Multiplexing Linear and Nonlinear Bragg Diffractions through Volume Gratings Fabricated by Femtosecond Laser Writing in Lithium Niobate Crystal

Abstract

The femtosecond-laser-writing technique provides a flexible method for fabrication of nonlinear photonic crystals in three dimensions, providing structures that enable efficient complex nonlinear wave interactions and modulation for applications including nonlinear holography, nonlinear beam shaping, and waveguide-integrated wavelength conversion. However, the tightly focused laser pulse inevitably causes structural modification and then changes the local refractive index, resulting in additional linear modulation of the interacting waves. Here, we use the same periodic distributions of the refractive index and the second-order nonlinear coefficient for grating arrays engineered in lithium niobate crystals by femtosecond laser writing to achieve polarization-dependent linear and nonlinear Bragg diffractions simultaneously. The experimental results show that the linear and nonlinear diffraction efficiencies range up to 31% and 2.9 × 10⁻⁵, respectively, for grating arrays with dimensions of 100 μm (x) × 100 μm (y) × 100 μm (z). This work paves the way toward the realization of the multiplexing of linear and nonlinear optical modulations in a single structure for potential applications that include multidimensional optical data storage and optical coding.

1. Introduction

The quasi-phase matching (QPM) scheme, proposed by Bloembergen et al. in 1962, provides a unique method to compensate for phase mismatch among nonlinear interacting waves and perform efficient three-wave mixing, which is on the basis of the periodically modulated second-order nonlinear coefficient (χ⁽²⁾) in nonlinear crystals [1]. The QPM scheme not only overcomes the walk-off effect that occurs in the birefringent phase matching process, but also extends the nonlinear frequency conversion range from ultraviolet up to terahertz waves [2,3]. With the successful fabrication of periodically-poled lithium niobate (LN), lithium tantalite (LT), and potassium titanyl phosphate (KTP) crystals, the QPM scheme has been widely used to generate wavelength-tunable classical coherent sources and quantum sources [4,5,6,7,8]. What follows was the nonlinear photonic crystal (NPC) proposed by Berger in 1998. This pioneer concept led to a further generalization of the in-plane QPM scheme, in which the availability of the noncollinear QPM process (also called nonlinear Bragg diffraction) enriched the nonlinear interaction processes [9]. Take the second harmonic generation (SHG) for example, in the nonlinear Bragg diffraction process, the outputs of generated second harmonic (SH) wave at a different angle from the incident fundamental beam are owing to the involvement of noncollinear reciprocal vectors in the QPM process. This phenomenon looks like the traditional Bragg diffraction that constructive interference occurs at a certain diffraction order based on Bragg’s law. Because the two-dimensional (2D) NPCs were fabricated using an electrical poling technique, nonlinear cascaded-frequency conversion, nonlinear conical beam generation, and path-entanglement quantum source generation were demonstrated successively using the nonlinear Bragg diffraction process [10,11,12,13]. In addition, NPCs have also been used to control nonlinear wavefronts for beam shaping and holography [14,15,16]. However, the conventional electric poling technique only permits 2D modulations of χ⁽²⁾ in NPCs. Therefore, efficient nonlinear wavefront shaping is limited to one dimension based on the on-axis hologram scheme, as one dimension along the optical axis is required to aid the QPM process [17]. To attain high conversion efficiency in nonlinear beam shaping, two additional dimensions perpendicular to the optical axis are necessary for full wavefront modulations. Consequently, previous studies on nonlinear beam shaping or holography have largely encountered low conversion efficiency [18].

In recent years, the fabrication of three-dimensional (3D) NPCs via the advanced femtosecond-laser-writing technique was successfully demonstrated, extending nonlinear Bragg diffraction into the 3D space [19,20]. The fabrication schemes used can be divided into two types: χ⁽²⁾ erasure through amorphization in uniform nonlinear crystals such as LN [21] and χ⁽²⁾ inversion through domain poling in nonuniform crystals such as barium calcium titanate (BCT) [22]. The former is caused by a physical change in structure resulting from multi-photon absorption under ultra-strong focusing pulses with a repetition rate of kilohertz, while the latter is mainly attributed to thermoelectric-field poling induced by a temperature gradient created by focusing pulses with a repetition rate of megahertz. Therefore, the femtosecond-laser-writing technique allows the fabrication of more complex 3D non-periodic structures, in addition to periodic ones. The incremental third dimension in comparison to 2D NPC enables efficient nonlinear beam shaping and multiplexing nonlinear holography [23,24,25,26]. For example, holographic arrays in Lithium niobate crystals have been used to efficiently generate SH orbital angular momentum of light and Hermite-Gaussian beams via nonlinear Bragg diffractions, in which the efficiency was found to be two orders of magnitude higher than that achieved using 2D NPCs [27]. In addition, nonlinear multiplexing holography has been achieved by engineering the nonlinear Ewald sphere in reciprocal space. Using a structure with only 16 periods along the optical axis, six images were reconstructed via irregular nonlinear Bragg diffractions [28]. Recently, the poling scheme was extended to uniform LN and BCT crystals to fabricate high-quality NPCs [29,30], while the erasure scheme has been applied to piezoelectric quartz crystals to provide efficient deep-ultraviolet SHG [31]. What’s more, the 3D checkerboard NPC has been theoretically shown to be capable of generating nonlinear vortex solitons [32]. However, the structural amorphization that inevitably occurs under irradiation by a tightly focused pulse leads to refractive index change, which then becomes more severe in the χ⁽²⁾ erasure scheme [33]. The undesired nonuniform refractive index distribution in LN and silica NPCs based on the χ⁽²⁾ erasure scheme caused considerable diffraction and scattering losses and affected the nonlinear conversion efficiencies [21,27,28,34]. Therefore, researchers have made efforts to either suppress the refractive index change in the fabrication of NPC [30] or utilize it for constructing NPC waveguides [25].

Femtosecond has been demonstrated as a powerful tool for creating complex micro-optical devices with high precision and accuracy in crystals or glasses based on refractive index change on the order of 10⁻² to 10⁻⁴ induced by structural changes [33]. These devices include 3D optical waveguides for spatial field guiding [35,36], volume Bragg gratings for optical filtering [37], complex photonic crystals for beam shaping and multiplexing holography [38,39]. While devices fabricated using Complementary Metal-Oxide-Semiconductor compatible techniques offer stronger optical field confinement and manipulation due to a higher refractive index change, their fabrication is less flexible and complex volume structures are difficult to realize. In contrast, the use of simpler 3D fabrication techniques enables the realization of larger, more complex structures that can enhance optical field manipulation via longer interaction length, despite the weaker optical field confinement resulting from a smaller refractive index change. The volume Bragg grating is a representative device with an absolute diffraction efficiency that can reach 90% for a 90 μm-long structure in fused silica [40].

Here, we have used a femtosecond-laser-writing system to fabricate volume gratings in an LN crystal in which the refractive index and χ⁽²⁾ share the same space-dependent periodic function, and then characterize the linear/nonlinear Bragg diffractions and the multiplexing of these diffractions via a single structure. As Berger indicates in his pioneering work on NPCs, the nonlinear Bragg law for SHG can be reduced to the well-known Bragg law if the medium has no dispersion [9]. This analogy indicates that it is possible to realize linear and nonlinear Bragg diffractions in structures with both space-dependent periodic refractive index and χ⁽²⁾ distributions. Our work brings the freedom of linear optical field modulation to the laser-engineered NPC, thus providing a new platform for the multiplexing of linear and nonlinear beam shaping or holography applications [41,42,43].

2. Materials and Methods

2.1. Principle

Figure 1a shows a schematic of the volume grating arrays, which present as square lattices in the x-y plane with periods of Λ_x and Λ_y, and as elongated cylinders with length L_z along the z-axis. These arrays could be engineered by the tightly focused laser pulses to have a refractive index of n_b and a second-order nonlinear coefficient of ${\chi }_b^{\left(2\right)}$ to distinguish them from the original values of n_a and ${\chi }_a^{\left(2\right)}$, respectively, in the non-engineered region. The reciprocal space of the gratings provides different orders of the reciprocal vectors:

$$\overrightarrow{G_{p,q}}=p\frac{2\pi }{{\Lambda }_x}\widehat{x}+q\frac{2\pi }{{\Lambda }_y}\widehat{y},$$

(1)

where p and q are arbitrary integers, while $\widehat{x}$ and $\widehat{y}$ refer to unit vectors along x and y directions, respectively. These vectors coexist in the linear and nonlinear reciprocal space to realize the nonlinear Bragg SHG diffraction shown in Figure 1b, i.e., the noncollinear QPM [44]:

$$\overrightarrow{k_{2\mathsf{\omega }}}-2\overrightarrow{k_{\mathsf{\omega }}}-\overrightarrow{G_{p,q}}=0,$$

(2)

and the traditional linear Bragg diffraction shown in Figure 1c [40], i.e.,

$$\overrightarrow{{k\prime }_{\omega }}-\overrightarrow{k_{\mathsf{\omega }}}-\overrightarrow{G_{p,q}}=0.$$

(3)

$\overrightarrow{k_{\mathsf{\omega }}}$, $\overrightarrow{{k\prime }_{\omega }}$, and $\overrightarrow{k_{2\mathsf{\omega }}}$ in Equations (2) and (3) refer to the wavevectors for the input fundamental beam, diffracted fundamental beam, and generated SH beam. Therefore, the linear and nonlinear Bragg diffractions can be seen as closed triangles that are formed by the wavevectors and the reciprocal vectors as shown in Figure 1b,c. Assuming that the incident fundamental beam has an angle φ relative to the reciprocal vectors $\overrightarrow{G_{p,q}}$, there is:

$$\overrightarrow{k_{\mathsf{\omega }}}·\overrightarrow{G_{p,q}}=\left|\overrightarrow{k_{\mathsf{\omega }}}\right|\left|\overrightarrow{G_{p,q}}\right|sin\phi .$$

(4)

Therefore, the QPM condition in Equation (2) can be rewritten as:

$${\lambda }_{2\omega }=\frac{2\pi n_{2\omega }}{\left|\overrightarrow{G_{p,q}}\right|}\left[\sqrt{{\left(1-\frac{n_{\omega }}{n_{2\omega }}\right)}^2{\mathit{sin}}^2\phi }-\frac{n_{\omega }}{n_{2\omega }}\mathit{cos}\phi \right],$$

(5)

while the linear Bragg diffraction in Equation (3) can be abbreviated as:

$${\lambda }_{\omega }=-\frac{4\pi n_{\omega }}{\left|\overrightarrow{G_{p,q}}\right|}\mathit{cos}\phi$$

(6)

Here, ${\lambda }_{\omega }$ and ${\lambda }_{2\omega }$ are the fundamental and SH wavelengths, respectively; $n_{\omega }$ and $n_{2\omega }$ are the corresponding refractive indexes; $\left|\overrightarrow{k_{\mathsf{\omega }}}\right|$ and $\left|\overrightarrow{G_{p,q}}\right|$ are the modulus of $\overrightarrow{k_{\mathsf{\omega }}}$ and $\overrightarrow{G_{p,q}}$, respectively. Equations (5) and (6) provide the relationship between the wavelengths and angles corresponding to specific reciprocal vectors for nonlinear and linear Bragg diffractions, respectively. The orders of $\overrightarrow{G_{p,q}}$ usually differ in the linear and nonlinear processes because of the different numbers of involving wavevectors and the material dispersion in the nonlinear one. The interacted waves have extraordinary polarization in Figure 1b to use the maximum nonlinear coefficient ${\chi }_{33}$ of LN and have ordinary polarization in Figure 1c to utilize the larger ordinary refractive index change [45]. These features enable the linear and nonlinear diffracted beams to be enhanced simultaneously via the involvement of different $\overrightarrow{G_{p,q}}$. As shown in Figure 1d, when the fundamental beam contains both extraordinary and ordinary components, both linear and nonlinear Bragg diffractions will be produced at an optimal wavelength and a specific angle of incidence.

2.2. Sample Fabrication and Experimental Setup

Sample grating arrays were fabricated using an in-house-built femtosecond-laser-writing system. An ultrafast amplifier laser system (Astrella-Tunable-V-USP-1k, Coherent, Santa Clara, CA, USA) served as the writing source, operating at a wavelength of 800 nm with a pulse duration of 35 fs, a repetition rate of 1 kHz, and maximum output power of approximately 1 W. The laser was focused using an objective lens (OPLNFL40X, N.A. = 0.75, Olympus, Tokyo, Japan) inside a 5% MgO-doped LN crystal along the z direction with a spot diameter of 1.4 μm. The LN crystal was mounted on a triaxial piezoelectric nanometer platform (E727, Physik Instrumente, Karlsruhe, Germany) to realize high-precision 3D movement for laser writing. A combination of a half-wave plate and a polarization beam splitter in the optical path was used to adjust the engineering power inside the LN crystal. The writing energy was set at 370 nJ before the objective and the line-scanning speed along the z-axis from depth to surface was 40 μm/s. Although the focusing spot will be distorted for engineering in a large depth owing to the aberration and the birefringence, it has little influence on the line-scanning scheme along the depth. The fabricated sample had dimensions of approximately 100 μm (L_x) × 100 μm (L_y) × 100 μm (L_z) and periods of Λ_x = 3.35 μm and Λ_y = 3.35 μm, corresponding to periodic numbering of 30 × 30 in the x-y plane, as the microscopy images shown in Figure 2a. The periods were designed based on Equations (4)–(6). Figure 2a depicts microscopy images obtained using the dark-field mode, where bright regions correspond to laser-engineered areas, and dark region correspond to the unprocessed area. The x-z plane image shows homogenous grating arrays, while the x-y plane image shows periodic distribution along the x and y directions. The homogeneity of the grating arrays in the x-z plane suggests that the femtosecond-laser-writing technique can be used to create highly precise and uniform structures, while the periodic distribution in the x-y plane confirms the periodic nature of the volume grating arrays.

The fabricated sample was fixed on a manual multi-axis stage with the crystal axis along the z-axis to characterize the linear and nonlinear diffraction effects. As shown in Figure 2b, a high-repetition-rate femtosecond laser (Mai Tai, Spectrum Physics, Milpitas, CA, USA), featuring a wavelength tunable between 690 nm and 1040 nm, a pulse duration of 105 fs, and a repetition rate of 80 MHz, served as the source to generate the fundamental input beam. The input power was controlled using a combination of a half-wave plate and a polarizing beam splitter, followed by the other half-wave plate to control the polarization state inside the sample. A lens with a focusing distance of 75 mm converged the fundamental beam into the sample, resulting in the beam waist of about 40 μm inside the grating arrays to increase the power density. The sample could be rotated around the z-axis to choose the optimal angles for the linear and nonlinear Bragg diffractions at certain input wavelengths. The angle between the propagation direction of fundamental beam and the normal direction of the crystal surface was defined as the angle of incidence and labeled as θ. We assume that the coordinate system is fixed with the LN crystal, so the wavevector of the incident fundamental beam inside the crystal is given by

$$\overrightarrow{k_{\mathsf{\omega }}}=\frac{2\pi n_{\omega }}{{\lambda }_{\omega }}\left(\sqrt{1-{\left(\frac{sin\theta }{n_{\omega }}\right)}^2}\widehat{x}+\frac{sin\theta }{n_{\omega }}\widehat{y}\right),$$

(7)

taking refraction into account at the air-LN crystal interface. Based on Equations (1), (4), (5) and (7), we can numerically obtain the relationship between θ and the fundamental wavelength for both linear and nonlinear Bragg diffractions. The linear and nonlinear diffraction patterns generated by the grating arrays were then projected onto a white scattering screen and recorded in the dark using a charge-coupled device (CCD) camera. Here, it should be noticed that the signals of SHG in Figure 3 were obtained after the fundamental beam was filtered out by a short-pass filter. A power meter was used to measure the powers of the different diffraction orders for calculating conversion efficiencies in linear and nonlinear diffraction processes.

3. Results and Discussion

The fundamental beam with a power of 1 W and its polarization along the z-axis of the LN crystal, i.e., the extraordinary wave, was focused into the gratings to produce an enhanced SHG via nonlinear Bragg diffraction. The angle of incidence was set to be θ = 8°, corresponding to theoretical QPM wavelengths of 813 nm, 841 nm, and 869 nm under the help of compensatory reciprocal vectors of $\overrightarrow{G_{-2,1}}$, $\overrightarrow{G_{-1,1}}$, and $\overrightarrow{G_{1,1}}$, respectively. The experimental QPM wavelengths were measured by tuning the fundamental wavelength and then observing the diffracted powers of the SHG. As shown in Figure 3a, the SHG has peaks at fundamental wavelengths of 810 nm, 838 nm, and 872 nm with the assistance of the above reciprocal vectors, resulting in enhanced diffraction orders of −2, −1, and 1 along the x-axis, respectively. These peaks provide direct evidence for the satisfaction of a non-collinear QPM process, i.e., nonlinear Bragg diffraction [21]. Based on the peak powers from left to right in Figure 3a, the calculated SHG conversion efficiencies are 2.152 × 10⁻⁵, 2.979 × 10⁻⁵, and 1.087 × 10⁻⁵, respectively. Figure 3b shows the corresponding diffracted patterns at the three QPM wavelengths, in which the enhanced spots obtained via nonlinear Bragg diffraction are pointed out by white arrows to indicate the involving orders of reciprocal vector. They are much stronger than those obtained with the phase mismatch. The intensity patterns of SHG are captured by the CCD camera after the fundamental beam was filtered out by a short-pass filter. To keep the contrast between enhanced and non-enhanced spots and guarantee the pattern visibility, the camera was set to be a bit over-exposure, which leads to color differences varying from blue for phase-mismatch spots (normal-exposure) to white (over-exposure) for the enhanced diffracted spots and the zero-diffraction order, as shown in Figure 3b. The phase-mismatch spots could be suppressed by increasing the interaction length or using fundamental beams with a narrow wavelength bandwidth.

The polarization of fundamental beam was then rotated by 90° to be located at the x-y plane, i.e., as an ordinary wave, to characterize the linear Bragg diffraction. The input power was set at 0.8 W to guarantee high visibility at the near-infrared wavelengths. The incident fundamental wavelengths were chosen to be 817 nm, 841 nm, and 869 nm based on the theoretical Bragg angles of 7°, 7.2° and 7.4°, respectively, under the assistance of reciprocal vector of $\overrightarrow{G_{1,0}}$. The theoretical value was calculated according to the Equations (1), (4), (6) and (7). These angles are insensitive to wavelengths in comparison to that in the nonlinear process. Figure 4a shows the experimental results, in which the powers of the zero-diffraction order and the target first-diffraction order vary with the angle of θ in large scale. All of the first-diffraction power curves have peaks at around θ = 5°, indicating that the Bragg law is satisfied. This angle deviates from the theoretical ones, which is partially caused by the misalignment between the grating arrays along x-axis and the normal surface of LN. Furthermore, because the incident angle of θ was measured using the reflected fundamental beam from the LN surface based on the setup shown in Figure 2b, it inevitably induces an experimental error. The experimental Bragg angle becomes more insensitive to wavelength owing to the relatively small sample size and the large bandwidth of fundamental beam. This feature relaxes the conditions required to realize the linear and nonlinear Bragg diffractions simultaneously. From Figure 4a, the Bragg diffraction efficiencies are calculated to be 31.0%, 26.3%, and 23.9% at 817 nm, 841 nm, and 869 nm, respectively. Figure 4b shows the corresponding diffraction patterns for these three wavelengths at the angle of incidence of θ = 7°. The diffracted spots at the 1st diffraction order are obviously comparable to the zero-diffraction orders and those at the −1st diffraction order are invisible, providing evidence to support the involvement of the reciprocal vector of $\overrightarrow{G_{1,0}}$.

To demonstrate the hybrid linear and nonlinear Bragg diffractions, the polarization of the fundamental beam was fixed at 45° relative to the z-axis to contain both the extraordinary and ordinary components. The QPM wavelengths in the nonlinear process were utilized to characterize the hybrid linear and nonlinear diffraction processes. The reason is that linear diffracted powers are not significantly affected by the change of wavelength and the angle bandwidth for linear Bragg diffraction is large enough to overlap with the angle for nonlinear Bragg diffraction. As a result, for the experimental QPM wavelengths of 810 nm, 838 nm, and 872 nm given by Figure 3a, the diffracted patterns in Figure 5a show that the enhanced SHG occurs at different diffraction orders because of the involvement of different reciprocal vectors, although the linear Bragg diffraction spots remain unchanged. The nonlinear Bragg diffraction is more sensitive to the wavelength because of the effects of the material dispersions and the SH intensity squarely dependent on the interaction length. Figure 5b presents the schematics of linear and nonlinear Bragg conditions in reciprocal space to clearly illustrate the results shown in Figure 5a. It should be noticed that the extraordinary fundamental beam will also produce linear diffraction, but the tenfold smaller refractive index change makes it much weaker than that induced by the ordinary component.

4. Conclusions

We have experimentally demonstrated that grating arrays fabricated in LN crystals using the femtosecond-laser-writing technique could realize both linear and nonlinear Bragg diffractions because they use the same space-dependent distributions of the refractive index and second-order nonlinearity. Furthermore, the linear and nonlinear Bragg diffractions are polarization-dependent, so that the two processes could be realized either separately or simultaneously through change of polarization state combining with careful design of the structural parameters and optimization of the incidental angle and the fundamental wavelength. The linear and nonlinear conversion efficiencies are still low but could be improved by increasing the size of the structure and optimizing the grating parameters. Furthermore, in combination with theories on wave shaping or holography, our scheme is potential to realize more complex nonlinear wave mixing for multidimensional optical data storage and optical multiplexing applications [24,46]. Last but not least, LN crystal whose light transmission range covers 400 nm to 5500 nm boasts excellent piezoelectric effect, electro-optic effect, and nonlinear optical effect, making it an outstanding candidate for optical modulators, laser frequency converters, and quantum light sources [2,39]. Thanks to the successful preparation of LN thin film and the development of technically challenging nanofabrication, researchers have been able to demonstrate an on-chip LN photonic device, which has propelled the development of low-power and high-efficiency nonlinear and quantum devices [47,48,49]. Therefore, the extension of the work on the LN thin film platform is expected to further enrich the functions of integrated LN devices and explore brand-new applications in the future.