Bound States in the Continuum (BIC) Guided Modes in Photonic Integrated Circuits (PICs)

Introduction

Photonic integrated circuits (PICs) are miniaturized optical devices that integrate multiple photonic functions onto a single chip. They are essential components in various applications, including telecommunications, sensing, and quantum computing. A critical aspect of PIC design is achieving efficient light propagation within waveguides while minimizing losses. Traditionally, this has been achieved by using high-index contrast waveguides, but recent advancements in the field have introduced a novel concept called bound states in the continuum (BICs) to achieve low-loss waveguiding.

What are BICs?

In the realm of wave dynamics, a bound state in the continuum (BIC) refers to a localized wave state that coexists within a continuous spectrum of propagating states. Typically, such a scenario would lead to energy dissipation or exchange due to radiative coupling. However, BICs defy this convention by minimizing or even eliminating the mode overlap with the radiative states. This concept, first introduced by von Neumann and Wigner in 1929, has found applications in various wave phenomena, including acoustic waves, water waves, elastic waves in solids, and, notably, optical waves.

In the context of optics, the precise control over geometry at the wavelength scale allows for the engineering of the desired coupling strength as a function of wavelength, mode order, and polarization. Recent progress in integrated photonics nanofabrication has enabled the exploration of diverse BICs in various nanophotonic structures, including 2D planar structures, gratings, and waveguides. The unique behavior of BICs, characterized by sharp spectral features and long-lived bound states in the absence of other dissipation mechanisms, has been harnessed in applications such as lasing and sensing.

BICs in Waveguides

BICs in the form of 1D waveguide modes were first theoretically predicted in 1978 and subsequently demonstrated experimentally in radiofrequency and photonic waveguides. A BIC waveguide mode can be realized by combining a conventional ridge waveguide with a nearby slab waveguide, where the effective index of the ridge mode is lower than that of the slab modes. However, previous demonstrations of photonic BIC guided modes have been limited to orthogonal polarizations between the guided mode and the slab modes within specific platforms like polymer ridge or rib waveguides.

The absence of a systematic and general description has hindered the full exploration of BIC waveguiding applicability to different modal polarizations and spatial geometries in PIC platforms using silicon and silicon nitride, as well as various electro-optic and nonlinear materials and their combinations. BIC waveguides offer new possibilities in photonics design by providing guided modes with unconventional field distributions, such as concentrating mode fields away from the highest index materials. These modes can be highly beneficial if their radiation losses are suppressed to levels comparable to conventional guided modes.

BICs in Heterogeneously Integrated Thin-Film Lithium Niobate Platform

In a heterogeneous integrated thin-film lithium niobate (TFLN) platform, BICs can exist in single, double, or multiple-ridge waveguide configurations. Let's consider a single rectangular silicon nitride (Si3N4) buried channel waveguide in proximity to a continuous thin-film lithium niobate (LN) slab. This configuration leads to the emergence of a lossless guided LN bound TE00 mode within the slab, while other hybridized modes overlap with the slab mode continuum and become lossy through radiation into the slab modes.

The ridge waveguide mode experiences the strongest coupling to the modes propagating at an angle θ to the waveguide, satisfying the phase-matching condition. In the cross-section normal to the propagation direction, the slab mode is a laterally oscillating wave, while the ridge modes have a lateral structure determined by the waveguide width for each polarization and transverse modal order. Due to the oscillating structure of the continuum modes, their mode overlap integral with the waveguide mode contains multiple zero-crossings as a function of the waveguide geometry. At these zero-crossings, the hybrid ridge mode becomes a BIC, decoupled from the radiating slab continuum, and lateral dissipation is prohibited.

Fig. 1: Bound States in Continuum Modal Properties (a) Illustrates a single ridge waveguide mode (Si3N4) interacting with a nearby slab waveguide mode (LiNbO3) with 3D FDTD simulation results showing the electric field (Ez) distributions at a leaky resonant condition. The slab waveguide mode, phase-matched with the hybridized ridge waveguide mode, efficiently couples and dissipates energy. (b) Lateral wave dissipations are blocked when a lateral phase shift (kszw) of the slab mode satisfies the BIC condition, resulting in zero overlap between the ridge and slab modes. (c) Shows the simulated effective refractive index (neff) and corresponding electric field (E^2) of various modes at specific dimensions and wavelength (1550 nm). The shaded region indicates the existence of the continuous spectrum of TE slab modes, with the width of the red shaded regions along the refractive index curves indicating the imaginary part of the effective indices, scaled for each mode.

Mathematical Modeling of BICs

The interaction between a bound mode and radiating slab modes can be quantitatively modeled using coupled-mode theory (CMT). This theory couples an oscillator (bound mode) with adjacent multiple radiating modes (slab modes). The resulting power propagation exponential loss length L can be expressed mathematically, and this equation indicates that the BIC condition is satisfied at multiple waveguide widths where the cosine term in the numerator is zero, corresponding to zero mode overlap.

Experimental Demonstration of BICs

In the experimental demonstration, an x-cut TFLN with a horizontal configuration of Au electrode on top is used to exploit the highest electro-optic coefficient (r33). A defect-free TFLN layer bonding is achieved using a thin atomic layer deposited (ALD) Al2O3 as an intermediate layer. The experimentally measured TE10 BIC mode propagation losses agree with the finite element eigenmode analysis simulation and the theoretical prediction, clearly showing the minima for the waveguide widths satisfying the BIC condition. This confirms that the main loss channel of the BIC mode is radiative coupling to the slab continuum.

The modeling indicates that the propagation loss of a TE quasi-BIC mode at wavelengths around 1550 nm can be less than 3 dB/cm over a wavelength bandwidth of up to approximately 100 nm and waveguide width variation of up to approximately 100 nm.

Fig. 2: Demonstration of TE10 BIC Mode in Heterogeneously Integrated Lithium Niobate Hybrid Waveguide (a) Illustration of the fabricated waveguide showing input light injected into the silicon nitride waveguide, meeting the lithium niobate bonded area where the refractive index is modulated by an external electric field. (b) Propagation loss for different waveguide widths: the black solid line represents coupled mode theory, orange circles indicate finite element eigenmode numerical analysis, and red triangles show experimental results. Multiple quasi-BIC conditions are observed with a period of approximately 2.2 µm. The 20 dB/cm dashed line indicates the highest experimentally measurable propagation loss. (c) False-colored scanning electron microscopy image of the polished cross-section of the fabricated waveguide, showing a defect-free bonding interface. (d-f) Numerically calculated electric field (Ez) profiles for the first BIC condition, leaky resonant condition, and the second BIC condition. (g) Numerically calculated propagation loss as a function of waveguide width and optical vacuum wavelength. The red color indicates the condition where the propagation loss is at 3 dB/cm. (h) Stitched optical microscope image of the fabricated device, showing the lithium niobate bonded in the middle of the chip with a nominal length of 17 mm, while the rest of the area is free from lithium niobate. The gold electrode has a nominal thickness of 900 nm and a nominal gap of 2.85 µm.

BIC-Based Mach-Zehnder Amplitude Modulator

To showcase the potential of BICs, a Mach-Zehnder interferometer (MZI) is realized with a simple straight silicon nitride waveguide with a bonded lithium niobate slab. The waveguide mode experiences strong modal perturbation at the first edge of the lithium niobate slab, leading to coupling into a new basis set of waveguide eigenmodes. Most power couples into either the fundamental TE00 LN bound mode (LN channel) or the quasi-BIC TE10 waveguide mode (BIC channel), while the remaining power becomes scattering loss.

The two non-interacting normal mode channels can coexist in the same geometry without an additional spatially separate waveguide due to the engineering of the low-loss BIC guided mode uncoupled from the radiative slab continuum. The abrupt transitions at the bonding edges serve as simple and compact directional couplers, avoiding conventional design challenges in heterogeneous integration of TFLN.

The BIC-based MZI (BIC-MZI) is functionally analogous to an MZI with asymmetric channels: the fundamental TE00 mode for the upper channel and the quasi-BIC waveguide mode for the lower channel. These waveguide eigenmodes originate from the hybridization of the coupled buried channel and slab TE modes, resulting in an apparent electric field beating pattern as the modes propagate. The beating pattern indicates the interference between two propagating guided modes of different effective indices: the BIC TE10 mode and the TE00 mode.

Electro-optic amplitude modulation is achieved by differential EO phase modulation of the two normal modes forming the MZI, due to their different electric field overlap with the lithium niobate layer. The BIC mode experiences weaker modulation compared to the LN mode. The highest transmission at an optimized waveguide width and optical wavelength is measured, equivalent to the intrinsic modulator insertion loss. The propagation losses of the BIC modes for various waveguide widths are extracted from the measured transmission using an analytic model of the MZI.

The electro-optic response of the BIC-MZI is characterized over a modulation frequency range, and the 3 dB bandwidth is measured. In summary, engineering a low-loss BIC mode enables the use of a single Si3N4 ridge to form the entire MZI amplitude modulator, avoiding aligned patterning and etching of the LN layer or designing and fabricating physically separate directional couplers.

Fig. 3: Mach–Zehnder Modulator Based on Quasi-BIC (a) Schematic of the BIC-MZI modulator: At the edge of the bonded lithium niobate slab, input light is coupled into two waveguide normal modes due to the abrupt transition in the modal bases, serving as directional couplers. The electric field profile shows a beating pattern formed by the fields of the two overlapping, non-interacting normal modes, with power oscillating between lithium niobate and silicon nitride regions. (b) Equivalent BIC-MZI model: Two asymmetric normal mode channels with different effective refractive indices but the same physical length. The LN mode (TE00) has power localized in lithium niobate, while the BIC mode (TE10) is localized in the Si3N4 ridge. Insets show numerically calculated mode power distributions. Directional couplers represent the bonding edges in the LN slab. Output lensed fiber collects light from the Si3N4 ridge waveguide. (c) Measured extinction ratios and propagation losses: Orange circles (left y-axis) show extinction ratios over transmission spectra for various waveguide widths, and blue circles (right y-axis) show corresponding propagation losses. Balanced powers from the two channels result in a high extinction ratio. (d) Measured total transmission: Includes fiber-to-chip coupling efficiency as a function of waveguide width and wavelength. The inset shows the transmission spectrum of the device with the highest transmission. (e) Measured optical transmission by electro-optic modulation: A sawtooth waveform was applied as a driving RF signal to evaluate a half-wave voltage, with the inset showing a log scale transmission indicating a 25 dB extinction ratio. (f) Measured electro-optic response: Characterized with an RF spectrum analyzer, showing a 3 dB bandwidth measured at 36 GHz.

Double Waveguide BIC

In the case of a double or multiple waveguide configuration, BICs can be achieved for specific supermodes by varying the waveguide spacings between adjacent waveguides. The individual waveguide widths no longer need to satisfy the BIC condition when multiple waveguides are coupled via the slab. Instead, for weak slab coupling, these waveguides can be considered individual oscillators leaking into a common dissipating channel (the lithium niobate layer).

The radiating waves from the left-side and right-side waveguides can destructively interfere at the outer edges, leading to double waveguide BICs. The BICs are satisfied when the phase shift along the channel between two waveguides becomes a multiple of π. These conditions are Fabry-Perot type BICs and show good agreement with simulated and experimental results. The effective indices oscillate as the waveguide spacing increases, and there are two distinct regimes for coupling: evanescent wave coupling for waveguides in proximity and coupling through the channel for larger spacings. The BIC conditions are located where the effective refractive indices of two waveguide supermodes become equal.

Fig. 4: Double Waveguide BIC Modal Properties (a) Schematic of two ridge waveguides (Si3N4) with a 180° out-of-phase condition interacting with a nearby slab waveguide (LiNbO3). Dissipating waves from the left (L) and right (R) ridge waveguides interfere, resulting in destructive interference outside the edges. Complete destructive interference (BIC condition) occurs when the phase shift along the z-axis (kszd) satisfies 2mπ for an anti-symmetric mode and (2m + 1)π for a symmetric mode. (b, c) Propagation losses and effective refractive indices: Anti-symmetric and symmetric modes calculated by coupled mode theory (black solids), eigenmode analysis (red and blue triangles), and experimental results (green circles) for w = 2.5 µm. BICs result from interactions of two oscillators coupled through radiating channels, showing oscillating patterns in effective indices and propagation losses. Experimental results represent on-chip insertion loss without fiber-to-chip losses (y-axis on the right). Uncertainties are smaller than symbol size. Other data and curves are plotted with the y-axis on the left. (d-f) z-component of the electric field (Ez) profiles for the anti-symmetric BIC condition, symmetric leaky resonant condition, and symmetric BIC condition.

Conclusion

This tutorial has provided an overview of bound states in the continuum (BICs) in photonic integrated circuits (PICs). We have discussed the concept of BICs, their mathematical modeling, and their experimental demonstration in both single and double waveguide configurations. The use of BICs in designing a Mach-Zehnder amplitude modulator has also been highlighted, showcasing the potential of BICs in advancing PIC technology.

Reference

[1] K. Han, T. W. LeBrun, and V. A. Aksyuk, "Bound-state-in-continuum guided modes in a multilayer electro-optically active photonic integrated circuit platform," *Optica*, vol. 11, no. 5, pp. 706-713, May 2024. doi: 10.1364/OPTICA.516044.