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Pump condition dependent Kerr frequency comb generation in mid-infrared
Results in Physics 15 (2019) 102789
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Pump condition dependent Kerr frequency comb generation in mid-infrared Shuxiao Wang
a,b
, Qing Wang
a,b
c
a
a,c
, Wei Wang , Xi Wang , Mingbin Yu , Qing Fang
c,d,⁎
a,⁎
, Yan Cai
T
a
State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Changning Road 865, Shanghai 200050, China University of the Chinese Academy of Sciences, Yuquan Road 19, Beijing 100049, China c Shanghai Industrial µTechnology Research Institute, Chengbei Road 235, Shanghai 201800, China d College of Science, Kunming University of Science and Technology, Kunming 650500, China b
ARTICLE INFO
ABSTRACT
Keywords: Kerr frequency combs Microring resonator Mid-infrared Ternary-pump Silicon photonics
We propose and simulate the mid-infrared (MIR) frequency comb generation based on the Kerr nonlinear effect in a silicon microring resonator (MRR). The MRR is formed by the suspended ridge waveguides which are designed to have the anomalous group-velocity dispersion. Taking the thermal effect into consideration, the modified Lugiato–Lefever equation (LLE) is used to simulate the frequency comb generation. The stable dissipative cavity soliton and mode-locked combs in the MIR wavelength range of 2.5 μm–5.2 μm can be obtained under different pump conditions. It is found that the ternary-pump scheme has wider bandwidth and flatter frequency combs as compared with the dual-pump scheme. The lowest total pump power is required for the ternary-pump approach to produce the same comb span in these three pump conditions.
Introduction The invention of the optical frequency combs provides a revolutionary tool for the optical arbitrary waveform generation [1], facilitating the development of the optical clockwork [2,3], the celestial spectrum calibration [4,5], and the precision spectroscopy [6,7]. MIR optical frequency combs are of significant interest for the molecular spectroscopy application due to the large absorption of molecular vibrational modes in this spectral region [8–10]. Kerr optical frequency combs based on the micro-resonator have attracted lots of attention because of small size, low power consumption, large quality-factor (Q) and mature production process. The optical frequency combs based on micro-resonators have been widely investigated on various materials such as calcium fluoride [11], magnesium fluoride [12,13], silica [6,14,15], aluminum nitride [16], diamond [17], silicon [18], and silicon nitride [19–24]. Silicon-on-insulator (SOI) has emerged as a particularly attractive platform for the chip-scale frequency comb generation due to the availability of CMOS fabrication technology, which allows for the integration of electronic and optical elements in a compact, robust and portable device. Silicon is a promising material for MIR wavelength range application since it has the relatively large nonlinearity, the high refractive index and a moderately broad transparency window up to
around the wavelength of 7 μm [18,25]. Moreover, there is no twophoton absorption (TPA) effect which prevents the generation of combs at the wavelength range longer than 2.2 μm [26]. The nonlinear loss by the three-photon absorption (3PA) is negligible at the wavelength longer than 3.3 μm [27]. The nonlinear loss beyond 3.3 μm is even lower due to the absence of 2PA and 3PA effects, where the dominant nonlinear loss is from four-photon absorption (4PA) [28]. The influence of the 4PA effect is small, and a comb can be generated without the need of a positive-intrinsic-negative (PIN) structure [29]. However, the implementation of the SOI platform in the mid-infrared wavelength range is restricted due to the high absorption of buried oxide (BOX) layer [30]. The microring based on the suspended silicon ridge waveguides are designed to reduce the absorption loss. The formalisms based on the coupled mode equations [31–33] and the driven and damped nonlinear Schrödinger (NLS) equation [34–38], which is also known as the LLE, are two main theoretical formalisms to describe the frequency comb generation and temporal dynamics in micro-resonator devices. The LLE can simulate ultra-broadband combs such as octave spanning frequency combs because of the high computing speed. When pumped with the laser light, the micro-cavity will collect a high-intensity light field in a small mode volume, which changes the micro-cavity temperature. Therefore, we add the phase shift caused by the introduction of the thermal effect to modify the LLE
Corresponding authors at: College of Science, Kunming University of Science and Technology, Kunming 650500, China (Qing Fang); State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Changning Road 865, Shanghai 200050, China (Yan Cai). E-mail addresses: [email protected] (Q. Fang), [email protected] (Y. Cai). ⁎
https://doi.org/10.1016/j.rinp.2019.102789 Received 2 October 2019; Received in revised form 1 November 2019; Accepted 1 November 2019 Available online 05 November 2019 2211-3797/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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model. Previous works on the MIR frequency comb generation have primarily focused on the experimental setup under single CW laser pump condition. In addition to the single CW laser pump, here we also investigate an alternative pump configuration where the cavity is simultaneously pumped at two or three different frequencies. The use of dual pumps provides an additional degree of freedom in the form of the modulation frequency. There have been some researches on the dualpump micro-resonator Kerr frequency combs in the near infrared (NIR) wavelength range. Strekalov and Yu proposed a comb generation technique based on the use of dual optical pumps in a magnesium fluorite resonator, which demonstrated to have a better efficiency for the comb generation process [39]. Hansson and Wabnitz made a study of the nonlinear dynamics of the bichromatically pumped micro-resonator Kerr frequency combs by the truncated four-wave model [40]. Other experimental and theoretical results on the dual-pump inputs have also been studied in both the normal and the anomalous dispersion regime [41–45]. But no similar work has been done in the MIR wavelength range yet. In this work, we study the pump condition dependent Kerr frequency comb generation in the MIR wavelength range. The silicon MRR is formed by the suspended ridge waveguides which are designed to have the anomalous group-velocity dispersion in the MIR wavelength range. The evolution of the cavity solitons and frequency combs are studied by a modified LLE model, which taking the thermal effect into consideration. The mode-locked combs in the MIR wavelength range of 2.5 μm–5.2 μm are obtained when a single CW pumped laser couples into the resonator. The nonlinear dynamics of the MRR Kerr frequency combs under different pump conditions are investigated in the MIR wavelength range. Compared with the single-pump condition, the frequency combs with multi pumps can be generated through thresholdless non-degenerate four-wave-mixing (FWM) and the intervals of the frequency combs can be controlled. It is found that the ternary-pump scheme has wider bandwidth and flatter frequency combs as compared with the dual-pump scheme. The lowest pump power is required for the ternary-pump approach to produce the same comb span in these three pump conditions.
the pump’s angular frequency and the n th angular resonance frequency that is pumped. The thermal effect in the MRR induced by the traveling light beam can change the refractive index and geometry of the MRR, which causes the drift of the resonant wavelength of the MRR. In the process analysis of the Kerr optical frequency comb generated in the MRR, this wavelength drift can be attributed to the phase shift. The phase shift of the light wave caused by the thermal effect after passing the unit length is expressed as, t
where KTHO is the thermo-optic coefficient of the core material, T (t ) is temperature alteration of the MRR, k 0 is the wave vector in vacuum. n1 is the amount of change of the refractive index of the waveguide core layer, which we use to indicate the amount of change of the effective refractive index of the waveguide. T (t ) can be derived from the dynamical thermal behaviour of the micro-cavity [46]. Based on the energy conservation principle, the net heat delivered to the cavity is the heat that goes in (qin ) subtracted by the heat that goes out () [46]:
Cp
i
k
E+i k 2
E (m + 1) (0, ) =
Ein +
i
k!
k
E + i |E|2 E
E (m) (L , ) ei
1
E (t , ) = t
2
2
i
0
k
+ iL k 2
k!
i
tR
+ iL |E|2 E +
(
1 p
0 (1 + a T )
/2
)
2
K T (t ) +1
E (t , ) = t
2
2
k
i 0 + iL k 2
k!
i
k
+ iL |E|2 + iLk 0 KTHO T (t ) E +
Ei
(6)
T (t ) is calculated by using the thermal dynamic Eq. (5) of the MRR. Eq. (6) can be efficiently simulated using the split-step Fourier method: i.e. Eq. (6) is split into the following two equations: tR
E (t , ) = t
2
2
i
0
k
+ iL k 2
k!
i
k
+ iLk 0 KTHO T (t ) E (7)
tR
E (t , ) = (iL |E|2 ) E + t
Ei
(8)
The Eqs. (7) and (8) are solved by the Fast Fourier Transform (FFT) method. Considering the nonlinear dynamics of the dual-pump micro-resonator Kerr frequency combs, the input pump light field can be expressed as:
(2)
k
= Ih
dn
where i is the linear absorption coefficient inside the resonator, is the power transmission coefficient, τ is the time, L is the roundtrip length of the resonator, and 0 is the linear phase accumulated by the intracavity field with respect to the pump field over one roundtrip. Other resonator parameters include the nonlinear coefficient , the k th dispersion coefficient k , and the CW driving field Ein . Combining the NLSE in Eq. (1) and the boundary conditions in Eq. (2), the LLE that describes the dynamics of a MRR is deduced as,
tR
dt
coefficient of the resonant wavelength is a = + dT / n 0 , which contains both the thermal expansion and the thermal index change. Therefore, the LLE model can be modified with the thermal effect as follows [47]:
(1)
0
dqout (t )
Here Cp is the heat capacity (J/°C) and K (J/(s°C)) is the thermal conductivity between the cavity mode volume and the surrounding.Ih is the actual optical power in the thermal cavity, 0 is the cold cavity resonant wavelength, and p is the pump wavelength. The temperature
The driven nonlinear Schrödinger equation (NLSE) describes the evolution of the electric field envelope in the waveguide. The periodic boundary condition describes the relationship between the continuous circular light field and the input pump light field [34,35]. The NLSE and the periodic boundary condition are as follows:
2
dqin (t ) d T (t ) = dt dt
(5)
Model and principle
E (z , ) = z
(4)
= k 0 n1 = k 0 KTHO T (t )
Ein = Ein1 + Ein2 =
Pin1 ei
1t
+
Pin2 e i
2t
(9)
where Pin1 and Pin2 are the powers of the two pump light fields, 1 and 2 are the angular frequencies of the two pump light fields. In the simulation, we set 2 = 1 + n FSR , which means that the two pump frequencies are separated by n times of the FSR of the MRR. The structure of the suspended ridge waveguide on the SOI substrate is shown in Fig. 1(a). The thicknesses of the waveguide (H), slab (HS), and buried oxide are 500 nm, 100 nm, and 3 μm, respectively. The upper cladding layer is the air. The BOX layer under the waveguide can be removed locally by the wet etching through the holes in the neighbouring supporting slab. The designed edge-to-edge distance between the waveguide and the hole is 2.5 μm to avoid interacting with the optical mode transmitted in the waveguide. A flat dispersion curve in an
Ei (3)
where tR is the round-trip time of the optical field in the MRR, E (t , ) and Ei are the intracavity field and the input field, t and τ are the slow and fast time. α is the cavity loss, equalling to ( i + )/2 . 0 equals to tR ( n 0 ) which is the cavity phase detuning, where 0 and n are 2
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Fig. 1. (a) A schematic of a suspended silicon waveguide. (b) Dispersion properties of the waveguide (H = 500 nm, HS = 100 nm) with different width. (c) The group index curve of the waveguide (H = 500 nm, HS = 100 nm, W = 2.2 μm) versus wavelength. The inset shows the electromagnetic field profile for the fundamental TE mode.
anomalous regime is essential to excite the frequency combs. The dispersion curves for a suspended waveguide with different widths are calculated [Fig. 1(b)]. Considering the effect of the bending radius on the dispersion, we choose the width of 2.2 μm to realize the near-zero dispersion around 3.5 μm wavelength. The group index of the waveguide (H = 500 nm, HS = 100 nm, W = 2.2 μm) can be extracted by the Finite Difference Eigenmode (FDE) solver in Lumerical [Fig. 1(c)]. The group index is 3.7 and the effective cross section area is 1.21 μm2 at 3.5 μm wavelength. In this design, the radius of the MRR is 100 μm. Considering the scattering loss, we assume the total waveguide propagation loss is 0.3 dB/cm, which corresponds to an intrinsic quality factor of 224000. The field coupling coefficient is set to be 0.018. The nonlinear Kerr coefficient of Silicon at 3.5 μm is 3.6 × 10−18 m2/W [48]. The thermal expansion coefficient of Silicon is 2.63 × 10−6 K−1. The FSR of the MRR is 128 GHz. In the simulation we neglect the higher-order dispersion, the self-steepening effect, and other higher-order nonlinearities.
combs are shown in Fig. 3(c)–(j), respectively, corresponding to the four different situations in Fig. 2(c). In SMI, the power in the cavity is rapidly increased as the detuning increases. When the power in the MRR exceeds the threshold of the optical parametric oscillation, the modulation instability (MI) produces primary sidebands in its gain spectrum. Due to the increase of power in the MRR, a gain is also generated around the primary sidebands of the optical comb, and the mini-combs are generated in the gain spectrum around the primary sidebands in UMI. The rapid decrease of the power in the cavity causes the parametric gain of the random pulse to decrease, and gradually disappears in the cavity because of the absorption and attenuation in the cavity. Finally, the power in the cavity is stabilized, and a soliton is stable in the time domain. In the frequency domain, the frequency comb line is full of the entire spectrum bandwidth and the comb line power in all modes changes smoothly. As shown in Fig. 3(j), we demonstrate the mode locked frequency combs in the mid-IR wavelength range of 2.5 μm–5.2 μm with the comb-line power lager than −70 dB. In the simulation, we adopt the method of adjusting the initial detuning amount, which corresponds to linearly adjusting the pump wavelength in the experiment to achieve the purpose of making the MRR quickly enter the thermal equilibrium state. One of the resonance wavelengths of the cold cavity is 3.502 μm. Considering the process from blue detuning to red detuning during the generation of the stable optical frequency combs. The initial detuning is set to be −0.0065. The detuning in the simulation linearly increases at the speed of 4 × 10−3/ns. The initial detuning is varied by changing the simulation roundtrip number. The total roundtrip is set to be 15,000, and the amount of detuning is adjusted from the first roundtrip to the 5335th roundtrip. The detuning remains to be 0.1602 for the remainder of the simulation time. Kerr effect causes red shift of resonant wavelength. Considering the thermal effect, the resonant wavelength will red shift after the temperature rises due to the positive thermo-optic coefficient of the silicon material. The modified LLE model shows that the detuning need to compensate for the phase shift introduced by not only the nonlinear effects but also the thermal effect in order to meet the phase balance conditions, which is described in Fig. 4(a) and (b). The variation of the temperature in the MRR is shown in Fig. 4(c). The primary sideband generation mechanism for a single-pump MRR is the MI of the pump mode [40]. The primary sidebands are usually located multiple FSRs away from the frequency of the pumpmode. The primary sidebands can in turn interact with the pump mode to create additional sidebands through the FWM process. With the increments of the external pump intensity and detuning, comb generation can further occur through the cascaded FWM process that creates new
Simulation and analysis Simulation of the single pump condition We pump the cavity at the resonant wavelength adjacent to 3.5 μm, with a CW laser power of 0.75 W. By adjusting the detuning between the wavelength of the pump and the resonant wavelength of the MRR, the evolution process of the time domain cavity solitons, the frequency domain Kerr frequency combs and the simulation results of the cavity power variation are shown in Fig. 2a, b, and c, respectively. The adjustment of the detuning in the simulation is from effective blue-detuning to effective red-detuning. Under the combined effects of the dispersion and the nonlinear modulation, the random jitter waveform first evolves into oscillatory solitons, and then gradually converges to a stable dissipative cavity soliton. The cavity solitons and the intra-cavity comb spectrum evolution with different detuning are shown in Fig. 3 with spectral and temporal snapshots. The initial time domain and the frequency domain input pump light are shown in Fig. 3(a) and (b), respectively. According to the change process of the time domain waveform in the MRR, the formation process of the dissipative cavity soliton can be divided into four consecutive stages [13,36,49]: Stable Modulation Instability (SMI) situation, Unstable Modulation Instability (UMI) situation, Unstable Cavity Soliton (UCS) situation, Stable Cavity Soliton (SCS) situation. Their time domain solitons and frequency domain optical frequency 3
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Fig. 2. (a) The evolution of the cavity soliton. (b) The evolution of Kerr optical frequency combs. (c) The normalized intra-cavity power curve. (Roundtrip is the number of turns in the simulation.)
sidebands even further away from the pump mode. At last the combs can span a full octave, as shown by the simulation above.
comb generation in single- and dual-pump cavities. The dual pumps make the frequency interval of the combs adjustable by the FSR of the MRR, and the number of solitons in the cavity can be controlled. We define the primary pump power Pin1 and secondary pump power Pin2 of the dual pumps. The same naming scheme for the ternary pumps and so on. Taking n = 8 as an example, which means that the two pump
Simulation of the dual-pump and ternary-pump scheme It is instructive to study the different mechanisms for the frequency
Fig. 3. Time domain solitons (a, c, e, g, i) and frequency domain optical frequency combs (b, d, f, h, j) at different stages. 4
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S. Wang, et al.
Fig. 4. (a) The mechanism of the detuning of the modified LLE with the thermal effect. (b) The mechanism of the detuning of the LLE without the thermal effect. (c) The variation of the temperature in the MRR. T is the temperature increment in the MRR (The solid blue curve).dT is the temperature increment in every roundtrip (The red dotted line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. (a) The evolution of cavity solitons (n = 8). (b) The evolution of Kerr optical frequency combs.
Fig. 6. (a) The input light field in the time domain by dual pumps (n = 8). (b) The input light field in the frequency domain by dual pumps. (c) Time domain solitons of the output. (d) Frequency domain optical frequency combs.
frequencies are separated by 8 times of the FSR of the MRR, we study the nonlinear dynamics of dual-pump Kerr frequency combs in the MRR. The evolution of the time domain solitons and the frequency
domain Kerr frequency combs are shown in Fig. 5(a) and (b) respectively, with the pump power of Pin1 = 0.5 W and Pin2 = 0.05 W . Compared with the single-pump condition, the primary comb 5
Results in Physics 15 (2019) 102789
S. Wang, et al.
generation mechanism of the dual-pump is no longer the MI process, but the nondegenerate FWM of the two pump modes. The precondition for the occurrence of FWM is to satisfy the conservation of energy and the conservation of momentum. The two pump sources are phaselocked and the spectral broadening satisfies the phase matching condition. In the case of eight peak waveforms, the peak fluctuation is gradually compressed into eight pulse waveforms under the modulation of the intracavity dispersion and the nonlinear effects. The eight pulses are equally spaced within the MRR. The eight large-width pulse waveforms are split into several smaller-width pulses. These small-width pulses collide in close proximity during the transmission, so that the pulse energy transfer forms a new pulse. This collision process is in the transmission process and it continues to occur until several pulses of the small-width pulses evolve into a very high energy pulse. At last, the eight pulse waveforms are formed. This evolutionary mechanism determines the number of solitons retained in the cavity. The number of solitons is equal to the initial number of input pulse peaks, which means that the n-value in the dual-pump frequency interval(n FSR ) . The cavity solitons and intra-cavity comb spectrum can be found in Fig. 6. The time domain and the frequency domain input dual-pump light are shown in Fig. 6(a) and (b). We can finally see eight solitons in the cavity and the interval of the optical frequency combs is eight times of the FSR of the MRR. To investigate the evolution mechanism of the dual-pump comb generation, we simulate the comb generation under different primary pump powers of 0.2 W, 0.5 W and 0.7 W, corresponding to three different states of the frequency comb generation. For the small pump amplitude, the combs will be generated through threshold-less nondegenerate FWM, resulting in a narrow bandwidth, as shown in Fig. 7(a). When the pump amplitude is larger, the comb generation is governed by threshold-less non-degenerate FWM process with a larger frequency comb bandwidth than in Fig. 7(a), as depicted in Fig. 7(b). As the pump amplitude crosses the MI threshold, the comb generation takes place via both non-degenerate FWM and intensity dependent MI processes, possessing secondary comb lines, as shown in Fig. 7(c). The dual-pump frequency combs that are generated below the MI threshold are stable [40]. In order to create a variable FSR comb, one should ensure that the pump amplitude must be smaller than the MI threshold. We also simulate the ternary-pump frequency comb generation and the dual-pump frequency comb generation under the same total pump power. The power of the ternary pumps are Pin1 = 0.3W, Pin2 = 0.05W, Pin3 = 0.05W and the power of the dual pumps are Pin1 = 0.3W, Pin2 = 0.1W . The input pump frequencies are separated by the same eight FSRs around the wavelength of 3.5 μm.
From the results of the simulation in Fig. 8(b) and (d), the ternary-pump condition can result in the frequency combs with a wider bandwidth and a flatter spectrum compared with the dual-pump condition. In order to find the reason of this phenomenon, we simulate the thresholdless non-degenerate FWM process of these two different mechanisms. The power of the dual pumps are Pin1 = 0.1W, Pin2 = 0.02W and the power of the ternary pumps are Pin1 = 0.1W, Pin2 = 0.01W, Pin3 = 0.01W . The results can be seen in Fig. 9. One set of FWM process occurs (green arrow) in the dual-pump mechanism while two sets of FWM process occur (green arrow and yellow arrow) in the ternary -pump mechanism under the same total pump power. This can attribute to a more efficient non-degenerate FWM process for the primary combs generation during the ternary-pump condition. The pump power required to produce the same comb span wavelength range of 2.5 μm–5.2 μm under three different pump conditions is shown in Table 1, which shows that the ternary-pump approach requires least total pump power. Conclusion In conclusion, we have shown a silicon MRR with an anomalous dispersion that are formed by suspended ridge waveguides. The evolution of the mid-infrared octave-spanning frequency combs in the frequency domain and the cavity solitons in the time domain are simulated by the modified LLE model adding the thermal effect. The stable dissipative cavity soliton and mode-locked combs spanning the wavelength range of 2.5 μm–5.2 μm are achieved by the simulation. We make a study of the nonlinear dynamics of dual-pump and ternarypump MRR Kerr frequency combs with variable FSRs, which means that the frequency interval of the combs is adjustable by controlling the frequencies of the pumps. Compared with the single-pump condition, the frequency combs with multi-pump can be generated through threshold-less non-degenerate FWM and the intervals of the frequency combs can be controlled. We find that the ternary-pump scheme has wider bandwidth and flatter frequency combs compared with the dualpump scheme. The ternary-pump approach requires least pump power which is required to produce the same comb span in these three pump conditions. This simulation work is very instructive for the experimental generation and realization of the MIR optical frequency combs in the silicon based MRR, which is important for spectroscopy applications and biosensors. CRediT authorship contribution statement Shuxiao Wang: Conceptualization, Methodology, Software, Writing - original draft. Qing Wang: Validation, Investigation. Wei Wang: Writing - review & editing, Visualization. Xi Wang: Investigation, Project administration. Mingbin Yu: Funding acquisition, Data curation. Qing Fang: Supervision, Resources. Yan Cai: Supervision, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by the National Key Research and Development Program of China (Grant No. 2018YFB2200500), the Shanghai Municipal Science and Technology Major Project (Grant No. 2017SHZDZX03) and the National Natural Science Foundation of China (No. 61764008).
Fig. 7. Frequency domain optical frequency combs under different pump Pin1 = 0.2 W, Pin2 = 0.01 W ; power conditions (n = 8) (a) (b) Pin1 = 0.5 W, Pin2 = 0.01 W ; (c) Pin1 = 0.7 W, Pin2 = 0.01 W ; 6
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Fig. 8. Time domain solitons and frequency domain optical frequency combs (n = 8). (a, b) The ternary-pump condition of Pin1 = 0.3 W, Pin2 = 0.05 W, Pin3 = 0.05 W (c, d) The dual-pump condition of Pin1 = 0.3 W, Pin2 = 0.1 W . [6] Del'Haye P, Schliesser A, Arcizet O, Wilken T, Holzwarth R, Kippenberg TJ. Optical frequency comb generation from a monolithic microresonator. Nature 2007;450:1214–7. [7] Gerginov V, Tanner CE, Diddams SA, Bartels A, Hollberg L. High-resolution spectroscopy with a femtosecond laser frequency comb. Opt Lett 2005;30:1734–6. [8] Guo H, Herkommer C, Billat A, Grassani D, Zhang C, Pfeiffer MHP, Weng W, Bres CS, Kippenberg TJ. Mid-infrared frequency comb via coherent dispersive wave generation in silicon nitride nanophotonic waveguides. Nat Photonics 2018;12:330. [9] Soref R. Mid-infrared photonics in silicon and germanium. Nat Photonics 2010;4:495–7. [10] Fedeli J-M, Nicoletti S. Mid-infrared (Mid-IR) silicon-based photonics. Proc. IEEE 2018;106:2302–12. [11] Savchenkov AA, Matsko AB, Ilchenko VS, Solomatine I, Seidel D, Maleki L. Tunable optical frequency comb with a crystalline whispering gallery mode resonator. Phys Rev Lett 2008;101. [12] Liang W, Savchenkov AA, Matsko AB, Ilchenko VS, Seidel D, Maleki L. Generation of near-infrared frequency combs from a MgF2 whispering gallery mode resonator. Opt Lett 2011;36:2290–2. [13] Herr T, Brasch V, Jost JD, Wang CY, Kondratiev NM, Gorodetsky ML, et al. Temporal solitons in optical microresonators. Nat Photonics 2014;8:145–52. [14] Li J, Lee H, Chen T, Vahala KJ. Low-pump-power, low-phase-noise, and microwave to millimeter-wave repetition rate operation in microcombs. Phys Rev Lett 2012;109. [15] Yi X, Yang Q-F, Yang KY, Suh M-G, Vahala K. Soliton frequency comb at microwave rates in a high-Q silica microresonator. Optica 2015;2:1078–85. [16] Jung H, Fong KY, Xiong C, Tang HX. Electrical tuning and switching of an optical frequency comb generated in aluminum nitride microring resonators. Opt Lett 2014;39:84–7. [17] Hausmann BJM, Bulu I, Venkataraman V, Deotare P, Loncar M. Diamond nonlinear photonics. Nat Photonics 2014;8:369–74. [18] Griffith AG, Lau RKW, Cardenas J, Okawachi Y, Mohanty A, Fain R, et al. Siliconchip mid-infrared frequency comb generation. Nat Commun 2015;6. [19] Joshi C, Jang JK, Luke K, Ji X, Miller SA, Klenner A, et al. Thermally controlled comb generation and soliton modelocking in microresonators. Opt Lett 2016;41:2565–8. [20] Foster MA, Levy JS, Kuzucu O, Saha K, Lipson M, Gaeta AL. Silicon-based monolithic optical frequency comb source. Opt Express 2011;19:14233–9. [21] Brasch V, Geiselmann M, Herr T, Lihachev G, Pfeiffer MHP, Gorodetsky ML, et al. Photonic chip-based optical frequency comb using soliton Cherenkov radiation. Science 2016;351:357–60. [22] Saha K, Okawachi Y, Shim B, Levy JS, Salem R, Johnson AR, et al. Modelocking and femtosecond pulse generation in chip-based frequency combs. Opt Express 2013;21:1335–43. [23] Huang SW, Yang J, Lim J, Zhou H, Yu M, Kwong DL, et al. A low-phase-noise 18 GHz Kerr frequency microcomb phase-locked over 65 THz. Sci Rep 2015;5. [24] Xue X, Xuan Y, Liu Y, Wang P-H, Chen S, Wang J, et al. Mode-locked dark pulse Kerr combs in normal-dispersion microresonators. Nat Photonics 2015:594. [25] Lin H, Luo Z, Gu T, Kimerling LC, Wada K, Agarwal A, et al. Mid-infrared integrated photonics on silicon: a perspective. Nanophotonics 2018;7:393–420. [26] Liu X, Osgood Jr. RM, Vlasov YA, Green WMJ. Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides. Nat Photonics 2010;4:557–60. [27] Pearl S, Rotenberg N, van Driel HM. Three photon absorption in silicon for 2300–3300 nm. Appl Phys Lett 2008;93. [28] Gai X, Yu Y, Kuyken B, Ma P, Madden SJ, Van Campenhout J, et al. Nonlinear absorption and refraction in crystalline silicon in the mid-infrared. Laser Photonics Rev 2013;7:1054–64. [29] Liu M, Wang L, Sun Q, Li S, Ge Z, Lu Z, et al. Influences of multiphoton absorption and free-carrier effects on frequency-comb generation in normal dispersion silicon microresonators. Photonics Res 2018;6:238–43. [30] Xia Y, Qiu C, Zhang X, Gao W, Shu J, Xu Q. Suspended Si ring resonator for mid-IR
Fig. 9. Frequency domain optical frequency combs (n = 8). (a) The dual-pump condition of Pin1 = 0.1 W, Pin2 = 0.02 W. (b) The ternary-pump condition of.Pin1 = 0.1 W, Pin2 = 0.01 W, Pin3 = 0.01 W. Table 1 The total pump power under three pump conditions.
Total power
Single-pump
Dual-pump
Ternary-pump
0.75 W
0.55 W
0.4 W
Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.rinp.2019.102789. References [1] Jiang Z, Huang C-B, Leaird DE, Weiner AM. Optical arbitrary waveform processing of more than 100 spectral comb lines. Nat Photonics 2007;1:463–7. [2] Diddams SA, Udem T, Bergquist JC, Curtis EA, Drullinger RE, Hollberg L, et al. An optical clock based on a single trapped Hg-199(+) ion. Science 2001;293:825–8. [3] Newbury NR. Searching for applications with a fine-tooth comb. Nat Photonics 2011;5:186–8. [4] Diddams SA, Hollberg L, Mbele V. Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb. Nature 2007;445:627–30. [5] Steinmetz T, Wilken T, Araujo-Hauck C, Holzwarth R, Haensch TW, Pasquini L, et al. Laser frequency combs for astronomical observations. Science 2008;321:1335–7.
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S. Wang, et al. application. Opt Lett 2013;38:1122–4. [31] Hansson T, Modotto D, Wabnitz S. On the numerical simulation of Kerr frequency combs using coupled mode equations. Opt Commun 2014;312:134–6. [32] Matsko AB, Savchenkov AA, Strekalov D, Ilchenko VS, Maleki L. Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion. Phys Rev A 2005;71. [33] Chembo YK, Yu N. Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators. Phys Rev A 2010;82. [34] Matsko AB, Savchenkov AA, Liang W, Ilchenko VS, Seidel D, Maleki L. Mode-locked Kerr frequency combs. Opt Lett 2011;36:2845–7. [35] Coen S, Randle HG, Sylvestre T, Erkintalo M. Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model. Opt Lett 2013;38:37–9. [36] Lamont MRE, Okawachi Y, Gaeta AL. Route to stabilized ultrabroadband microresonator-based frequency combs. Opt Lett 2013;38:3478–81. [37] Chembo YK, Menyuk CR. Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators. Phys Rev A 2013;87. [38] Hansson T, Modotto D, Wabnitz S. Dynamics of the modulational instability in microresonator frequency combs. Phys Rev A 2013;88. [39] Strekalov DV, Yu N. Generation of optical combs in a whispering gallery mode resonator from a bichromatic pump. Phys Rev A 2009;79. [40] Hansson T, Wabnitz S. Bichromatically pumped microresonator frequency combs. Phys Rev A 2014;90.
[41] Peccianti M, Pasquazi A, Park Y, Little BE, Chu ST, Moss DJ, et al. Demonstration of a stable ultrafast laser based on a nonlinear microcavity. Nat Commun 2012;3. [42] Pasquazi A, Peccianti M, Little BE, Chu ST, Moss DJ, Morandotti R. Stable, dual mode, high repetition rate mode-locked laser based on a microring resonator. Opt Express 2012;20:27355–62. [43] Wang W, Chu ST, Little BE, Pasquazi A, Wang Y, Wang L, et al. Dual-pump Kerr micro-cavity optical frequency comb with varying FSR spacing. Sci Rep 2016;6. [44] Okawachi Y, Yu M, Luke K, Carvalho DO, Ramelow S, Farsi A, et al. Dual-pumped degenerate Kerr oscillator in a silicon nitride microresonator. Opt. Lett. 2015;40:5267–70. [45] Hong S, Ledee F, Park J, Song S, Lee H, Lee YS, et al. Mode-locking of all-fiber lasers operating at both anomalous and normal dispersion regimes in the C- and L-Bands using thin film of 2D perovskite crystallites. Laser Photonics Rev 2018;12. [46] Carmon T, Yang L, Vahala KJ. Dynamical thermal behavior and thermal self-stability of microcavities. Opt Express 2004;12:4742–50. [47] Mingfang H, Kaixin C, Zhefeng H. Research on Kerr optical frequency comb based on micro-ring resonator with thermal effect. Laser Optoelectron Prog 2018;11:358–68. [48] Agarwal AM, Kimerling LC, Michel J. Nonlinear Group IV photonics based on silicon and germanium: from near-infrared to mid-infrared. Nanophotonics 2014;3:247–68. [49] Erkintalo M, Coen S. Coherence properties of Kerr frequency combs. Opt Lett 2014;39:283–6.
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Results in Physics journal homepage: www.elsevier.com/locate/rinp
Pump condition dependent Kerr frequency comb generation in mid-infrared Shuxiao Wang
a,b
, Qing Wang
a,b
c
a
a,c
, Wei Wang , Xi Wang , Mingbin Yu , Qing Fang
c,d,⁎
a,⁎
, Yan Cai
T
a
State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Changning Road 865, Shanghai 200050, China University of the Chinese Academy of Sciences, Yuquan Road 19, Beijing 100049, China c Shanghai Industrial µTechnology Research Institute, Chengbei Road 235, Shanghai 201800, China d College of Science, Kunming University of Science and Technology, Kunming 650500, China b
ARTICLE INFO
ABSTRACT
Keywords: Kerr frequency combs Microring resonator Mid-infrared Ternary-pump Silicon photonics
We propose and simulate the mid-infrared (MIR) frequency comb generation based on the Kerr nonlinear effect in a silicon microring resonator (MRR). The MRR is formed by the suspended ridge waveguides which are designed to have the anomalous group-velocity dispersion. Taking the thermal effect into consideration, the modified Lugiato–Lefever equation (LLE) is used to simulate the frequency comb generation. The stable dissipative cavity soliton and mode-locked combs in the MIR wavelength range of 2.5 μm–5.2 μm can be obtained under different pump conditions. It is found that the ternary-pump scheme has wider bandwidth and flatter frequency combs as compared with the dual-pump scheme. The lowest total pump power is required for the ternary-pump approach to produce the same comb span in these three pump conditions.
Introduction The invention of the optical frequency combs provides a revolutionary tool for the optical arbitrary waveform generation [1], facilitating the development of the optical clockwork [2,3], the celestial spectrum calibration [4,5], and the precision spectroscopy [6,7]. MIR optical frequency combs are of significant interest for the molecular spectroscopy application due to the large absorption of molecular vibrational modes in this spectral region [8–10]. Kerr optical frequency combs based on the micro-resonator have attracted lots of attention because of small size, low power consumption, large quality-factor (Q) and mature production process. The optical frequency combs based on micro-resonators have been widely investigated on various materials such as calcium fluoride [11], magnesium fluoride [12,13], silica [6,14,15], aluminum nitride [16], diamond [17], silicon [18], and silicon nitride [19–24]. Silicon-on-insulator (SOI) has emerged as a particularly attractive platform for the chip-scale frequency comb generation due to the availability of CMOS fabrication technology, which allows for the integration of electronic and optical elements in a compact, robust and portable device. Silicon is a promising material for MIR wavelength range application since it has the relatively large nonlinearity, the high refractive index and a moderately broad transparency window up to
around the wavelength of 7 μm [18,25]. Moreover, there is no twophoton absorption (TPA) effect which prevents the generation of combs at the wavelength range longer than 2.2 μm [26]. The nonlinear loss by the three-photon absorption (3PA) is negligible at the wavelength longer than 3.3 μm [27]. The nonlinear loss beyond 3.3 μm is even lower due to the absence of 2PA and 3PA effects, where the dominant nonlinear loss is from four-photon absorption (4PA) [28]. The influence of the 4PA effect is small, and a comb can be generated without the need of a positive-intrinsic-negative (PIN) structure [29]. However, the implementation of the SOI platform in the mid-infrared wavelength range is restricted due to the high absorption of buried oxide (BOX) layer [30]. The microring based on the suspended silicon ridge waveguides are designed to reduce the absorption loss. The formalisms based on the coupled mode equations [31–33] and the driven and damped nonlinear Schrödinger (NLS) equation [34–38], which is also known as the LLE, are two main theoretical formalisms to describe the frequency comb generation and temporal dynamics in micro-resonator devices. The LLE can simulate ultra-broadband combs such as octave spanning frequency combs because of the high computing speed. When pumped with the laser light, the micro-cavity will collect a high-intensity light field in a small mode volume, which changes the micro-cavity temperature. Therefore, we add the phase shift caused by the introduction of the thermal effect to modify the LLE
Corresponding authors at: College of Science, Kunming University of Science and Technology, Kunming 650500, China (Qing Fang); State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Changning Road 865, Shanghai 200050, China (Yan Cai). E-mail addresses: [email protected] (Q. Fang), [email protected] (Y. Cai). ⁎
https://doi.org/10.1016/j.rinp.2019.102789 Received 2 October 2019; Received in revised form 1 November 2019; Accepted 1 November 2019 Available online 05 November 2019 2211-3797/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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model. Previous works on the MIR frequency comb generation have primarily focused on the experimental setup under single CW laser pump condition. In addition to the single CW laser pump, here we also investigate an alternative pump configuration where the cavity is simultaneously pumped at two or three different frequencies. The use of dual pumps provides an additional degree of freedom in the form of the modulation frequency. There have been some researches on the dualpump micro-resonator Kerr frequency combs in the near infrared (NIR) wavelength range. Strekalov and Yu proposed a comb generation technique based on the use of dual optical pumps in a magnesium fluorite resonator, which demonstrated to have a better efficiency for the comb generation process [39]. Hansson and Wabnitz made a study of the nonlinear dynamics of the bichromatically pumped micro-resonator Kerr frequency combs by the truncated four-wave model [40]. Other experimental and theoretical results on the dual-pump inputs have also been studied in both the normal and the anomalous dispersion regime [41–45]. But no similar work has been done in the MIR wavelength range yet. In this work, we study the pump condition dependent Kerr frequency comb generation in the MIR wavelength range. The silicon MRR is formed by the suspended ridge waveguides which are designed to have the anomalous group-velocity dispersion in the MIR wavelength range. The evolution of the cavity solitons and frequency combs are studied by a modified LLE model, which taking the thermal effect into consideration. The mode-locked combs in the MIR wavelength range of 2.5 μm–5.2 μm are obtained when a single CW pumped laser couples into the resonator. The nonlinear dynamics of the MRR Kerr frequency combs under different pump conditions are investigated in the MIR wavelength range. Compared with the single-pump condition, the frequency combs with multi pumps can be generated through thresholdless non-degenerate four-wave-mixing (FWM) and the intervals of the frequency combs can be controlled. It is found that the ternary-pump scheme has wider bandwidth and flatter frequency combs as compared with the dual-pump scheme. The lowest pump power is required for the ternary-pump approach to produce the same comb span in these three pump conditions.
the pump’s angular frequency and the n th angular resonance frequency that is pumped. The thermal effect in the MRR induced by the traveling light beam can change the refractive index and geometry of the MRR, which causes the drift of the resonant wavelength of the MRR. In the process analysis of the Kerr optical frequency comb generated in the MRR, this wavelength drift can be attributed to the phase shift. The phase shift of the light wave caused by the thermal effect after passing the unit length is expressed as, t
where KTHO is the thermo-optic coefficient of the core material, T (t ) is temperature alteration of the MRR, k 0 is the wave vector in vacuum. n1 is the amount of change of the refractive index of the waveguide core layer, which we use to indicate the amount of change of the effective refractive index of the waveguide. T (t ) can be derived from the dynamical thermal behaviour of the micro-cavity [46]. Based on the energy conservation principle, the net heat delivered to the cavity is the heat that goes in (qin ) subtracted by the heat that goes out () [46]:
Cp
i
k
E+i k 2
E (m + 1) (0, ) =
Ein +
i
k!
k
E + i |E|2 E
E (m) (L , ) ei
1
E (t , ) = t
2
2
i
0
k
+ iL k 2
k!
i
tR
+ iL |E|2 E +
(
1 p
0 (1 + a T )
/2
)
2
K T (t ) +1
E (t , ) = t
2
2
k
i 0 + iL k 2
k!
i
k
+ iL |E|2 + iLk 0 KTHO T (t ) E +
Ei
(6)
T (t ) is calculated by using the thermal dynamic Eq. (5) of the MRR. Eq. (6) can be efficiently simulated using the split-step Fourier method: i.e. Eq. (6) is split into the following two equations: tR
E (t , ) = t
2
2
i
0
k
+ iL k 2
k!
i
k
+ iLk 0 KTHO T (t ) E (7)
tR
E (t , ) = (iL |E|2 ) E + t
Ei
(8)
The Eqs. (7) and (8) are solved by the Fast Fourier Transform (FFT) method. Considering the nonlinear dynamics of the dual-pump micro-resonator Kerr frequency combs, the input pump light field can be expressed as:
(2)
k
= Ih
dn
where i is the linear absorption coefficient inside the resonator, is the power transmission coefficient, τ is the time, L is the roundtrip length of the resonator, and 0 is the linear phase accumulated by the intracavity field with respect to the pump field over one roundtrip. Other resonator parameters include the nonlinear coefficient , the k th dispersion coefficient k , and the CW driving field Ein . Combining the NLSE in Eq. (1) and the boundary conditions in Eq. (2), the LLE that describes the dynamics of a MRR is deduced as,
tR
dt
coefficient of the resonant wavelength is a = + dT / n 0 , which contains both the thermal expansion and the thermal index change. Therefore, the LLE model can be modified with the thermal effect as follows [47]:
(1)
0
dqout (t )
Here Cp is the heat capacity (J/°C) and K (J/(s°C)) is the thermal conductivity between the cavity mode volume and the surrounding.Ih is the actual optical power in the thermal cavity, 0 is the cold cavity resonant wavelength, and p is the pump wavelength. The temperature
The driven nonlinear Schrödinger equation (NLSE) describes the evolution of the electric field envelope in the waveguide. The periodic boundary condition describes the relationship between the continuous circular light field and the input pump light field [34,35]. The NLSE and the periodic boundary condition are as follows:
2
dqin (t ) d T (t ) = dt dt
(5)
Model and principle
E (z , ) = z
(4)
= k 0 n1 = k 0 KTHO T (t )
Ein = Ein1 + Ein2 =
Pin1 ei
1t
+
Pin2 e i
2t
(9)
where Pin1 and Pin2 are the powers of the two pump light fields, 1 and 2 are the angular frequencies of the two pump light fields. In the simulation, we set 2 = 1 + n FSR , which means that the two pump frequencies are separated by n times of the FSR of the MRR. The structure of the suspended ridge waveguide on the SOI substrate is shown in Fig. 1(a). The thicknesses of the waveguide (H), slab (HS), and buried oxide are 500 nm, 100 nm, and 3 μm, respectively. The upper cladding layer is the air. The BOX layer under the waveguide can be removed locally by the wet etching through the holes in the neighbouring supporting slab. The designed edge-to-edge distance between the waveguide and the hole is 2.5 μm to avoid interacting with the optical mode transmitted in the waveguide. A flat dispersion curve in an
Ei (3)
where tR is the round-trip time of the optical field in the MRR, E (t , ) and Ei are the intracavity field and the input field, t and τ are the slow and fast time. α is the cavity loss, equalling to ( i + )/2 . 0 equals to tR ( n 0 ) which is the cavity phase detuning, where 0 and n are 2
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Fig. 1. (a) A schematic of a suspended silicon waveguide. (b) Dispersion properties of the waveguide (H = 500 nm, HS = 100 nm) with different width. (c) The group index curve of the waveguide (H = 500 nm, HS = 100 nm, W = 2.2 μm) versus wavelength. The inset shows the electromagnetic field profile for the fundamental TE mode.
anomalous regime is essential to excite the frequency combs. The dispersion curves for a suspended waveguide with different widths are calculated [Fig. 1(b)]. Considering the effect of the bending radius on the dispersion, we choose the width of 2.2 μm to realize the near-zero dispersion around 3.5 μm wavelength. The group index of the waveguide (H = 500 nm, HS = 100 nm, W = 2.2 μm) can be extracted by the Finite Difference Eigenmode (FDE) solver in Lumerical [Fig. 1(c)]. The group index is 3.7 and the effective cross section area is 1.21 μm2 at 3.5 μm wavelength. In this design, the radius of the MRR is 100 μm. Considering the scattering loss, we assume the total waveguide propagation loss is 0.3 dB/cm, which corresponds to an intrinsic quality factor of 224000. The field coupling coefficient is set to be 0.018. The nonlinear Kerr coefficient of Silicon at 3.5 μm is 3.6 × 10−18 m2/W [48]. The thermal expansion coefficient of Silicon is 2.63 × 10−6 K−1. The FSR of the MRR is 128 GHz. In the simulation we neglect the higher-order dispersion, the self-steepening effect, and other higher-order nonlinearities.
combs are shown in Fig. 3(c)–(j), respectively, corresponding to the four different situations in Fig. 2(c). In SMI, the power in the cavity is rapidly increased as the detuning increases. When the power in the MRR exceeds the threshold of the optical parametric oscillation, the modulation instability (MI) produces primary sidebands in its gain spectrum. Due to the increase of power in the MRR, a gain is also generated around the primary sidebands of the optical comb, and the mini-combs are generated in the gain spectrum around the primary sidebands in UMI. The rapid decrease of the power in the cavity causes the parametric gain of the random pulse to decrease, and gradually disappears in the cavity because of the absorption and attenuation in the cavity. Finally, the power in the cavity is stabilized, and a soliton is stable in the time domain. In the frequency domain, the frequency comb line is full of the entire spectrum bandwidth and the comb line power in all modes changes smoothly. As shown in Fig. 3(j), we demonstrate the mode locked frequency combs in the mid-IR wavelength range of 2.5 μm–5.2 μm with the comb-line power lager than −70 dB. In the simulation, we adopt the method of adjusting the initial detuning amount, which corresponds to linearly adjusting the pump wavelength in the experiment to achieve the purpose of making the MRR quickly enter the thermal equilibrium state. One of the resonance wavelengths of the cold cavity is 3.502 μm. Considering the process from blue detuning to red detuning during the generation of the stable optical frequency combs. The initial detuning is set to be −0.0065. The detuning in the simulation linearly increases at the speed of 4 × 10−3/ns. The initial detuning is varied by changing the simulation roundtrip number. The total roundtrip is set to be 15,000, and the amount of detuning is adjusted from the first roundtrip to the 5335th roundtrip. The detuning remains to be 0.1602 for the remainder of the simulation time. Kerr effect causes red shift of resonant wavelength. Considering the thermal effect, the resonant wavelength will red shift after the temperature rises due to the positive thermo-optic coefficient of the silicon material. The modified LLE model shows that the detuning need to compensate for the phase shift introduced by not only the nonlinear effects but also the thermal effect in order to meet the phase balance conditions, which is described in Fig. 4(a) and (b). The variation of the temperature in the MRR is shown in Fig. 4(c). The primary sideband generation mechanism for a single-pump MRR is the MI of the pump mode [40]. The primary sidebands are usually located multiple FSRs away from the frequency of the pumpmode. The primary sidebands can in turn interact with the pump mode to create additional sidebands through the FWM process. With the increments of the external pump intensity and detuning, comb generation can further occur through the cascaded FWM process that creates new
Simulation and analysis Simulation of the single pump condition We pump the cavity at the resonant wavelength adjacent to 3.5 μm, with a CW laser power of 0.75 W. By adjusting the detuning between the wavelength of the pump and the resonant wavelength of the MRR, the evolution process of the time domain cavity solitons, the frequency domain Kerr frequency combs and the simulation results of the cavity power variation are shown in Fig. 2a, b, and c, respectively. The adjustment of the detuning in the simulation is from effective blue-detuning to effective red-detuning. Under the combined effects of the dispersion and the nonlinear modulation, the random jitter waveform first evolves into oscillatory solitons, and then gradually converges to a stable dissipative cavity soliton. The cavity solitons and the intra-cavity comb spectrum evolution with different detuning are shown in Fig. 3 with spectral and temporal snapshots. The initial time domain and the frequency domain input pump light are shown in Fig. 3(a) and (b), respectively. According to the change process of the time domain waveform in the MRR, the formation process of the dissipative cavity soliton can be divided into four consecutive stages [13,36,49]: Stable Modulation Instability (SMI) situation, Unstable Modulation Instability (UMI) situation, Unstable Cavity Soliton (UCS) situation, Stable Cavity Soliton (SCS) situation. Their time domain solitons and frequency domain optical frequency 3
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Fig. 2. (a) The evolution of the cavity soliton. (b) The evolution of Kerr optical frequency combs. (c) The normalized intra-cavity power curve. (Roundtrip is the number of turns in the simulation.)
sidebands even further away from the pump mode. At last the combs can span a full octave, as shown by the simulation above.
comb generation in single- and dual-pump cavities. The dual pumps make the frequency interval of the combs adjustable by the FSR of the MRR, and the number of solitons in the cavity can be controlled. We define the primary pump power Pin1 and secondary pump power Pin2 of the dual pumps. The same naming scheme for the ternary pumps and so on. Taking n = 8 as an example, which means that the two pump
Simulation of the dual-pump and ternary-pump scheme It is instructive to study the different mechanisms for the frequency
Fig. 3. Time domain solitons (a, c, e, g, i) and frequency domain optical frequency combs (b, d, f, h, j) at different stages. 4
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Fig. 4. (a) The mechanism of the detuning of the modified LLE with the thermal effect. (b) The mechanism of the detuning of the LLE without the thermal effect. (c) The variation of the temperature in the MRR. T is the temperature increment in the MRR (The solid blue curve).dT is the temperature increment in every roundtrip (The red dotted line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. (a) The evolution of cavity solitons (n = 8). (b) The evolution of Kerr optical frequency combs.
Fig. 6. (a) The input light field in the time domain by dual pumps (n = 8). (b) The input light field in the frequency domain by dual pumps. (c) Time domain solitons of the output. (d) Frequency domain optical frequency combs.
frequencies are separated by 8 times of the FSR of the MRR, we study the nonlinear dynamics of dual-pump Kerr frequency combs in the MRR. The evolution of the time domain solitons and the frequency
domain Kerr frequency combs are shown in Fig. 5(a) and (b) respectively, with the pump power of Pin1 = 0.5 W and Pin2 = 0.05 W . Compared with the single-pump condition, the primary comb 5
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generation mechanism of the dual-pump is no longer the MI process, but the nondegenerate FWM of the two pump modes. The precondition for the occurrence of FWM is to satisfy the conservation of energy and the conservation of momentum. The two pump sources are phaselocked and the spectral broadening satisfies the phase matching condition. In the case of eight peak waveforms, the peak fluctuation is gradually compressed into eight pulse waveforms under the modulation of the intracavity dispersion and the nonlinear effects. The eight pulses are equally spaced within the MRR. The eight large-width pulse waveforms are split into several smaller-width pulses. These small-width pulses collide in close proximity during the transmission, so that the pulse energy transfer forms a new pulse. This collision process is in the transmission process and it continues to occur until several pulses of the small-width pulses evolve into a very high energy pulse. At last, the eight pulse waveforms are formed. This evolutionary mechanism determines the number of solitons retained in the cavity. The number of solitons is equal to the initial number of input pulse peaks, which means that the n-value in the dual-pump frequency interval(n FSR ) . The cavity solitons and intra-cavity comb spectrum can be found in Fig. 6. The time domain and the frequency domain input dual-pump light are shown in Fig. 6(a) and (b). We can finally see eight solitons in the cavity and the interval of the optical frequency combs is eight times of the FSR of the MRR. To investigate the evolution mechanism of the dual-pump comb generation, we simulate the comb generation under different primary pump powers of 0.2 W, 0.5 W and 0.7 W, corresponding to three different states of the frequency comb generation. For the small pump amplitude, the combs will be generated through threshold-less nondegenerate FWM, resulting in a narrow bandwidth, as shown in Fig. 7(a). When the pump amplitude is larger, the comb generation is governed by threshold-less non-degenerate FWM process with a larger frequency comb bandwidth than in Fig. 7(a), as depicted in Fig. 7(b). As the pump amplitude crosses the MI threshold, the comb generation takes place via both non-degenerate FWM and intensity dependent MI processes, possessing secondary comb lines, as shown in Fig. 7(c). The dual-pump frequency combs that are generated below the MI threshold are stable [40]. In order to create a variable FSR comb, one should ensure that the pump amplitude must be smaller than the MI threshold. We also simulate the ternary-pump frequency comb generation and the dual-pump frequency comb generation under the same total pump power. The power of the ternary pumps are Pin1 = 0.3W, Pin2 = 0.05W, Pin3 = 0.05W and the power of the dual pumps are Pin1 = 0.3W, Pin2 = 0.1W . The input pump frequencies are separated by the same eight FSRs around the wavelength of 3.5 μm.
From the results of the simulation in Fig. 8(b) and (d), the ternary-pump condition can result in the frequency combs with a wider bandwidth and a flatter spectrum compared with the dual-pump condition. In order to find the reason of this phenomenon, we simulate the thresholdless non-degenerate FWM process of these two different mechanisms. The power of the dual pumps are Pin1 = 0.1W, Pin2 = 0.02W and the power of the ternary pumps are Pin1 = 0.1W, Pin2 = 0.01W, Pin3 = 0.01W . The results can be seen in Fig. 9. One set of FWM process occurs (green arrow) in the dual-pump mechanism while two sets of FWM process occur (green arrow and yellow arrow) in the ternary -pump mechanism under the same total pump power. This can attribute to a more efficient non-degenerate FWM process for the primary combs generation during the ternary-pump condition. The pump power required to produce the same comb span wavelength range of 2.5 μm–5.2 μm under three different pump conditions is shown in Table 1, which shows that the ternary-pump approach requires least total pump power. Conclusion In conclusion, we have shown a silicon MRR with an anomalous dispersion that are formed by suspended ridge waveguides. The evolution of the mid-infrared octave-spanning frequency combs in the frequency domain and the cavity solitons in the time domain are simulated by the modified LLE model adding the thermal effect. The stable dissipative cavity soliton and mode-locked combs spanning the wavelength range of 2.5 μm–5.2 μm are achieved by the simulation. We make a study of the nonlinear dynamics of dual-pump and ternarypump MRR Kerr frequency combs with variable FSRs, which means that the frequency interval of the combs is adjustable by controlling the frequencies of the pumps. Compared with the single-pump condition, the frequency combs with multi-pump can be generated through threshold-less non-degenerate FWM and the intervals of the frequency combs can be controlled. We find that the ternary-pump scheme has wider bandwidth and flatter frequency combs compared with the dualpump scheme. The ternary-pump approach requires least pump power which is required to produce the same comb span in these three pump conditions. This simulation work is very instructive for the experimental generation and realization of the MIR optical frequency combs in the silicon based MRR, which is important for spectroscopy applications and biosensors. CRediT authorship contribution statement Shuxiao Wang: Conceptualization, Methodology, Software, Writing - original draft. Qing Wang: Validation, Investigation. Wei Wang: Writing - review & editing, Visualization. Xi Wang: Investigation, Project administration. Mingbin Yu: Funding acquisition, Data curation. Qing Fang: Supervision, Resources. Yan Cai: Supervision, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by the National Key Research and Development Program of China (Grant No. 2018YFB2200500), the Shanghai Municipal Science and Technology Major Project (Grant No. 2017SHZDZX03) and the National Natural Science Foundation of China (No. 61764008).
Fig. 7. Frequency domain optical frequency combs under different pump Pin1 = 0.2 W, Pin2 = 0.01 W ; power conditions (n = 8) (a) (b) Pin1 = 0.5 W, Pin2 = 0.01 W ; (c) Pin1 = 0.7 W, Pin2 = 0.01 W ; 6
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Fig. 9. Frequency domain optical frequency combs (n = 8). (a) The dual-pump condition of Pin1 = 0.1 W, Pin2 = 0.02 W. (b) The ternary-pump condition of.Pin1 = 0.1 W, Pin2 = 0.01 W, Pin3 = 0.01 W. Table 1 The total pump power under three pump conditions.
Total power
Single-pump
Dual-pump
Ternary-pump
0.75 W
0.55 W
0.4 W
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