Mode Spectrum and Temporal Soliton Formation in Optical Microresonators

The formation of temporal dissipative solitons in optical microresonators enables compact, high-repetition rate sources of ultrashort pulses as well as low noise, broadband optical frequency combs with smooth spectral envelopes. Here we study the influence of the microresonator mode spectrum on temporal soliton formation in a crystalline ${\mathrm{MgF}}_2$ microresonator. While an overall anomalous group velocity dispersion is required, it is found that higher order dispersion can be tolerated as long as it does not dominate the resonator’s mode structure. Avoided mode crossings induced by linear mode coupling in the resonator mode spectrum are found to prevent soliton formation when affecting resonator modes close to the pump laser frequency. The experimental observations are in excellent agreement with numerical simulations based on the nonlinear coupled mode equations. The presented results provide for the first time design criteria for the generation of temporal solitons in optical microresonators.

Figure 1

(a) Measured optical spectrum with smooth ${\mathrm{sech}}^2$-shaped spectral envelope (red line) of a single temporal soliton generated in a continuous wave laser driven crystalline high-finesse ${\mathrm{MgF}}_2$ microresonator. The spectral 3-dB width of 13 nm (1.62 THz) corresponds to a soliton pulse duration of 194 fs (full width at half maximum). The pump power is 30 mW at a wavelength of 1552 nm. The inset shows the resolution bandwidth (RBW) limited rf signal at a frequency of 14.09 GHz corresponding to the soliton pulse repetition rate. (b) Frequency resolved optical gating (FROG) of the ultrashort pulses outcoupled from the resonator. The pulses are separated by 71 ps corresponding to the inverse repetition rate (The FROG measurement uses second harmonic generation in a beta barium borate crystal. The second harmonic frequency ${\omega }_{\mathrm{SHG}}/2\pi$ is centered around 384 THz. After attenuation of the pump laser, the remaining soliton spectrum has been amplified to approximately 30 mW of the average optical power. Conventional fiber is used for dispersion compensation.

Figure 2

(a) Mode structure of a ${\mathrm{MgF}}_2$ resonator with linewidths in the range of 50–500 kHz and an approximate FSR of 14.09 GHz, as measured by frequency comb assisted diode laser spectroscopy [35]. Dots forming a continuous line represent a particular mode family. Different free spectral ranges correspond to different slopes of the lines, whereas dispersion and variation of the free spectral range show as a curvature of the lines (convex and concave curvatures correspond to anomalous and normal GVD, respectively). The dispersion can be strongly affected by mode crossings. Four specific mode families have been numbered by yellow labels. The color codes the measured resonance depth and helps to track particular mode families. (b) Comparable measurement of the fundamental TM11 mode in a ${\mathrm{Si}}_3{\mathrm{N}}_4$ microresonator with a resonator linewidth of 350 MHz and approximate FSR of 76 GHz (consisting of a 800 nm high and $2\mu \mathrm{m}$ wide ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide embedded in ${\mathrm{SiO}}_2$). The mode family shows signs of mode coupling to other mode families. (c) Illustration of higher order dispersion with $D_3>0$. The gray line indicates anomalous GVD described by $D_2$ only. (d) Illustration of mode coupling induced mode frequency shift altering the dispersion properties locally. A simple parametrization using magnitude $a$ and position $b$ of the avoided mode crossing can be used for numerical modeling.

Figure 3

Experimental investigation of soliton formation in different dispersion scenarios in a ${\mathrm{MgF}}_2$ microresonator. (a) and (d) Soliton formation is observed in resonance families 1 and 4 of Fig. 1, which show an almost ideal $D_2>0$ dominated anomalous GVD. The soliton formation is detected by the observation of the step signature in the converted light signal (inside the dashed grey box). (b) and (c) Soliton formation is not observed in mode families 2 and 3 where strong deviations from $D_2$ dominated dispersion are present. The coupled pump power is $\mathcal{O}(1\mathrm{mW})$ at a wavelength of 1552 nm.

Figure 4

Numerical investigation of soliton formation in different dispersion scenarios. (a) The ratio $P_{\text{peak}}/P_{\mathrm{avg}}$ of peak to average intracavity power is used as an indicator of soliton formation (which results in high peak power) for different combinations of $D_2$ and $D_3$. (b) Optical spectra for different parameters simulated in panel (a) (1, 2, 3) show a narrow spectral width for high values of $D_2$ and asymmetric spectra, as well as dispersive wave phenomena (peak at $\mu =-65$) for nonzero $D_3$. (c) Temporal field envelope inside the microresonator corresponding to the spectra shown in panel (b). The soliton pulse duration lengthens for larger values of $D_2$. Oscillatory features in the background field appear for nonzero $D_3$. (d) The high ratio of peak to average power is used as an indicator of soliton formation for different situations characterized by an avoided mode crossing that is parametrized by magnitude $a$ and distance $b$ from the pump laser [cf. Fig. 2]. (e) Optical spectra for different simulation parameters in panel (d) (1, 2, 3) show the characteristic “up-down” feature induced by an avoided mode crossing. (f) Temporal field envelope inside the microresonator corresponding to panel (e). The pulses have the same duration but oscillatory features in the background field appear in the presence of avoided mode crossings.