1. Introduction

As an emerging wide bandgap semiconductor material, beta-phase gallium oxide (β-Ga₂O₃) has attracted considerable attentions in catalysis [1,2], gas sensors [3], power electronics [4] and potential optical devices such as waveguides and resonators. Due to its wide bandgap (4.85 eV), β-Ga₂O₃ possesses a board transparent spectrum (>300 nm) from UV to visible wavelengths. Moreover, β-Ga₂O₃ is compatible with III-N material system [5]. This indicates β-Ga₂O₃ optical devices can integrate with III-nitride based visible light sources [6–8], detectors [9,10], and solar based applications [11,12]. Therefore, β-Ga₂O₃ is also a promising candidate for integrated photonics system, especially at UV and visible regime. In order to understand and improve the performances of various β-Ga₂O₃ devices, the comprehensive investigation on the material properties of β-Ga₂O₃ is of crucial importance. Intensive studies have been reported on the electronic [13], optical [14–16], and thermal properties [17] of β-Ga₂O₃. In terms of the optical properties of β-Ga₂O₃, previous researches mainly focused on the transmission [14], and refractive index [15,16]. However, the nonlinear optical properties of β-Ga₂O₃ have not been investigated yet, not to mention in UV and visible regime.

In UV and visible spectral range, under high optical power density, the performance of optical waveguides and resonators are mainly degraded by two-photon absorption (TPA) process [18,19]. For resonators that operate by critical coupling from bus waveguides, the refractive index shifting is governed by Kerr nonlinear refractive index at high power density and decreases the coupling efficiency especially in ultra-high quality factor resonators [20]. Therefore, in order to demonstrate high performance β-Ga₂O₃ based optical devices, it’s critical to characterize the nonlinear optical properties of β-Ga₂O₃ including TPA coefficient and Kerr nonlinear refractive index.

To investigate and evaluate the nonlinear optical properties of β-Ga₂O₃ in visible spectral range, we performed a typical Z-scan characterization to study the TPA coefficient and Kerr nonlinear refractive index of β-Ga₂O₃. It’s found that β-Ga₂O₃ has much smaller two photon absorption coefficient (20 times smaller) and Kerr nonlinear refractive index (4-5 times smaller) [18] than GaN, which is ideal for waveguides and resonators. Furthermore, due to the highly asymmetric crystalline structure of β-Ga₂O₃, the optical nonlinearity is also highly anisotropic. Relative higher in-plane nonlinear optical anisotropy was observed on ($\overline{2}$01) plane than on (010) plane. These result can serve as references for the design of photonic devices based on β-Ga₂O₃.

This paper is organized as the following: In Section 2, we describe methods used in this work including experimental setup and theoretical models. In Section 3, we show our experimental results and make a discussion mainly on its significance in integrated photonics system. In section 4, we will draw the conclusion of this work.

2. Methods

The unintentionally doped (UID) β-Ga₂O₃ samples were provided by Tamura Corporation with a carrier concentration on the order of ~10¹⁷ cm⁻³ and a thickness of ~500 µm. The backside of the samples was polished by hand-grinding method with diamond lapping film of 0.5 µm grade. The polishing process was carefully controlled so that the thickness of the samples after polishing was reduced by less than 100 µm.

Figure 1(a) schematically shows the experimental setup of the Z-scan measurement [19–22] used in this study. Figure 1(b) depicts the crystal structure of β-Ga₂O₃, (010) and ($\overline{2}$01) planes are also indicated. The light source was an ultrafast titanium-sapphire laser operating at 808 nm with 100 fs pulse width and 82 MHz repetition rate. A 𝜒² crystal was used to generate second harmonic wave at the wavelength of 404 nm. The average power of input second harmonic beam was kept at 12 mW, which corresponds to ~7 × 10⁹ W/cm³ peak power density and ~0.7 mJ/cm³ single pulse energy density. Due to the lack of reference, no laser-induced damage threshold (LIDT) of β-Ga₂O₃ was found. As investigated in Ref [23], the LIDTs of dielectric materials are correlated to bandgap energy. To roughly estimate LIDT of β-Ga₂O₃, the LIDTs of two widely recorded materials, GaN (3.4 eV) and HfO₂ (5.2 eV), were checked. The threshold energy are ~5.4 J/cm³ for GaN at 400 nm with 120 fs pulse width [24], and ranging from 0.5 J/cm³ to 20 J/cm³ for HfO₂ [25,26]. Those LIDTs are more than 500 times higher than the operation pulse energy density used in this work. Therefore, we neglected the laser-induced damage effects.

 

Fig. 1 (a) Experimental setup used in this work. AT indicates the attenuator, λ/2 the half-wavelength plate, PL the polarizer, OB the optical objective lens, A the aperture, PM the power meter. L1, L2, L3 are lenses. (b) Crystalline structure of β-Ga2O3 and planes that characterized in this study. (c) The field intensity at focal point estimated, which agree with our open aperture fitting.

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Half wavelength plate working on 404 nm was placed after 𝜒² crystal for light polarization tuning. Beam was expanded by a set of lens before sending into optical objectives in order to fully utilize the numerical aperture. The samples were positioned between two optical objectives. An aperture was implemented in front of power meter to perform open and closed aperture testing.

To test the reliability of the setup, the beam size of the output light was used as an indicator. Based on Gaussian optics, with an incident beam diameter of 4 mm and a numerical aperture (NA) of 0.25, the calculated diameter of the output light is ~5 um (at 1/e²) as shown in Fig. 1(c). This value is consistent with the beam diameter we extracted from open aperture Z-scan measurement in the next section.

The TPA coefficient and Kerr nonlinear refractive index can be obtained from open/closed aperture scanning based on Eqs. (1)–(3).

3. Experimental results and discussions

Figure 2 shows the open/closed aperture scanning results for (010) and ($\overline{2}$01) β-Ga₂O₃. Fitting the experimental data with Eqs. (1)–(3) yielded the TPA coefficients and Kerr nonlinear refractive index, which was summarized in Table 1. (010) β-Ga₂O₃ had a TPA coefficient α_TPA = 1.2 cm/GW and n_kerr = –2.1 × 10⁻¹⁵ cm²/W when $\overrightarrow{E}$ $\perp$ [102], while α_TPA of 0.6 cm/GW and n_kerr of –2.9 × 10⁻¹⁵ cm²/W were obtained on ($\overline{2}$01) β-Ga₂O₃ when $\overrightarrow{E}$ $\parallel$ [102]. The ($\overline{2}$01) β-Ga₂O₃ has smaller TPA coefficient than (010) β-Ga₂O₃. This result indicates ($\overline{2}$01) β-Ga₂O₃ waveguides and resonators will have less loss induced by TPA in visible spectral regime. On the other hand, ($\overline{2}$01) β-Ga₂O₃ has larger Kerr nonlinear refractive index. In addition, the output beam diameters were also obtained, which were in good agree with theoretical calculations based on Gaussian optics. It is worth noting that besides the transition from valance band (VB) to conduction band (CB), β-Ga₂O₃ exhibits up to three additional transition channels [1] that may contribute to the optical transition. Photon energies corresponding to the channels are: UV (3.2–3.6 eV), blue (2.8–3.0 eV), and green (2.4 eV). The green transition channel is mainly introduced by Be, Ge, and Sn dopants, therefore it is not a main factor in this work. The blue transition channel is originated from oxide vacancies and correlated to the n-type conductivity of sample. Those intermediate states may enhance the TPA process and introduce saturable one photon absorption [32]. But this process should have minimum effect in this study as the sample used in this work is UID bulk β-Ga₂O₃ with high crystalline quality. The UV transition channel is resulted from recombination between free electrons and self-trapped holes. The electrons deep inside VB are required to be excited in order to contribute to the absorption process, which is relatively less efficient than transition from VB to oxide vacancy states.

 

Fig. 2 Typical open aperture scanning curve obtained in this work. (a) Scan curve for (010) sample, (b) for ($\overline{2}$01) sample. (c) and (d), closed aperture Z-scan measurement of (010) and ($\overline{2}$01) samples, respectively.

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Table 1. TPA coefficient, Kerr nonlinear refractive index obtained in this work for (010) and ($\overline{2}$01) samples. Beam diameter from fitting parameter is also shown.

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From Fig. 2, it can be clearly observed that closed aperture measurement exhibits more intense noise comparing with open aperture curve. This can be mainly attributed to two reasons: firstly, the testing wavelength was located at 404 nm which is the second harmonic beam from Ti:S laser. Therefore, comparing with our previous work [19], the laser power was ~80 times lower. It is the low power that makes the noise from beam nonidealities more significant, which results in the “bump” and “dip” features in Figs. 2(c) and 2(d) at Z~0.2 mm. Secondly, according to Ref [31], n_kerr is supposed to be zero near ~0.7E_g photon energy, which corresponds to 3.4 eV for β-Ga₂O₃. Since the testing photon energy at 404 nm is ~3.1 eV, the n_kerr is supposed to be in a relative small magnitude. This also increase testing difficulties throughout the experiment. To verify the closed aperture data, relation Z_pv $\approx$ 1.7Z₀ was implemented [21], the Z_pv represents peak-valley distance in closed aperture scan and Z₀ was obtained from Eqs. (1) and (2). Fitting of Figs. 2(a) and 2(b) yielded Z₀ of 0.15 mm and 0.13 mm, corresponding to Z_pv of 0.25 mm and 0.22 mm, respectively. These are consistent with the measured (0.22 mm and 0.23 mm) Z_pv shown in Figs. 2(c) and 2(d).

It also worth noting that below 3.4 eV photon energy, using the widely implemented fitting model [31] developed by Sheik-Bahae et al. the n_kerr is supposed to be positive, while the n_kerr obtained in this work is negative. Such negative n_kerr value has profound physics behind. As suggested in [31], the n_kerr is contributed from TPA effect, Raman effect, linear stark effect, and quadratic stark effect. For n_kerr obtained at relative long wavelength, fitting model implementing only TPA effect is sufficient. However, at short wavelength (~0.7 E_g photon energy), contribution from TPA effect is approaching zero and quadratic stark effect contributes significant negative phase change which in turn gives n_kerr. This explains why negative n_kerr was obtained in this work.

There are several physical models that can be implemented to understand the higher optical nonlinearity at (010) plane [33–38]. Quantum mechanical approach [33,34] calculates third order susceptibility accurately but requires intensive computing resources. Therefore it is only utilized in simple crystal structures [34]. A simplified bond-orbital model developed in [35–38] successfully explained the optical nonlinearities for various kind of materials including metal-oxide crystals [38]. We employed this model in this work to qualitatively understand the anisotropic nonlinearity of (010) and ($\overline{2}$01) β-Ga₂O₃. From bond orbital model, optical anisotropy of crystal is mainly contributed from bonding electrons between adjacent atoms [35]. While for other electrons that screening around individual atoms, the contribution to the optical anisotropy is less significant. As investigated comprehensively in [39], β-Ga₂O₃ is constructed by two types of gallium ions and three types of oxygen ions shown in Figs. 3(a) and 3(b). Type I gallium ion Ga_(I) is surrounded by a distorted tetrahedron of oxygen ions with atom distance from 1.80 A to 1.85 A (1.83 A in average), type II gallium ion Ga_(II) is surrounded by a distorted octahedron with atom distance from 1.95 A to 2.08 A (2.00 A in average). On (010) plane of β-Ga₂O₃, the bonds are not orderly orientated and therefore electrical field in (010) plane are more likely to interact with most of the bonds. However, on ($\overline{2}$01) plane, bonds between Ga_(I) and O_(III) are highly oriented along (010) direction. Such bonds have relative weak interaction with electric field polarized in ($\overline{2}$01) plane. The consequence is that the optical nonlinearity on ($\overline{2}$01) is relatively weaker than that on (010) plane.

 

Fig. 3 Schematic view for the arrangement of ions inside β-Ga₂O₃, different types of ions are marked out by numbers. (a) Schematic view for (010) sample, (b) for ($\overline{2}$01) sample.

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We previously estimated TPA coefficients and Kerr nonlinear refractive index for GaN [19] at same wavelength. The TPA coefficient of (010) and ($\overline{2}$01) β-Ga₂O₃ are ~10 and ~20 times smaller than that of GaN at 404 nm, respectively. Since the bandgap energy of β-Ga₂O₃ and GaN are different, we compare their nonlinear optical coefficient at same relative photon energy which is defined as E_p/E_g (0.63 for β-Ga₂O₃ tested at 404 nm). At E_p/E_g = 0.63 relative photon energy, TPA coefficient of (010) and ($\overline{2}$01) β-Ga₂O₃ samples are ~15 and ~30 times smaller than that of GaN. The high TPA coefficient observed in GaN might due to its exciton effects [22]. This result implies that β-Ga₂O₃ material is more capable to handle high optical power density applications in visible wavelength spectral range. Furthermore, Kerr nonlinear refractive index of β-Ga₂O₃ are also 4–5 times smaller at 404 nm and 5–7 times smaller at same relative photon energy than that of GaN. For optical applications that requires critical coupling, e.g. coupling from bus waveguide to ring or disk resonators, Kerr effect modifies the coupling efficiency at high optical power density, especially in those high resonance quality factor resonators [20]. Therefore, β-Ga₂O₃ based resonators will exhibit extreme coupling stability under high power operation comparing with GaN based resonators.

Figure 4 represents the polarization dependences of transmission minimum of (010) and ($\overline{2}$01) β-Ga₂O₃ during open aperture testing. Since β-Ga₂O₃ has a monoclinic crystal structure (Fig. 1), 41 independent nonzero elements are required to fully describe its third order nonlinearity d_il. Therefore, it is very difficult and inconvenient to find explicit expression for d_eff using polarization angle and individual nonlinear component d_il [21,22,27,28]. To describe the in-plane anisotropic nonlinearity clearly, we use $\Delta$T_max/$\Delta$T_min in this work. Relative higher in-plane nonlinear optical anisotropy was found on ($\overline{2}$01) β-Ga₂O₃ with $\Delta$T_max/$\Delta$T_min of 1.93 while (010) β-Ga₂O₃ had a $\Delta$T_max/$\Delta$T_min of 1.29.

 

Fig. 4 Polarization dependence of minimum transmittance obtained in this work. (a) Polarization dependence for (010) sample, (b) for ($\overline{2}$01) sample.

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Figure 5 shows the wavelength dependence of TPA coefficient using Eq. (4) for (010) and ($\overline{2}$01) β-Ga₂O₃. For (010) β-Ga₂O₃, E$\perp$ has larger TPA coefficient that E$\parallel$. For ($\overline{2}$01) β-Ga₂O₃, E$\perp$ has smaller TPA coefficient that E$\parallel$. Highest TPA absorption coefficient is observed in (010) β-Ga₂O₃ when electric field is perpendicular to [102] direction, while lowest TPA coefficient is obtained on ($\overline{2}$01) β-Ga₂O₃ when electric field is parallel to [102] direction. For each plane, the polarization dependence of TPA coefficient is a function of multiple physical parameters [38] such as bond iconicity, covalent radii, etc. It is extremely difficult to give a quantitative explanation and out of the scope of this work. Future experimental and theoretical studies are undergoing to clarify its polarization dependence. For both samples, the maximum TPA absorption occurs at ~360 nm wavelength. For wavelength above 360 nm, TPA coefficient decreases as wavelength increases. It can be attributed to the decreased excitation energy with increasing wavelength. While for wavelength below 360 nm, the TPA coefficient increase with increasing wavelength as the transition approaches its resonance wavelength.

 

Fig. 5 The estimated wavelength dependence of TPA coefficient for (010) and ($\overline{2}$01) samples. “E$\perp$” indicates that the electrical field intensity is perpendicular to [102] direction, while “E$\parallel$” indicates that field intensity is parallel to [102] direction

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4. Conclusions

We characterized the TPA coefficient and Kerr nonlinear refractive index of both (010) and ($\overline{2}$01) β-Ga₂O₃. TPA coefficient of Ga₂O₃ was found to be 10 to 20 times smaller than that of GaN at 404 nm. The Kerr nonlinear refractive index of Ga₂O₃ was 4 to 5 times lower than that of GaN. Therefore, due to its ultra-low TPA coefficient and its small Kerr nonlinear refractive index, β-Ga₂O₃ has the potential to serve as a more efficient platform for integrated photonic applications in UV and visible spectral range. Furthermore, the optical nonlinearities of β-Ga₂O₃ is highly anisotropic due to the asymmetric crystal structure of β-Ga₂O_3. These results can serve as guidelines for designing β-Ga₂O₃ based integrated photonics system.