1.

Introduction

Photonic integrated circuits (PICs) have recently become a mature and powerful technology. It allows for a variety of applications including optical interconnects,¹ microwave photonics,² and quantum information.³ Waveguide arrays are the cornerstones for integrated photonics, which play an ever-increasing role in various functional applications, such as wavelength-division multiplexers,⁴ space-division multiplexers,⁵ mode division multiplexers,⁶ and chip-scale optical interconnections.⁷ Waveguide arrays possess all the basic characteristics of the photonic crystal structure, including Brillouin zones, allowed bands, and forbidden bands, and therefore, support wave dynamics that are equivalent to the dynamics of electron transport in semiconductors.⁸ Periodic modulation of a photonic lattice opens up novel opportunities for diffraction management,⁹ manipulating the vortex light,¹⁰ and controlling nonlinear interactions of light.¹¹

It remains a challenge to significantly improve the comprehensive performance of dense photonic-integrated components to fully meet the stringent requirements of a large-scale photonic integration. Substantial research efforts have been made by incorporating superlattices,^12,13 inverse design,¹⁴ artificial-gauge-bending designs,^15–17 extreme-skin-depth,^18,19 etc. However, these approaches have mainly focused on silicon-on-insulator (SOI) wafers with air cladding, which have a large refractive index contrast. They are not implementable on large-scale optical systems, fundamentally limiting the integration density and scalability of optical chips for practical applications, because the absence of a solid upper cladding destroys the mirror symmetry of unetched waveguides and is incompatible with most metal back-end-of-the-line (BEOL) processes. This incompatibility complicates integration with critical photonic devices, such as high-speed modulators and photodetectors (PDs).

The silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) photonics,^20–24 as a promising photonic integration platform, has facilitated the development of a diverse range of low-loss ($<1\mathrm{dB}/\mathrm{m}$) planar-integrated devices and chip-scale solutions. This platform offers unprecedented transparency across a wide wavelength range (400 to 2350 nm) and enables fabrication through wafer-scale processes. Serving as a complementary platform to SOI,^25,26 LNOI,²⁷ and III-V photonics,²⁸ ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide technology introduces a new era of system-on-chip applications that cannot be achieved solely with other platforms.²² Nevertheless, the low-refractive index contrast and the relatively immature fabrication technology of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides pose obstacles to directly transplant the existing dense photonic schemes into the ${\mathrm{Si}}_3{\mathrm{N}}_4$ platform.

In this paper, we propose an alternative approach to realize ultra-broadband (experimentally more than 500 nm and theoretically more than 1000 nm) nearly zero crosstalk transmission by waveguide superlattices with artificial gauge field (AGF) using 800-nm-thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides. The ultradense waveguide arrays with silica upper cladding feature negligible insertion losses and record large bandwidth from 1200 to 1700 nm at a minimum gap (400 nm) with $<-24\mathrm{dB}$ crosstalk. Finally, 112 Gbit/s signals encoded on each channel are successfully transmitted along the ultracompact waveguide arrays with high-fidelity signal-to-noise ratio (SNR) profiles and bit error rates (BERs) below the 7% hard-decision forward error correction threshold. The strong coupling suppression of the field-induced $n$-“photon” resonances, introduced by the curved trajectory of waveguide superlattices, results in an ultra-broadband and dense PIC, which facilitates substantial on-chip footprint reduction and opens up possibilities for high-density heterogeneous integration. This work, transferable to other platforms, holds the potential to advance device performance, such as half-wavelength-pitched optical phased array (OPA), high-density energy-efficient modulators, and ultradense wavelength-division multiplexers.

2.

Theory and Design Principles

We first start from a standard 2D model of a binary waveguide array with periodically curved trajectory of frequency along the light-propagation direction, $z$. We typically apply a sinusoidal modulation profile $x_0(z)=A\sin (\mathrm{\Omega }z)$, where $A$ and $P=2\pi /\mathrm{\Omega }$ are the amplitude and period of the trajectory, respectively. The array consists of a series of equally spaced alternating waveguides of varying width $w_1$ and $w_2$, and the center-to-center separation between waveguides is $a/2$. In such a structure shown in Fig. 1, light propagation is determined by the overlap between the modes of adjacent waveguides, similar to wave dynamics in discrete lattices.²⁹ Different colors are used to distinguish the waveguides of different widths. The waveguide superlattices with AGF, as a general one-dimensional periodic optical structure, are analyzed in the framework of the Floquet–Bloch (FB) analysis.³⁰ It predicts that the spectrum of propagation constants for the eigenmodes of the array, known as FB waves, is partitioned into bands with gaps between them where no propagating modes exist. In the nearest-neighbor tight-binding approximation and assuming that the lowest Bloch band of the array is excited, the light propagation in a modulated binary array can be fully described based on the coupled-mode theory governing the modal amplitudes ${\mathrm{\Psi }}_n$ of light waves confined in the individual waveguides number $n$:³¹

Fig. 1

Schematic of the waveguide superlattices with AGF.

Fig. 2

(a) Band-gap diagram (first two minibands) of a typical straight waveguide array for $a=2.65\mu \mathrm{m}$, $w_1=1\mu \mathrm{m}$, $w_2=0.85\mu \mathrm{m}$, $\mathrm{\Delta }n=0.31$, $\lambda =1.55\mu \mathrm{m}$, and $n_0=1.450$, where the propagation constant is plotted as a function of the Bloch wave number and the shaded regions represent the bands. (b) Ninth-order ($J_9$) and tenth-order ($J_{10}$) Bessel modulation of the first kind. $J_{10}$ and $J_9$ have the same sign in the marked red region.

The quantum mechanical description of the light propagation dynamics in the proposed waveguide array, as given by Eq. (1), when combined with the modulation described by Eq. (3), i.e., a sinusoidal driving field coupled with a periodic width modulation, is analogously equivalent to the motion of a charged particle within a two-site crystalline potential, subject to the action of an external field of frequency $\mathrm{\Omega }=2\pi /P$. The condition $E_g=n\mathrm{\Omega }$ when the ratio $n$ is an integer, where $E_g=2\mathrm{\Delta }\beta$ represents the width of the gap in the band-gap diagram, corresponds to field-induced $n$-“photon” resonances between the two minibands. Light dynamics in the modulated waveguide array can be captured by the effective equations under this condition of n-“photon” resonances:

Then, we focus on the dispersion of coupling to investigate the wavelength dependence of waveguide superlattices with a well-defined AGF:

Intuitively, the effective coupling dispersion can be fully suppressed as long as $J_n$ and $J_{n-1}$ share the same signs. The unusual modulation-induced dispersion properties of this engineered waveguide array enable the elimination of wavelength dependence in crosstalk reduction. It is worth noting that the insensitivity to wavelength also indicates robust performance against changes in key structural parameters, such as waveguide spacing, the amplitude of the trajectory, and the widths of the waveguides. We calculate the derivative of effective coupling $c_{\mathrm{eff}}$ concerning $a$, the period of the binary array, as an example to clarify the mechanism of robustness:

Equation (6) verifies that this modulation can also mitigate the structural sensitivity to waveguide separation, provided that $J_n$ and $J_{n-1}$ have the same sign.

Based on these analyses, we design the AGF-enabled densely packed waveguide superlattices on a standard 800-nm-thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ platform with a silica upper cladding. We choose the waveguide widths of 1300, 1150, 1000, 850, and 1400 nm constrained by the single-mode condition at 1550 nm. The waveguide superlattices follow a sinusoidal curve, with modulation period $P$ fixed at $25\mu \mathrm{m}$, which satisfies the field-induced n-“photon” resonances of $n=10$. Figure 2(b) shows the $J_9$ and $J_{10}$ functions, and they indeed have the same sign in the marked region. It is found that $J_9$ and $J_{10}$ are positive integers arbitrarily close to 0 when the normalized modulation depth $\xi$ is in the dark red area. Near the zero effective coupling condition, i.e., $J_{10}\approx 0$, $J_9$ is also almost zero, which indicates that the intrinsic waveguide coupling and coupling dispersion are fully suppressed. Therefore, this design provides flexibility for engineering coupling coefficient and dispersion, thereby achieving broadband and robust zero-coupling performance for densely packed waveguides.

3.

Characterizations of the Waveguide Superlattices with AGF

We simulate light evolution in the modulated and normal (i.e., without modulation) waveguide array in Fig. 3, respectively, to make a clear comparison. The straight waveguide superlattice has the same width parameters as the AGF-enabled superlattice. The normal AGF waveguide array has been optimized to the best level, and the width of the single-mode waveguides is set to 1000 nm, the width of a standard single-mode waveguide. Due to the lower refractive index, the widths of single-mode ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides are larger than that of SOI waveguides at the same wavelength, thus their pitches are also larger. For the AGF-enabled superlattice in Fig. 3(d), the light remains localized in the input waveguide with almost no coupling to other waveguides at all three wavelengths (1310, 1550, and 1650 nm), owing to the bending-induced resonances between narrow and wide waveguides. The nearly isolated guidance indicates zero crosstalk between adjacent waveguides. It can be seen that our modulated waveguides exhibit high through transmission across a very broad bandwidth from the O-band to extend beyond the C-band. In contrast, if either form of the modulation (the width modulation or sinusoidal trajectory modulation) is removed, the light will couple with other waveguides, and the crosstalk between them will drastically increase. The crosstalk in normal AGF waveguides in Fig. 3(c) could be suppressed to some extent at 1550 nm, while the useable bandwidth is significantly limited (see Fig. S2 in the Supplementary Material). The conventional one in Fig. 3(a) exhibits a discrete diffraction phenomenon and drastic changes in coupling length attributed to the strong dispersion. Although the conventional superlattice and AGF waveguide array have successfully demonstrated low crosstalk on the standard SOI platform with air cladding at C-band, the deleterious effects of manufacturing defects are exacerbated by the high-refractive index contrast between silicon and air. This exposure becomes a barrier to heterogeneous integration, significantly limiting the scalability and integration of the photonic chip. In addition, the use of silica upper cladding could reduce phase errors, which is greatly needed in phase-sensitive applications, such as optical-phased arrays and high-speed modulators.³² The above-mentioned crosstalk suppression mechanisms in Figs. 3(a)–3(c) are not strong enough when meeting the platform with a low refractive index contrast and usually work only for a particular band, showing the weaknesses of these schemes.

Fig. 3

Simulated normalized field evolution in (a) normal waveguides, (b) ordinary superlattice, (c) AGF waveguide array, and (d) AGF-enabled superlattice, where gap = 400 nm, $L=200\mu \mathrm{m}$. Schematic top views of the simulated structures are located at the corners of each subimage.

The simulated transmission of the proposed dense waveguide array is depicted in Fig. 4. Most of the previous designs work for one polarization only (TE or TM polarization); the calculated effective indices of the fundamental TE and TM modes of 800-nm thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides are close enough to support dual polarizations (see Fig. S1 in the Supplementary Material). Although a slight increase in crosstalk can be observed at longer wavelengths, the crosstalk for the fundamental TE and TM modes remains $<25\mathrm{dB}$ in the spectrum of 700 to 1700 nm, opening up possibilities for colorless and low-crosstalk ultrahigh-density photonic integration.

Fig. 4

Simulated crosstalk in the designed waveguide array (gap = 400 nm, $\lambda =700$ to 1700 nm) when (a) fundamental TE mode and (b) fundamental TM mode are launched, respectively. The pink planes correspond to $-28$ and $-25\mathrm{dB}$, respectively.

In experiments, we characterize the proposed waveguide arrays with different pitches, i.e., gap = 400, 500, and 600 nm. The sample length is $200\mu \mathrm{m}$ for a gap of 400 nm and more than 1 mm for gaps larger than 400 nm. Figures 5(a) and 5(b) show the microscope image and the scanning electron microscope (SEM) image of a cross section of the fabricated device, respectively, where the width and sinusoidal trajectory variation can be observed. The edge couplers are used for light input and output, which could achieve broadband and dual-polarization operation. After the light is coupled into the chip (see Fig. S15 in the Supplementary Material), it propagates along the single-mode waveguide and enters the dense waveguide array horizontally. Figure 5(d) displays the experimentally captured optical propagations for the modulated samples at 850 nm. The red light was injected into the five input ports of the proposed waveguide array one at a time to test its functionality at 850 nm. The representative images of three light paths were selected to demonstrate that each channel can operate independently without coupling into the neighboring waveguides at this wavelength. Measured normalized transmission spectra of the AGF-enabled superlattice for gap = 400 and 500 nm are displayed in Figs. 6(a) and 6(b), respectively. Crosstalk suppression occurred over an ultrawide wavelength range from 1200 to 1700 nm, where the crosstalk is $<-24\mathrm{dB}$ in this entire 500 nm bandwidth. It can be seen that the crosstalk was decreasing as the waveguide array became sparse. Although the crosstalk suppression was strong enough for the most practical applications, it could be further reduced by increasing the bending radius of interfaces between the separated waveguides and the dense waveguide array. Moreover, the measured bandwidth of the device was limited by the device instead of the proposed AGF-enabled superlattice.

Fig. 5

(a) The top-view microscope image of the fabricated device on the ${\mathrm{Si}}_3{\mathrm{N}}_4$ platform ($\text{gap}=400\text{n}\mathrm{m}$, $L=200\mu \mathrm{m}$). (b) SEM image of the cross section of a waveguide ($\text{gap}=400\text{n}\mathrm{m}$, $L=200\mu \mathrm{m}$). (c) The top-view microscope image of the fabricated device on the ${\mathrm{Si}}_3{\mathrm{N}}_4$ platform ($\text{gap}=500\text{n}\mathrm{m}$, $L=1\mathrm{mm}$). (d) Experimentally recorded light trajectory in modulated samples for 850 nm.

Fig. 6

(a) Measured transmission spectra of the AGF-enabled superlattice ($\text{gap}=400\text{n}\mathrm{m}$). (b) Measured transmission spectra of the AGF-enabled superlattice ($\text{gap}=500\text{n}\mathrm{m}$). The pink planes correspond to $-24$ and $-28\mathrm{dB}$, respectively.

To further clarify the mechanism of robustness, we simulate the transmission spectra of the AGF-enabled superlattices with the variations of $A$ and $w$ at 50 and 25 nm in Figs. 7 and 8, respectively. The results show that despite introducing significant structural parameter changes, the waveguide arrays still exhibit excellent crosstalk suppression and negligible insertion loss across the entire frequency spectrum for all channels. The deliberately modified devices can keep crosstalk below $-20\mathrm{dB}$ with a bandwidth exceeding 500 nm, demonstrating the robustness of our scheme. This robustness ensures a large tolerance to dimensional uncertainties in the fabrication process, allowing scalability to large-scale circuits.

Fig. 7

Simulated crosstalk for AGF-enabled superlattice with (a) $\mathrm{\Delta }A=-50\mathrm{nm}$ and (b) $\mathrm{\Delta }A=+50\mathrm{nm}$. The pink planes correspond to $-20\mathrm{dB}$.

Fig. 8

Simulated crosstalk for AGF-enabled superlattice that widths of all waveguides (a) increase by 25 nm and (b) decrease by 25 nm. The pink planes correspond to $-20\mathrm{dB}$.

Alongside the measurements of loss and crosstalk, we also verified the ability of the proposed waveguide array to perform high-speed signal routing (see Note S8 in the Supplementary Material for more details on the experimental setup and digital signal processing techniques). The SNR profiles and the bit allocation were obtained and plotted in Figs. 9(a) and 9(b). The SNR profiles of the entire transmission link include the AWG, the electrical driver, the waveguide array, the modulator, the PD, and the digital sampling oscilloscope. The transmission through each channel of the waveguide array achieves nearly identical SNR performance to the BtB case, indicating excellent signal quality. The roughly unchanged end-to-end SNR profiles, combined with extensively suppressed SNR performance in the adjacent waveguides (see Fig. S12 in the Supplementary Material), suggest that the transmitted signals in the excited waveguide do not affect the neighboring waveguides, which is indicative of good isolation between waveguides and effective crosstalk suppression in the spectrum. In optical communication systems, it is desirable for each waveguide (channel) to operate independently without interfering with each other. SNR measurements can verify the signal isolation performance and spectral integrity of the proposed waveguide array, ensuring that each channel in a multichannel system can reliably transmit signals. The SNR performance was used to generate the DMT signal through the bit allocation algorithm, as shown in Fig. 9(b). The maximum bit allocation is 5, corresponding to 32-QAM. The constellations in the inset of Fig. 9(b) indicate good signal quality for high-order modulation formats (see Fig. S13 in the Supplementary Material). In addition, we conducted the high-speed 112 Gbit/s PAM4 signal transmission experiment through the AGF-enabled waveguide superlattice (see Fig. S14 in the Supplementary Material for details). All the waveguide channels present almost identical BER and SNR performances relative to the BtB case, and demonstrating the proposed low-crosstalk and ultracompact waveguide array is capable of high-speed on-chip transmission.

Fig. 9

(a) Measured SNR profiles of the system after transmission through the waveguide array. (b) The bit allocation of 112-Gbit/s DMT signals. Inset: constellations of different subcarriers.

4.

Discussion

The waveguides are perhaps the most fundamental building blocks in integrated photonics. However, coupling between closely packed waveguides has been a long-term challenge due to significant crosstalk at small pitches. The high-density waveguide arrays demonstrated in this study can effectively address this issue. The engineering capability of the exceptional coupling is presented both theoretically and experimentally, and it provides a new flexible toolbox for densely packed power-efficient photonic components. For example, shrinking the spacing of an OPA to half wavelength could lead to superior beam characteristics with a 180-deg field of view.³³ Dense waveguide arrays enable the development of highly efficient thermo-optic phase shifters and compact switches.³⁴ In addition, the penalty-free transmission of high-speed signals and the support of advanced modulation formats through the proposed array can stimulate new directions in broadband, compact, and high-speed optical modulators.

Unlike straight waveguides, the sinusoidal profiles of waveguides are prone to light leakage, causing additional propagation losses. As with other bent waveguides, the propagation loss generally increases when the bending radius of the waveguide decreases. The minimum bending radius of the AGF waveguides we used is calculated to be $57\mu \mathrm{m}$. It was reported that when the bending radius exceeds $50\mu \mathrm{m}$, the bending loss of one 90-deg bending waveguide basically remains below 0.0092 dB.³⁵ In sinusoidal waveguides, the curvature changes continuously, and both the first-order and second-order derivatives are also continuous. This characteristic results in a lower propagation loss compared to equivalent arc-shaped waveguides. Furthermore, it is important to highlight that the minimum bending radius occurs specifically at the inflection point of the sinusoidal profile. At this point, the average propagation loss of AGF waveguides with continuously varying curvature should be much smaller and acceptable (see Note S4 in the Supplementary Material for details). The sinusoidal curve is used here because it can introduce the AGF in a simple and easy-to-understand mathematical form. Specialized artificial gauge fields can indeed be constructed using alternative curve shapes, such as Bezier curves.³⁶ In fact, the optimized Bezier bend waveguide has lower propagation loss than the sinusoidal waveguide at the same small bending radius,³⁷ which may generate AGF for better performance in future works. Other curve shapes can also be optimized for various trajectories, potentially in collaboration with intelligent algorithms, such as inverse design methods¹⁴ and deep learning.³⁸ This can create AGFs that stimulate new directions in scientific studies at the subwavelength scale.

It is worth noting that the dimensional control requirements of the proposed approach, unlike the nanohole metamaterials,³⁹ are well within the capabilities of state-of-the-art foundries, making it amenable to a large-scale production.⁴⁰ The millimeter-level lengths of the waveguide array in this work are sufficient for most applications, such as ultradense wavelength-division multiplexers,⁵ on-chip spectrometers,⁴¹ and OPA.^42,43 The specially designed superlattice structure, combined with an AGF, enables precise control over the light propagation characteristics, achieving colorless and low-crosstalk optical transmission and laying the foundation for the miniaturization and cost-effectiveness of advanced photonic systems.

5.

Conclusion

In summary, we have experimentally demonstrated an ultra-broadband, low-crosstalk dual-polarization ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide superlattice leveraging AGF. The physical principle is attributed to the bending-induced resonances between the narrow and wide waveguides, corresponding to the field-induced n-“photon” resonances between the two mini bands, which results in a broadband zero-coupling effect. This coupling mechanism achieves a working bandwidth exceeding 500 nm for the crosstalk below $-24\mathrm{dB}$. Given its compatibility with BEOL processes, this well-designed waveguide array already serves as a robust solution for dense waveguide integration. When combined with the potential for transferability to other platforms, it will be an attractive strategy to significantly improve the waveguide density limit and performance capabilities of various active and passive photonic devices and systems, such as half-wavelength spacing OPAs, high-density on-chip optical interconnects, and energy-efficient modulators. Our proposed scheme also offers a versatile arrangement to study a range of fascinating scientific phenomena, including Rabi oscillations.