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Abstract
Electro-optic modulators (EOMs) are essential devices of optical communications and quantum computing systems. In particular, ultra-compact EOMs are necessary for highly integrated photonic chips. Thin film lithium niobate materials are a promising platform for designing highly efficient EOMs. However, EOMs based on conventional waveguide structures are at a millimeter scale and challenging to scale down further, greatly hindering the capability of on-chip integration. Here, we design an EOM based on lithium niobate valley photonic crystal (VPC) structures for the first time. Due to the high effective refractive index introduced by the strong slow light effect, the EOM can achieve an ultra-compact size of 4 μm×14 μm with a half-wave voltage of 1.4 V. The EOM has a high transmittance of 0.87 in the 1068 nm because of the unique spin-valley locking effect in VPC structures. The design is fully compatible with current nanofabrication technology and immune to fabrication defects. Therefore, it opens a new possibility in designing lithium niobate electro-optic modulators and will find broad applications in optical communication and quantum photonic devices.
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1. Introduction
Electro-optic modulators (EOMs) modulate optical waves using high-speed electrical signals, which are key components in optical communications networks [1,2], microwave photonics [3,4], and quantum photonics [5,6]. With the development of integrated photonic chips (PICs), it is required to have EOMs with broad working bandwidth, ultra-compact size, high efficiency, and low energy consumption [7,8]. EOMs based on various material platforms have been demonstrated, including silicon [9,10], indium phosphide [11], lithium niobate, and plasmonic metallic structures [12]. A strong electro-optic (EO) response is required for ultra-compact and energy-efficient devices to realize an optical modulation with a short optical path and a low half-wave voltage (${V_\pi }$). In addition, low linear absorption is necessary for obtaining high transmittance. Among different materials, EOMs made of lithium niobate (LN) have the advantages of relatively small size, broad bandwidth, low loss, good linearity, and a high extinction ratio [13]. Meanwhile, the strong EO (Pockels) effect in the LN causes its refractive index to vary linearly with the applied voltage, which allows for pure phase modulation. In contrast, silicon, most commonly used in integrated photonic chips, does not allow for pure phase modulation because of residual amplitude modulation due to free carrier-induced losses [14].
Conventional LN EOMs based on bulk LN crystals typically have a large size and high half-wave voltage (${V_\pi }$) due to the low refractive index contrast inducing weak field confinement [15–17]. The recent development of lithium niobate on insulator (LNOI) enables the fabrication of high-performance waveguides capable of better light confinement, which can be applied in integrated nanophotonic devices [18–24]. In particular, there is a significant reduction in the half-wave voltage, with the low half-wave voltage being 1.4 V [25]. However, most of the LNOI electro-optic modulators still have a size of a few millimeters, and it is challenging to reduce the size further. To the best of our knowledge, the smallest ultra-compact LNOI EOM is 1.2 mm [26]. In addition, during the propagation of light waves in the modulator, there may be a mismatch between the spatial distribution of the modulated light field and the applied electric field due to the difference between the group velocities of the light and radio frequency (RF) waves for electrical input [24], which is more profound in large devices. The mismatch further decreases the efficiency of the modulator. Therefore, this is a stringent requirement for designing LNOI EOMs based on new working principles.
In comparison, photonic devices based on photonic crystal (PC) structures feature high design flexibility, miniaturization, and low energy consumption [27–30]. Both one- and two-dimensional PC cavities have been demonstrated [31,32] based on the LNOI platform for achieving resonant EOMs [33]. This mechanism has the advantage of reduced electrode size and can work at high RF frequencies. However, conventional PC structures have a high scattering loss from defects, resulting in low transmission efficiency. In contrast, edge states in valley photonic crystal (VPC) structures derived from the optical quantum valley Hall effect (QVHE) have been proposed to design topological waveguides for robust unidirectional light transmission based on the unique spin-valley locking effect [34,35], which can achieve near unity high transmittance [36]. High-performance functional devices based on VPC structures with different materials have been demonstrated at various working wavelengths [37–40]. However, there is no demonstration of EOMs based on VPC structures, which could potentially further improve modulators’ compactness and efficiency.
This paper reports the first LN EOM based on VPC structures. Here, the unique slow light effect in the VPC structure is used to achieve strong electro-optic modulation within a short optical path of 4 $\mu m$×14 $\mu m$ with a low half-wave voltage of 1.4 V. In addition, the spin-valley locking effect achieves an ultrahigh transmittance of 0.87 in 1068 nm. Therefore, our EOM outperforms most of the demonstrated EOMs and takes an essential step toward miniaturizing LN EOMs. The design is fully compatible with current nanofabrication technology and immune to fabrication defects. It opens a new possibility in designing LN EOMs and will find broad applications in optical communication and quantum photonic devices.
2. Design of the electro-optic modulator based on VPC structures
The conceptual demonstration of the EOM based on VPC structure is schematically shown in Fig. 1(a), which is a free-standing LN thin film (220 nm thick) on a SiO2 substrate. The structure is a topological waveguide (along the $y$-direction, indicated by the blue shadow in Fig. 1(a)) composed of two VPCs with mirror symmetry with gold electrodes deposited on both sides. By applying voltage to the electrodes, the effective refractive index of the waveguide can be strongly modulated to introduce a phase modulation (π-phase modulation) within a short optical path. The geometry of the EOM is shown in Fig. 1(a). Fig. 1(b) shows the simulated electric field distributions (simulated using the finite element method with commercial software (COMSOL)), which present a strong electric field across the EOM to achieve maximum electric-optic modulation. The electrodes are placed close to the topological waveguide to increase the field strength of ${E_Z}$ at a given voltage by minimizing the gaps (Fig. 1(b)) due to the relationship as ${E_Z}$ = V/s, where V is external voltage. The strong EO effect of LN can generate a refractive modulation under external voltage, which can produce a phase modulation on the incident light. The design is to modulate the transverse electric (TE) polarized incident light with the electric field along the $z$-direction which is parallel to the external modulation electric field (Fig. 1(b)). The LN thin film (220 nm thick) is chosen to be $x$-cut to utilize the largest EO coefficient (${r_{33}}$) [33]. LN exhibits an EO response with a direct current (DC) electric field due to the Pockels effect [41], which can be formulated as:
Fig. 1. Design of LN EOM. (a) LN device concept diagram and cross-sectional view of the device. LN photonic crystal layer thickness (metal electrode thickness) d = 220 nm, the gap between electrodes is s = 8 $\mu m$, the width of the electrode is t = 3 $\mu m$. (b) Electric field distributions inside the device due to introducing external voltage, where the arrows indicate the vectorial direction of the electric field.
The electric field distribution in the device is shown in Fig. 1(b), which is simulated using the finite element method (COMSOL). It can be seen that the direction of the electric field inside the LN film is along the $z$-axis direction. In this case, the amount of refractive index modulation due to the Pockels effect varies linearly with the external electric field ${E_Z}$ from the external voltage (Eq. (1)). To quantify the Pockels effect, we calculate the local field factor inside the device according to the calculation in [42,43]. In the case of LN PC structure we define the effective EO effect as Eq. (2).
where ${f_i}({i = 1,2,3} )$ is the local field factor of the three waves involved in the second-order process. $v_g^{bulk}$ is the group velocity of bulk LN material without nanostructures, which is $v_g^{bulk} = c/{n_e}$. Meanwhile, $v_g^{PC}$ is the group velocity of the modes inside the PC structure, which is $v_g^{PC} = d\omega /dk$. From Eq. (2) and (3) one can see that it is essential to minimize the group velocity to maximize the EO coefficient $r_{33}^{PC}$, since $v_g^{bulk}$ is a constant only depending on the material. The corresponding refractive index modulation is formulated as [42]:Therefore, there are two ways to achieve the required refractive index modulation. With a given $r_{33}^{PC}$, $\Delta n$ is controlled by ${E_z}$ according to the applied voltage. On the other hand, with a given ${E_z}$, the $\Delta n$ can be controlled by $r_{33}^{PC}$. In practical applications, the phase modulation is defined as:
Slow light effects have been demonstrated in various conventional PC structures via engineering the photonic band structures [44,45]. However, due to the strong scattering loss, those designs suffer low transmission efficiency in the slow light wavelength region. In comparison, VPC structures can achieve high transmission efficiency in the slow light region due to the scatter immune unidirectional transmission base on the spin-valley locking effect [46,47]. Therefore, we propose an EOM based on VPC structures in LN material.
We first design a honey PC structure with triangular shape lattices (${r_A}$=${r_B}$=90 nm, which is defined as the distance between the vertex and geometric center of the triangle) with ${C_{6V}}$ rotation symmetry (Fig. 2(a)), which shows a Dirac cone structure in the photonic band diagram of transverse electric (TE) polarized mode (blue curves in Fig. 2(b)). The lattice constant of the structure is $a$=440 nm. Then a photonic bandgap (red curves in Fig. 2(b)) is created by simultaneously increasing ${r_A}$ (${r_A}$=180 nm) and decreasing ${r_B}$ (${r_B}$=0 nm) to degenerate K and K’ by reducing the ${C_{6V}}$ symmetry to ${C_{3V}}$. The gap width is$\; 2|{\lambda_{{\varepsilon_r}}^P} |$, which is decided by the area contrast of triangles A and B in the relationship of $|{\lambda_{{\varepsilon_r}}^P} |\propto \left|{\mathop \smallint \nolimits_B {\varepsilon_r}dS - \mathop \smallint \nolimits_A {\varepsilon_r}dS} \right|$ [48], where ${\varepsilon _r}$ is the relative permittivity. Therefore, ${r_B}$ is reduced to 0 nm to achieve the largest bandgap. The resulted bandgap is between 0.396-0.428 a/λ (1028 nm-1111 nm). VPC1 is defined when ${r_A}$=0 nm and ${r_B}$=180 nm, and VPC2 is defined when ${r_A}$=180 nm and ${r_B}$=0 nm, which is mirror symmetric to VPC1.
Fig. 2. (a) Schematic of the VPC structure based on LN material. (b) Photonic band diagrams of the original honeycomb PC and VPC structures. The light cone is marked by the grey shadow. Edge state plots of the beard (c) and zigzag (d) shape boundary structures.
Due to the QVHE effect, the sign of Berry curvatures at the K and K’ valleys are opposite. The topological indices [49,50] of the K and K’ valleys are ${C_K} = 1/2$ and ${C_{K^{\prime}}} ={-} 1/2$, calculated as ${C_{\tau z}} = {\tau _z}sgn({\lambda_{{\varepsilon_r}}^P} )/2$. Thus, even if the total Chern number is equal to 0, the valley Chern number is ${C_V} = {C_K} - {C_{K^{\prime}}}$ which describes the topological properties of the VPCs [51]. The valley Chern numbers of VPC1 and VPC2 are 1 and -1, respectively. A topological phase transition occurs when the valley Chern number at the interface is not 0. Thus, it is possible to obtain topological edge states with a spin-valley locking effect by constructing the boundaries using VPCs with different valley Chern numbers (VPC1 and VPC2).
We show the plots of beard and zigzag-shaped boundaries in Fig. 2(c) and (d), which can be applied as waveguides in our EOM. The ${v_g}$ and ${n_g}$ of the edge states are calculated as ${v_g} = d\omega /dk$ and ${n_g} = c/{v_g}$ [52]. Therefore, ${v_g}$ is the slope of the edge states, which can be zero at the maximum and minimum points on the edge state plots. The resulted plots of ${v_g}$ and ${n_g}$ of different boundaries are shown in Fig. 3. As we expected the edge states of beard-shaped boundary can achieve a ${v_g}$ of zero at the point of Kx = 0, which is the minimum point in principle corresponding to a standing wave mode. However, in real applications the point can not be used since the mode can not propagate through the waveguide. Therefore, we choose a point near the zero ${v_g}$ point at 1068 nm to achieve a relative low group velocity and high transmission. The group velocity ${v_g}/c$ is 0.001613 (Fig. 3(a)) and the group refractive index ${n_g}$ is 620 (Fig. 3(b)). Considering ${r_{33}} = 32\; pm/V$, the EO coefficient in the VPC is equal to $r_{33}^{PC} = 157216\; pm/V$.
Fig. 3. Plots of normalized group velocity (${v_g}/c$) (a) and group index (${n_g}$) (b) versus Kx and wavelength of the beard-shaped boundary. Plots of normalized group velocity (${v_g}/c$) (c) and group index (${n_g}$) (d) versus Kx and wavelength of the zigzag-shaped boundary.
3. Performance of the electro-optic phase modulator
The designed EOM is schematically shown in Fig. 4(a), which is a waveguide based on the beard-shaped boundary according to the above discussion. The length of the EOM is 4 $\mu m$, and the corresponding half-wave voltage is 1.4 V. The phase distribution of the light wave in the EOM is shown in Fig. 4(a), which generates phase modulation at half-wave voltage at the wavelength of 1068 nm. The design is significantly smaller in size than other thin-film LN EOMs [24,26]. The corresponding electric field intensity distribution at 1068 nm is shown in Fig. 4(b), which is quite uniform throughout the EOM structure, suggesting a high transmittance (0.87). The corresponding forward transmittance spectrum is shown in Fig. 4(c), with a maximum value of 0.98 in 1063 nm, confirming the capability of the topological waveguide constructed using VPC structures. We choose designing the device at 1068 nm because this wavelength maintains high transmittance with low group velocity. The blue region represents the unidirectional transmission working bandwidth, where the forward transmittance is higher than 0.5. As a result, we have achieved an ultracompact EOM design based on LN material with high transmission efficiency.
Fig. 4. (a) Schematic of the EOM design based on VPC structures and the phase modulation under different voltages (0 V and 1.4 $V$) at 1068 nm. (b) Electric field intensity distributions in the EOM at the wavelength of 1068 nm. (c) Forward transmittance spectrum of the EOM.
In addition, the electric field from the applied voltage can be controlled by the gap width (s) between the electrode in the relationship of ${E_Z}$ = V/s. Therefore, for required ${E_Z}$ the half-wave voltage can be further reduced by decreasing the gap width. The limitation of the minimum gap width is the width of the topological waveguide. Further decrease of the gap will increase the propagation loss of the waveguide mode due to the excitation of surface plasmon of the gold electrodes [24].
With a given electrode gap width (s), the required half-wave voltage is inversely proportional to the length of the EOM ($L$), according to Eq. (5). The results are plotted in Figs. 5(a) and (b), when the gap width is 8 $\mu m$. Different colored lines indicate different device lengths. On the other hand, when the device length is fixed at 4 $\mu m$, ${E_Z}$ is linearly proportional to the gap width. The results are shown in Figs. 5(c) and (d).
Fig. 5. (a) Required external voltages as a function of required phase modulations of the EOM with different lengths when the gap between electrodes is fixed at 8 $\mu m$. (b) Plot of half-wave voltage versus the length of EOM. (c) Required external voltages as a function of required phase modulations of the EOM with different gaps between electrodes when the EOM length is fixed at 4 $\mu m$. (d) Plot of half-wave voltage versus the gap between the electrode.
Additionally, we further demonstrate the defect-immune capability of our design by simulating devices with random defects that could be introduced in the fabrication process (details can be found in Supplement 1). From the simulation, we can conclude that the manufacturing imperfection only slightly compromises the overall performance of the EOM, confirming the high feasibility in the experimental realization of the designed structures. Furthermore, we study the effect of the varying thickness of the LN film (Supplement 1) on the performance of the EOM and found that the transmittance spectrum remains essentially unchanged. In conclusion, our design is compatible with current nanofabrication technology [53] and has substantial tolerance to fabrication errors and defects.
In terms of applications, light in the short-wave near-infrared band between 750 nm and 1100 nm can be applied to optical communications [54,55], LIDAR [56], spectral analysis [57] and optical imaging [58,59], including optical coherence tomography (OCT) and near-infrared imaging [60]. Therefore, the designed LN EOM can potentially be applied in those areas, offering the advantages of low-loss and ultracompact design. In addition, according to the scaling law of PCs, the working wavelength of PCs, including VPCs, can be tuned by the lattice constant and the size of the lattice. Therefore, it is possible to enlarge the lattice constant and the lattice size to move the working wavelength into the telecommunication wavelength, which is longer than the working wavelength of the current work. In comparison, the lattice constant and lattice size could be reduced for working in the visible wavelength range (shorter than the working wavelength in this work). Meanwhile, lattice constant and lattice size optimisations may be performed to achieve optimal performance at the desired working wavelength, which is also decided by the refractive index. So the working principle and our design method can be generally applied to design LN VPC EOM devices in a broad wavelength range for various applications.
4. Conclusion
In conclusion, we demonstrated the first LN EOM based on VPC structures. The unique slow light effect in the VPC structure is used to achieve strong EO modulation with an ultracompact design of 4 $\mu m$, the footprint of the device is 4 $\mu m$×14 $\mu m$. The device can work with a low half-wave voltage of 1.4 V. In addition, the EOM achieve a high transmittance of 0.87 in modulation wavelength (1068 nm) due to the spin-valley locking effect. Therefore, our EOM achieves an ultracompact design and outperforms most of the demonstrated EOMs, taking an essential step toward miniaturizing LN EOMs for on-chip integration. The design is fully compatible with current nanofabrication technology and immune to fabrication defects. It opens a new possibility in designing LN EOMs for optical communication and quantum photonic devices.
Funding
National Key Research and Development Program of China (2022YFA1404201); Australian Research Council (DP220100603, FT220100559); National Natural Science Foundation of China (U23A20375).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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