SHG enhancement in these two waveguides could be larger if the difference in the SH loss is considered.
4. Discussion
Since the insertion loss is known, the propagation loss of our QPM type-II waveguide can be estimated by identifying other loss components in the system. The transmission loss of each objective we used in the experiment is about 1.2 dB. When the laser beam is coupled into and out of the sample, the Fresnel reflection loss at each end facet is 0.62 dB of the FW. The mode-mismatch induced coupling loss in the front facet is about 4.5 dB, which is calculated by an overlap intergral between the simulated mode profile of the TM fundamental mode in the type-II waveguide and incident mode. Based on the focusing condition, the diameter of the focused point is calculated to be about 20 $\mu m$. By using an objective with larger NA, the focus point can be further suppressed and the coupling loss will be lower. As a result, the loss of the FW in the QPM type-II waveguide is about 11.82 dB, corresponding to an average propagation loss of 13 dB/cm. Besides, the propagation loss of SH is assumed to be 1.5 times of that at FW, as the previous works has indicated [27,33].
With the consideration of the propagation loss, the SHG process in the QPM type-II waveguide can be described by the nonlinear coupled wave equation with the depleted pump approximation [43], as shown below:
In the unmodified region, we assume that $M=1$. After numerical solving the coupled equation and comparing the result with the experimental conversion efficiency calculated with the peak power of both the FW and SH, the modulation depth M is determined to be 0.89. It is noteworthy this result is the compared with the previous work [27,33]. By optimizing the fabrication parameters, both the modulation depth and the propagation loss could be lower. The fabrication error in the QPM period could also be improved by an optimized focusing condition of fs laser beam. With a more compact type-II waveguide for tight mode confinement and the consideration of the geometrical dispersion in the design of the QPM structure, a higher conversion efficiency is expected.
To further improve the performance of QPM type-II waveguide, the match of the waveguide core and the nonlinearity modification regime in the cross section is also necessary. For a 1064 $\textrm{nm}$ laser focused in the LN crystal, the critical power for self-focusing is about 0.82 $\textrm{MW}$. In experiment, the peak power we used to inscribe the waveguide is 1.72 $\textrm{MW}$, and that for nonlinearity modification is only 0.4 $\textrm{MW}$, which is below the threshold to produce the filament. Therefore, the cross section area of the QPM grating is smaller than that of the waveguide mode, resulting an inefficient mode overlap between the FW and SH, as Fig. 3 has shown. To solve this problem, a multi-scan technology can be employed at different depths in the modification of the nonlinearity, thus a larger interaction area of the QPM enhanced SHG can be ensured.
5. Conclusion
In this paper, a fs laser direct writing QPM type-II waveguide is demonstrated. Compared with the previous works that using type-III cladding waveguides for QPM SHG, our type-II waveguide features simple structure and naturally support the most efficient ee-e type SHG on LN. In experimental, an enhanced SHG with a normalized conversion efficiency of 8.76 %$\textrm{W}^{-1}\textrm{cm}^{-2}$ is obtained, corresponding to a modulation depth of 0.89. There is still a large space to optimize the processing parameters and details for a better modulation quality. Our work has certain significance for all-optically fabricated QPM waveguides with type-II configurations, and its unique open structure may be valuable for the further development of the nonlinear coupled waveguide array and reconfigurable QPM devices based on LN.
Funding
National Key Research and Development Program of China (2019YFB2203501); National Natural Science Foundation of China (12134009, 91950107); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01-ZX06); Shanghai Jiao Tong University (21X010200828).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data available on request from the authors.
References
1. C. Wang, M. Zhang, X. Chen, M. Bertrand, A. Shams-Ansari, S. Chandrasekhar, P. J. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at cmos-compatible voltages,” Nature 562(7725), 101–104 (2018). [CrossRef]
2. L. Cai, A. Mahmoud, and G. Piazza, “Low-loss waveguides on y-cut thin film lithium niobate: towards acousto-optic applications,” Opt. Express 27(7), 9794–9802 (2019). [CrossRef]
3. C. Wang, X. Xiong, N. M. Andrade, V. Venkataraman, X. Ren, G. Guo, and M. Lončar, “Second harmonic generation in nano-structured thin-film lithium niobate waveguides,” Opt. Express 25(6), 6963–6973 (2017). [CrossRef]
4. R. Luo, Y. He, H. Liang, M. Li, and Q. Lin, “Highly tunable efficient second-harmonic generation in a lithium niobate nanophotonic waveguide,” Optica 5(8), 1006 (2018). [CrossRef]
5. R. Luo, Y. He, H. Liang, M. Li, J. Ling, and Q. Lin, “Optical parametric generation in a lithium niobate microring with modal phase matching,” Phys. Rev. Appl. 11(3), 034026 (2019). [CrossRef]
6. D. Sun, Y. Zhang, D. Wang, W. Song, X. Liu, J. Pang, D. Q. Geng, Y. Sang, and H. Liu, “Microstructure and domain engineering of lithium niobate crystal films for integrated photonic applications,” Light: Sci. Appl. 9(1), 197 (2020). [CrossRef]
7. J. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]
8. J. Lu, J. B. Surya, X. Liu, A. W. Bruch, Z. Gong, Y. Xu, and H. X. Tang, “Periodically poled thin-film lithium niobate microring resonators with a second-harmonic generation efficiency of 250,000%/W,” Optica 6(12), 1455–1460 (2019). [CrossRef]
9. J. Zhao, C. Ma, M. Rüsing, and S. Mookherjea, “High quality entangled photon pair generation in periodically poled thin-film lithium niobate waveguides,” Phys. Rev. Lett. 124(16), 163603 (2020). [CrossRef]
10. C. Wang, C. Langrock, A. Marandi, M. Jankowski, M. Zhang, B. Desiatov, M. M. Fejer, and M. Lončar, “Ultrahigh-efficiency wavelength conversion in nanophotonic periodically poled lithium niobate waveguides,” Optica 5(11), 1438 (2018). [CrossRef]
11. Y. Niu, L. Yang, D. Guo, Y. Chen, X. Li, G. Zhao, and X. Hu, “Efficient 671 nm red light generation in annealed proton-exchanged periodically poled linbo3 waveguides,” Chin. Opt. Lett. 18(11), 111902 (2020). [CrossRef]
12. X.-H. Tian, W. Zhou, K. Ren, C. Zhang, X. Liu, G.-T. Xue, J.-C. Duan, X. Cai, X. Hu, Y.-X. Gong, Z. Xie, and S.-D. Zhu, “Effect of dimension variation for second-harmonic generation in lithium niobate on insulator waveguide [invited],” Chin. Opt. Lett. 19(6), 060015 (2021). [CrossRef]
13. M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]
14. T. Wang, P. Chen, C. Xu, Y. Zhang, D. Wei, X. Hu, G. Zhao, M. Xiao, and S. ning Zhu, “Periodically poled linbo3 crystals from 1d and 2d to 3d,” Sci. China: Technol. Sci. 63(7), 1110–1126 (2020). [CrossRef]
15. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]
16. C. Wang, H. Huo, M. Johnson, M. Shen, and E. Mazur, “The thresholds of surface nano-/micro-morphology modifications with femtosecond laser pulse irradiations,” Nanotechnology 21(7), 075304 (2010). [CrossRef]
17. J. Burghoff, H. Hartung, S. Nolte, and A. Tünnermann, “Structural properties of femtosecond laser-induced modifications in linbo3,” Appl. Phys. A 86(2), 165–170 (2006). [CrossRef]
18. L. Ge, H. Jiang, Y. Liu, B. Zhu, C. Lu, Y. Chen, and X. Chen, “Quality improvement and mode evolution of high-q lithium niobate micro-disk induced by “light annealing”,” Opt. Mater. Express 9(4), 1632 (2019). [CrossRef]
19. Y. Ju, C. Liu, Y. Liao, Y. Liu, L. Zhang, Y. Shen, D. Chen, and Y. Cheng, “Direct fabrication of a microfluidic chip for electrophoresis analysis by water-assisted femtosecond laser writing in porous glass,” Chin. Opt. Lett. 11(7), 072201 (2013). [CrossRef]
20. T. Jiang, S. Gao, Z.-N. Tian, H. zhuang Zhang, and L. Niu, “Fabrication of diamond ultra-fine structures by femtosecond laser,” Chin. Opt. Lett. 18(10), 101402 (2020). [CrossRef]
21. J. Burghoff, S. Nolte, and A. Tünnermann, “Origins of waveguiding in femtosecond laser-structured linbo3,” Appl. Phys. A 89(1), 127–132 (2007). [CrossRef]
22. K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014). [CrossRef]
23. F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond-laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014). [CrossRef]
24. J. Burghoff, C. Grebing, S. Nolte, and A. Tünnermann, “Efficient frequency doubling in femtosecond laser-written waveguides in lithium niobate,” Appl. Phys. Lett. 89(8), 081108 (2006). [CrossRef]
25. L. Li, C. Romero, J. R. V. de Aldana, L. Wang, Y. Tan, and F. Chen, “Efficient quasi-phase-matching in fan-out ppslt crystal waveguides by femtosecond laser direct writing,” Opt. Express 27(25), 36875–36885 (2019). [CrossRef]
26. L. Wang, X. Zhang, L. Li, Q. Lu, C. Romero, J. R. V. de Aldana, and F. Chen, “Second harmonic generation of femtosecond laser written depressed cladding waveguides in periodically poled mgo:litao3 crystal,” Opt. Express 27(3), 2101–2111 (2019). [CrossRef]
27. J. U. Thomas, V. Hilbert, R. Geiss, T. Pertsch, A. Tünnermann, and S. Nolte, “Quasi phase matching in femtosecond pulse volume structured x-cut lithium niobate,” Laser Photonics Rev. 7(3), L17–L20 (2013). [CrossRef]
28. X. Chen, P. Karpinski, V. Shvedov, K. Koynov, B.-X. Wang, J. Trull, C. Cojocaru, W. Krolikowski, and Y. Sheng, “Ferroelectric domain engineering by focused infrared femtosecond pulses,” Appl. Phys. Lett. 107(14), 141102 (2015). [CrossRef]
29. J. Imbrock, H. Hanafi, M. Ayoub, and C. Denz, “Local domain inversion in mgo-doped lithium niobate by pyroelectric field-assisted femtosecond laser lithography,” Appl. Phys. Lett. 113(25), 252901 (2018). [CrossRef]
30. B. Zhu, H. Liu, Y. Chen, and X. Chen, “High conversion efficiency second-harmonic beam shaping via amplitude-type nonlinear photonic crystals,” Opt. Lett. 45(1), 220–223 (2020). [CrossRef]
31. B. Zhu, H. Liu, Y. Liu, X. Yan, Y. Chen, and X. Chen, “Second-harmonic computer-generated holographic imaging through monolithic lithium niobate crystal by femtosecond laser micromachining,” Opt. Lett. 45(15), 4132–4135 (2020). [CrossRef]
32. B. Zhu, X. Liu, H. Liu, Y. Liu, X. Yan, Y. Chen, and X. Chen, “Dual-wavelength focusing through fresnel zone plate fabricated in lithium niobate crystal by femtosecond laser micromachining,” Phys. Status Solidi RRL 14(8), 2000213 (2020). [CrossRef]
33. S. Kroesen, K. Tekce, J. Imbrock, and C. Denz, “Monolithic fabrication of quasi phase-matched waveguides by femtosecond laser structuring the χ(2) nonlinearity,” Appl. Phys. Lett. 107(10), 101109 (2015). [CrossRef]
34. D. Wei, C.-W. Wang, H. Wang, X. Hu, D. M. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. ning Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12(10), 596–600 (2018). [CrossRef]
35. D. Wei, C.-W. Wang, X. Xu, H. Wang, Y. Hu, P. Chen, J. Li, Y. Zhu, C. Xin, X. Hu, Y. Zhang, D. Wu, J. Chu, S. ning Zhu, and M. Xiao, “Efficient nonlinear beam shaping in three-dimensional lithium niobate nonlinear photonic crystals,” Nat. Commun. 10(1), 4193 (2019). [CrossRef]
36. S. Liu, L. M. Mazur, W. Krolikowski, and Y. Sheng, “Nonlinear volume holography in 3d nonlinear photonic crystals,” Laser Photonics Rev. 14(11), 2000224 (2020). [CrossRef]
37. P. Chen, C.-W. Wang, D. Wei, Y. Hu, X. Xu, J. Li, D. Wu, J. Ma, S. Ji, L. Zhang, L. Xu, T. J. Wang, C. Xu, J. Chu, S.-D. Zhu, M. Xiao, and Y. Zhang, “Quasi-phase-matching-division multiplexing holography in a three-dimensional nonlinear photonic crystal,” Light: Sci. Appl. 10(1), 146 (2021). [CrossRef]
38. X. Xu, T. Wang, P. Chen, C. Zhou, J. Ma, D. Wei, H. Wang, B. Niu, X. Fang, D. Wu, S.-D. Zhu, M. Gu, M. Xiao, and Y. Zhang, “Femtosecond laser writing of lithium niobate ferroelectric nanodomains,” Nature 609(7927), 496–501 (2022). [CrossRef]
39. J. Imbrock, L. Wesemann, S. Kroesen, M. Ayoub, and C. Denz, “Waveguide-integrated three-dimensional quasi-phase-matching structures,” Optica 7(1), 28–34 (2020). [CrossRef]
40. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
41. P. Török, P. Varga, Z. J. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995). [CrossRef]
42. H. Nguyen, A. Ródenas, J. R. V. de Aldana, J. Martínez, F. Chen, M. Aguiló, M. C. Pujol, and F. Díaz, “Heuristic modelling of laser written mid-infrared linbo3 stressed-cladding waveguides,” Opt. Express 24(7), 7777–7791 (2016). [CrossRef]
43. R. W. Boyd, Nonlinear Optics (Academic Press, 2020).
