Integrated photonics has shown great advances in the field of optics by enabling the development of compact and high-performance photonic devices through a scalable nanofabrication process [1–5]. These advancements leverage the integration of optical components, including waveguides, modulators, and detectors, to perform complex functions with reduced size, stable operation, and low power consumption. One of the most crucial components in integrated photonic devices is the integrated interferometer, which allows the precise control of optical signals through light interference [6,7]. For example, the Mach–Zehnder interferometer (MZI) is widely used as an integrated intensity modulator, and an array of MZIs can facilitate the unitary transformation of input light in an arbitrary manner [8–10]. While realizing bulk optical systems with even a few interferometers poses significant difficulties due to the inherent sensitivity of the interference, complex interferometric systems on a chip can be implemented with enhanced stability thanks to their small footprint and sophisticated nanofabrication processes. Specifically, the beam splitter (BS) is considered one of the key building blocks of integrated photonic devices, forming the integrated MZI along with the integrated phase modulator [11].

Scalability is crucial for quantum photonic technologies because it determines the ability to increase the complexity and functionality of quantum systems. Integrated photonics is expected to offer a promising route to achieve such scalability [12–14]. Among various applications, the integrated quantum interferometer, a key building block in photonic quantum technologies, provides a promising route to manipulate and measure high-fidelity quantum states with high precision [9,15–17]. However, quantum integrated photonic devices typically require considerable improvements in building blocks, such as negligible optical loss, broad bandwidth, robustness to fabrication errors, and high stability. The BS is one of the most fundamental building blocks of quantum photonic devices. For example, the BS induces Hong–Ou–Mandel (HOM) interference by causing indistinguishable photons to bunch at the output ports [18]. Arguably, HOM interference is fundamental to quantum photonic operations. As a result, great efforts have been put into implementing HOM interference in myriad integrated photonic circuits. In fact, on-chip BSs using various material platforms such as Si, SiN, AlN, and GaN have been extensively explored for measuring on-chip HOM interference [19–23]. Regarding the design of BSs, a multimode interference (MMI) coupler and a directional coupler (DC) are the most common configurations. While the MMI coupler offers robustness and tolerance to fabrication imperfections, it typically causes a large footprint and non-negligible loss [24]. The optical loss of the MMI couplers is particularly critical for quantum photonics applications, limiting their use in large-scale integrated interferometers. On the other hand, DCs are compact and efficient, but they are highly sensitive to wavelength variations and fabrication inaccuracies. Although low-loss DCs are widely used to implement integrated interferometers, the imperfect operation of DCs often leads to performance inconsistencies and requires post-fabrication adjustments [16,17].

Adiabatic couplers are promising candidates for high-performance BSs thanks to their robustness and low loss, but they often lead to large footprints due to the inherent nature of the adiabatic process [25,26]. To overcome the limitations of the conventional adiabatic devices, the concept of rapid adiabatic couplers (RACs) has been recently proposed as a compelling solution [27–31]. The design principle of RACs is based on the well-established coupled-local-mode theory for waveguide optics, optimizing conventional adiabatic designs with suppression of unwanted coupling between the spatial modes in the waveguides. As a result, the footprint of the adiabatic device can be minimized without sacrificing performance. RACs have been realized in recent experiments, uniquely enabling ultra-compact, broadband, and low-loss Mach–Zehnder interferometers (MZIs) with balanced splitting ratios. However, RACs have only been implemented on silicon photonics platforms [27–29]. More importantly, HOM interference through RACs has not been demonstrated despite their superior optical properties. Thin-film lithium niobate (TFLN) is one of the most promising material platforms in integrated photonics due to its wide transparent window (from 0.4 to 5 μm), strong electro-optic effect, high second-order nonlinearity, and ferroelectric, piezoelectric, and photoelastic effects [4,5,32,33]. The TFLN platform has enabled various integrated quantum photonic devices with unprecedented performance or footprint, including bright photon-pair sources, squeezed light generation, quantum frequency converters, and quantum transducers, just to name a few [5,32,33]. Recently, HOM interference has been observed on TFLN platforms using conventional DCs [34]. However, these devices suffer from limited visibility, likely due to the performance limitations of the DCs. To our knowledge, the concept of the RAC has not been applied to TFLN photonic devices. Large intrinsic birefringence and slanted etching profiles of the waveguide present the most significant hurdles to the design of the TFLN RACs. If these challenges can be managed, TFLN RACs are expected to achieve superior optical properties, robustness, and a small footprint, similar to implementations in other material platforms. In fact, the TFLN RACs hold great potential to be utilized in a unit cell of photonic quantum processors that require ultralow-loss and broadband operation as well as fabrication tolerance and scalability [35,36].

In this paper, we theoretically investigate and experimentally demonstrate TFLN-based RACs. First, we show how the design rules of RACs can be extended to TFLN photonic devices by fully considering their unique characteristics through analytical and numerical analysis. Then, we experimentally demonstrate an RAC-based MZI with a broadband high extinction ratio in the wavelength range from 1500 to 1600 nm and measure the power-splitting ratio of a single RAC. Finally, we experimentally show that RACs enable the observation of HOM interference with a visibility over 99.2%. These results demonstrate the potential of diverse rapid adiabatic devices in a wide range of applications for TFLN photonic devices.

Figure 1(a) schematically shows the concept of the RAC, which gradually controls adiabatic mode evolution along the propagation direction. The RAC-based BS is designed for implementation onto a $Z$-cut 600 nm thick TFLN layer, a 2 μm thick buried silicon oxide layer, and a 525 μm thick silicon substrate. The RAC-based BS is divided into three sections denoted by regions 1–3. In each region, the top widths of the waveguides or the transverse positions vary along the propagation direction.

In region 1, the gap is tapered to support the minimum gap required for the adiabatic transition. Specifically, the gap between $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ narrows from 400 μm to 550 nm, with the widths of $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ fixed at 500 nm and 800 nm, respectively. Region 2 plays a crucial role in the RACs with the tapered waveguide widths accompanied by angle variations, which differentiates it from conventional adiabatic couplers. This design minimizes unwanted coupling between the local modes. In this region, the gap between the tops of $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ is fixed at 550 nm. The cross-section schematics of the waveguides in the region 2 are depicted in Figs. 1(b) and 1(c) with design parameters. The sidewall angle of the waveguides is 67°. In region 3, unwanted coupling between the modes is avoided as the gap between two waveguides increases from 550 nm to 400 μm. The 650 nm wide $\mathrm{W}{\mathrm{g}}_3$ and $\mathrm{W}{\mathrm{g}}_4$ remain unchanged, while the 550 nm gap between the two waveguides widens up to 400 μm.

For comprehensive explanation of the concept of the RAC, we should explain the design of region 2. The RACs, like the conventional adiabatic couplers, are also composed of two tapered waveguides in which the adiabatic mode transition mainly occurs because of the narrow gap size. The widths of $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ are linearly varied along the coupler length from 500 to 650 nm and from 800 to 650 nm, respectively. Thanks to the considerable coupling between the two waveguides, the input mode of the narrow $\mathrm{W}{\mathrm{g}}_1$ (wide $\mathrm{W}{\mathrm{g}}_2$) adiabatically evolves into the odd-like (even-like) mode. The widths of the output waveguides are identical to support even and odd modes at the output ports. As a result, the symmetry of the two modes leads to the balanced splitting ratio close to 50%:50%. To investigate such mode evolution, the effective mode indices of the eigenmodes are calculated as a function of the $W_1$. The effective indices of the even- and odd-like TM modes are plotted in Fig. 1(d). Furthermore, electric field profiles of the eigenmodes at the input and output ports of region 2 are plotted in Fig. 1(e). In particular, the corresponding eigenmodes and the electric field profiles are denoted by the colored circles and the edges. Thus, the slowly tapered width allows the adiabatic transition from the inputs of $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ to the odd and even modes, respectively.

RACs leverage rotation of the waveguides as an additional degree of freedom in design to satisfy the rapid adiabatic transition. The carefully designed and locally variable waveguide rotation suppresses the unwanted coupling between the odd- and even-like eigenmodes shown in Fig. 1(e). Figure 2(a) schematically shows the definition of the rotation angle, ${\theta }_r$. The coupled-local-mode theory constitutes the foundations of the RAC, describing the coupling between eigenmodes hosted by waveguides [37]. In the coupled-local-mode theory, the amplitude propagation of the each eigenmode in lossless media can be written as $$\frac{\mathrm{d}{a}_m}{\mathrm{d}z}=\sum _{n\ne m}{C}_{mn}(z)a_{n}z{e}^{-j{\int }_0^z\mathrm{\Delta }{\beta }_{mn}(z^{\prime })\mathrm{d}{z}^{\prime }},$$(1)where $a_n$ and ${\beta }_n$ are the amplitude and the propagation constants of a mode. Also, $\mathrm{\Delta }{\beta }_{mn}={\beta }_m-{\beta }_n$ and the subscripts indicate the modal indices of the eigenmodes. In particular, $C_{mn}$ in Eq. (1) is the coupling coefficient and written by $$C_{mn}=-\sqrt{\frac{{ϵ}_0}{{\mu }_0}}\frac{k_0}{4\mathrm{\Delta }{\beta }_{mn}}{\int }_A{E}_m^{*}·E_n\frac{\partial {ϵ}_r}{\partial z}\mathrm{d}A,$$(2)where $E_n$ and $k_0$ are the normalized electric field of the $n$th mode and the free-space propagation constant at the operating wavelength, respectively. In addition, ${ϵ}_0$, ${\mu }_0$, and ${ϵ}_r$ are vacuum permittivity, vacuum permeability, and relative dielectric permittivity, respectively.

The odd-like and even-like eigenmodes shown in Fig. 1(e) are indexed as 1 and 2, respectively. We can just consider the coupling between these two modes because the coupling coefficients with other modes are negligible due to the large $|\mathrm{\Delta }{\beta }_{mn}|$. Although the coupling coefficient is dictated by the changes of ${ϵ}_r$ in Eq. (2), one cannot easily control the spatial distribution of ${ϵ}_r$ but geometry of the structures including waveguide widths, heights, and positions. As a result, it is convenient to rewrite Eq. (2) by using the generalized step-index coupled mode theory overlaps for a step-index distribution, in which case the partial derivative with respect to $z$ can be expanded by the chain rules. The geometry of the waveguides is exploited for giving more freedom to the RAC-based design, by which we can employ the generalized step-index coupled mode theory [38,39]. As a result, the coupling coefficients can be calculated by the electromagnetic fields of the four shifting sidewalls. Figure 2(b) schematically shows the four slanted shifting walls of the TFLN waveguides. Furthermore, the perturbation theory of the anisotropic materials should be carefully applied into the calculation of the coupling coefficients because the TFLN waveguides uniquely possess severe birefringence and the slanted sidewall angles close to 67° [40,41]. As LN is a uniaxial birefringent crystal, we denote ordinary and extraordinary permittivities by ${ϵ}_o$ and ${ϵ}_e$, respectively. With detailed derivations in Appendix A, the coupling coefficients in Eq. (2) can be written by $$\begin{gathered} C_{12}=-\sqrt{\frac{{ϵ}_0}{{\mu }_0}}\frac{k_0\sin ({\theta }_s)}{4\mathrm{\Delta }{\beta }_{12}} \\ \sum _{p=1}^4\frac{\partial x_p}{\partial z}\int F_{1,p}^{*}(\tau ({ϵ}_{\mathrm{LN}})-\tau ({ϵ}_c))·F_{2,p}h\mathrm{d}l, \end{gathered}$$(3)where $F_{i,p}^{*}=(D_{i,\perp ,p},E_{i,||,p},E_{i,z})$, and $i=1$ or 2. $\tau ({ϵ}_{\mathrm{LN}})$ and $\tau ({ϵ}_c)$ can be written by $$\tau ({ϵ}_{\mathrm{LN}})=\left(\begin{array}{ccc}-\frac{1}{{ϵ}_r+\mathrm{\Delta }{ϵ}_r\cos (2{\theta }_s)}& 0& \frac{\mathrm{\Delta }{ϵ}_r\sin (2{\theta }_s)}{{ϵ}_r+\mathrm{\Delta }{ϵ}_r\cos (2{\theta }_s)}\\ 0& {ϵ}_r-\mathrm{\Delta }{ϵ}_r& 0\\ \frac{\mathrm{\Delta }{ϵ}_r\sin (2{\theta }_s)}{{ϵ}_r+\mathrm{\Delta }{ϵ}_r\cos (2{\theta }_s)}& 0& \frac{{ϵ}_r^2-\mathrm{\Delta }{ϵ}_r^2}{{ϵ}_r+\mathrm{\Delta }{ϵ}_r\cos (2{\theta }_s)}\end{array}\right),$$(4)$$\tau ({ϵ}_c)=\left(\begin{array}{ccc}-\frac{1}{{ϵ}_c}& 0& 0\\ 0& {ϵ}_c& 0\\ 0& 0& {ϵ}_c\end{array}\right),$$(5)where ${ϵ}_r$ and $\mathrm{\Delta }{ϵ}_r$ are $\frac{1}{2}({ϵ}_e+{ϵ}_o)$ and $\frac{1}{2}({ϵ}_e-{ϵ}_o)$, respectively. In Eq. (3), $\frac{\partial x_p}{\partial z}$ is $\tan ({\theta }_p)$, which is the rotation angle of the $p$th sidewalls shown in Fig. 2(b). ${\theta }_p$ is determined by ${\theta }_r$ and the width and length of the tapering. Given the tapering structure, ${\theta }_r$ should be carefully chosen at every cross-section to achieve ${\sum }_p^4\tan ({\theta }_p)\int F_{m,p}^{*}(\tau ({ϵ}_{\mathrm{LN}})-\tau ({ϵ}_c))·F_{n,p}h\mathrm{d}l\simeq 0$. Furthermore, the electromagnetic fields and material properties at the four shifting walls are only required.

We should note recent works on RAC devices based on isotropic materials and perpendicular sidewalls [30,31]. Like previous works, the proposed RAC devices leverage the ${\theta }_r$ to minimize the unwanted coupling in the adiabatic transition. However, our theoretical works in Eqs. (1)–(3) and Appendix A generalize the coupled local-mode theory to calculate exact coupling coefficients between the modes for the devices that have slanted sidewall angles and highly anisotropic material properties.

To find the optimal values of ${\theta }_r$ in every cross section, we calculate coupling coefficient ($C_{12}$) as a function of ${\theta }_r$. Also, we intentionally set the length of RAC to 150 μm as the relatively long length can reduce the scattering loss of the device. As a result, $C_{12}$ is calculated as a function of ${\theta }_r$ and depicted in Fig. 2(c). The vertical axis in the 2D mapping image of calculated $C_{12}$ is the tilting angle (${\theta }_r$). The red dots indicate the ${\theta }_r$ at which the $C_{12}$ is minimized. Figure 2(d) is the zoom-in version of Fig. 2(c) for clarifying.

We numerically verify performance of the RAC, using eigenmode expansion (EME) simulation (see Appendix B for details). The calculated transmission spectra of the 150 μm long RAC design shown in Fig. 2(d) are plotted in Fig. 3(a). While the device is designed for the gap size of 550 nm in region 2, we vary the gap sizes from 550 to 650 nm to test the fabrication tolerance. In Fig. 3(a), splitting ratios of the RAC with a 550 nm gap (black) are plotted in black curves and are better than 49%:51% over the wavelength range from 1381.82 to 2290.91 nm. The black curves simultaneously show all four possible transmission spectra for two input and two output ports. Furthermore, the splitting ratios of 600 nm (650 nm) gap size are also plotted in red (blue) curves and better than 49%:51% over the wavelength range from 1436.36 (1563.64) to 1690.91 (1618.18) nm. Moreover, the splitting ratios are better than 48%:52% (46%:54%) over the wavelength range from 1400 (1490.91) to 2254.55 (2254.55) nm for the case of 600 nm (650 nm) gap size. In the wavelength region longer than 2254.55 nm, the splitting ratios of RACs with 600 and 650 nm gap become unbalanced, which could be attributed to the coupling between two waveguides induced by considerably long wavelength. But we can check that the splitting ratios of RACs are insensitive to the gap size between two waveguides in the broadband. This means that the proposed RACs have excellent fabrication tolerance.

As shown in Fig. 3(b), the splitting ratio of the RAC with a 550 nm gap is also investigated in accordance with the length of region 2. As the length increases, the splitting ratios show convergence close to 50%:50%. When the length is longer than the designed length of 150 μm, the splitting ratio at 1550 nm is better than 51%:49%. The fast convergence mainly results from the optimal bending orientation shown in Fig. 2(d). The rapid adiabatic convergence of the proposed device with the optimal bending outperforms those of a conventional adiabatic design without the bending and a design with an opposite bending orientation (see Appendix C for details).

Figure 3(c) shows the calculated transmission loss as a function of RAC length in region 2. For both inputs through $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$, the transmission loss of the RAC less than $-40\mathrm{dB}$ is maintained down to the 50 μm length of region 2. Furthermore, we can calculate the optimal bending required for the 50 μm long device (see Appendix D for details).

It is worth clearly noting the advantages of the proposed RACs over conventional BSs such as the MMI couplers and DCs. First of all, the low-loss operation of the RAC outperforms the conventionally designed TFLN MMI coupler [42] and a low-loss MMI coupler in silicon photonics [30]. To compare the performance of the RACs with that of the DCs, we plot the calculated transmission spectra of the RACs and DCs in Fig. 3(d). In contrast to the broadband operation of the RAC, the DC with the 550 nm gap only shows a balanced splitting ratio better than 51%:49% at 1545.45 nm wavelength. Overall, the DCs suffer from narrow operating bandwidth and high sensitivity to the fabrication tolerance (see Appendix E for detailed comparisons). The distinct performance mainly stems from the different operating mechanisms of the RACs and DCs. While the RACs rely on the adiabatic mode transition, the DCs employ the interference between the super modes.

To experimentally investigate the proposed RAC devices, we design and fabricate the unbalanced MZIs composed of two identical RAC devices. The interference visibility of the unbalanced MZIs allows for robust evaluation of the splitting ratio of the low-loss BSs in general [8]. Figure 4(a) shows the optical microscopy image of the fabricated MZI array. Using a 600 nm thick $Z$-cut lithium niobate layer, we fabricate the devices with conventional nanofabrication techniques (see Appendix B for details on the fabrication procedures). Figure 4(b) schematically shows the unbalanced $2\times 2\mathrm{MZI}$, which includes two identical 150 μm long RACs with the 550 nm gap. To measure the optical properties of the unbalanced MZIs, we use a continuous-wave telecom laser and couple the TM-polarized light into the chip through a polarization controller and a lensed fiber. The transmitted telecom light is detected by another lensed fiber and InGaAs photodetector. Figures 4(c) and 4(d) show transmission spectra of the unbalanced MZI. In particular, the extinction ratio of MZI, obtained from $\mathrm{W}{\mathrm{g}}_1$ and $\mathrm{W}{\mathrm{g}}_2$ to $\mathrm{W}{\mathrm{g}}_3$ and $\mathrm{W}{\mathrm{g}}_4$ ports, is over 20 dB in the broad wavelength range from 1500 to 1600 nm. For both input waveguides, a comparable extinction ratio and transmission loss are found at the both outputs at $\mathrm{W}{\mathrm{g}}_3$ and $\mathrm{W}{\mathrm{g}}_4$. In other words, the devices show the splitting ratio better than 45%–55% with a bandwidth greater than 100 nm. We expect that the extinction ratio of 20 dB is mainly limited by the polarization contrast of the fiber-based polarization controllers in the setup. Furthermore, the maximum extinction ratio is higher than 40 dB, indicating the splitting ratio close to 49.5%–50.5%, showing good agreement with the simulated results shown in Fig. 3(b).

To show the potentials of the proposed RAC devices in a quantum optics experiment, we perform the two-photon interference measurement. First, we fabricate an array of RAC-based BSs and measure the beam splitting performance in the telecom band (see Appendix B for details). The transmission spectra are shown in Figs. 4(e) and 4(f). The balanced splitting ratio is observed over a broad wavelength range from 1500 to 1600 nm. While the transmission spectra in Figs. 4(e) and 4(f) are sensitive to the facets due to the cleaving process, the splitting ratio obtained from the input port $\mathrm{W}{\mathrm{g}}_1$ ($\mathrm{W}{\mathrm{g}}_2$) to $\mathrm{W}{\mathrm{g}}_3$ and $\mathrm{W}{\mathrm{g}}_4$ is 55.45%:44.55% (52.50%:47.50%) at the 1556 nm wavelength, which is the center wavelength of an off-chip photon-pair source generated by the type-II spontaneous parametric down conversion in 10 mm long periodically poled KTP crystal. Using a conventional fiber beam splitter, we verify indistinguishability between the prepared photon pairs by the HOM interference visibility of 99.93% (see Appendix B for details). Figure 5(a) schematically shows the on-chip HOM interference. Two single photons are coupled to the on-chip RAC-based BS, and the time delay between the two photons, denoted by $\mathrm{\Delta }t$ in Fig. 5(a), is adjusted by the off-the-shelf translation stage (Thorlabs KDC101). The single photons coming out at the outputs are collected by lensed fibers and the superconducting nanowire single-photon detector (SNSPD). The coincidence counts that make the single-photon pair bunch at the same output ports are registered with the homemade coincidence counting unit (CCU). The measured single and coincidence counts in two channels are measured by moving the time-delay controller and plotted in Fig. 5(b). Clearly, the bunching effects known as HOM interference, are observed with the visibility of 99.25% with a triangle fit after the subtraction of accidental coincidence counts. It is worth noting that the measured visibility is not only almost comparable to that of the conventional bulk fiber beam splitter (see Appendix B for details), but also one of the highest measured visibilities among numerous on-chip HOM interferences. This high visibility experimentally verifies the balanced splitting ratio. Finally, we also fabricate a 50 μm long RAC device with a 550 nm gap. We analyze the transmission spectra of the 50 μm long RAC-based BS with which HOM experiments are performed (see Appendix D for details.) We should note that the measured visibility is 98.95% with a triangle fit after the subtraction of accidental coincidence counts.

The theory and design presented in this paper generalize the design principles of the existing RAC device. By considering factors such as birefringence and slanted etching profiles, our generalized design methodology can enhance the performance and scalability of RACs across diverse non-centrosymmetric and unconventional photonic material platforms including lithium niobate, barium titanate, gallium nitride, and aluminum nitride [5,22,23,43,44]. In particular, such materials are of great interest in the field of nonlinear or quantum photonics because of their exceptional nonlinearity, tunability, and wide band-gap.

While this work focused on the application of RACs to $Z$-cut lithium niobate, extending the concept to $X$-cut designs requires more careful consideration. Unlike $Z$-cut designs, where the eigenmodes are unaffected by local rotation, $X$-cut designs can alter the eigenmodes themselves through rotation. This necessitates a more sophisticated design approach to maintain optimal performance. Future exploration of $X$-cut designs could lead to the development of advanced modulators and other photonic components, further expanding the applicability of RACs.

The successful demonstration of RACs on the TFLN platform opens up numerous possibilities for their application in integrated photonics. Indeed, the concept of the rapid adiabacity is not limited to the fundamental spatial eigenmode in the waveguide, but applicable to high dimensions. By extending the concept to the high-order spatial modes or multiple operating wavelengths, we can readily design various rapid adiabatic photonic components including spatial mode multiplexers and demultiplexers [45–47], polarization rotator splitters [48,49], polarization beam splitters [50], and adiabatic wavelength division multiplexers [51,52]. These devices require sophisticated control of coupling between spatial eigenmodes of the neighboring waveguides.

In conclusion, we have demonstrated the potential of rapid adiabatic couplers (RACs) on a thin-film lithium niobate (TFLN) platform for integrated quantum photonics. Our RAC-based Mach–Zehnder interferometer (MZI) achieved a high extinction ratio over a broad wavelength range (from 1500 to 1600 nm), and we observed 99.25% visibility in the Hong–Ou–Mandel (HOM) interference experiment. These experimental results highlight the significant advancements in integrating RACs into TFLN photonic devices, offering high performance, low loss, and broad bandwidth, which are challenging to achieve simultaneously with conventional beam splitters. This work clearly showcases the potentials of the rapid adiabatic TFLN photonic devices in classical and quantum integrated interferometers.