Impact of Non-Vertical Sidewalls on Bandgap Properties of Lithium Niobate Photonic Crystals

Abstract

We investigate the influence of non-vertical sidewall angles on the band structure characteristics of thin-film lithium niobate (LN) photonic crystals (PhCs), considering both suspended LN membranes and LN on insulator (LNOI) configurations. Utilizing the gap-to-midgap ratio as a figure-of-merit, we observe a 34% reduction for a suspended LN PhC with 60° sidewall angles compared to the one with vertical sidewalls and a more substantial 73% reduction for LNOI PhCs with 70° sidewall angles. We address this challenge through the optimization of geometrical parameters of PhC unit cells with non-vertical sidewalls, taking fabrication feasibility into account. Our work provides a design guideline for the development of realistic LN PhC devices for future large-scale LN photonic circuits.

1. Introduction

Lithium niobate (LN) stands out as a highly promising material platform for integrated photonics [1,2], which possesses a compelling combination of characteristics, including a broad transparency window, large electro-optic and nonlinear effects, and low absorption [3]. In recent years, the substantial development in thin-film LN photonics [4,5,6] has been noteworthy. This has led to the demonstration of low passive waveguide loss down to dB/m [7,8] and fast active electro-optical (EO) modulators up to 100 GHz [9,10,11], underscoring the great potential of LN photonics for large-scale and high-performance photonic integrated circuits (PICs).

Despite significant progress in LN photonics, a notable challenge in constructing scalable LN circuits lies in the substantial device size, which partially stems from the preference for ridge waveguides in many cases due to the need for electrodes in EO modulators [9,10,11,12], the difficulty in etching LN films [13], and the requirements for low-loss waveguides [7,8]. In this context, photonic crystal (PhC) structures, guiding optical modes through the photonic bandgap rather than total internal reflection, offer strong light confinement and would thus be beneficial for shrinking the devices. LN PhC structures on thin-film platforms are preferred over conventional thin-film lithium niobate photonic structures primarily due to their exceptional capability to control and tightly confine light at the sub-wavelength scale [14,15]. This strong confinement leads to a significantly enhanced electro-optic interaction and tuning efficiency, enabling devices with a tiny electro-optic modal volume and consequently high energy efficiency. Furthermore, this tight confinement can result in a broad modulation bandwidth that can reach the photon-lifetime limit of the cavity, making photonic crystal structures highly advantageous for miniaturized, high-speed, and energy-efficient integrated photonic circuits [16]. Representative works on LN platforms include PhC cavities on a wavelength scale [16,17], sharp waveguides bends [18], and modulators with enhanced EO interactions [14,19,20].

Most LN fabrication techniques yield a sidewall angle ($\theta$) less than 90°, including diamond dicing [21,22], chemical mechanical polishing [8], a focused ion beam [23,24,25,26], and dry etching [7,9,10,11,12,14,16,17,18,19].

The sidewall angle typically falls within the range of 40° to 80° [1,11,16,17,18,27], depending on the specific fabrication methods. This often results in cone-like holes in fabricated LN PhCs, as illustrated in Figure 1a,b. To address this, the limitation of slanted sidewalls has been considered in the design of the PhCs [17]. Limited aspect ratios due to the conical shapes of holes are typically observed when PhC structures are produced in annealed proton exchanged (APE) waveguides by focused ion beam (FIB) milling [28]. In contrast, advanced dry etching techniques such as inductively coupled plasma (ICP) etching and atomic layer etching (ALE) have been explored to achieve steeper sidewalls and smoother surfaces [29], requiring precise control over plasma parameters and etching conditions. The slanted sidewalls of these holes [16,23,25,27] may lead to performance degradation of the PhCs. For PhC waveguides built from slanted holes, they may also lead to mode hybridization and mode conversion. Also, the cone-like holes may impede the fabrication of PhCs with a small lattice constant due to structural interference. However, there have been few reports on the impact of non-vertical sidewalls on LN PhC devices, and the potential mitigation of this issue with practical sidewall angles remains unexplored. Moreover, the LN PhCs with slanted sidewalls offer a novel avenue for studying bound states in the continuum (BICs), which are peculiar states enabling light trapping in thin photonic structures [30,31,32]. The slanted geometry of the etched sidewalls facilitates the creation of intrinsic BICs, which can be excited under vertical incidence conditions [33]. Thus, the role of sidewall angles in LN PhC devices is believed to be important in many practical application scenarios.

In this paper, we study the impact of sidewall angle on the bandgap characteristics for thin-film LN PhCs with a hexagonal lattice for suspended LN slabs and LN on insulator (LNOI) slabs, as shown in Figure 1a,b. It is found that the sidewall angle critically determines the bandgap of LN PhCs. In some cases, particularly for LNOI PhC devices, the bandgap, $∆\omega$, may even be closed. The gap-to-midgap ratio, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$, is utilized as a figure-of-merit [34] to evaluate the bandgap properties in PhCs with slant sidewalls. The central wavelength of the bandgap, a size-dependent characteristic, can be tuned by adjusting the main geometrical parameters of PhCs. This value exhibited a range of variation from 1338 nm to 1717 nm under different parameter modification conditions and for configurations explored in our work. For suspended LN PhCs, reducing the sidewall angle to $\theta$ = 60° results in a 34% reduction in the gap–midgap ratio compared to that with vertical sidewalls. Similarly, for LN PhCs on silica, decreasing $\theta$ to 70° leads to a 73% reduction in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$. To address this decrease, we propose compensative solutions, such as increasing the central hole radius, r_c, or reducing the lattice constant, a, and the slab thickness, h. Especially, we note that a reduction in h by 150 nm increases ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ by 34% for a suspended LN PhC with $\theta$ = 60° and h = 400 nm. Similar optimization in h for an LNOI PhC slab with $\theta$ = 70° yields a ∼200% increment. Our study establishes design guidelines for PhC devices on LN, bridging practical fabrication needs with optical performance demands. This facilitates energy-efficient, compact LN devices robust to process variations, enabling scalable photonic integrated circuits where optimized architectures align with achievable fabrication precision. The guidelines advance high-performance LN photonics by embedding manufacturability constraints directly into device design paradigms.

2. Design and Characterization

Utilizing hexagonal array of holes in LN slabs, we studied TE-like bandgap properties by changing sidewall angles. This structure was selected as the representative of all other possible structures as the bandgap size in the considered structure is a large value, which enables us to show bandgap properties controlled by sidewall angles in detail. On the other hand, considering the popularity of compact LN photonic devices based on TE mode structures, the TE-like band structure was selected as the reference for evaluation. First, we investigated semi-2D LN PhC slabs. Due to the absence of the bandgap for LN PhC slabs thinner than 1 μm with a square lattice, we focused on the hexagonal lattice. The unit cell comprises an air hole within the LN thin film for the both suspended LN in air and LNOI configurations, as shown in Figure 1a and Figure 1b, respectively. To determine the band structure within the Brillouin zone for unit cells in these LN PhCs, a 3D eigenmode solver based on the finite-element method was employed. The model was surrounded with symmetric boundaries in periodic directions and the perfectly matched layer (PML) at cladding area boundaries. Fine mesh size (as small as one-eighth of effective wavelength) was applied to the model to obtain proper accuracy in the extracted eigenmode resolution. Quasi-TE/TM modes were distinguished via analyzing electric field components. A custom algorithm was utilized for detecting TE-like bandgaps and calculating the gap-to-midgap ratio from exported eigenvalues and their corresponding dominant field component. Finally, the eigenvalues were plotted over the normalized wave vector. a, h, and r_c are adjusted so that the bandgap is around the telecom bands. Here, the slanted sidewalls impose geometrical limitations on the design of PhCs. When $\theta$ < 90°, the hole radius at the top of a unit cell, r_t, is larger than r_c, potentially causing overlap between air holes if r_t > ${\displaystyle \frac{\sqrt{3}}{2}}$a. Consequently, the geometrical parameters of unit cells (a, h, and r_c) must be carefully adjusted to avoid structural interference caused by slanted sidewalls.

The band structures for suspended LN PhC and LNOI PhC are obtained for different sidewall angles in Figure 2a and Figure 2b, respectively. Both TE-like and TM-like bands are presented. Distinct sets of parameters a, h, and r_c are employed for LN PhC and LNOI PhC to align the bandgap around the telecom bands. By setting a = 620 nm, r_c = 180 nm, and h = 300 nm in the suspended LN PhC slab, the bandgap between the two lowest TE-like bands (labeled as Band 1 and Band 2 in Figure 2a is in the range of 1350 nm to 1530 nm. The bandgap is between 1540 nm and 1630 nm for a = 580 nm, r_c = 140 nm, and h = 450 nm for the case of the LNOI PhC slab (Figure 2b). These parameters remain constant for varied $\theta$, and we note that for $\theta$ = 60°, the LNOI PhC has structural interference. These two sets of parameters are used throughout the paper unless specified otherwise.

In Figure 2, the suspended LN PhC features a large TE-like bandgap (referred to as $∆\omega$) that is desirable for building PhC waveguides or cavities. In contrast, $∆\omega$ is greatly reduced in LNOI PhC slabs due to a reduced refractive index contrast between LN and the substrate. A higher-index substrate also leads to a downward shift in the light cone, bounded by $\omega$ = ${\displaystyle \frac{kc}{n_{silica}}}$, where $n_{silica}$ is the refractive index of silica, thereby imposing constraints in the potentially achievable bandgap. In both cases, $∆\omega$ is reduced when $\theta$ decreases, as shown by the yellow dashed lines in Figure 2a,b.

The gap–midgap ratio, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$, offers an independent metric across geometric scales to quantify the performance of PhCs with different sidewall angles, where ${\omega }_m$ is the angular frequency at the midpoint of the TE-like bandgap. A small ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is generally undesirable and indicates the degraded “light-confining capability” of PhCs. ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for varied $\theta$ values is annotated in Figure 2a,b and summarized in Figure 2c, revealing a notable reduction when $\theta$ < 90°. Specifically, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ drops from 0.124 at $\theta$ = 90° to 0.082 at $\theta$ = 60° for the suspended PhC slab, marking a reduction of ∼34%. For the LNOI PhC, and ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ decreases more severely by 73% from 0.056 at $\theta$ = 90° to 0.015 at $\theta$ = 70°. In addition, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is more sensitive to sidewall angle variation when $\theta$ is small, as in Figure 2c for both PhC configurations. In the suspended case, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ decreases by around ∼14% as $\theta$ decreases by 5° from $\theta$ = 65°, whereas ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is almost unchanged for $\theta >{85}^{\circ }$. This variation in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is more significant for LNOI PhCs. ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ decreases by >45% as $\theta$ decreases by 5° from $\theta$ = 75°, while the ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is reduced by ∼10% with $\theta$ from 90° to 85°. $∆\omega$ becomes 0 for sidewall angles smaller than 65°.

The decrease in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for non-vertical sidewalls is jointly caused by a slight decrease in the eigenfrequencies in Band 1, the dielectric band, and a large decrease in the eigenfrequencies in Band 2, the air band. This can be understood from the mode distributions for Band 1 at the K-point and Band 2 at the M-point in the unit cell for various sidewall angles, as shown in Figure 3. For both PhC configurations, when $\theta$ decreases from 90° to 70°, it is noted that the normalized electric field distribution is intensified around the bottom inside the air holes for Band 2, in comparison to a slight change for the electric field inside the dielectric area for Band 1. Therefore, one expects to observe a more pronounced variation in eigenfrequency for Band 2 compared to Band 1 as $\theta$ decreases. This unequal eigenfrequency shift of the two bands results in a reduced bandgap and a significant decrease in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for a small sidewall angle.

3. Manipulating the Impact of Sidewall Angles

We examine the effect of the sidewall angle on ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ in both types of LN PhCs for varied a, r_c, or h and optimize them to maximize ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for practical sidewall angles. Figure 4 shows that the overall ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ increases with r_c for varied $\theta$. For suspended LN PhC in Figure 4a, the changes in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ with r_c are shown for 60°≤ $\theta$ ≤ 90°. As r_c increases from 150 nm to 210 nm, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ increases from 0.069 to 0.167 for $\theta$ = 90°, marking a 142% rise. Similarly, for $\theta$ = 60°, the ratio increases from 0.041 to 0.113, reflecting a larger increment of 175%. These results indicate that ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ features a more pronounced increase with r_c at a smaller $\theta$. Figure 4b shows the ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for LNOI PhCs with 65°≤ $\theta$ ≤ 90° by changing r_c. Compared to Figure 4a, adding a silica sublayer to the PhC slab causes a remarkable decrement in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$. Despite this change, the overall variations in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ with r_c for LNOI PhCs are similar to those observed in suspending LN PhC slabs. ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ increases from 0.019 to 0.085 at $\theta$ = 90° when r_c increases from 120 nm to 160 nm. Similarly, in the same range of r_c, the ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ ratio shows an increment from 0.0003 to 0.054 at $\theta$ = 75°, showing a greater impact of r_c on ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ at smaller $\theta$, which is also similar to the observed behavior in suspended LN PhCs.

The increase in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ with r_c can be leveraged to counteract the impact of non-vertical sidewall on ${\displaystyle \frac{∆\omega }{{\omega }_m}}$. For example, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for LNOI PhC drops from 0.038 at ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ = 90° to 0.003 at ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ = 70° when r_c = 130 nm in Figure 4b, which can be improved to 0.037 with a larger r_c = 160 nm. In this sense, it is desirable to design the unit cell with a large r_c or, more precisely, r_c/a, for both suspended LN and LNOI PhC slabs. In the case of LNOI PhCs, it should be noted that r_c cannot be excessively large. This limitation arises because an overly large r_c would lead to Band 2 being positioned inside the light cone. In addition, the possible r_c value for a fixed lattice constant is limited in fabrication. This limitation becomes tighter for PhCs with slanted sidewalls, which limits the achievable ${\displaystyle \frac{∆\omega }{{\omega }_m}}$.

An increase in a, with fixed r_c and h values, results in a reduction in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for both suspended LN PhCs and LNOI PhCs, as shown in Figure 5. Specifically, Figure 5a shows that ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ decreases when a increases from 600 nm to 640 nm for 60°≤ $\theta$ ≤ 90° in suspended PhC slabs. The change in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ with a is slightly different at various $\theta$. For $\theta$ = 90°, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ falls by >17%, reducing from 0.136 for a = 600 nm to 0.112 for a = 640 nm, which is 5% more than the variation in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for $\theta$ = 60°, reducing from 0.089 for a = 600 nm to 0.076 for a = 640 nm. On the other hand, Figure 5b shows ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ in the LNOI configuration as a function of a in the range of 540 nm to 620 nm for different $\theta$. ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for a = 620 nm is half of that for a = 540 nm in the case of $\theta$ = 90°. For $\theta$ = 70°, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is reduced by more than 75% comparing a = 540 nm to a = 620 nm. The bandgap is almost closed for $\theta$ = 65°, leading to ${\displaystyle \frac{∆\omega }{{\omega }_m}}$≈ 0. Similar to suspended PhCs, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ is more sensitive to a at a smaller $\theta$.

Accordingly, by reducing a, the PhC with a small $\theta$ could have a comparable ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for a PhC with a vertical sidewall angle and a large a. For example, comparing ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ values in Figure 5a, a suspended PhC slab in the air with $\theta$ = 80° and a = 630 nm can be replaced by one with $\theta$ = 70° and a = 600 nm to obtain the same ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ if the working wavelength is flexible. Similarly, for the LNOI PhC in Figure 5b, a can be reduced by 60 nm from a = 600 nm to compensate for the 15° reduction in $\theta$ from 90° to 75° to obtain the same ${\displaystyle \frac{∆\omega }{{\omega }_m}}$.

By modifying h while maintaining a constant r_c and a, both the dielectric area and the air holes are affected. This leads to a more intricate variation in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ with h, as shown in Figure 6a,b for the suspended LN PhC slab and the LNOI PhC slab, respectively. The most notable feature observed in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ concerning h is the distinct variations it exhibits for $\theta$ values within different ranges. For suspended LN PhC slabs shown in Figure 6a, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ increases monotonically with h for 80°≤ $\theta$ ≤ 90°, showing a slower rate of increase towards h = 400 nm. For a smaller $\theta$ in the range of 60°≤ $\theta$ ≤ 75°, it is very interesting that ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ of the PhC features an optimized value at a specific h. For $\theta$ = 65°, it appears evident that the commonly used 300 nm thick LN slab yields the largest ${\displaystyle \frac{∆\omega }{{\omega }_m}}$, thus showcasing a superior light-confining capability for constructing functional devices. In addition, Figure 6a highlights the increased sensitivity of ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ to sidewall angle when thicker slabs are adopted.

For h = 200 nm, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ decreases from 0.095 to 0.078, marking an approximately 18% decrement, while this decrement rises to ∼54% for h = 400 nm. Similarly, ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ for LNOI PhCs in Figure 6b increases monotonically with h between 300 nm and 600 nm for 85°≤° and shows a maximum value at a specified h for a smaller $\theta$, i.e., 65°≤ $\theta$ ≤ 80°. In the case of $\theta$ = 65°, the bandgap tends to diminish, existing only for 300 nm ≤ h ≤ 420 nm, with a maximum value of 0.0065 achieved around h = 360 nm. Figure 6a,b could serve as a reference for optimizing h with the constraints of fabrication-limited $\theta$. This is particularly crucial for LNOI PhCs due to the substantial variations in ${\displaystyle \frac{∆\omega }{{\omega }_m}}$ concerning $\theta$.

The optimal geometrical parameters for achieving photonic bandgaps within the communication band (around $\lambda$ = 1550 nm) are not unique and can be systematically selected based on fabrication limitations or application-specific design criteria. For suspended LN PhC unit cell, a bandgap encompassing the communication band is achieved with the following parameters: h = 400 nm, r_c = 180 nm, a = 620 nm, and sidewall angle $\theta$ = 60° which can be proposed as one of optimized configurations. In contrast, the LNOI PhC unit cell exhibits comparable performance with distinct optimized geometrical characteristics, with h = 450 nm, r_c = 165 nm, a = 580 nm, and $\theta$ = 70°. Both configurations maintain sidewall angles within experimentally feasible ranges, underscoring their practical viability for scalable device integration.

4. Conclusions

In this work, we investigated the impact of non-vertical sidewalls on the bandgap characteristics of thin-film LN PhCs. The gap–midgap ratio is used as a metric for comparing the performance of PhCs with different sidewall angles. This study underscores the inherent drawback associated with the formation of slanted sidewalls in LN PhC devices, especially for LNOI PhCs. Furthermore, we conducted a comprehensive geometrical optimization of PhC unit cells to counteract the reduction in the gap–midgap ratio induced by angled sidewalls and provide a guideline for the fabrication-aware design of LN PhC devices, extending the scalability of LN photonics.