1. Introduction

Lithium niobate (LN) crystal, an outstanding material platform for achieving high-performance electro-optic (EO), acousto-optic (AO), and nonlinear optic devices, is embracing a significant revolution from traditional wafer to novel thin film on insulator wafer [1,2]. Sub-micrometer-thickness crystalline LN thin films bonded on insulator (usually silica), i.e., lithium niobate on insulator (LNOI), not only inherits the excellent characteristics of LN crystal, but also enable compact high-index-contrast waveguides, making it a promising platform for the development of advanced, high-density EO photonic integrated circuits (PICs) [1–4].

Waveguides, especially the single-mode waveguides (SMWs), are the backbone of various LNOI PICs. Nowadays, hybrid waveguides and monolithic waveguides are two main types of the LNOI waveguides [2,4]. Hybrid waveguides are based on rib-loading or heterogeneous bonding and implemented by etching the easier etched material [5–8], which exempt the necessity of LN etching. However, only a portion of optical field in hybrid waveguides is confined in the LN film, resulting in weaker nonlinear and EO effects. In addition, the devices using hybrid waveguides are quite sensitive to the change of temperature due to different thermal expansion coefficients between the LN and the loaded material [2]. Thus, the hybrid waveguides are not optimal option for the LNOI PICs. Monolithic waveguide, however, is implemented by etching the LN film into rib structure, which can achieve strong confinement and low loss for the light wave traveling inside and, hence, is considered an optimal option for the LNOI PICs [2,9]. Up to now, various LNOI PICs realized with rib waveguides, such as, EO modulator [10–13], EO tunable interleaver [14], ring resonators [15–17], optical amplifier [18], tunable Bragg grating filter [19], second-harmonic generator [20], and entangled photon pair generator [21] have been demonstrated. However, a systematic and in-depth investigation of single-mode conditions (SMCs) for the LNOI rib waveguides has not been carried out. Recently, Y. Li et al. reported their investigation for the SMCs of the LNOI rib waveguides based on the leakage from the core modes to the same polarized slab modes [22]. However, the leakage to the orthogonally polarized slab modes, which has also important impacts on the SMCs, does not involved in their work. In comparison, the SMCs for SOI rib waveguides, another important building block based on silicon (Si) material platform for PICs, have been widely and deeply investigated [23–25]. Unlike SOI rib waveguides formed with isotropic Si crystal, the LNOI rib waveguides formed with anisotropic LN crystal exhibit larger material and waveguide birefringence, which, in conjunction with the unique lateral leakage that rib waveguides possess [26], leads to quite complicated SMCs. Considering that SMW is essential building block in PICs, it is quite necessary to further investigate SMCs for the LNOI rib waveguides.

In this paper, we deeply investigate the SMCs for the shallowly etched x-cut, y-propagation LNOI rib waveguides, a type of waveguides most widely used in the LNOI PICs, using a commercial mode solver (COMSOL) based on the finite element method (FEM). For simplicity, the LNOI rib waveguides in the following refer to this shallow etch type except where otherwise stated. Compared with the work in [22], our investigation for the SMCs of the LNOI rib waveguides involves the above two types of leakage. To our best knowledge, it is the first time to take lateral leakage losses of the higher order modes into consideration when investigating SMCs of the LNOI rib waveguides. The impacts of the morphological parameters of the LNOI rib waveguides and the operation wavelengths on the SMCs are also investigated and presented.

2. Waveguide structure

The cross-sectional view of our investigated LNOI rib waveguide is shown schematically in Fig. 1, which consists of one ridge and two side slabs on both sides of the ridge. Such waveguide structure supports two types of guided modes. One is the modes confined in the ridge (ridge modes), labeled conventionally as E^x_ij or E^z_ij (i, j = 1, 2, 3 …), where subscript indicates the mode order and superscript the direction of the maximum electric field component, the other is the modes confined in the slab (slab modes), labeled conventionally as TE_m or TM_m (m = 0, 1, 2, …), where TE indicates transverse electric field and TM transverse magnetic field. According to the coordinate system shown in Fig. 1, the E^z_ij and TE_m modes have the same polarization along z direction (TE polarization), while the E^x_ij and TM_m modes have the same polarization along x direction (TM polarization). When the ridge mode E^x_ij or E^z_ij leaks into the slab, it will be converted into slab modes TM_m or TE_m. The morphological parameters of the investigated LNOI rib waveguide are also shown in Fig. 1, in which h is the waveguide height (i.e., the thickness of the initial LN thin film), w the top width of the ridge, h_e the etch depth, h_s the side-slab height, h_c the thickness of the cladding layer, θ the side wall angle. Usually, the SiO₂ buffer layer is thick enough (3-5 µm) to avoid any field penetration into the underlying substrate. The refractive indices of the materials in the core, buffer, and cladding layers are denoted as n₁, n₂, and n₃, respectively, as shown in Fig. 1, and at the most commonly used 1550 nm wavelength, n₁ = 2.1376 for the TE polarization and 2.2111 for the TM polarization, and n₂ = n₃ = 1.444 for both polarizations. It should be pointed out that the waveguide structure shown in Fig. 1 facilitates placement of the tuning electrodes on the two sides of the ridge to utilize efficiently the maximum EO coefficient γ₃₃. And on this basis, the maximum EO interaction can be achieved by optimizing the morphological parameters of the waveguide [2]. Thus, such waveguide structure is widely used in the LNOI PICs.

 

Fig. 1. Cross-sectional view of the LNOI rib waveguide investigated in this paper.

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3. Method

To achieve single-mode propagation in the LNOI rib waveguides, two mechanisms can be employed. The first one is by adopting deep etching to achieve the cut-off of the first order modes E^x₂₁ and E^z₂₁, just like the single-mode mechanism used in rectangular waveguide. The second one is by adopting shallow etching to make the ridge to accommodate the fundamental modes E^x₁₁ and E^z₁₁ and the first modes E^x₂₁ and E^z₂₁, but the waveguide parameters will be controlled precisely to achieve the leakage of the E^x₂₁ and E^z₂₁ modes into the two side slabs, and ultimately, only the fundamental modes of the ridge in the waveguide. It needs to be pointed out that due to the anisotropy of LN two types of leakage, including the leakage to the same polarized modes and to the orthogonally polarized modes, could happen in the LNOI rib waveguides. Obviously, compared with the first mechanism, the second one is capable of realizing not only a lower coupling loss between the waveguide and the standard SMF due to a larger waveguide cross section, but also a lower propagation loss induced by the side-wall roughness due to a smaller side-wall area, and therefore most widely adopted in the design of the LNOI PICs. In view of this, our investigation for the SMCs focuses on these shallowly etched LNOI rib waveguides. Note that the word “shallow” and “deep” in this paper are just used to differentiate between the above two mechanisms, does not mean that the etch depth is really shallow.

The maximum etching depth h_emax that can still realize single-mode propagation in the shallowly etched LNOI rib waveguides are investigated with the FEM (COMSOL). Here, two cases need to be considered, one is the unetched slab (corresponding to thickness h) is thick enough to accommodate the first order modes TE₁ and TM₁, the other is the unetched slab only accommodates the fundamental modes TE₀ and TM₀. For the former, single-mode propagation requires that the TE₁ and TM₁ modes of the unetched slab can leak into the etched slab (corresponding to thickness h_s). For the latter, single-mode propagation requires that the ridge modes E^z₂₁ and E^x₂₁ can leak into the etched slab. In view of the fact that the effective indices of the E^z₂₁ and E^x₂₁ modes are related to the ridge width w, to get h_emax for this case, a quite narrow width of w = 400 nm is used here. At last, in consideration that the above leakage of the high-order mode is due to the fact that its effective index is equal to that of the same polarized fundamental mode of the etched slab, then h_emax can be obtained.

Figure 2 presents the calculated h_emax with the FEM for different h from 400 to 1000 nm under the condition that h_c= 200 nm, the operation wavelength is 1550 nm, and the air cladding is also taken into account. For typical thickness h = 400, 600, and 900 nm, the calculated h_emax are 357, 503, and 626 nm for the TE polarization, respectively, and 299, 520, and 546 nm for the TM polarization, respectively. According to the above analysis, for the LNOI rib waveguides with h_e < h_emax, single-mode propagation is possible, but whether that happens depends on the leakage losses of the E^x₂₁ and E^z₂₁ modes determined by the w, h and the polarization state of the light wave, which will be discussed in the section 4.

 

Fig. 2. Calculated h_emax for LNOI rib waveguides corresponding to the TE and TM polarizations at different h.

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The above analysis only involves the leakage to the same polarized modes. However, shallowly etched rib waveguides have a unique characteristic of the lateral leakage due to the coupling of the two orthogonally polarized modes at the ridge boundary [26]. Because of the high index contrast, the ridge mode has a strong longitudinal electric field component along the waveguide axis, while the orthogonal slab mode can propagate at any angle relative to the waveguide axis. If the effective index of the ridge mode is smaller than that of the orthogonal slab mode, quite strongly phase-matched coupling between the ridge mode and the orthogonal slab mode may happen, depending on the propagation angle of the orthogonal slab mode. For the SOI rib waveguides, only the TM-like guided mode (here the label of the mode follows that used in references) can be coupled to the TE slab mode, resulting in lateral leakage, which has been investigated deeply in [27,28]. However, for the LNOI rib waveguides, due to its strong materials and waveguide birefringence the E^x₂₁ or E^z₂₁ mode could be coupled, respectively, to the orthogonal TE₀ or TM₀ mode, and hence, become leaky [29].

Obviously, the lateral leakage will result in large losses of the ridge modes. However, at special widths, resonant cancellation of lateral leakage can be realized and hence dramatically reduce the lateral leakage losses of the ridge modes, these special widths are called magic widths [26]. The lateral leakage losses of the fundamental ridge modes have been investigated in design of LNOI waveguide [29] and polarizer [30]. Practically, similar principle can also be utilized to design single-mode LNOI rib waveguide by engineering the lateral leakage losses of the E^x₂₁ and E^z₂₁ modes. If the morphological parameters of the LNOI rib waveguide make the lateral leakage happen and the leakage loss is larger than a certain level, such as 1 dB/mm used in this paper, then single-mode propagation in rib waveguide can be considered to have been achieved. Such a loss level is need to determine whether the losses of the higher order modes are large enough to make them disappear completely after propagating a certain length inside the waveguide. The loss level can be adjusted in accordance with the specific applications.

The complex propagation constant β of the ridge mode in the LNOI rib waveguides can be calculated with FEM by applying perfectly matched layer (PML) boundaries, and then the lateral leakage loss can be obtained by [29]

4. Results and Discussion

In this section, the SMCs for the LNOI rib waveguides are investigated by combining modal dispersion curves with the lateral leakage losses of the E^z₂₁ and E^x₂₁ modes. As aforementioned, the lateral leakages of the ridge modes E^z₂₁ and E^x₂₁ are related to the effective indices of the fundamental TE₀ and TM₀ slab modes, respectively. In view of this, the dispersion curves of the TE₀ and TM₀ modes of the LNOI slab waveguides were calculated with FEM at 1550 nm wavelength for different h_s and fixed h_c= 0, 200, and 400 nm, respectively. The calculated results are shown in Fig. 3. It can be seen that for different h_c, the two polarizations have almost the same effective index around h_s= 660 nm, and for the h_s less (larger) than this value the effective index of the TM₀ is less (larger) than that of the TE₀. Thus, for the LNOI rib waveguides formed with h > 660 nm, both lateral leakage from the E^z₂₁ mode to the TM₀ mode and from the E^x₂₁ mode to the TE₀ mode are possible, but which happens depends on the etch depth h_e. However, for the LNOI rib waveguides formed with h < 660 nm, only lateral leakage from the E^x₂₁ mode to the TE₀ happens. In addition, from Fig. 3, for the cases of h_s > 660 nm, three curves of the TM₀ mode corresponding to h_c= 0, 200, and 400 nm are very close to one another, and for the cases of h_s < 660 nm, three curves of the TE₀ mode also show similar feature, indicating that the cladding thickness h_c only has a slight impact on the SMCs.

 

Fig. 3. Effective indices of the TE₀ and TM₀ modes of the LNOI slab waveguides as a function of the slab thickness h_s at different h_c = 0, 200, 400 nm, respectively.

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In consideration of the impact of the slab thickness h_s on the lateral leakage, we select three typical LN film thicknesses of h = 400, 600, and 900 nm as examples to investigate the SMCs of the LNOI rib waveguides under the condition of h_c= 200 nm, θ = 22°, and λ = 1550 nm. Obviously, for the LNOI rib waveguides with h_e < h_emax, the achievement of single-mode propagation depends mainly on the ridge width w. Thus, for the sake of clarity, we define w(h_e) as the critical width to describe the SMCs of the LNOI rib waveguides having etch depth h_e. Here w(h_e) corresponds to the ridge width with which both the E^x₂₁ and E^z₂₁ modes are leaky and the corresponding leakage loss is larger than 1 dB/mm.

For the case of h = 900 nm, h_emax are 626 and 546 nm for the TE and TM polarizations, respectively. Here two cases of h_e = 540 and 90 nm are presented to illustrate how to get w(h_e). For the case of h_e= 540 nm, the dispersion curves of the ridge and slab modes involved are calculated at different w from 0.42 µm to 3.0 µm and shown in Fig. 4(a). Meanwhile, the leakage losses of the E^z₂₁ and E^x₂₁ modes are also calculated at different w from 0.42 µm to 1.0 µm and shown in Fig. 4(b). It needs to be pointed out that although the LNOI rib waveguide in this case accommodate the E^z₁₂ and E^x₁₂ modes, both have leakage losses much larger than 1 dB/mm (not shown in Fig. 4(b)) and, hence, are not taken into consideration here. From Fig. 4(a), the E^z₂₁ mode is cut-off at w < w_z = 0.74 µm, i.e., leaks to the same polarized TE₀ mode, as indicated by the mode field distribution of the E^z₂₁ mode shown in the inset of Fig. 4(a), while the E^x₂₁ mode is cut-off at w < w_x = 0.42 µm, i.e., leaks to the same polarized TM₀ mode. Therefore, if only the leakage to the same polarized modes are taken into consideration, the SMC require w < w_x (= 0.42 µm). However, when the leakage to the orthogonally polarized modes, i.e, the leakage from the E^x₂₁ mode to the TE₀ mode, are taken into consideration, the SMC will be determined by the cut-off width of the E^x₂₁ mode, and the SMW width will be enlarged to w_z (= 0.74 µm), as shown in Fig. 4(a). Further, at w = w_z = 0.74 µm, the effective index of the E^x₂₁ mode is smaller than that of the TE₀ mode as shown in Fig. 4(a), meeting the cut off condition, but the E^x₂₁ mode has a quite low loss of 3.27×10⁻⁴ dB/mm at this width because w_z is very close to the magic width w_m (= 0.763 µm) of the E^x₂₁ mode, as shown in the inset of Fig. 4(b). Thus, to achieve the cut-off of the E^x₂₁ mode based on a larger loss of more than 1 dB/mm, the width w should be smaller than w_x1 (= 0.66 µm). At last, based on the above facts, w(h_e = 540 nm) = 0.66 µm can be obtained.

 

Fig. 4. Calculated (a) and (c) modal dispersion curves, (b) and (d) modal leakage losses at different w, (a) and (b) h_e= 540 nm, (c) and (d) h_e= 90 nm, as well as (e) w(h_e) and the fitting curve of w(h_e) at different h_e, of the LNOI rib waveguides formed with 900-nm thick LN film at θ = 22°, h_c = 200 nm, and λ = 1550 nm.

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Next, for the case of h_e= 90 nm, the calculated dispersion curves and leakage losses of the modes involved are shown in Figs. 4(c) and 4(d), respectively. Over the range from 0.5 µm to 3.0 µm, the E^z₂₁ mode is cut-off and the E^z₁₁ mode leaks to the TM₀ slab mode as shown in Fig. 4(c). Meanwhile, the E^x₂₁ mode is cut-off when w < w_x =2.42 µm and its leakage loss is far less than the given level of 1 dB/mm when w > w_x, as show in Fig. 4(d). In view of these facts, w(h_e= 90 nm) = 2.42 µm can be obtained. Note that although the E^z₁₁ mode is leaky as shown in Fig. 4(c), it has different leakage losses from 0.15 to 1.68 dB/mm over the width range from 0.5 to 2.42 µm due to the magic width, and at some special width, such as around 0.5 µm as shown in Fig. 4(d), the propagation of the E^z₁₁ mode with a quite low loss is possible. Thus, the LNOI SMW at these widths can support both the E^z₁₁ and E^x₁₁ modes propagation with low loss.

With the same method, w(h_e) for other etch depth h_e can be obtained, and the fitting curve formed by all of these w(h_e) that separates multi-mode region and single-mode region, as shown in Fig. 4(e), can be obtained. Obviously, below each w(h_e), there are a group of SMWs that have the same h_e but different and smaller w.

Next, for the case of h = 400 nm, h_e= 40 nm is selected as an example and the calculated dispersion curves and leakage losses of the modes involved are shown in Figs. 5(a) and 5(b), respectively. It can be seen that if only the leakage to the same polarized modes are taken into consideration, the SMC require w < w_x (= 1.72 µm). However, in consideration that over the range from 0.5 µm to 3 µm, the E^x₂₁ modes leaks to the orthogonal TE₀ slab mode, the SMW width can be enlarged to w_z (= 2.52 µm), corresponding to the cut-off width of the E^z₂₁ mode, as shown in Fig. 5(a). At the same time, the leakage losses of the E^x₂₁ mode are greater than 1 dB/mm when w < w_x1 (= 2.8 µm) as shown in Fig. 5(b). Thus, w(h_e= 40 nm) = 2.52 µm can be obtained. It needs to be pointed out here that over the whole single-mode region from 0.5 µm to 2.52 µm, the E^x₁₁ mode leaks to the TE₀ mode with different leakage losses at different width w and has a maximum loss at w = 0.9 µm and a minimum loss at w = 2.3 µm, i.e., the magic width of the E^x₁₁ mode. Based on these facts, the LNOI SMW supporting both the E^z₁₁ and E^x₁₁ modes or only supporting the E^z₁₁ mode can be obtained by choosing suitable ridge width w.

 

Fig. 5. Calculated (a) and (d) modal dispersion curves, (b) and (e) modal leakage losses at different w, (a) and (b) h_e= 40 nm, (d) and (e) h_e= 20 nm, as well as (c) w(h_e) and the fitting curve of w(h_e) at different h_e, of the LNOI rib waveguides formed with 400-nm thick LN film at θ = 22°, h_c = 200 nm, and λ = 1550 nm.

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Further, by applying the above method, we calculated w(h_e) and investigated the SMCs for other etch depth h_e for this case of h = 400 nm. The results are shown in Fig. 5(c), which exbibits unique SMC characteristics, that is the w(h_e) decrease firstly and then increase in the range of h_e = 20 to 36 nm. The reason that causes this phenomenon can be accounted by an example of h_e = 20 nm. Figures 5(d) and 5(e) show respectively the calculated dispersion curves and leakage losses of the modes involved at h_e = 20 nm. It can be seen that the E^x₂₁ mode leaks to the TE₀ slab mode in the range of w =2.61 to 4.0 µm, as shown in Fig. 5(d). but the leakage losses are between ∼0 dB/mm (at magic width w_m = 3.25 µm) and 0.66 dB/mm (at w_x = 2.61µm). In view of the fact that these losses are less than the given level of 1 dB/mm, the LNOI rib waveguide is multi-mode in this region in accordance with the previous stipulation. Meanwhile, from Fig. 5(d), E^x₂₁ mode will be cut-off when w < 2.61 µm, therefore, w(h_e = 20 nm) should be somewhat smaller than 2.61 µm for this situation. Thus we set w(h_e = 20 nm) = 2.6 µm here.

For the case of h = 600 nm, the calculated dispersion curves and leakage losses of the modes involved at h_e= 240 nm are shown in Figs. 6(a) and 6(b), respectively. As shown in Fig. 6(a), the leakage to the same polarized modes require w < w_x (= 0.72 µm), while the leakage from the E^x₂₁ mode to the orthogonally polarized TE₀ mode leads to w < w_z (= 0.94 µm). If the given loss level of 1 dB/mm is taken into account, then from the leakage losses of the E^z₂₁ and E^x₂₁ modes shown in Fig. 6(b), w(h_e= 240 nm) = w_z1= 1.09 µm can be obtained. However, because the magic width of the E^x₂₁ mode is at w_m = 0.77 µm and the leakage losses of the E^x₂₁ mode are smaller than 1 dB/mm from w_x (= 0.72 µm) to w_x1 (= 0.86 µm), as shown in Fig. 6(b) and its inset, the range from w = 0.72 µm to 0.86 µm need to be excluded from single-mode region. With the same method, we can get w(h_e) and the fitting curve of w(h_e) at different h_e from 30 to 450 nm, above the fitting curve of w(h_e) is multi-mode region, while below it is single-mode region except a belt of multi-mode region, as shown in Fig. 6(c).

 

Fig. 6. Calculated (a) modal dispersion curves, (b) modal leakage losses at different w, and (c) w(h_e) and the fitting curve of w(h_e) at different h_e of the LNOI rib waveguides formed with 600-nm thick LN film at θ = 22°, h_c = 200 nm, and λ = 1550 nm.

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The above three examples indicate that because of the lateral leakage loss and the magic width the LNOI rib waveguides have unique and complex SMCs, determined by the LN thin film thickness h, the etch depth h_e, and the ridge width w. It is difficult to describe SMCs of the LNOI rib waveguides by using an inequality, as used for SOI waveguides in [23–25]. Thus, to obtain the SMC of the LNOI rib waveguide with given parameters, one need to investigate the modal dispersion curves and lateral leakage losses of the waveguide specifically.

At last, for the case of h = 600 nm, the impact of parameters h_c, θ, and λ on the w(h_e) of the LNOI rib waveguides are also investigated with the above method over the etch depth range from 30 to 450 nm. The calculated values of w(h_e) for the cases of h_c = 0 (no SiO₂ cover layer), 200, and 400 nm at fixed θ = 22° and λ=1550 nm are shown in Fig. 7(a), which indicates that with the increase of h_e, the impact of the thickness h_c of the SiO₂ cover layer on the w(h_e) becomes gradually noteworthy. This is because the increases in the h_e and h_c lead to the decline in the confinement of the waveguide for the light wave. Meanwhile, the calculated w(h_e) for the cases of θ = 18°, 20° and 22° at h_c = 200 nm and λ=1550 nm indicates that these three cases of θ have almost the same impact on the SMCs, as shown in Fig. 7(b). Similarly, the calculated w(h_e) for the cases of λ=1530, 1550, 1610 nm at fixed h_c = 200 nm and θ = 22° indicate that the above operation wavelengths only have a slight impact on the SMCs of the waveguide, as shown in Fig. 7(c).

 

Fig. 7. Calculated w(h_e) for the LNOI rib waveguides formed with 600-nm thick LN film for different cases, (a) h_c = 0, 200, 400 nm, respectively, and θ = 22°, λ = 1550 nm, (b) θ = 18°, 20°, 22°, respectively, and h_c= 200 nm, λ = 1550 nm, (c) λ = 1530, 1550, 1610 nm, respectively, and θ = 22°, h_c= 200 nm.

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5. Conclusion

We have investigated and obtained the SMCs of the LNOI rib waveguides formed with three typical-thickness LN film by combining the modal dispersion curves with the lateral leakage losses of the first order modes. Our results indicate that the single-mode propagation in shallowly etched LNOI rib waveguides can be realized by controlling the waveguide parameters precisely so as to achieve the leakage of the first order ridge modes into the two side slabs. And the SMCs of these LNOI rib waveguides are mainly determined by the thickness of the LN thin film, the etch depth, and the width of the ridge, while the thickness of the cladding layer, the side wall angle, and the operation wavelength within a certain range only have slight impact on the SMCs. Moreover, it is quite difficult to describe the SMCs of these LNOI rib waveguides with a mathematical equation because they exhibit quite complex modal dispersions and leakage characteristics determined by the strong birefringence of LN and the specific structural parameters of the waveguide. One has to get the SMCs of each LNOI rib waveguide by investigating its modal dispersions and leakage losses. We believe that the above method and results can provide a guidance in designing low-loss LNOI SMW and high-performance PICs.