A Brief Review on Nonlinear Photonic Crystals Induced by Direct Femtosecond Laser Writing

Abstract

Direct femtosecond laser writing or inscription is a useful technique, and it has been employed to engineer various materials in many applications including nonlinear photonic crystals, which are of periodically patterned second-order nonlinearity to get and control the coherent light at new frequencies. By manipulation of second-order nonlinearity, either erased or poled, quasi-phase matching has been achieved in several crystals, especially three-dimensional nonlinear photonic crystals have been originally proposed and proved to be truly three-dimensional. Here we shortly review on the recent advances in the research field of nonlinear photonic crystals inscribed by femtosecond laser, as well as look into the future in this field.

1. Introduction

Frequency conversion processes are employed for the generation of new, desired frequencies, which has attracted many research interests due to its existing and potential applications in nonlinear photonics, including new laser sources [1,2], nonlinear optical microscopies [3,4], optical communications [5,6], nonlinear holograms [7], optical tweezers [8,9,10], and quantum information [11,12]. Several processes are available to obtain nonlinear frequency conversion, including second harmonic generation (SHG), difference-frequency generation (DFG), sum-frequency generation (SFG), optical parametric oscillation (OPO), and optical parametric amplification (OPA) [13]. Among them, SHG is the most commonly used for frequency doubling. In 1962, Armstrong et al. revealed that the SHG process is dominated by phase mismatch ($\Delta \overrightarrow{k}$), which is the difference between the fundamental wave vector ${\overrightarrow{k}}_F$ and the second harmonic (SH) wave vector ${\overrightarrow{k}}_{SHG}$ [14]. Commonly, $\Delta \overrightarrow{k}\ne 0$ due to the dispersion of nonlinear crystal, and electromagnetic energy oscillates between the fundamental wave and second harmonic wave with coherent length $l_C$ due to the dispersion of crystal. In order to obtain a good-quality and high-efficiency SH emission, phase mismatch must be eliminated.

There are generally two methods to overcome the dispersion of nonlinear crystals. One way is using birefrigent crystal, attaining birefrigence phase matching (BPM), which was first experimentally demonstrated in 1961 [15]. By choosing proper polarizations of the fundamental wave and using optical anisotropy of birefrigent crystals, the phase-matching condition is fulfilled, i.e., $\Delta \overrightarrow{k}=0$ [13]. However, BPM is limited by the small birefrigence of some popular nonlinear crystals including LiNbO${}_3$ and restricted to some specific wavelengths. Additionally, the conversion efficiency of BPM is highly limited by walk-off angle [13]. The other way is to periodically pole second-order nonlinearity (${\chi }^{\left(2\right)}$) to obtain a quasiphase matching (QPM) [14,15,16]. Via QPM, not only can the largest nonlinear coefficient be utilized, but also can the spatial walk-off angle be avoided. As long as ${\chi }^{\left(2\right)}$ is spatially modulated, a reciprocal vector lattice (RVL) ${\overrightarrow{G}}_m$ is produced, which contributes to interacting wave vectors [17]. Consequently, the quasiphase matching condition is ${\overrightarrow{k}}_F-2{\overrightarrow{k}}_{SHG}-{\overrightarrow{G}}_m=0$. For example, a simple 1D grating (Figure 1a) gives $G_m=\frac{2m\pi }{\Lambda }$, where m is an integer, and $\Lambda =2l_C$ is the poling period. According to the Fourier transform, the intensity of the corresponding RVL is $\frac{2}{m\pi }$ when the duty cycle is 0.5. The first-order RVL with $m=1$ has the largest Fourier coefficient $\frac{2}{\pi }$.

One traditional QPM technique is to pole ferroelectric domains by electric field [18,19,20], and the most common method is periodically poled lithium niobate (PPLN) [21,22,23]. Other methods, including chemical indiffusion [24,25], crystal-growing [26], scanning force microscopic poling [27], electron-beam poling [28,29], and UV-light poling [30,31], have been developed for specific circumstances. In 1998, the 1D QPM configuration was extended to 2D and 3D by Berger, and a concept of nonlinear photonic crystal (NPC) was introduced, where ${\chi }^{\left(2\right)}$ is spatially periodically modulated without changing the refractive index [32]. It is analogous to photonic crystal (PC), but with a periodic controlled refractive index [33,34]. Generally, the phase-matching condition is modified as ${\overrightarrow{k}}_{SHG}-2{\overrightarrow{k}}_F-{\overrightarrow{G}}_{m,n,l}=0$, where ${\overrightarrow{G}}_{m,n,l}$ is a 3D RVL (Figure 2c), with n and l being integers. Correspondingly, 3D RVL is expressed as ${\overrightarrow{G}}_{m,n,l}=m\frac{2\pi }{{\Lambda }_x}\widehat{x}+n\frac{2\pi }{{\Lambda }_y}\widehat{y}+l\frac{2\pi }{{\Lambda }_z}\widehat{z}$; ${\Lambda }_x,{\Lambda }_y$ and ${\Lambda }_z$ are the poling periods along the $x,y$ and z axes respectively. So, the second harmonic emission could be versatilely generated by choosing different combinations of $m,n,l$, and ${\Lambda }_x,{\Lambda }_y,{\Lambda }_z$ [35,36,37].

Regarding 3D, NPCs have plenty of unique applications, such as the frequency doubling of femtosecond laser pulses [38], 3D nonlinear beam shaping [39,40,41], and high-dimensional entanglement [42,43,44]. However, the electric field required for poling in PPLN must be applied along the axis of spontaneous polarization, so these kinds of NPCs always have 1D or 2D structures [45,46,47]. Therefore, NPC was restricted to 1D and 2D for a long time, because no technique was able to manipulate ${\chi }^{\left(2\right)}$ in 3D before the direct femtosecond laser inscription of ${\chi }^{\left(2\right)}$ technique was developed. It is a material–light interaction, either erasing or poling ${\chi }^{\left(2\right)}$, introduced by a tightly focused near-infrared femtosecond laser. For the first approach, the crystalline structure of dielectric crystals is partially destroyed, resulting in a local reduction in ${\chi }^{\left(2\right)}$ (Figure 1b,d). The second approach is to invert the spontaneous polarization either using a thermoelectric or a pyroelectric field. Benefiting from femtosecond laser inscription, one can manipulate ${\chi }^{\left(2\right)}$ in 3D inside nonlinear crystals as desired. In addition to applications in the field of photonic crystals and on-chip devices [48,49], the extension of this technique to nonlinear crystals paved the way to achieve the monolithic fabrication of 3D nonlinear integrated photonic devices for all-optical communication and on-chip signal processing. In this review, we focus on full-optical ${\chi }^{\left(2\right)}$-engineering techniques. We describe both the ${\chi }^{\left(2\right)}$-erasing and the ${\chi }^{\left(2\right)}$-poling techniques, demonstrating their recent progresses and applications in NPCs, and review typical femtosecond laser incription parameters.

2. ${\mathbf{\chi }}^{\left(\mathbf{2}\right)}$-Erasing Technique Using Femtosecond Laser

Femtosecond laser inscription is a popular technique used to fabricate structures inside transparent materials by engineering its bulk refractive index [50]. Tightly focused pulses are nonlinearly absorbed and confined into the focal volume before excited free electrons dissipate to the surrounding area due to the short period of femtosecond pulses [51]. Multiphoton absorption [52] localized on micro and submicro scales leads to bond breaking and rebonding, constituents’ migration, or thermomechanical effects [53,54,55,56,57,58,59,60,61]. Therefore, the refractive index is changed [62,63,64], which is used for many applications such as 3D optical storage [65], surface modification [66], photonic devices [67], terahertz (THz) devices [68], and optical fluid [69], especially in the field of waveguides [70,71,72,73,74,75,76].

Normally, two types of refractive index changes are induced [54,55,77], including the Type I modification and Type II modification. For the first type, $\Delta$n ($\Delta$n>0) is in the range of ${10}^{-4}-{10}^{-3}$ due to a slight change or a phase transition of the crystal structure [63]. The second one refers to a large $\Delta$n ($\Delta$n<0), because that there are voids brought in by microexplosions [78]. Recently, Type I modification has been developed to engineer ${\chi }^{\left(2\right)}$ to achieve QPM, which is called laser-induced quasiphase matching (LiQPM). Instead of being periodically switched, ${\chi }^{\left(2\right)}$ is selectively erased via optimized laser engineering parameters. Generally, ${\chi }^{\left(2\right)}$ is partially destroyed and the residue is marked as d, defining a ratio $\nu =\frac{d}{d_{max}}$ ($0\le \nu \le 1$), where $d_{max}$ is the crystal’s second nonlinearity. $\nu$ = 0 when ${\chi }^{\left(2\right)}$ is totally destroyed, while $\nu$ = 1 when it is unchanged. The periodic erasing of ${\chi }^{\left(2\right)}$ also introduces a RVL, and the intensity of 1D RVL is $\frac{1-\nu }{m\pi }$ if the duty cycle is 0.5 (Figure 1b). The intensity of LiQPM is maximized as $\frac{1}{m\pi }$ when $\nu =0$, while it is $\frac{2}{m\pi }$ for ${\chi }^{\left(2\right)}$-poling QPM (Figure 1a,c).

Recently, the ${\chi }^{\left(2\right)}$-erasing technique has attracted a lot of interests since Thomas et al. produced the first 1D LiQPM structure (Figure 3a–c) in an x-cut LiNbO${}_3$ in 2013 [79]. Employing this structure, a very poor conversion efficiency of ∼0.015% was observed. Imbrock et al. demonstrated a 1D grating integrated with cladding waveguides (Figure 3d–f) in a z-cut LiNbO${}_3$ in 2015, which was the first demonstration of integration two types of crystal modifications induced by a femtosecond laser [80]. By reducing possible diffraction lost, the energy conversion efficiency was increased to 5.72%. Additionally, they developed a chirped structure for broadband SHG and a cascaded single-period structure for generating parallel multiwavelength SHG [81]. In 2018, Wei et al. reported a 3D NPC by selectively erasing ${\chi }^{\left(2\right)}$ in a z-cut Mg-doped LiNbO${}_3$ (Figure 4), which was the first experimental demonstration of 3D NPC using the ${\chi }^{\left(2\right)}$-erasing technique [82]. The conversion efficiency is considerably increased by the versatile phase-matching configuration in 3D NPC. They also proposed nonlinear beam shaping with a 3D NPC (Figure 5) fabricated via this technique [83]. Zhu et al. extended this technique to engineer ${\chi }^{\left(2\right)}$ on the surface of a BPM crystal and achieved high-efficiency nonlinear beam shaping [84], where the phase matching condition is fulfilled through the birefrigence in the longitudinal direction and the nonlinear Raman–Nath condition in the transverse direction (Figure 6). Furthermore, they proposed a nonlinear holography in a monolithic lithium niobite crystal [85]. Shao et al. demonstrated 1D QPM in z-cut quartz as well as in x-cut LiNbO${}_3$ crystals [86]. Notably, they proposed the generation of deep-ultraviolet wave of $177.3$ nm with a quartz sample with a conversion efficiency of $0.107\%$. They also demonstrated an angular engineering strategy of a LiQPM structure for widely tunable SHG and experimentally demonstrated it in a quartz crystal [87]. The output wavelength covered 221 to 332 nm, and the conversion efficiency at the peak power was more than $1\%$. Very recently, they developed a metacrystal of quartz and increased the emission conversion efficiency at 177.3/167.8 nm to 0.23%, which is close to that with RBBF [88]. Yuan et al. demonstrated a QPM structure inside a Type II waveguide in LiNbO${}_3$ crystal (Figure 3g); the normalized conversion efficiency was enhanced to 8.76% W${}^{-1}$cm${}^{-2}$ [89]. Notably, the author did not point out which type of modification of the refractive index changed, but the darker line indicated a Type II modification (Figure 3g).

3. ${\mathbf{\chi }}^{\left(\mathbf{2}\right)}$-Poling Technique by Femtosecond Laser

Laser light is employed in ferroelectric domain inversion engineering in two ways: indirectly or directly. Indirect poling of ferroelectric crystals occurs prior to the application of an external homogeneous electric field, which is enhanced by laser light to reduce the coercive field [90,91,92]. Therefore, the selective uniform illumination of a femtosecond laser on ferroelectric crystal results in spatially selective domain inversion, eliminating the need for structured electrodes or crystal heating during voltage application. Several groups have reported related studies with different ferroelectric crystals using this technique [93,94,95,96]. It has been improved as a two-step voltage application under ultraviolet (UV) light either with the help of a mask [97] or a beam-writing technique [98] to realize a 2D domain inversion in MgO-doped LiNbO${}_3$. Later, it was reported that UV light and an electric field do not need to be applied simultaneously [99]. However, the electric field must be applied along the polar axis of the crystal for domain inversion; consequently, the UV-assisted poling technique can never be employed in three dimensions. Direct poling with laser light was proposed in 1994 [100]. It was suggested that the strong electric field generated by the propagating ultrashort pulses could accelerate ions in the ferroelectric crystal. As a result, the kinetic energy of the ions is increased so that they are able to switch between their two stable positions, leading to spontaneous polarization flipping. Not until 2005 was it experimentally demonstrated that short ultraviolet pulses laser could promote nanoscale-domain inversions on the surface of LiNbO${}_3$ crystal due to a strong absorption of UV light [101]. Later, it was discovered that the inversion of the ferroelectric domain only happens under a certain intensity of laser light, and the concept of domain reversion window was proposed [102]. It is an energy range of a continuous laser pulse that can induce domain inversion, which has been theoretically and experimentally confirmed [103]. However, there was no systematic study of the inverted structure in either Ref. [102] or Ref. [103].

Recently, several full-optical domain-inversion methods induced by femtosecond laser inscription have been approached [104,105,106,107,108], where the mechanisms to invert spontaneous polarization are totally different from the one proposed in Ref. [100]. Basically, ferroelectric-domain inversion is caused either by a thermoelectric field [109] or pyroelectric field [110] induced by femtosecond laser inscription or thermal annealing. Therefore, the poling mechanisms are categorized into two types: primary domain inversion due to a thermoelectric field and secondary domain inversion due to a pyroelectric field. As long as the employed femtosecond laser meets some certain requirements, ${\chi }^{\left(2\right)}$ is inverted instead of erased at an arbitrary place and along any orientation, with the crystalline undamaged. Consequently, ${\chi }^{\left(2\right)}$ is modified in such a way that there is no scattering loss; thus, the conversion efficiency is significantly increased for this type of QPM compared with that of LiQPM.

3.1. Primary Domain Inversion

Multiphoton absorption of a laser beam at the focal point leads to the local temperature being increased rapidly [111,112]. Consequently, the temperature gradient within the laser focal region and the surrounding area results a 3D thermoelectric field. Taking LiNbO${}_3$ as an example, only the z component of this electric field is related to the domain inversion (Figure 7) [105,113], which is bipolar in the focal region: one half is antiparallel to the spontaneous polarization, while the other half is parallel to the spontaneous polarization. Additionally, ionic conductivity is increased via laser heating, leading to a local reduction in the coercive field to make it easier to invert the spontaneous polarization at the laser focal point [114]. When the thermoelectric field surpasses the coercive field, domain inversion happens in half of the focal volume, where the thermoelectric field is opposite to spontaneous polarization. In this model, the pyroelectric field is ignored within the laser focal region, because it is one order smaller than the thermoelectric field [107]. Normally, inverted domains have an elongated form due to the spherical aberration induced by refractive index differences when the laser beam enters the crystal from air [104]. The laser intensity decays along the propagating direction because its absorption is intensity-dependent. Meanwhile, the laser undergoes birefringence, which introduces multiple focuses [115,116]. Therefore, the inverted domain has a larger cross-section in the incoming direction of the laser than that in the outgoing direction [105].

The model of the primary domain inversion described above is first proposed by Sheng et al. and they have reported a series of works based on it [104,105,113]. Starting dealing with LiNbO${}_3$, they demonstrated a 2D QPM structure with an engineering accuracy of about 1.5 μm [104]. Then integrating a grating structure with a waveguide, they managed to get a conversion efficiency of 17.45% for doubling a fundamental wavelength of 815 nm [105]. In contrast, the highest reported conversion efficiency via LiQPM is 5.72% [80]. Furthermore, they applied this technique to invert ferroelectric domains in a tetragonal perovskite ferroelectric crystal of barium calcium titanate (BCT) and captured a $\theta$-like domain structure directly introduced by the 3D thermoelectric field (Figure 7). It is clear that there is a central horizontal line representing the domain wall, perpendicular to the spontaneous polarization of the BCT. Furthermore, they demonstrated a 3D photonic crystal of simple tetragon fabricated using this technique and obtained two types of SHG: a square lattice of SH spots located at the center, which was standard nonlinear Raman–Nath diffraction [35]; and peripheral spots distributed as homocentric rings situated relatively far from the center, which provided direct evidence of the 3D characteristics of the nonlinear interaction.

They keep studying on this topic and reported several applications of nonlinear beam shaping employing calcium barium niobate (CBN) crystals. Firstly, they demonstrated an experiment of nonlinear wavefront shaping through a 3D NPC [117]. In this report, the 3D nonlinear structure of three layers including a fork and linear and circular gratings was proposed and employed for parallel 3D nonlinear wave shaping. Those three patterns were transversely overlapped inside a CBN crystal and simultaneously converted a fundamental Gaussian beam into three pairs of second-harmonic vortices, Gaussians, and conical beams. In addition, the 3D nonlinear wavefront shaping was dynamically controlled for the first time by adjusting the distance between two adjacent patterns. Secondly, they promoted a curved fork pattern (Figure 8) to approach a perfect vortex second harmonic, of which the diameter was insensitive to the topological charge [118]. This was the first demonstration of perfect vortices via the nonlinear beam-shaping method, which has been realized in linear optics [119,120]. Thirdly, they studied a real 3D nonlinear volume hologram of high quality [121], where a vortex second harmonic with zero-intensity in the center was generated effectively at the resonance frequency. Fourthly, they also achieved 2D and 3D NPCs of hexagonal patterns in single ferroelectric-domain CBN crystals [122]. In this work, multiple-ferroelectric-domain CBNs were made into single-ferroelectric-domain crystals before patterns were inscribed. Subsequently, those samples were used in an SHG experiment via nonlinear Raman–Nath diffraction, and they obtained a similar SHG pattern, as shown in Ref. [113]. The generated second-harmonic emission possessed hexagonal symmetry, with its intensity increasing monotonically along the propagation. Comparing those SH emissions via 3D NPC and 2D NPC, the one generated via 3D PNC was much higher, as expected. In addition, the 2D NPC pattern was also produced in a multidomain CBN crystal, and the SH emission via the single-domain sample was higher. The poled domains appeared to be stable and can be completely erased by thermal annealing above the Curie temperature for 30 min, as were the single-ferroelectric domain.

A second model of primary domain inversion was proposed by Xu et al. [107]. They demonstrated a nonreciprocal femtosecond-laser-writing technique in LiNbO${}_3$ with a resolution of 30 nm, where the 3D domain inversion is rewritable. Noticeably, a prefabricated drain is required in this model to effectively guide the electric charges produced during the poling process, without which the laser inscription of the same parameters can never produce an inversion of the domain. This engineering process can be properly simulated, assuming that the input laser has an intensity of Gaussian profile. Thus, it could be expressed as: $I=I_{0}G(x,y,x)$, where $I_0$ is a constant current, and $G(x,y,z)$ is a 3D Gaussian function. Since LiNbO${}_3$ crystal has strong absorption at 318 nm and the wavelength of the femtosecond laser utilized here is 800 nm, the nonlinear absorption is a three-photon process [111,112]. Thus, the generated heat is

$$Q=\gamma I^3,$$

(1)

where $\gamma$ is the three-photon absorption coefficient of LiNbO${}_3$. Now, a solid heat-transfer model is considered as follows:

$$\rho C_p▿T\mathbf{u}-\kappa {▿}^{2}T=Q,$$

(2)

where $\rho$ is the density, $C_p$ is the heat capacity, and $\kappa$ is the thermal conductivity of LiNbO${}_3$ crystal. T is the temperature, and $\mathbf{u}$ is the velocity vector. Suppose S is the Seebeck coefficient, the temperature gradient between the hot laser focal point and the cold surrounding area induces an electric field:

$$\mathbf{E}=S\Delta T.$$

(3)

The simulated 3D thermoelectric field $\mathbf{E}$ is shown in Figure 9a, and its z components are presented in Figure 9c, with ${\mathbf{E}}_1$ parallel and ${\mathbf{E}}_2$ antiparallel to the spontaneous polarization. Within the enclosed ellipsoidal area denoted by a and b, the thermoelectric field exceeds and flips the spontaneous polarization when antiparallel to it. For example, in Figure 9c, only ${\mathbf{E}}_2$ flips the local spontaneous polarization. Therefore, this poling technique strongly depends on the laser-writing direction and is rewritable. As shown in Figure 9d, the laser both comes and writes along the $+z$ direction, which is also the spontaneous polarization direction. Figure 9d,g indicate that when the laser beam moves along the $+z$ direction, ${\mathbf{E}}_2$ switches the polarization back regardless of the direction. If it is flipped, the laser acts as an eraser; if it is normal, the laser does not work at all. In the contrast, when the laser moves along the $-z$ direction, ${\mathbf{E}}_1$ overtakes ${\mathbf{E}}_2$, and the domain is inverted (Figure 9e) with a width of a. In another case, as shown in Figure 9f, the laser moves along the y axis of the crystal and half of the laser focal area is inverted by ${\mathbf{E}}_1$ with a width of b. Therefore, with a proper design, a focused laser beam can be used as a laser pen or eraser to fabricate arbitrary 3D ferroelectric-domain structures.

A third model following the same principle with the primary domain inversion was developed by Wang et al. [108]. It is a two-step poling technique: in the first step, laser marking, a tiny seed is created by a single pulse illuminated along the z-axis either on the surface or inside the LiNbO${}_3$ crystal, shorted for LM; in the second step, laser inducing, a temperature gradient is induced by moving the focal point of the multiple laser pulses away from the seed, shortened as LI. This high temperature gradient induces a 3D thermoelectric field, which not only makes the domain inverted but also drives inverted domain growth when it surpasses the coercive field. LMs, which are Type II modifications, can be inscribed not only on the surface but also inside the crystal, requiring different pulse energies to compensate for the slight loss during propagation. Generally, the seeds have uncontrollable, random quality. Therefore, oversized or downsized seeds make it harder to pole the ferroelectric domains; unfortunately, this mechanism is not understood yet. Furthermore, because the coercive field varies at each point of the crystal, correspondingly, the minimum energy of the multiple laser pulses for inducing the poling varies from 600 nJ to 700 nJ. It must be mentioned that a single LI can simultaneously invert ferroelectric domains along thousands of seeds within a region of about 200 $\mathsf{\mu }$m; thus, this method is highly efficient and energy-saving.

3.2. Secondary Domain Inversion

The model of secondary domain inversion is also a two-step process: laser inscription and ten heat annealing, which is similar with the one demonstrated in previous reports. Laser-induced filaments act as seeds to induce domain inversion in subsequent thermal treatment, which was first demonstrated by Imbrock et al. [106].

A pyroelectric field instead of a thermoelectric field plays an important role here. More specifically, the sum of the depolarizating field ${\mathbf{E}}_{dep}$ due to the spontaneous polarization, screening field ${\mathbf{E}}_{scr}$, and thermally activated bulk charges is taken into account to explain the mechanism of domain inversion. In some certain ferroelectric crystals such as LiNbO${}_3$, domain back switching and domain wall motion can be promoted by a, temperature change. Heating or cooling temporarily changes the magnitude of spontaneous polarization; therefore ${\mathbf{E}}_{dep}$ is temperature-dependent. Normally, the pyroelectric field ${\mathbf{E}}_{pyr}$ can be expressed as:

$${\mathbf{E}}_{pyr}={\mathbf{E}}_{dep}+{\mathbf{E}}_{scr}.$$

(4)

At room temperature, ${\mathbf{E}}_{dep}$ is fully screened by bulk and surface charges; thus, ${\mathbf{E}}_{pyr}=0$ (see Figure 10a). When the crystal is being heated, the polarization decreases, leading to a reduction in ${\mathbf{E}}_{dep}$. Therefore, ${\mathbf{E}}_{pyr}$ is not zero any more, and the net value is parallel with ${\mathbf{E}}_{scr}$ (Figure 10b), which can be expressed as:

$${\mathbf{E}}_{pyr}=-\frac{1}{ϵ{ϵ}_0}\left(p\right)\Delta T,$$

(5)

where $\Delta T$ is the temperature change, and p is the pyroelectric coefficient. Meanwhile, thermal heating activates charges to drift into the pyroelectric field, resulting a new field of charges ${\mathbf{E}}_{cha}$ parallel to ${\mathbf{E}}_{dep}$ to compensate for ${\mathbf{E}}_{pyr}$ (Figure 10c). Now, the pyroelectric field is zero again and expressed as

$${\mathbf{E}}_{pyr}={\mathbf{E}}_{dep}+{\mathbf{E}}_{scr}+{\mathbf{E}}_{cha}$$

(6)

The concentration of the charges is higher inside the filament than that in the unexposed area, which leads to a larger charge field and a larger domain inside the filament [123]. Subsequently, the spontaneous polarization increases, and the ${\mathbf{E}}_{dep}$ enlarges as well in the process of cooling down, resulting in a pyroelectric field antiparallel to ${\mathbf{E}}_{scr}$ (Figure 10d). When the value of this pyroelectric field in the exposed area is larger than the threshold electric field, the spontaneous polarization is inverted.

This technique is suitable for ferroelectric crystals with high pyroelectric coefficients, and one of them is LiNbO${}_3$, which has a dielectric as large as 28 and a pyroelectric coefficient $p=-0.38\times {10}^{-4}$ C/(m${}^2$K) [124]. A unidirectional domain inversion along the z axis of MgO-doped lithium niobate crystal was proposed in Ref. [106] (Figure 11). Filaments locate either on the surface or in the middle of the z axis, and the corresponding domain inversions are extended either to the other surface or to both sides. The width of the domain inversion depends on the fluence of the femtosecond laser and the length of the filament. Moreover, the domain can have a high aspect length/width ratio of up to 800 $\mathsf{\mu }$m/1 $\mathsf{\mu }$m with all domain boundaries being straight over the entire domain length. Furthermore, the effect of temperature during the heating treatment and the lattice periods on the number and size of inverted domains were investigated in a 2D NPC [123]. A threshold temperature was identified as $190{}^{\circ }$C, and an optimal temperature regime was $220{}^{\circ }$C–$300{}^{\circ }$C, within which all domains could be inverted in a 2D rectangular lattice with periods of 15 $\mathsf{\mu }$m × 6.3 $\mathsf{\mu }$m. Above $300{}^{\circ }$C, the already inverted domains switched back. Smaller lattice periods and lower temperatures resulted in fewer inverted domains, while the average of the central domains was 2.38 $\mathsf{\mu }$m, which was almost independent of the temperature and lattice periods. In addition, the normalized effective nonlinear coefficients $|G_{10}{|}^2$ of the 2D NPC under study were calculated in the order of 1%. This limitation is due to the small duty cycle restricted by this method.

3.3. Two Types of Domain Inversion Simultaneously Occurring

PT-relaxor ferroelectrics not only exhibit excellent piezoelectric but also show large second-order nonlinearity; therefore, they are good candidates for NPCs. One of them is 0.62 Pb (Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$-0.38 PbTiO${}_3$ (PMN-38PT), which is a single-domain crystal and has a coercive field of 8.25 kV cm ${}^{-1}$ [125]. As a comparison, the coercive field for intrinsic LiNbO${}_3$ crystal is about 5420 kV cm${}^{-1}$ and, for the congruent LiNbO${}_3$ crystal, it is 210 kV cm${}^{-1}$ [126,127]. Therefore, domain inversion is relatively easier to attain in PMN-38PT crystal than in LiNbO${}_3$. Recently, primary and secondary domain inversions were simultaneously achieved in PMN-38PT crystal by Sheng et al. [128]. The nonlinear absorption of a femtosecond laser not only leads to a thermoelectric field, inducing a primary domain inversion at the focal region, but also generates free carriers inside it, leading to a secondary domain inversion around it (Figure 12c), depending on the laser beam power. Here, the secondary domain inversion is based on the same mechanism as proposed in Ref. [106], which as described in the previous subsection. However, there are two significant differences: it has a hollow cylindrical structure (Figure 12) and grows only on one side of the focus. This is because that the redistribution of the free carriers results in a cylindrically symmetric space charge field; the pyroelectric field is unidirectional outside the focal region, while the thermoelectric field is bipolar inside the laser focal region. Furthermore, the laser has an effect of erasure on the secondary domain inversion. The thermoelectric field can switch back the flipped spontaneous polarization induced by the pyroelectric field in the overlapping area when the interval distance between two patterns is small enough (Figure 12e,f). This mechanism is similar to that described in Ref. [107]. Sheng et al. also developed a 1D QPM structure and firstly demonstrated a quasiphase matched collinear SHG in PMN-38PT crystal [129]. Three different processes of nonlinear interaction were observed by controlling the linear polarization of the fundamental beam. In addition, the relative strength of all second-order nonlinearity tensor elements were determined by comparing the emission power of the SHG at QPM resonances, and the absolute value of the $d_{31}$ was computed as 1.3 pm/V, which was the first quantitative measurement of nonlinear properties of a PMN-38PT crystal.

4. Laser Writing Parameters

For both the ${\chi }^{\left(2\right)}$-erasing or ${\chi }^{\left(2\right)}$-poling techniques, the laser-writing parameters including pulse energy and scanning speed strongly depend on the properties of the applied crystals and specific structures. Generally, mode-locked Ti:Sapphire femtosecond laser systems with regenerative amplifiers are employed as light sources due to their stabilities. Normally, the centered wavelength is 800 nm, the FWHM ranges from 75 fs to 180 fs [104,106,107], and the applied pulse energy of the laser is significantly dependent on the application purpose of the laser beam. Some typically detailed inscription parameters are listed in

Table 1. Selected laser-writing parameters.

${\mathit{\chi }}^{\left(2\right)}$-Erasing	Reference	Nonlinear Crystal	Repeat Frequency	Bandwidth	Wavelength	Pulse Energy	Scan Speed	N.A.
	Ref. [79]	LiNbO${}_3$	100 kHz	170 fs	800 nm	650 nJ	1 mm/s	0.5
	Ref. [80]	LiNbO${}_3$	1 kHz	120 fs	800 nm	60–72 nJ	80 $\mathsf{\mu }$m/s	0.8
	Ref. [82]	LiNbO${}_3$	1 kHz	104 fs	800 nm	100–200 nJ	55–100 $\mathsf{\mu }$m/s	0.8
	Ref. [84]	MgO-doped LiNbO${}_3$	1 kHz	500 fs	1030 nm	900 nJ	280 $\mathsf{\mu }$m/s	0.5
	Ref. [86]	LiNbO${}_3$	200 kHz	350 fs	1040 nm	16/20 nJ	1 mm/s	0.3
		Quartz				8/12 nJ
${\mathbf{\chi }}^{\left(\mathbf{2}\right)}$-Poling	References	Nonlinear Crystal	Repeat Frequency	Bandwidth	Wavelength	Pulse Energy	Scan Speed	N.A.
	Ref. [104]	LiNbO${}_3$	76 MHz	180 fs	800 nm	0–5 nJ	10 $\mathsf{\mu }$m/s	0.65
	Ref. [113]	BCT	76 MHz	180 fs	800 nm	∽6 nJ	10 $\mathsf{\mu }$m/s	0.65
	Ref. [117]	CBN	76 MHz	180 fs	800 nm	3.6–6.6 nJ	10 $\mathsf{\mu }$m/s	0.65
	Ref. [128]	PMN-38PT	80 MHz	180 fs	800 nm	0–5 nJ		0.4
Seeds	Ref. [106]	MgO-doped LiNbO${}_3$	1 kHz	100 fs	800 nm	50–500 nJ		0.8
LM	Ref. [108]	MgO-doped LiNbO${}_3$	1000 kHz	170 fs	1026 nm	150 nJ		0.42
LI			500 kHz			300–900 nJ

.

More specifically, for the ${\chi }^{\left(2\right)}$-erasing progress, small structures with a cross-section of several tens of micrometers in diameter have been inscribed by low repeat frequency of 1 kHz with scan speeds varying from 50 to 100 $\mathsf{\mu }$m/s (Figure 3f,g) [80,82]. Pulse energy here varies from 100 nJ to 200 nJ depending on the depth and specific crystal. Large-scale patterns with a cross-section of several hundreds of micrometers in diameter were fabricated with a medium repeat frequency of a hundred kilohertzs, scan speed of 1 mm/s, and single pulse energy as large as 650 nJ (Figure 3c) [79]. Additionally, compact ytterbium-doped diode-pumped ultrafast amplified lasers were also employed as light sources; a laser with an FWHM as large as 500 fs and a wavelength of 1030 nm was used to erase ${\chi }^{\left(2\right)}$[84]. In this report, the repeat frequency was 1 kHz with a scan speed of 280 $\mathsf{\mu }$m/s, and the single-pulse energy was 900 nJ.

Generally, pulse energy used for directly inducing poling of ${\chi }^{\left(2\right)}$ is smaller than that used in LIQPM, because the crystal structure must not be destroyed in this case. For example, in Ref. [104], the repeat frequency employed was around 76 MHz and the corresponding pulse energy was no larger than 5 nJ, which was optimized by depth. Generally, the scan velocity is as slow as 10 $\mathsf{\mu }$m/s, which is time-consuming. Notably, in both the two-step primary domain inversion [107,108] and two-step secondary domain inversion techniques [106,123], the first-step seed inscribed by a femtosecond laser is a Type II modification on the crystal. In this step, the pule energy, period distance, or after-heating temperature could be factors affecting the sizes of the inverted domains in the next step. For example, in Ref. [106], seeds were induced with a repetition rate of 1 kHz with energies from 50 to 550 nJ; the inverted domain width roughly increased with pulse energy (Figure 13). It was also demonstrated that annealing temperature does not seem to affect domain size (Figure 14) [123].

5. Conclusions

In conclusion, recent advances and achievements in the field quasiphase matching structures using direct femtosecond laser writing techniques were briefly overviewed. This review first briefly introduced the concept of nonlinear photonic crystal and then focused on the quasiphase matching structures achieved by full-optical methods, either erasing or poling ${\chi }^{\left(2\right)}$, which produce efficient SH emission. Direct femtosecond laser writing on ${\chi }^{\left(2\right)}$ has enormous potential owing to its unique 3D microfabrication ability, which is unmatched by other methods. Moreover, 3D QPM technology has been showing great advantages and will play a significant role in the field of nonlinear photonics, providing a promising technique for versatile integrated optical applications including frequency conversion, nonlinear Talbot imaging [130], on-chip entangled light sources [42,43], THz devices [68], 3D nonlinear holography, and beam shaping [83,84]. Nevertheless, there are some challenges to be further overcome; for example, the quality of the emission needs to be improved, as does the conversion efficiency. Additionally, it is necessary to investigate the affects of related parameters. Overall, full-optical engineering on ${\chi }^{\left(2\right)}$, especially the ${\chi }^{\left(2\right)}$-poling technique, is just starting application in 3D NPCs; its multifunctional applications in integrated on-chip photonics need to be explored in the future.