⇦Chinese Physics Letters, 2021, Vol. 38, No. 6, Article code 064201 Broadband Sheet Parametric Oscillator for $\chi^{(2)}$ Optical Frequency Comb Generation via Cavity Phase Matching Xin Ni (倪鑫)1, Kunpeng Jia (贾琨鹏)1, Xiaohan Wang (汪小涵)1, Huaying Liu (刘华颖)1, Jian Guo (郭健)1, Shu-Wei Huang (黄书伟)2, Baicheng Yao (姚佰承)3, Nicolò Sernicola1,4, Zhenlin Wang (王振林)1, Xinjie Lv (吕新杰)1*, Gang Zhao (赵刚)1*, Zhenda Xie (谢臻达)1*, and Shi-Ning Zhu (祝世宁)1 Affiliations 1National Laboratory of Solid State Microstructures, School of Electronic Science and Engineering, School of Physics, and College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China 2Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder, Boulder, CO 80301, USA 3Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education), University of Electronic Science and Technology of China, Chengdu 611731, China 4Institute for Optics, Information and Photonics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Schloßplatz 4, Erlangen 91054, Germany Received 3 February 2021; accepted 24 March 2021; published online 25 May 2021 Supported by the National Key Research and Development Program of China (Grant Nos. 2019YFA0705000 and 2017YFA0303700), the Key R&D Program of Guangdong Province (Grant No. 2018B030329001), the Leading-Edge Technology Program of Jiangsu Natural Science Foundation (Grant No. BK20192001), and the National Natural Science Foundation of China (Grant Nos. 51890861, 11690031, 11621091, and 11674169).
*Corresponding authors. Email: xiezhenda@nju.edu.cn; zhaogang@nju.edu.cn; lvxinjie@nju.edu.cn Citation Text: Ni X, Jia K P, Wang X H, Liu H Y, and Guo J et al. 2021 Chin. Phys. Lett. 38 064201    Abstract We demonstrate a broadband optical parametric oscillation, using a sheet cavity, via cavity phase-matching. A 21.2 THz broad comb-like spectrum is achieved, with a uniform line spacing of 133.0 GHz, despite a relatively large dispersion of 275.4 fs$^{2}$/mm around 1064 nm. With 22.6% high slope efficiency, and 14.9 kW peak power handling, this sheet optical parametric oscillator can be further developed for $\chi^{(2)}$ comb. DOI:10.1088/0256-307X/38/6/064201 © 2021 Chinese Physics Society Article Text Coherent radiation with pristine frequency spacings, or an optical frequency comb, is a measure of optical frequency, facilitating a revolutionary precision metrology, with potential for use in a variety of applications.[1–11] Such applications would benefit from miniaturized comb sources for portable devices with high beat-note frequencies, which are not available with conventional mode-locked laser sources, but whose requirements may potentially be met by virtue of the recent development of micro-optical parametric oscillators ($\mu$OPOs). To date, research has tended to focus on the micro-ring $\mu$OPOs with third-order nonlinearity, $\chi^{(3)}$.[12–18] A high-quality factor $Q$ is key to the enhancement of the relatively weak third-order parametric processes in these micro-ring $\mu$OPOs, in order to lower the parametric threshold. However, such high $Q$ also enforces strict phase-matching conditions, because the effective nonlinear interaction length increases as the $Q$ value scales up. Therefore, the dispersion needs to be perfectly engineered over a large comb span, which is normally a challenge for $\mu$OPOs, and must limit the central wavelength for a given material. On the other hand, parametric oscillation can also be generated using second-order nonlinearity, $\chi^{(2)}$,[19–27] which is much stronger than $\chi^{(3)}$, and represents a novel approach to $\mu$OPO combs. Our experimental and other earlier theoretical works have shown that cavity phase matching (CPM)[28–34] can be realized in doubly-resonant $\chi^{(2)}$ Fabry–Pérot micro-cavities,[35] where the strict material phase-matching is not necessary, as the phase mismatch is corrected by the total reflection on the cavity mirrors for each round trip. Although the effective nonlinear interaction length can still be greatly extended while $Q$ increases, the CPM condition is only dependent on cavity length, and results in a large bandwidth. In this work, such a large CPM bandwidth plays a crucial role in achieving broadband parametric oscillation with a comb-like spectrum. Two types of sheet optical parametric oscillators (SOPOs) are fabricated, with different cavity lengths. Broad bandwidth can be achieved from these SOPOs by means of a $\chi^{(2)}$ CPM optical parametric down-conversion process, which exceeds 21.2 THz and 14.3 THz, with spectral line spacings of about 133.0 GHz and 514.0 GHz, respectively. By virtue of a strong second-order nonlinearity, such sub-millimeter SOPOs are capable of 22.6% slope efficiency, and 14.9 kW peak power. Despite a normal dispersion of 275.4 fs$^{2}$/mm, the line spacings are measured to be equidistant within the accuracy of a high-performance wavelength meter, which is less than the transform-limited linewidth. These results reveal the line-to-line nonlinear interactions that automatically lock the line spacings, and are verified by our simulation results. This simulation also exhibits lower noise under a cw pump, which may be used for optical frequency comb generation. As first proposed by Armstrong et al. in 1962,[28] and our earlier experimental demonstration,[34] a sub-coherence-length Fabry–Pérot nonlinear cavity can generate a second-order nonlinear process by cavity phase-matching. The $\pi$ phase shift gained from the reflection on the cavity mirrors compensates for the phase-mismatch of the light waves when they recirculate inside the cavity,[32,34] as the pump light is always in phase with the signal and idler when they recirculate inside the sheet cavity. Efficient frequency conversion can be achieved if the following first-order CPM condition is satisfied: $$ l_{\rm cav}\leqslant l_{\rm coh}=\frac{\pi}{k_{\rm p}-k_{\rm s}-k_{\rm i}},~~ \tag {1} $$ where $l_{\rm cav}$ is the cavity length, $l_{\rm coh}$ is the coherence length of the three-wave coupling, ${k}_{\rm p}$, ${k}_{\rm s}$ and ${k}_{\rm i}$ are the wave vectors of pump, signal, and idler, respectively. We introduce the effective nonlinear coefficient, $d_{\rm eff}$, to describe the strength of the nonlinear coupling, in place of the nonlinear coefficient, $d_{31}$, of the material, which is given by $$ d_{\rm eff}=d_{31}\Big|{\rm sinc}\Big(\frac{\pi l_{\rm cav}}{2l_{\rm coh}}\Big)\Big|.~~ \tag {2} $$ In a doubly resonant SOPO, both CPM and longitudinal mode-matching conditions need to be satisfied with respect to the resonance modes. Despite the large CPM bandwidth, the longitudinal mode-matching results in strong frequency mode selectivity in the parametric beams. As shown in our previous work, tunable single-longitudinal-mode parametric oscillation can be achieved for SOPO via nondegenerate signal and idler modes, either in polarization or in frequency. In this study, however, degenerate type-I CPM SOPO is selected, so that more than one pair of modes can be excited at the same time. As shown in Fig. 1(a), two types of SOPOs are fabricated from fine-polished monolithic crystal $y$-cut MgO-doped lithium niobate crystal, at thicknesses of 140 µm and 485.25 µm, respectively. The type-I CPM process is phase-matched in the $y$-cut SOPOs so that larger coherence length can be achieved. The surface areas are both $5{\,\rm mm} \times 5{\,\rm mm}$ in the $xz$ plane. The two end faces of these crystal sheets are both anti-reflection coated for 532 nm (reflectivity $R < 1$%), and high-reflection coated in the 900–1150 nm range, with $R>99.8$% for the input surface, and $R=97.0$% for the output surface. The $Q$ values measure $6.6\times 10^{4}$ and $\mathrm{1.9\times 1}0^{5}$ for the 140 µm and 485.25 µm SOPOs, respectively. The type-I CPM process is designed to be phase-matched between the $z$-polarized pump at 532 nm, with $x$-polarized parametric beams centered around 1064 nm. Based on the temperature-dependent Sellmeier equation,[36–38] simulations show that the above parametric down-conversion process can be naturally phase-matched at $114^{\circ}C$, and CPM can be used to extend the phase-matching bandwidth, a prerequisite for comb generation. Figure 1(b) shows the calculated coherence length in comparison to the cavity lengths (denoted by red and blue lines) over a wavelength range from 900 nm to 1300 nm. It also includes the $d_{\rm eff}$ calculation for both SOPOs, based on Eq. (2). According to Eq. (1), CPM bandwidths of 351 nm and 183.1 nm can be expected for the above two SOPOs, respectively.
cpl-38-6-064201-fig1.png
Fig. 1. The schematic of the broadband SOPO generation. (a) Design of SOPOs. Two types of SOPOs are fabricated, with cavity lengths of 140 µm and 485.25 µm, respectively. Inset: an image of our SOPO sample, with aluminum mount. (b) Coherence length and $d_{\rm eff}$ calculation. In the upper figure, the solid blue and red lines mark the CPM condition, where $l_{\rm cav}=l_{\rm coh}$ for different cavity lengths. In the lower figure, blue and red lines indicate $d_{\rm eff}$ as a function of signal/idler wavelengths for $l_{\rm cav}=140$ µm and 485.25 µm.
In our experiment, the SOPOs are pumped by a single-longitudinal-mode frequency-doubled yttrium-aluminum-garnet laser (Powerlite, Continuum, Santa Clara, CA), with a pulse duration of 10 ns, and a repetition rate of 10 Hz. The average power drift of the YAG laser over eight hours, where $\Delta T=\pm 3\,^{\circ}\!$C, is 5%. A solid pinhole is put inside the laser cavity to purify the transverse mode. The output is focused onto another pinhole to enable further selection of the TEM$_{00}$ mode, and is imaged onto the SOPO with a beam full width at half maximum of 500 µm. The SOPO is embedded in a temperature-controllable oven with an accuracy better than 10 mK. The natural phase-matching temperature for our SOPO measures $111\,^{\circ}\!$C, which is in good agreement with the theoretical value. We scan the temperature around $111\,^{\circ}\!$C; parametric oscillation output can be observed once the pump frequency is matched with the cavity modes for the signal and idler. A single-shot spectrum of the output is captured by a CCD-camera spectrometer (Acton 500, Princeton Instruments). As shown in Figs. 2(a), 2(b) and 2(c), the output beams exhibit a comb-like spectrum for both SOPOs, with spans exceeding 54 nm and 80 nm, or 14.3 THz and 21.2 THz, for the 140 µm and 485.25 µm SOPOs, respectively. The line spacings measure 514.0 GHz and 133.0 GHz, respectively, which are in good agreement with cavity lengths of 140 µm and 485.25 µm. Although the 140 µm SOPO has a larger CPM bandwidth in theory, its higher threshold prevents us from achieving a broader spectral span. This could potentially be improved by further increasing the $Q$ in future studies.
cpl-38-6-064201-fig2.png
Fig. 2. SOPO spectrum and output pulse energy. Spectra of (a) 140 µm SOPO at 1.03 mJ pump. (b) 485.25 µm SOPO at 540 µJ pump, and (c) at 830 µJ pump. (d) Output pulse energy as a function of pump pulse energy. Output power of 130 µJ is measured, with a high slope efficiency of 22.6%.
The thresholds for parametric oscillation are 860 µJ and 320 µJ for the 140 µm and 485.25 µm cavities, with maximum output pulse energies of 42 µJ and 113 µJ at pump energies of 1.03 mJ and 830 µJ, respectively. We measure the output pulse energy as a function of that of the pump for the 485.25 µm SOPO. The result is shown in Fig. 2(d). The maximum conversion and slope efficiency are calculated to be 13.6% and 22.6%, respectively. Under the condition of maximum output, a peak power of 14.9 kW can be calculated, taking into account the measured 5.5 ns pulse width. Compared to the type-II CPM SOPO in our previous work,[34] the maximum conversion and slope efficiency are calculated to be 18.6% and 21.2%, and the spectral width is 14.6 THz. The uniformity of the comb line spacing is a key feature of an optical frequency comb, and our SOPO also exhibits this uniformity within the transform-limited linewidth of this pulse-pumped case, for both experimental and theoretical approaches. Figure 3(a) shows a schematic of the line-to-line measurement setup. The output parametric beam is collimated, and directed to a high-resolution double-pass grating filter, so that each single spectral line can be filtered out individually. This grating filter comprises a high-efficiency ruled grating and a silver mirror, which reflects the first-order diffraction beam back to the grating for the second diffraction. By tuning the mirror angle, we can couple any line into the fiber for measurement. The optical frequency of each line is measured by a high-performance wavelength meter (WS-7, High finesse) with an absolute accuracy of 40 MHz. For simplicity, only signal spectral lines with higher frequency than degeneracy are measured, and we number these starting from #0 at the degenerate point. Our SOPO has a normal dispersion of 275.4 fs$^{2}$/mm[36] at 1064 nm; however, the measured lines tend to be uniformly spaced, within the accuracy limit of the wavelength meter, as shown in Fig. 3(b). The deviations of these lines from the equal-spacing distribution measure 7.6 MHz in root-mean-square value. This is in agreement with the sub 10 MHz relative accuracy of the wavelength meter, and is much smaller than the linewidth of a single line, which is transform-limited to about 200 MHz in such a pulse-pumped case. This experimental result also shows good agreement with that given by our simulation. We perform the simulation for such an optical parametric effect in the SOPO; only the second-order nonlinear optical effects are included in this simulation. In both experiment and simulation, the line spacing can deviate by up to 1.049 GHz from the natural resonance of the SOPO cavity. More interestingly, some lines (#21–24th modes from the degeneracy) on the side of spectrum oscillate in the high-order transverse modes to follow the equal spacing, and remain within the cavity resonances. In such a case, we must couple them into a multi-mode fiber, and sacrifice the accuracy to 160 MHz. This line-pulling effect can only be interpreted as a result of line-to-line nonlinear interaction. Such a line-to-line interaction is well-known in the $\chi^{(3)}$ $\mu$OPOs, where two adjacent lines can seed into the next adjacent resonance modes via four-wave mixing. Here, this interaction may be interpreted as a result of cascaded $\chi^{(2)}$ processes,[21–27] which is confirmed by our simulation, with only second-order nonlinearity. As shown in Fig. 3(c), starting from line pair, #$+$/$-i$ ($\omega_{\pm i}=\frac{\omega_{\rm p}}{2}\pm i\varOmega$, where $\varOmega$ is the line spacing), and its closest adjacent line pair, #$+$/$-(i+1)$, the cavity-enhanced sum frequency generation (SFG) process between $\omega_{-i}$ and $\omega_{i+1}$ can be cavity phase-matched to generate $\omega_{\rm p}+\varOmega$. Difference frequency generation (DFG) occurs between this SFG light and line #$-(i+1)$, with an output frequency of $\omega_{i}+2 \varOmega$ into the #($i+$2) resonance. This DFG process can seed into the optical parametric amplification (OPA) process, with pump light to lock the #($i+2$) line, and cascade over the whole comb span to lock the comb at one uniform spacing.
cpl-38-6-064201-fig3.png
Fig. 3. Line-to-line measurement of the SOPO. (a) Experimental setup to measure the optical frequencies of individual spectral lines. (b) Spectral lines and $\mu$OPO resonance frequency deviations from equal spacing. Error bars show the instrument limit of wavelength meter. Lines #22–24 are in high-order transverse modes, and can only be detected via a multi-mode fiber, which results in reduced accuracy. (c) Schematic of the cascaded line-to-line interaction. The red, blue and green pairs denote the lines #$+$/$-i$, #$+$/$-(i+1)$, and #$+$/$-(i+2)$. Starting from spectral line pairs #$+$/$-i$ and #$+$/$-(i+1)$, line pair #$+$/$-(i+2)$ can be generated via cascaded SFG, DFG, and OPA processes. Such processes can be repeated for a coherent spectrum over the whole span.
cpl-38-6-064201-fig4.png
Fig. 4. Simulation of a 485.25 ${µ}$m SOPO. (a) Noise level as a function of pump detuning at different pump powers ($C$: cross correlation). Here the comb noise can be characterized by the cross correlation between the temporal waveform inside one repetition and that of another repetition. A balance between noise and bandwidth can be achieved at 22 MHz pump detuning and 1.8 mJ pump, where the spectral and temporal behaviors are studied in detail. (b) Temporal profiles of 26 repetitions. (c) Relative intensity noise over 26 repetitions. The first repetition is used as reference and relative intensity noise is the difference between the intensity of the first repetition and others relative to the average intensity. (d) Spectrum of the SOPO. (e) Beat-note spectrum of the rf peak around the repetition frequency under a cw pump.
We perform a simulation using the split-step Fourier method, for the purpose of comparison with the experimental results. As shown in Fig. 4(a), we scan the pump, detuning $\upsilon$ at different pump levels to search for the lowest intensity noise, where $\upsilon =\omega_{\rm p}-2\varOmega_{0}$, and $\varOmega_{0}$ is the central cavity resonance frequency. The noise intensity shows similar behavior to the detuning changes at different pump power, and the lowest noise can be achieved when $\upsilon =22$ MHz at 1.8 mJ pump. We plot 26 repetitions of the temporal waveforms under the above conditions; the results are shown in Fig. 4(b), and the relative noise plot in Fig. 4(c). The SOPO spectrum can be simulated via a Fourier transform of the temporal waveform, as shown in Fig. 4(d). It is interesting to compare the line spacings to the free spectrum range (FSR) of the cavity resonances in such a normal-depression SOPO cavity. Figure 4(e) shows the simulation result for the rf beat-note at a repetition frequency of around 133.06 GHz, and the linewidth is fit to be 1.67 MHz. Such linewidths and side band floors are limited by our simulation capability of 600 ns waveform length, as proven by a comparison with the theoretical beat-note spectrum in the red line, which can be achieved by choosing a stable repetition in time domain and reproducing a new periodic repetitive signal. The pulse profile limits temporal uniformity, as well as the spectral resolution. With a cw pump, the temporal and spectral noise may be further reduced, as confirmed by our simulation. The spectrum noise is much lower than that for a pulse pump, and can be achieved when $\upsilon =-193$ MHz at 12 kW pump (for details, see the Supplementary Information). A high cw oscillation threshold of 10 kW for a bulk crystal SOPO makes the cw pump impractical. However, the cw-pumped SOPO is possible for higher-$Q$ SOPO or waveguide devices. Our estimates show that a 100 W and 1 W threshold for cw comb generation can be achieved with a $Q=6\times 10 ^{6}$ SOPO and a $Q=3.5\times 10 ^{5}$ titanium diffusion waveguide,[38–40] respectively. In summary, we have demonstrated a new approach to $\chi^{(2)}$ frequency comb generation in a sheet optical parametric oscillator via cavity phase-matching. Cavity phase-matching plays a key role in mitigating the stringent phase-matching condition imposed by a high-$Q$ cavity, facilitating the achievement of a broader bandwidth of up to 21.2 THz. The slope efficiency and peak output power exceed 22.6% and 14.9 kW, respectively. SOPO lines are measured to be equally spaced within the accuracy of the wavelength meter, i.e., 40 MHz (or 160 MHz in for lines #21–24), suggesting the mechanism for an actual comb generation. The experimental results are supported and confirmed by our computer simulations, which also reveal the capacity of this approach to achieve a broad-band low-noise spectrum under cw pump. Therefore, this new $\chi^{(2)}$ SOPO platform represents a promising candidate for portable integrated optical frequency comb generation. 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