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.
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.
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.
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.
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.
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