Mode-locking is a process in which different modes of an optical resonator establish, through nonlinear interactions, stable synchronization. This self-organization underlies light sources that enable many modern scientific applications, such as ultrafast and high-field optics and frequency combs. Despite this, mode-locking has almost exclusively referred to self-organization of light in a single dimension -time. Here we present a theoretical approach, attractor dissection, for understanding three-dimensional (3D) spatiotemporal mode-locking (STML). The key idea is to find, for each distinct type of 3D pulse, a specific, minimal reduced model, and thus to identify the important intracavity effects responsible for its formation and stability. An intuition for the results follows from the "minimum loss principle," the idea that a laser strives to find the configuration of intracavity light that minimizes loss (maximizes gain extraction). Through this approach, we identify and explain several distinct forms of STML. These novel phases of coherent laser light have no analogues in 1D and are supported by experimental measurements of the three-dimensional field, revealing STML states comprising more than 10 7 cavity modes. Our results should facilitate the discovery and understanding of new higher-dimensional forms of coherent light which, in turn, may enable new applications.The supplementary material in this document is organized as follows.Section 1 provides a description of spatiotemporal mode-locking (STML) in the frequency domain, in terms of the resonant frequencies of the cavity's modes.Section 2 details our primary numerical models, based on propagation of the laser field through the gain medium using generalized nonlinear Schrödinger equations (NLSEs), plus additional effects. These models include most relevant effects, but our use of them mainly focuses on the computationally-efficient case where only a small number of transverse modes are considered. These models are generalizations of the most widely used models for describing ultrafast lasers and nonlinear pulse propagation in the modern literature.Section 3 details the treatment of individual effects within the cavity in the simplified description of STML outlined in the paper. Specifically, we develop the nonlinear projection operations for each relevant effect and show the calculation of the attractors for that effect through Eqn. 2 in the main article.Section 4 describes reduced models which incorporate a subset of the effects considered in the primary numerical models, by combining components described in Section 3. Extending the approach from Section 3, these models are quite approximate as whole-laser simulations but are computationally compact enough to be applied to conditions relevant to our experiments in a 90transverse mode fiber, and simple enough to be interpreted easily. Accordingly, they help bridge the gap between experiments and the nonlinear wave physics of STML studied primarily in the few-mode case.Section 5 summarizes relevant findin...

Narrow linewidth visible light lasers are critical for atomic, molecular and optical (AMO) physics including atomic clocks, quantum computing, atomic and molecular spectroscopy, and sensing. Stimulated Brillouin scattering (SBS) is a promising approach to realize highly coherent on-chip visible light laser emission. Here we report demonstration of a visible light photonic integrated Brillouin laser, with emission at 674 nm, a 14.7 mW optical threshold, corresponding to a threshold density of 4.92 mW μm−2, and a 269 Hz linewidth. Significant advances in visible light silicon nitride/silica all-waveguide resonators are achieved to overcome barriers to SBS in the visible, including 1 dB/meter waveguide losses, 55.4 million quality factor (Q), and measurement of the 25.110 GHz Stokes frequency shift and 290 MHz gain bandwidth. This advancement in integrated ultra-narrow linewidth visible wavelength SBS lasers opens the door to compact quantum and atomic systems and implementation of increasingly complex AMO based physics and experiments.

Laser stabilization sits at the heart of many precision scientific experiments and applications, including quantum information science, metrology, and atomic timekeeping. Many of these systems narrow the laser linewidth and stabilize the carrier by use of Pound–Drever–Hall (PDH) locking to a table-scale, ultrahigh quality factor (Q), vacuum spaced Fabry–Perot reference cavity. Integrating these cavities to bring characteristics of PDH stabilization to the chip scale is critical to reducing their size, cost, and weight, and enabling a wide range of portable and system-on-chip applications. We report a significant advance in integrated laser linewidth narrowing, stabilization, and noise reduction by use of a photonic integrated 4.0 m long coil resonator to stabilize a semiconductor laser. We achieve a 36 Hz 1 / π -integral linewidth, Allan deviation of 1.8 × 10 − 13 at 10 ms measurement time, and a 2.3 kHz/s drift—to the best of our knowledge, the lowest integral linewidth and highest stability demonstrated for an integrated waveguide reference cavity. This performance represents over an order of magnitude improvement in integral linewidth and frequency noise over previous integrated waveguide PDH stabilized reference cavities and bulk-optic and integrated injection locked approaches, and over two orders of magnitude improvement in frequency and phase noise than integrated injection locked approaches. Two different wavelength coil designs are demonstrated, stabilizing lasers at 1550 nm and 1319 nm. The resonator is bus-coupled to a 4.0 m long coil, with a 49 MHz free spectral range, mode volume of 1.0 × 10 10 µ m 3 , and 142 million intrinsic Q , fabricated in a CMOS compatible, ultralow loss silicon nitride waveguide platform. Our measurements and simulations show that the thermorefractive noise floor for this particular cavity is reached for frequencies down to 20 Hz in an ambient environment with simple passive vibration isolation and without vacuum or thermal isolation. The thermorefractive noise limited performance is estimated to yield an 8 Hz 1 / π -integral linewidth and Allan deviation of 5 × 10 − 14 at 10 ms, opening a stability regime that heretofore has been available only in fundamentally non-integrated systems. These results demonstrate the potential to bring the characteristics of laboratory-scale stabilized lasers to the integrated, wafer-scale compatible chip scale, and are of interest for a number of applications in quantum technologies and atomic, molecular, and optical physics, and with further developments below 10 Hz linewidth, can be highly relevant to ultralow noise microwave generation.

We fabricate silicon waveguides in silicon-on-insulator (SOI) wafers clad with either silicon dioxide, silicon nitride, or aluminum oxide and, by measuring their electro-optic behavior, we characterize the capacitively induced free-carrier effect. By comparing our results with simulations, we confirm that the observed voltage dependences of the transmission spectra are due to changes in the concentrations of holes and electrons within the semiconductor waveguides and show how strongly these effects depend on the cladding material that comes into contact with the waveguide. Waveguide loss is additionally found to have a high sensitivity to the applied voltage, suggesting that these effects may find use in applications that require low- or high-loss propagation. These phenomena, which are present in all semiconductor waveguides, may be incorporated into more complex waveguide designs in the future to create high-efficiency electro-optic modulators and wavemixers.

Optical resonator-based frequency stabilization plays a critical role in ultra-low linewidth laser emission and precision sensing, atom clocks, and quantum applications. However, there has been limited success in translating traditional bench-top stabilization cavities to compact on-chip integrated waveguide structures that are compatible with photonic integration. The challenge lies in realizing waveguides that not only deliver low optical loss but also exhibit a low thermo-optic coefficient and frequency noise stability. Given the problematic sources of frequency noise within dielectrics, such as thermorefractive noise, resonators with small thermo-optic response are desirable for on-chip reference cavities. We report the first demonstration of a Ta2O5 (tantala) waveguide core fabricated on a crystal quartz substrate lower cladding with TEOS-PECVD SiO2 upper cladding. This waveguide offers significant advantages over other waveguides in terms of its low thermo-optic coefficient and reduced thermorefractive-related frequency noise. We describe the waveguide structure and key design parameters as well as fabrication considerations for processing tantala on quartz waveguides. We report a waveguide thermo-optic coefficient of −1.14 × 10−6 RIU/K, a value that is over 6 times smaller in magnitude than that of SiO2-substrate tantala waveguides, with a propagation loss of 1.19 dB/cm at 1550 nm and <1.33 dB/cm across the 1525 nm–1610 nm wavelength range. Within a 1.6 mm radius ring resonator, we demonstrate a 2.54 × 105 intrinsic Q factor. With the potential for very low loss and the ability to control the thermal response, this waveguide platform takes a key step toward creating thermally stable integrated resonators for on-chip laser frequency stabilization and other applications.