The first half of the thesis analyzes optical solitons as a result of Kerr nonlinearity, including its universal scaling, its dynamics in the presence of laser feedback, the analytical properties of its relativistic counterpart, as well as its applications as a wavelength reference. The second half of the thesis focuses on stimulated Brillouin lasers and their linewidth performance, demonstrating new levels of Brillouin laser performance and two correction factors to its linewidth that have been established for semiconductor lasers.
INTRODUCTION
- Optical microresonators and their characterization
- The mode equation
- Kerr nonlinearities, frequency combs, and optical solitons
- Brillouin scattering and stimulated Brillouin laser
- Overview of chapters
Effectively, the external coupling creates another loss channel for the resonator, which adds to the intrinsic loss and determines the overall loss of the mode. Chapter 9 studies the linewidth enhancement factor (also known as Henry factor) of the Brillouin laser.
GREATER THAN ONE BILLION 𝑄 FACTOR FOR ON-CHIP MICRORESONATORS
The increasing dependence of the intrinsic 𝑄 factor with increasing resonator diameter has previously been observed in the wedge resonator structure and results from the round-trip loss. To perform this measurement, the pump line was filtered using a fiber Bragg grating, as shown in the setup.
UNIVERSAL ISOCONTOURS FOR DISSIPATIVE KERR SOLITONS
In parallel with the iso-power data point collection, the soliton pulse width was also measured by fitting the optical spectral envelope [15]. Detuning-dependent properties and dispersion-induced instabilities of transient dissipative Kerr solitons in optical microresonators.
INTEGRATED TURNKEY SOLITON MICROCOMBS
Main results
Due to the thermal hysteresis [27] and the abrupt flow discontinuity within the cavity at the transition to the soliton regime (Fig. 4.2c). And the stable soliton emission is further confirmed by monitoring the real-time evolution of the soliton repetition frequency signal (Fig. 4.2c). 4.2f).
Methods
We note that the equations effectively refer to the frequency of the injection-locked laser instead of the free-running laser, which will simplify the following discussions. Bottom panel: snapshots of the soliton field at development time𝜏 =350 (gray dashed line) and𝜏=380 (black solid line), also marked as white dashed lines in the top panel and black dashed lines in the middle panel.
Supplementary information: Additional measurements Different types of microcombs in the injection locking systemDifferent types of microcombs in the injection locking system
To further investigate the performance of the key-single microcomb system, the frequency of the pump laser is driven by a linear current scan (Fig. 4.9b,c). When the laser is scanned across the resonance, feedback locking occurs and pulls the pump laser frequency toward the resonance until the drive frequency is out of the lockband.
TOWARDS VISIBLE SOLITON MICROCOMB GENERATION
- Silica resonator design
- Soliton generation at 1064 nm
- Soliton generation at 778 nm
- Discussion
The measured frequency spectrum of the TM1 mode family in the 3.4 µm thick resonator is plotted in the figure. A magnification of the TM1 mode spectrum for 𝑡 = 1.47 µm with a fit that includes third-order dispersion (red curve) is shown in Fig. The electrical spectrum of the photo-detected soliton current is given in the inset of Fig.
Finally, the detected beat note of the soliton and dispersive wave is shown as the inset in Fig.
DIRAC SOLITONS IN OPTICAL MICRORESONATORS
Polarization mode coupling and coupled LLEs
Loosely speaking, the hyperbolic shape of the natural frequency creates an anomalous spreading window suitable for soliton generation. In general, and as noted in the introduction, the dispersion in this spectrum is only locally well defined, because the mode composition of the hybrid mode can change rapidly with respect to wavenumber. These rapid compositional changes in the hybridized modes redistribute the pulse energy in the frequency domain and produce a new contribution to the chirp within the pulse, manifesting as phase differences between the two mode components of the pulse.
The "mass" of the DS field is the inter-mode coupling, and the mode spectrum corresponds to a relativistic field theory.
Closed-form soliton solutions
Same as panel (a), but showing only the range for the soliton resonance line with an arbitrary fixed positive repetition rate shift, 𝑣. c) Real part (orange line) and imaginary part (green line) of the 𝐸1 component of the DS at𝛿𝜔=−0.9𝑔c. This phase twist within the pulse contributes to the chirping and shifting of the soliton pulse when they are coupled together, as discussed in the previous section. This phenomenon can be explained by different mode compositions at the different ends of the spectrum.
In the general case of 𝑣 ≠ 0, the total power spectrum is expected to become asymmetric around the center frequency of the soliton.
DS with dissipation and repetition rate shifts
The inset shows a comparison of the simulated (orange solid) and analytical (black dashed) pulse shape |𝐸1| at𝛿𝜔 =0. c) Graph of repetition rate shifts on the mode spectrum. For a fixed declaration 𝛿𝜔, the displacement of the repetition rate 𝑣 depends on the ratios of the nonlinear coefficients 𝑔11, 𝑔22 and 𝑔12. Coupled LLE simulations were also performed, showing that the simulated pulse shape and repetition rate shifts match the analytical solutions (Figure 6.3b).
DS provides a new and controllable way to tune the repetition rate of frequency combs.
Implementation of Dirac solitons
Since the repetition rate offset is a result of the imbalance of self-phase modulations, increasing the proportion of 𝑔12 leads to more stability in the repetition rate, while reducing the proportion of 𝑔12 allows more tuning. We note that depending on the nonlinear nature of the resonator material and the mode overlap, the cross-phase modulation can be larger or smaller than the self-phase modulation [52, 53]. In the vicinity of 𝛿𝜔 =−𝑔c, the repetition rate offset approaches zero, which is consistent with the local KS equivalence argument in the previous section.
On the other hand, if we explicitly break the reflection symmetry of the resonator by decreasing the wedge angle (𝛼 <90◦ ), the original modes will see an asymmetric change in the refractive index profile, causing them to couple.
Demonstration of Dirac solitons
Discussion
The two dashed lines indicate the second-order dispersion of the TE2 (left) and TM1 (right) fundamental modes without crossing. The blue dashed line shows the sech2 fit and the black dashed line the DS fit. e) Simulated modal spectrum for a wedge resonator with mode hybridization at about 532 nm. The inset shows the geometry of the resonator. f) Simulated DS spectrum at 532 nm generated in the wedge resonator in (e).
The solid line shows the spectrum when the system is pumped at the crossover center, while the dashed line shows the spectrum when the system is pumped 200 modes away from the crossover center.
Methods
A quick plot of the range (Fig. 6.2b) shows that the boundaries are tangent to the mode spectrum curves. Finally, the effective nonlinear coefficient 𝐺(𝛿 𝐷2 .. 1) can be calculated as a weighted average of the nonlinear coefficients, the weight being the power proportions on each mode derived above. To proceed further, we take the soliton ansatz as the exact solution of the DS derived earlier.
Inserting the wedge angle adds a dielectric triangle to the lower right and subtracts a dielectric triangle to the upper right (Fig. 6.6).
Supplementary information
This happens when |𝐸1|0and |𝐸2|0 are both zero or both non-zero. the two components in the background are completely in-phase or out-of-phase with respect to the mode coupling. The fields at each𝜃 correspond to a point in the diagram and follow a contour defined by constant ¯𝐻and ¯𝑁as𝜃varies. Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer.
55] Bao, C. et al. Observation of Fermi-Pasta-Ulam repetition induced by breathing solitons in an optical microresonator.
VERNIER SPECTROMETER USING COUNTERPROPAGATING SOLITON MICROCOMBS
Main results
The spectral ratio of doubly locked cw and ccw solitons reveals the inherent optical Vernier frequency (Figure 7.1A). In real time, the measured laser wavelength (Figure 7.1F) fluctuates within ±0.02 pm in a time interval of 5 ms. The measurement of linear frequency chirp (-12.4 THz/s) as well as frequency versus time at high resolution (subtracting the average linear frequency ramp) is shown in Fig.
The bottom panel of Figure 7.2B shows a higher resolution magnification of one of the stepped regions.
Supplementary information Acquisition of correlation signalAcquisition of correlation signal
If this is not the case, artifacts will appear in the spectrogram due to the distortion of the frequency grids (Fig. 7.5D). A portion of the laser signal is also measured in the MSS to determine its frequency during scanning. This accuracy results from the fundamental relative high frequency stability of the cw and ccw microcomb frequencies.
Continuing in this way for each pair of peaks in Fig. 7.6A allows 𝑛 to be determined, from which the frequency of the corresponding FMLL line can be determined.
TOWARDS MILLI-HERTZ LASER FREQUENCY NOISE ON A CHIP
- Background on Brillouin cascade
- Pump and SBL mode selection
- Cross-correlation method
- Frequency noise measurement
- Discussion
The Brillouin laser wave propagating in the opposite direction to the pumping is collected by a circulator. Inset: low-offset frequency portion of the frequency noise as measured by the phase noise analyzer. Using the basic linewidth formula for the Brillouin laser [6], the two-sided frequency noise of the laser is given by,.
The frequency noise of the pump laser (Newport, TLB-6728) was measured to be 90 Hz2/Hz at a lag frequency of 1 MHz.
LINEWIDTH ENHANCEMENT FACTOR IN A MICROCAVITY BRILLOUIN LASER
Main results
The SBL emission propagates against the direction of pumping due to the phase-matching condition. Finally, 𝛾 is determined by measuring the cavity width at each wavelength (equivalent, the total 𝑄 factor 𝑄T of the resonator). Both this and the linewidth contribution of the pump phase noise are discussed in Supplementary information.
The sign of 𝛼 can also be controlled via the sign of the frequency mismatch resolution.
Supplementary information
For a system with Lorentzian gain, the imaginary part of the gain-induced susceptibility can be written as With the Kramers-Kronig relations, 𝜒I necessarily leads to the real part of the susceptibility 𝜒R through the relation. Here we study the effect of the anti-Stokes process on the 𝛼 factor in a cascaded Brillouin laser system.
Using the techniques from Chapter 3, the linewidth of the SBL can be found, down to the zeroth order of 𝛾/Γ, axis.
PETERMANN-FACTOR SENSITIVITY LIMIT NEAR AN EXCEPTIONAL POINT IN A BRILLOUIN RING LASER
GYROSCOPE
Main results
Δ𝜔Sis measured as the beat frequency of the SBL laser signals at photodetection and the SNR is set by the laser linewidth. Because the laser linewidth can be understood as a result of diffusion of the phasor in Fig. 10.1e, the linewidth increases with operation near the EP. 𝑆𝜈/(2𝜏) where 𝑆𝜈 is the one-sided power spectral density of the white frequency noise plotted in panel (b). b) Measured white frequency noise of the beat signal determined using the Allan deviation measurement.
The center of the locking region is also shown and is shifted by the Kerr nonlinearity which varies as the SBL power difference.
Methods
The apparent divergence of the linewidth near the EP is an interesting feature of the present model and also one that agrees well with the data (at least in the measurement range). However, the limitations on this divergence imposed by the passive cavity loss linewidth are a topic of further study. The theoretical formula for the white frequency noise of the beat frequency far from the EP reads.
In the expression, 𝑁th and 𝑛th are the numbers of the thermal occupation of the SBL state and the phonon state, respectively.
Supplementary information
1i is reduced from the true squared amplitude hΨ|Ψi by a factor of the squared length of the right-hand eigenvector𝜓R. The equation of motion for 𝑐1reads 𝑖. 10.29). Taking into account the fluctuations, the equation of motion for 𝑐1 can be modified as follows,. 10.30), shows that the noise input to the amplitude of the right eigenvector field (𝑐1) is increased (relative to the noise input for cw or ccw fields alone) by a factor of the squared length of the left eigenvector h𝜓L.
The center of the locking band can be found by setting Δ𝜔D =0, which leads to Δ𝜔P =−(Γ/𝛾)Δ𝜔Kerr.
