Mode hybridization analysis in thin film lithium niobate strip multimode waveguides

Mode hybridization phenomenon in air-cladded X-cut Y-propagating and Z-propagating thin film lithium niobate strip multimode waveguides is numerically studied and a mathematical relation between structural parameters leading to hybrid modes is formulated. Dependence of hybrid modes on waveguide dimensions, sidewall angles and wavelength is also analyzed. The results obtained are used to design lithium niobate on insulator (LNOI) taper for converting fundamental TM mode to higher order TE mode, and an optimum length for achieving a high conversion efficiency of 99.5% is evaluated. Birefringent Y-propagating LN and isotropic Z-propagating LN tapers are compared in terms of length, figures of merit, and fabrication tolerance. Tapers exhibit a broad bandwidth of 200 nm with an extinction ratio less than − 18 dB. The results of mode hybridization analysis are useful in design optimization of adiabatic tapers, tunable time delays, optical interconnects, mode converters and demultiplexers for mode division multiplexing (MDM) applications.

Photonic devices on high-contrast platforms have guided modes that are inherently hybrid in nature 1 . They have all six field components (Ex, Ey, Ez, Hx, Hy, Hz) to be non-zero, unlike the planar waveguides which have pure transverse electric (TE) and transverse magnetic (TM) modes. However, there is a dominant E and a dominant H component for all the modes in general. Such modes are referred to as quasi-transverse electric (qTE) and quasi-transverse magnetic (qTM) modes. If high-contrast waveguides like silicon-on-insulator (SOI) and Lithium niobate on insulator (LNOI) have vertical asymmetry of index and/or horizontal asymmetry in structure, then there are some special modes where the dominant and non-dominant field components are almost equal 2,3 . Such modes cannot be classified as qTE or qTM modes and are called as "Hybrid modes". This phenomenon of mode hybridization happens when two orthogonally polarized modes with similar effective indices couple to each other 4 . Mode hybridization regions are used in designing tapers for TM-TE higher order mode conversions [5][6][7][8][9] in polarization splitter-rotators. These components are essential for the implementation of polarization diversity schemes, which circumvent the problem of polarization-dependent losses and dispersion in high-contrast waveguides 10 . They are also key components in optoelectronic integrated circuits for coherent optical system applications 11 . Hybrid modes are also reported to exhibit large group velocity dispersion compared to qTE and qTM modes, which could be useful in tunable time delays and optical signal processing applications 12 . While hybridization phenomenon is useful for mode converters and delays, the same could cause high-crosstalk in mode division multiplexing (MDM) applications [13][14][15] . Hybrid modes in SOI nanowires and LNOI are observed and reported in 4,9 . However, mathematical relation between structural parameters of waveguides that lead to hybrid modes and, its dependence on operating wavelength have not been explored yet.
In this paper, a complete analysis of mode hybridizations has been carried out in thin film lithium niobate (LN) strip multimode waveguides on X-cut Y-propagating and Z-propagating crystals. LNOI is an excellent platform for integrated photonics because of its broad transparency range from 350 to 5200 nm, strong electrooptic, acousto-optic, and thermo-optic effects which make it an excellent choice for active and non-linear devices as well [16][17][18][19] . Further, an attempt has been made to formulate a mathematical relation between structural parameters of waveguides that lead to hybrid modes. Dependence of this phenomenon on the operating wavelength and sidewall angles of waveguides are also studied. Based on the results obtained, tapers are designed on both birefringent and isotropic LN waveguides to convert TM 0 to TE 1 mode and, an optimum length is determined to achieve a large conversion efficiency of 99.5%. Device length and fabrication tolerance of mode converters www.nature.com/scientificreports/ that strongly depend on shift in hybrid points resulting from fabrication errors like sidewall angles and width deviations are also studied in detail.
The cross-section of the waveguide analyzed is shown in Fig. 1a, along with the chosen orientation for LN principal dielectric axes with respect to the waveguide. The dispersive nature of LN is modeled using Sellmeier equation 20 . Multimode waveguides are designed at a telecommunication wavelength λ of 1550 nm to support eight modes that include qTE, qTM, and hybrid modes at some special dimensions. The SiO 2 buried oxide (BOX) layer is generally thick enough (2-3 µm) to avoid any field penetration into the underlying substrate (not shown here). Sample results for variation of effective index with waveguide width at a height of 0.4 µm is shown in Fig. 1b. The anticrossings marked in black dotted circles, where the effective indices of two orthogonally polarized modes become almost equal indicate hybrid modes. Field profiles of hybrid modes between TM 0 and TE 1 at hybrid point A are shown in Fig. 1c,d.
The horizontal (E h ) and vertical (E v ) electric field components for hybrid mode at point A are shown in Fig. 2 and is observed that E h and E v are almost equal. The phenomenon of hybridization can occur due to perturbation in the form of asymmetry in waveguide geometry 11 , or dielectric perturbation with the upper and the lower clad having different refractive indices 3 . The two hybrid super modes couple with each other at the anticrossing points, with a coupling factor (k) given by Eq. (1), where β S1 and β S2 are the propagation constants of hybrid modes at the anticrossing point.
In X-cut Y-propagating LN, qTE modes see an extraordinary index (n e ), while qTM modes see an ordinary index (n o ), which results in birefringence or velocity difference between these modes. In X-cut Z-propagating LN, with Z being the optic axis of the crystal, both qTE and qTM modes see an ordinary index (n o ), which is when γ TE is close to 1, it represents a qTE mode, while a value close to 0 represents a qTM mode. TE polarization fraction of 0.5 represents strong hybridness of modes. In this analysis, a mode is considered hybrid if 0.4 < γ TE < 0.6. It is interesting to obtain physical intuitive insights into the nature of the hybrid modes as explained next. In order to excite hybrid modes, the non-dominant (minor) field component needs to be enhanced. Existing approaches to increase the non-dominant field component include the following: setting the waveguide width equal to waveguide height 1 , introducing bends in the waveguide 21 , using index asymmetry in the vertical direction 3 , and using structural asymmetry like one sided angled side wall 11 . In this work, index asymmetry is present in the vertical direction as the strip waveguide has the top air clad and the SiO 2 bottom clad. This can be treated as a buried waveguide (same top and bottom clad) with dielectric perturbation Δε.
Mode coupling between two modes occurs when the modes have similar effective indices, significant overlap between the mode profiles, and same symmetry for the individual field components. These conditions lead to a coupling factor 22 defined by Eq. (3), where indices i and j represent the modes being coupled.
Consider the example of a mode pair TM 1 and TE 2 , with the waveguide width 2.2 µm and height of 0.35 µm on X-cut Y-propagating LN. These two modes become fully hybrid supermodes when the width is 2.135 µm. The individual field components of TM 1, TE 1 , and TE 2 modes in this waveguide are shown in Fig. 3a-c. It can be observed that the respective field components of TM 1 and TE 2 have same symmetry. Hence for these two modes, the scalar products in Eq. (4) have an even symmetry with respect to y-axis (Fig. 3d), resulting in a www.nature.com/scientificreports/ non-zero coupling factor k. Next pair of modes TM 1 and TE 1 can also be observed for the possibility of mode coupling. It is seen that coupling factor k vanishes for this pair, as the scalar products in Eq. (4) exhibit odd symmetry with respect to y-axis about the waveguide center ( Fig. 3e). While perturbation theory can be used to understand the physics of hybrid modes, supermode theory should be used to find accurate coupling factors for larger values of Δε.
With perturbation in the form of index asymmetry along vertical direction, this structure can lead to hybrids between TM i and TE i+2 k+1 , where i and k are integers. A structure with perturbation in the horizontal direction 23 as in a L-waveguide can lead to hybrids between TM i and TE i+2 k as well. Another important criterion for mode coupling is the overlap factor 24 which represents the fraction of electromagnetic fields overlapping between the field profiles of two modes as defined in Eq. (5). For the pair of modes where the coupling was possible (TM i and TE i+2 k+1 ) in this waveguide structure, overlap factor was of the order of 10 −5 , while it was 10 −30 for the hybridization-incompatible mode pairs.

Results and discussion: mode hybridization in LN strip waveguides
The width 'w' of the strip waveguide is varied up to 4 µm and height 'h' between 0.3 and 0.6 µm to observe hybrid modes. Maximum height is restricted to 0.6 µm so as to have one peak in the vertical direction, with higher order modes supported along the horizontal direction. The dependence of TE polarization fraction on width and height of the X-cut Y-propagating LNOI waveguide for the first five modes at a wavelength of 1550 nm are shown in Fig. 4. The color bar shows that a qTE mode is represented by 1 (maroon at the top), qTM mode by 0 (navy blue at the bottom), and hybrid modes by cyan to yellow corresponding to 0.4 < γ TE < 0.6. Green on the color bar indicates strongest hybridness. Mode-1 is the fundamental TE mode and remains so over the entire range of width and height variation. This is indicated by a maroon block in Fig. 4a. Modes 2 and 3 represent TM 0 and TE 1 just before the beginning of hybridization. White-striped regions in Fig. 4c-e represent cutoff regions for the corresponding modes. TE polarization fraction plots for modes 3, 4, and 5 in Fig. 4c-e show alternate maroon and blue strips apart from the hybridization curve (in green-yellow), because of change in the order of modes as the waveguide dimensions change. This is explained for mode-4 in Table 1, where different sample pairs of (w, h) are considered in each region.
At point A, fourth mode is TE 1 which is indicated by γ TE in maroon color. At point B in white-striped region, mode-4 is cutoff, as its effective index is below the BOX layer index of 1.443. In the blue region where point C is located, mode-4 is TM 1 indicated by γ TE close to 0. On the strip which is green-yellow, the mode is hybrid (between TE 2 and TM 1 ) which is indicated by point D. In the maroon strip where point E is located, mode-4 is TE 2 . Next region is again blue (sample point H), and mode-4 is TM 0 . The second green-yellow curve indicates hybrid between modes 4 and 5 (TE 3 and TM 0 ) and a sample point I is shown on this curve. Finally, γ TE is maroon  www.nature.com/scientificreports/ in the patch where point J is located, and mode-4 is TE 3 here. Similar numerical analysis is carried out for X-cut Z-propagating LN waveguides and was observed that the hybrid region is narrow and steeper compared to the Y-propagating case. The points (w, h) that lead to TE polarization fraction in the range of 0.4 to 0.6 are fitted with a Matlab curve fitting tool, shown in Fig. 5. Hybridization curves for Y-propagating LN are fitted with a two-term power series model that uses non-linear least-squares algorithm. The expressions obtained for representatives of the first two hybrid curves are given by Eq. (6), while the values of coefficients with 95% confidence bounds are specified in Eqs. (7) and (8). The goodness of the fits was also evaluated and was found that the root mean squared error is no more than 0.0073.
The coefficients for the first curve representing hybrid between TM 0 and TE 1 are, The coefficients for the second curve representing hybrid between TE 2 and TM 1 are, Hybridization curves for Z-propagating LN are fitted with a linear polynomial of third order as shown in Fig. 5, expressions obtained for representatives of the first two hybrid curves are given by Eq. (9) and the coefficients are specified in Eq. (10), Coefficients (with 95% confidence bounds):  (9) h = f (w) = aw 3 + bw 2 + cw + d Table 1. Sample points from Fig. 4(d) of mode-4 indicating transition from qTE to qTM and then to hybrid mode.

Sample points (w,h) in μm
Mode-   www.nature.com/scientificreports/ rence of hybrid mode between TM 0 and TE 1 . At this particular height, hybrid region extended over a broadband of almost 50 nm as shown in Fig. 8a. However, there was no specific trend observed as the height was varied and other hybrid modes were evaluated.

Effect of angled sidewalls.
Lithium niobate is a difficult material to etch and angled sidewalls are a common feature of fabricated waveguides 25 . As a result, the points of hybridization (w H , h H ) tend to shift towards different dimensions compared to waveguides with vertical sidewalls. Such shift in hybrid point of a particular mode was evaluated in 26 . In this work, the shift has been found for all hybrid points in the specified range of dimensions, at a side wall angle 'θ' of 20 o and − 20° as shown in Figs. 9 and 10. As the sidewall angle increased at a fixed height, hybrid points shifted towards lower widths; at a fixed width, hybrid points shifted towards larger heights. Here, the top width is considered to be fixed, and the trapezoidal cross section of waveguide is formed by increase or decrease of bottom width. Increased bottom width leads to a positive sidewall angle and reduced bottom width leads to a negative sidewall angle. Angled sidewalls also increase the asymmetry of the waveguide and hence lead to stronger hybridness. This can be inferred from the increased density of points in left curve at θ = 20°, in each pair of the curves of Figs. 9 and 10. However, negative sidewall angle tends to reduce the hybrid points in both Y-and Z-propagating waveguides for heights lower than 0.4 µm. The effect of variation in wavelength on hybrid points in a waveguide with a sidewall angle of 20 o is shown in Fig. 8b, which indicates that band of hybrid points (cyan to yellow) shifted towards higher wavelengths. Recently, mode hybridization between fundamental TE and fundamental TM mode has been investigated in X-cut LNOI ridge waveguides, using its material birefringence 27 . Present work reports hybridization between TM i and TE i+2 k+1 (i and k are integers), which can be extended to hybrids between TM i and TE i+2 k by introducing bends 27 or using perturbation in the horizontal direction 23 .

Mode hybridization in SOI strip waveguides.
In order to compare the hybrid regions with that of LNOI, another popular platform of SOI is analyzed in this section. Silicon is isotropic and lacks intrinsic birefringence as opposed to LN. Mode hybridization regions are much narrower in SOI waveguides compared to the counterparts on X-cut LNOI. TE polarization plot for hybrid between TM 0 and TE 1 is shown in Fig. 11a and hybrid points between TM 0 -TE 1 (modes 2 and 3), and TM 1 -TE 2 (modes 4 and 5) are represented in Fig. 11b. The dense group of points forming a triangular shape on the top left corner majorly results from square-like core geometry. In this region, height of the waveguide approaches its width and results in modes with the horizontal and vertical electric field components becoming almost equal. Also, nodes increased to two in the vertical direction beyond a height of 270 nm.

Design of tapers as mode converters based on mode hybridization. Mode converters are impor-
tant for the implementation of mode division multiplexing schemes and polarization-independent devices. In this section, mode converters using tapers that utilize mode hybridization phenomena are designed on X-cut LNOI and analyzed numerically. The end widths of the taper are chosen such that w 1 < w H < w 2 , where w H is the waveguide width for a specific mode hybridization region, and w 1 , w 2 are taper end widths 6 . tM 0 to te 1 mode converter. A taper is designed for conversion of TM 0 to TE 1 with a height of 0.4 µm, w 1 < 1.725 µm < w 2 (for Y-propagating) and w 1 < 1.494 µm < w 2 (for Z-propagating) at a wavelength of 1550 nm,  www.nature.com/scientificreports/ whose schematic is shown in Fig. 12a. This choice of hybrid point (1.725 µm, 0.4 µm and 1.494 µm, 0.4 µm) was made from the data available in Fig. 5. Prominent figures of merit for this device are mode conversion efficiency (MCE), bandwidth (BW) and extinction ratio (ER). Mode conversion efficiency and extinction ratio 28 are defined in Eqs. (11) and (12) and are plotted as a function of operating wavelength in Fig. 12b. For an input TM mode, To achieve a mode conversion efficiency of 99.5%, length of the taper was obtained as 477 µm and 707 µm in Y-propagating and Z-propagating waveguides respectively. MCE variation along the length of the taper is depicted in Fig. 13, and it is noticed that taper length (L T ) that achieves required MCE comes down drastically with the increase in positive sidewall angle, due to stronger hybridization and coupling. From Figs. 9 and 10, it is known that hybrid points shift as the sidewall angle varies, and the change can be as large as 600 nm for a sidewall angle of 20° compared to vertical sidewall in both Y-propagating and Z-propagating LN. If the taper widths are chosen without considering shift in the hybrid points, there is an unnecessary increase in taper length to achieve the same efficiency, as tabulated in Table 2 (for 15° and 20°). Even small negative sidewall angle of 5° can almost double the taper length for desired efficiency. www.nature.com/scientificreports/ The device lengths are large compared to polarization converters on SOI counterparts, which has a length less than 100 µm 29 . This is obvious because of the large index contrast between core and clad in SOI waveguides. Very short device length of 15 µm is also reported 30 on SOI platform that uses bilevel taper. As the fabrication technology of LN matures and losses come down, such designs can be attempted in LN as well. Cubic curving tapers 5 could also be implemented to reduce the taper length. Moreover, LN comes with its own advantages of broad bandwidth of ~ 200 nm as compared to 100 nm in SOI 29 , and large fabrication tolerance as shown in Fig. 14. It is observed from Fig. 14a that MCE is stable for both Y-propagating and Z-propagating LN over a large width deviation (Δw) of − 100 to 100 nm. The difference between w 1 and w 2 is chosen to be 300 nm, and the hybrid width will remain well within this range even with Δw as large as ± 100 nm. It was observed that a smaller difference between w 1 and w 2 (δw) could lead to hybrid mode at the input end itself, as hybrid points are wide-spread in X-cut LNOI. Smaller δw can as well result in hybrid mode at the output if Δw is large due to fabrication errors. Figure 14a also shows the TE polarization fraction (γ TE ), that is indicative of purity of output TE 1 mode. At a Δw of − 100 nm, Y-propagating device has γ TE of 0.77, while Z-propagating device has 0.89 for TE 1 mode. Tolerance to sidewall angle is illustrated in Fig. 14b, and it is observed that both the configurations are stable for positive angles, but the effciency is highly sensitive to negative angles. To understand this better, dependence of TE polarization fraction of hybrid modes with the sidewall angle was evaluated at a fixed height as depicted in Fig. 15. Steep changes in TE polarization fraction at sidewall angles of − 5° and − 10° leaves a narrow choice of widths to choose from, for hybrid modes. This explains the reason for degradation of efficiency in Fig. 14b beyond an angle of − 4°.
Simulations were performed using 3D FDTD method to accurately model the propagation of light through Y-propagating taper of length 477 µm. With TM 0 mode launched through the input waveguide, TE 1 was obtained at the output waveguide. Mode profiles at the intermediate points along the propagation direction is shown in Fig. 16. Long tapers which do not support mode conversion can also be designed using the results from

Conclusion
Mode hybridization in air-cladded strip multimode waveguides on X-cut LNOI and SOI is analyzed numerically to obtain mathematical relation between structural parameters leading to hybrid modes. Relation between width and height that lead to TE polarization fraction of 0.4-0.6 (which is a measure of hybridization) is found to be a two-term Power series in Y-propagating and 3 rd order polynomial in Z-propagating LN strip waveguides. Table 2. Taper length for a MCE of 99.5% with different θ at w 2 -w 1 = 0.3 μm.

Sidewall angle θ ( o )
L T with shifts considered (μm) L T with shifts NOT considered (μm) L T with shifts considered (μm) L T with shifts NOT considered (μm) 0  477  -707  -5  328  304  329  342   10  216  181  197  185   15  135  208  142  185   20  102  212  104  www.nature.com/scientificreports/ Regions of hybridization were found to be wider in Y-propagating LN compared to Z-propagating LNOI and SOI counterparts. Hybridization regions occur at lower widths in the isotropic Z-propagating configuration comparaed to the birefringent Y-propagating configuration. Hybrid points tend to shift as the sidewall angle of the waveguide increased, and it could be as large as ~ 600 nm for a sidewall angle of 20° in comparison with vertical sidewalls. Waveguides with larger positive sidewall angles exhibited denser regions of hybridization as a result of enhanced asymmetry and higher coupling factors. Hybrid mode regions were utilized to design tapers that convert TM 0 to TE 1. To achieve MCE of 99.5%, length of the taper required is 477 µm in Y-propagating LN and 707 µm in Z-propagating LN, when the sidewalls are vertical. Tapers designed with the incorporation of shift due to angled sidewalls (> 10 o ) were found to achieve the specified MCE at a shorter length (reduced by 100 μm in Y-propagating and 84 μm in Z-propagating when θ = 20°). Even small negative sidewall angle of 5° can almost double the taper length for desired efficiency. Tapers on LN are found to exhibit a broad BW of ~ 260 nm with MCE greater than 96%, ~ 200 nm with extinction ratio less than − 18 dB; large fabrication tolerance of ± 100 nm and good extinction ratio of − 23 dB at design wavelength. These results of hybrid mode analysis are crucial for the design of tapers as mode converters, polarization rotators, multimode photonics, tunable time delays and optical signal processing applications. The results can as well be used to avoid hybrid modes, while designing multimode waveguides as optical interconnects and MDM demultiplexers in order to avoid unwanted mode coupling and crosstalk.

Methods
TE polarization fraction values and effective indices of thin film lithium niobate waveguides are obtained using full-vectorial finite-difference Eigen mode solver from a commercial simulation tool (Lumerical). Sellmeier equation is used to model the dispersive nature of LN with diagonal anisotropy being incorporated in the material. LN is a negative uniaxial material, with the ordinary index n o = 2.21116 and extraordinary index n e = 2.13755 at a wavelength of 1550 nm. Eigen mode expansion method (EME) is used to analyze mode conversion in the taper. EME is a frequency-domain method for solving Maxwell's equations that is ideal for simulating light propagation over long distances. This is chosen over FDTD for the benefits of speed and low computational cost. Simulations were also performed using 3D FDTD method to accurately model the propagation of light through taper that achieves desired mode conversion.