## 1 Introduction  
Optical delay lines (ODLs) are key components in many applications, such as optical signal processing and buffering in optical networks, beamforming and filtering in microwave photonic systems, biomedical sensing, and 3D light detection and ranging (LiDAR) [1; 2; 3; 4; 5; 6]. Tunable optical true-time delay lines allow flexible control of the propagation delay of an optical signal. Tunable ODLs can be implemented using bulk optics, switched fiber spools, or integrated photonic chips [5]. The attractive possibility of the on-chip integration of many optical functionalities into a compact module has been driving the photonic integrated circuits (PICs) to realize a collection of devices for optical synchronization, buffering, multiplexing, and true-time delay in versatile applications for the last two decades. Integrated on-chip ODLs exhibit multiple advantages compared to bulk optics or fiber-based ODLs, such as a reduced cost, size, weight, and power consumption [3; 4; 5; 6; 7; 8; 9; 10; 11; 12]. Ring resonators and photonic crystals are the most common elements used to construct controllable PICs, able to continuously tune the group delay up to several hundreds of picoseconds and even a few nanoseconds [4]. Ring resonators exploit their resonant spectral behavior to achieve compact-footprint devices at the expense of narrow bandwidth [13]. The miniaturization of ODL components also improves the tuning speed. In an integrated chip, the ODLs can be conveniently combined with other functional devices such as optical modulators, filters, lasers, and photodiodes, offering more powerful optical and microwave processing capabilities than a single component does [1]. The integrated optical delay lines have been demonstrated for a variety of applications [14; 15; 16; 17; 18; 19]. Many different integration platforms can be utilized to make ODLs. Basic trade-offs between the integration density and waveguide propagation loss need to be considered before choosing the proper platform. The silicon photonics platform based on the silicon-on-insulator (SOI) is regarded as one of the most promising technologies for large-scale high-density photonic integration because of its large index contrast, small bending radius, and use of compatible fabrication facilities in semiconductor foundries [5; 17]. Currently, there are several semiconductor foundries and research institutes providing the multi-project-wafer (MPW) process for SOI-based photonic integration, which can reduce the research and development cost for realizing tunable ODLs. Therefore, our design is based on the SOI platform. There are two commonly used approaches to tune the SOI-based ODLs: Tuning with thermal heating or carrier-induced effects. Both tuning mechanisms change the refractive index of the waveguide and then change the phase. However, the carrier-induced effect via carrier injection or depletion requires the formation of a PN junction in the waveguide, and the refractive index change is associated with a loss change. Therefore, thermal tuning is usually the choice for its simple device structure and low optical loss as long as the tuning speed is not the major concern. In fact, the thermal tuning speed can be as short as a few \(\upmu\)s for the local heating of a compact SOI waveguide [20]. The high thermal resistance of the silica cladding on a SOI structure can also reduce the thermal crosstalk and power consumption. Tunable ODLs based on ring resonators have been demonstrated by several groups [21; 22; 23; 24; 25]. Most of the designs are for optical signal processing or buffering. In these applications, the critical requirements include a large enough true delay, wide bandwidth, and low higher-order phase variations that will lead to dispersion and signal distortions. Therefore, multiple stages of ring resonators are employed to achieve the desired delay with large bandwidth and low distortion [4]. However, the manufacturing variations in the associated multiple rings will result in fluctuations in the device characteristics.