Chapter VI: Spatial-mode-interaction induced soliton dispersive wave
6.4 Phase-matching condition of dispersive wave in microresonator
When the soliton propagates in a optical fiber or waveguide, it emits a dispersive wave at the wavelength whose group velocity is identical to soliton group velocity.
At this wavelength, the momentum is conserved when a soliton photon is converted to a dispersive wave photon, and this greatly enhances the dispersive wave emission.
In the microresonator, phase matching between the soliton and the dispersive wave occurs when theµt h soliton line atωp+ωrepµis resonant with the µt hfrequency of the soliton-forming mode family, i.e.,ωp+ωrepµ=ω−(µ). Under this condition, the dispersive wave emission is enhanced by the resonator. The detailed derivation will be given in section 6.5. As an aside, the Kerr shift for modeµis much smaller than other terms in this analysis and is neglected in the phase matching condition. So that it is possible to use a graphical interpretation of the phase matching condition based on the relative-mode-frequency of fig. 6.1 (a),ω0+D1Aµis subtracted from both sides of the phase matching condition to give the following condition:
∆ω−(µ)=(ωrep−D1A)µ−δω, (6.2) whereδω≡ ω0−ωPis the detuning of the resonator relative to the pump frequency.
If the soliton repetition frequency equals the FSR at µ = 0 (i.e.,ωrep = D1A), then the r.h.s. of eq. (6.2) is the horizontal dashed black line in fig. 6.1 (a) (repeated in fig. 6.3 (a)). Under these circumstances the dispersive wave phase matches to the soliton pulse at the crossing of that line with the soliton-forming mode branch.
However, while the mode dispersion profile (∆ω−(µ)) is determined entirely by the resonator geometry and the dielectric material properties, the soliton repetition rate ωrep depends upon frequency offsets between the pump and the soliton spectral maximum. Defining this offset as Ω, the repetition frequency is given by the following equation (A. B. Matsko and Maleki, 2013; Jang, Erkintalo, Coen, et al., 2015):
ωrep = D1A+ D2A
D1AΩ. (6.3)
The offset frequency Ω can be caused by soliton recoil due to a dispersive wave and also by the Raman-induced soliton self-frequency shift (SSFS) (Milián et al., 2015; V Brasch et al., 2016; Karpov, Hairun Guo, Kordts, Victor Brasch, M. H.
Pfeiffer, et al., 2016; Xu Yi, Q.-F. Yang, Ki Youl Yang, Suh, et al., 2015; Xu Yi, Q.-F. Yang, Ki Youl Yang, and Kerry Vahala, 2016b). In this work,Ωis dominated by the Raman interaction, because the typical dispersive wave power is < 0.2% of the soliton power, causing a negligible dispersive wave recoil (recoil of less than one mode). Photo-thermal-induced change inD1A is another possible contribution that will varyωrep as pumping is varied (Del’Haye et al., 2008). However, the thermal tuning of D1A is estimated to be ∼ −4.5 kHz/mW (by measurement of resonant frequency photo-thermal shift of ∼ −40 MHz/mW). With total soliton power less than 1 mW (Xu Yi, Q.-F. Yang, Ki Youl Yang, Suh, et al., 2015), this photo-thermal- induced change in repetition frequency is negligible compared with that caused by the Raman self-frequency-shift.
-500 -600 -400
-300 -200
-100 0
0 -100 -200 -300
Repetition Rate - offset (kHz)
Measurement
Self-frequency shift Ω/2π (GHz) equation
-20 0 20
Power (20 dB / div)
RF + offset (MHz)
Figure 6.2: Measured soliton repetition rate (blue points) is plotted versus soliton self-frequency shift. The dashed blue line is a plot of eq. (6.3). The offset for the repetition rate vertical scale is D1A = 21.9733 GHz. A typical microwave beatnote of the photo-detected soliton and dispersive wave is shown in the inset (resolution bandwidth is 10 kHz).
Combining eqs. (6.2) and (6.3) gives the following phase matching condition:
∆ω−(µ)= µD2A
D1AΩ−δω. (6.4)
The Raman-induced SSFS is a negative frequency shift (Ω < 0) with a magnitude that increases with soliton bandwidth and average power. Accordingly, with in- creasing soliton power (and bandwidth), the plot of the r.h.s. of eq. (6.4) versus µ acquires an increasingly negative slope (green dashed line in fig. 6.3 (a)). The phase
Pump
188 190 192 194 196 198
-60 -40 -20
Power (dBm)
Optical frequency (THz)
-200 -100 0 100 200
0 500
Mode number (µ) 250
-250
Mode freq Comb freq w/o SSFS Comb freq w/ SSFS
(a)
(b) (c)
(ω−ω0−µD1A)/2π (MHz)
∆ω− 750
Relative power (50 dB / div)
198.0 198.3
Optical frequency (THz) 198.6
Figure 6.3: Dispersive wave phase matching condition. (a) Soliton and interaction mode family dispersion curves are shown with phase matching dashed lines (see eq.
(6.4)). The black line is the case where ωrep = D1A and the green line includes a Raman-induced change in ωrep. The intersection of the soliton branch with these lines is the dispersive wave phase matching point (arrows). (b) Soliton optical spectra corresponding to small (red) and large (blue) cavity-laser detuning (δω).
Sech2fitting of the spectral envelope is shown as the orange curves. (c) Dispersive wave spectra with cavity-laser detuning (soliton power and bandwidth) increasing from lower to upper trace.
matching mode number,µ= µDW, therefore also increases (i.e., the dispersive wave shifts to a higher optical frequency) with increasing soliton power. The two soliton spectra presented in fig. 6.3 (b) illustrate this effect (red spectrum is lower power and has the lower dispersive wave frequency). Fig. 6.3 (c) also shows a series of higher-resolution scans of the dispersive wave with soliton power increasing from the lower to upper scans and is, again, consistent with the prediction.
The frequency shift,Ω, repetition frequency,ωrep, and the dispersive wave frequency were measured for a series of soliton powers that were set by controlling the cavity- pump detuning frequency (δω) using the method in ref. (Xu Yi, Q.-F. Yang, Ki Youl Yang, Suh, et al., 2015; Xu Yi, Q.-F. Yang, Ki Youl Yang, and Kerry Vahala, 2016a).
ωrepwas measured using an electrical spectrum analyzer after photodetection of the resonator optical output. The offset frequency Ω was measured on an optical spectrum analyzer by fitting the center of optical spectrum (see fig. 6.1 (b)) to determine the spectral maximum and then measuring the wavelength offset relative
-500 -400
-300 -200
-100 0
dispersive wave peak (THz)
198.4
198.2
198.0
198.6 Measurement
equation
Self-frequency shift Ω/2π (GHz)
Figure 6.4: Measured dispersive-wave peak frequencies (red points) are plotted versus soliton self-frequency shift. The dashed red line uses eq. (6.4) to determine the dispersive wave frequency (≈ µDWD1A+ω0) as described in the text.
to the pump. This same spectral fitting also allows determination of the soliton pulse width, τs (Xu Yi, Q.-F. Yang, Ki Youl Yang, Suh, et al., 2015). Once the soliton pulsewidth is known, the pump-resonator frequency detuning operating point can be inferred using δω ≈ D2/2D21τs2 (Xu Yi, Q.-F. Yang, Ki Youl Yang, Suh, et al., 2015). δω/2π ranged between 7.8 to 21.1 MHz during the measurement.
The soliton repetition rate is plotted versus Ω in fig. 6.2 and is fitted using eq.
(6.3). The intercept closely agrees with D1A and the slope allows determination of D2A/2π = 14.7 kHz (in good agreement with 15.2 kHz from fitting to the measured dispersion curve in fig. 6.1 (a)). The dispersive wave frequency is also plotted in fig.
6.4 versusΩ and compared with a calculation using eq. (6.4). In this calculation,
∆ω−(µ)is approximated using a linear expansion in µnear µ= 200. Also, a−60 kHz offset is added toD1A/2πin∆ω−(µ)due to the calibration uncertainty (∼ ±100 kHz) of FSR (Jiang Li, Lee, Ki Youl Yang, et al., 2012). No other free parameters are used in the plot.