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Abstract
The influence of frequency detuning on the field in silicon microresonators with multiphoton absorption and FC effect is investigated. In this study, results show that frequency detuning facilitates soliton generation. With appropriate frequency detuning, not only bright solitons but also dark ones can be excited in silicon microresonators, which compensates for the absence of solitons with multiphoton absorption and FC. In particular, the larger the frequency detuning is, the wider is the combs spectrum with 2PA obtained. In order to excite the soliton efficiently, the regulation of frequency detuning with multiphoton absorption and FC effect is also studied. In regulating the frequency detuning process with 2PA, a progressively enhanced soliton can be formed in the region near zero detuning. In the tuning process, 3PA can generate bright and dark solitons respectively at various detuning intervals, and independent bright solitons can be observed in microresonators with 4PA. The research results are significant for studying the generation of solitons in silicon microresonators with multiphoton absorption and FC effect.
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1. Introduction
Microresonator-based optical frequency combs have attracted extensive interests due to their compactness, flexibility, low power consumption, and compatibility with CMOS integration [1,2]. Frequency combs are caused by the well-known four-wave mixing (FWM) due to the nonlinear effect of the strong power density inside microresonators [3]. A variety of materials, such as Si3N4 [4], CaF2 [5], MgF2 [6,7], aluminum nitride [8,9], and lithium niobate [10,11] are used for microresonators. With different operating parameters, the microresonators work in various states, including bright or dark solitons [12,13], Turing patterns [14], and chaos [15,16]. The soliton is significant for the numerous precision measurement fields, because of the broadband comb spectrum [17,18]. At present, the operating wavelength of optical frequency combs based on the microresonators are mainly concentrated in the near infrared range. For extensive applications, operating wavelengths are expected to be extended to the mid-infrared (MIR) range.
The silicon is transparent in the MIR range, therefore, it is a promising candidate for frequency combs generation [19]. Moreover, the silicon microresonators are generally cladded by SiO2 or air [20,21], and the huge refractive index difference between the core and cladding materials is favorable to the confinement of optical modes inside the microresonators. This condition also leads to effective nonlinearity because of high power density, which facilitate frequency combs generation. However, a critical disadvantage of silicon microresonator is multiphoton absorption and concomitant free-carrier (FC) effects, which limit the nonlinear efficiency [22]. Previous researches have proved that when the pump wavelength is approximately half the bandgap energy of the material, two-photon absorption (2PA) and FC can be effectively moderated [23]. But for the available MIR range, multiphoton absorption and FC cannot be neglected. It is shown that the frequency comb spectrum cannot be formed in the silicon microresonator due to the universal 2PA. And in an etchless silicon microresonator with operating wavelength 2.4 µm, the combination of three-photon absorption (3PA) and FC also results in only a single mode in the microresonator [24]. To solve this problem, a suitable control of the free-carrier time (FCT) provides a mechanism for soliton frequency combs in the MIR range with silicon microresonators [25]. However, the techniques used to control FCT, including material modification [26,27], increasing surface recombination [28–30], and most prominently carrier removal through integration of a PIN diode across the waveguide [31–33], are all achieved by changing the structural parameters and materials properties of the microresonators, which are all complicated. For microresonators design [34,35], in addition to ultra-high Q values and nonlinear effects, multi-photon absorption must also be considered. Therefore, a convenient method is required to excite solitons in silicon microresonators with multiphoton absorption and FC in the MIR range.
In previous studies, only the influences of multiphoton absorption and free-carrier effects on frequency combs are discussed [24,39]. In this study, which is different from previous studies, we focus on the regulation of field inside microresonators with multiphoton absorption and free-carrier effects. We investigate the influence of frequency detuning on optical field, which covers several wavelength ranges in silicon microresonators with multiphoton absorption and FC effects. Results show that regulation of the frequency detuning is beneficial for solitons generation. By selecting appropriate frequency detuning, not only bright solitons but also dark solitons can be excited in silicon microresonators under the combined action of multiphoton absorption, FC effect and frequency detuning. It makes up for the absence of solitons with multiphoton absorption and FC. To excite the soliton efficiently, we studied the regulation of frequency detuning with multiphoton absorption and FC effect. We illustrate the field evolution with frequency tuning under multiphoton absorption. In the process of detuning, the field experiences various forms, including the available solitons. It is proved that solitons can effectively generate in the microresonator by means of detuning regulation. The results of this study are important for studying the generation of soliton fields in silicon microresonators. And it provides a new approach to excite solitons with multiphoton absorption and FC in the MIR range.
2. Theoretical model
The evolution of optical field in a microresonator is usually described by the well-known Lugiato–Lefever Equation (LLE) [36–38]. When higher-order dispersion, multiphoton absorption, free-carrier effect, and self-steepening are introduced, the LLE is described as follows [25]:
This study investigates the waveguide structure microresonator, which has a radius of 100 µm. Based on Eq. (2), the FC effect occurs with multiphoton absorption simultaneously. Therefore, the FC effect is also considered along with multiphoton absorption. The initial field in the microresonator is assumed as a weak Gaussian pulse. By setting the multiphoton absorption coefficient, the influence of 2PA, 3PA, and 4PA effect along with the FC effect on field can be discussed separately.
3. Influence of frequency detuning on field with multiphoton absorption and FC effects
At present, the widely used silicon microresonators with the SiO2 cladding material take effect in the telecom range, when the pump wavelength is 1.56 µm [41]. In these circumstances, 2PA plays a major role in the microresonators. And in an oxide-clad, etchless silicon device, 3PA becomes the dominant nonlinear absorption process [21]. Additionally, the model showing the action of 4PA is an air-clad, etchless silicon microresonator [21]. The parameters and properties of each model are summarized in Table 1.
Table 1. Model parameters of microresonators with multiphoton absorption and free-carrier effects
For the silicon microresonators, when the pump power is 1 W, the stable fields and corresponding spectra in the microresonators with different frequency detuning δ0 are exhibited in Fig. 1. The black curves show the presence of 2PA and the blue curves indicate the absence of 2PA. In Fig. 1(a-1), when the frequency detuning is negative, the microresonator with 2PA forms a bright soliton, and the field in the microresonator without 2PA is slightly fluctuant. The negative frequency detuning approaches zero (Fig. 1(b-1)), the field with 2PA evolves into double pulses, and the comb spectrum shows a significant slow modulation. With the additional remark that the fluctuation amplitude of the blue curves in Figs. 1(a-1) and (b-1) is the order of 10−12 mW, the distributions of field can be approximated as the DC field. In Figs. 1(c-1) and (c-2), the frequency detuning is a positive small value, the field with 2PA is a dark soliton with relatively weak amplitude, and that without 2PA is absolute DC distribution. According to the spectrograms, it can be observed that in the absence of 2PA, the field in the microresonator has almost a single mode. Therefore, with a suitable frequency detuning, the 2PA facilitate broadband frequency combs. In addition, we also found that bright solitons produced by 2PA, can be generated in a certain range of frequency detuning. The spectra with various frequency detuning are also shown in Fig. 2. In the range of negative frequency detuning, the magnitude of the detuning affects the width of the spectrum. The larger the frequency detuning, the wider the spectrum broadening, and it is also beneficial for a stronger spectrum.
Fig. 1. Stationary fields and spectra in the absence (blue) and presence (black) of 2PA with different frequency detuning δ0. (a-1) (a-2) δ0 = −2.4; (b-1) (b-2) δ0 = −0.6; (c-1) (c-2) δ0 = 0.8.
In the following, a SiO2-cladding, etchless silicon microresonator with pump wavelength 2.4 µm, where 3PA is dominant, is discussed. The results of field evolution in the absence (blue) and presence (black) of 3PA with different frequency detuning δ0 are demonstrated in Fig. 3. When the microresonator without 3PA is in large positive or negative detuning state (Fig. 3(a) and (d)), an approximate Gaussian soliton generates. However, with the consideration of 3PA, the ruleless distribution which varies with propagation exists in the microresonator. On the contrary, in the interval of detuning near zero, 3PA can form a bright soliton (Fig. 3(b)) and dark soliton (Fig. 3(c)). In the corresponding spectra, except for the pump mode, the intensity of other modes is relatively weak. Only DC fields exist in the absence of 3PA. The result proves that large frequency detuning contributes to solitons generation, and in the weak detuning range, 3PA is conducive to bright or dark soliton.
Fig. 3. Stationary fields and spectra in the absence (blue) and presence (black) of 3PA with different frequency detuning δ0. (a-1) (a-2) δ0 = −3; (b-1) (b-2) δ0 = −0.2; (c-1) (c-2) δ0 = 0.6; (d-1) (d-2) δ0 = 3.
The operating wavelength is extended to a longer range, and an air-clad, etchless silicon microresonator, which is pumped at 4 µm, is employed. This device lacks 2PA and 3PA, and the primary nonlinear loss is four-photon absorption (4PA). Figure 4 shows the ultimate fields and spectra with 4PA. Evidently, appropriate nonlinear loss of 4PA results in the formation of bright (Fig. 4(a)) and dark (Fig. 4(b)) solitons in the range of detuning close to zero, compared with the DC distribution without 4PA. However, the relative intensity of the soliton pulse is extremely weak. In Fig. 4(a-1), the peak of the soliton pulse relative to background is approximately 2 nW. In Fig. 4(b-1), the detuning parameter is small, which is around 0. In addition, compared with β2PA and β3PA, β4PA is also smaller. Weak frequency detuning and 4PA result in extremely small pulse peak power. Consequently, the fluctuation of dark soliton in Fig. 4(b-1) is only 0.1 nW. The broadband spectrum is at a very low level because of the exceedingly weak pulse intensity. When the detuning parameter increases, minimal difference exists between the fluctuant fields with and without 4PA, and the two curves are almost coincident (Fig. 4(c-1)). Only a slight difference is observed in the spectrum near the 2 µm band (Fig. 4(c-2)). Thus, in the case of 4PA, the solitons can be excited in microresonators with the aid of detuning regulation.
Fig. 4. Stationary fields and spectra in the absence (blue) and presence (black) of 4PA with different frequency detuning δ0. (a-1) (a-2) δ0 = −0.2; (b-1) (b-2) δ0 = 0.6; (c-1) (c-2) δ0 = 2.8.
4. Detuning regulation of soliton with multiphoton absorption and FC effects
A temporal soliton inside a microresonator is important for obtaining a broadband comb spectrum. According to the preceding discussion, solitons can be excited only if frequency detuning is equal to particular values. Nevertheless, the linear frequency detuning δ0 = (ω0-ωp)tR/L [25], where ω0 expresses a resonant frequency of the microresonator, which is affected by various factors, such as dispersion, structure and operating temperature of the microresonator. Thus, directly setting the appropriate pump wavelength to excite solitons is a complex process. In general, frequency tuning, which is realized by the scanning pump wavelength, is regularly used in practice [42,43]. As a result, the detuning regulation with multiphoton absorption is also discussed.
First, the detuning regulation process of 2PA is explored. We set the frequency detuning to scan at a constant speed and the scanning range is −3 to 3. The entire process of detuning regulation only lasts 600 ns, and heat conduction does not occur during this time. Therefore, it is assumed that the temperature of the microresonator remains constant during the tuning process and the thermal effect can be ignored. The curve in Fig. 5(a) is the variation of optical power in the microresonator with scanning time, which indicates the variation of field distribution. The field evolution is divided into three distinct stages. In the initial period, the field does not exhibit a regular distribution owing to a strong negative detuning effect, and the power in the microresonator is also low. In the second phase, the power begins to increase rapidly, and the single soliton pulse is stable. During this stage the pulse changes as shown in Fig. 5(b). As the frequency detuning approaches zero, the pulse energy increases gradually. When the detuning parameter is scanned to a specific value, the balance among the nonlinear effect, 2PA and detuning in the microresonator is destroyed, which results in a drastic reduction of the power. The strong positive detuning leads to an irregular field. When the field evolves into a new status, a noticeable change in power occurs. Therefore, the operational region of the microresonator can be deduced by monitoring the output power.
Fig. 5. (a) Curve of power versus scanning time with 2PA and (b) evolution of soliton pulses in the second stage in (a).
Second, we observe the process of frequency detuning regulation with 3PA. The adjustment range and speed of detuning remain unchanged. The power variation curve is shown in Fig. 6, the shape of which is similar to that in Fig. 5(a), but the evolution experiences are completely different. In the initial stage of tuning, a distinct soliton pulse can be observed, which is inhibited by the reduced detuning. In the second period, the loss caused by detuning results in the low power in the microresonator, and the field becomes a faint fluctuation, the distribution of which is not unique. When the power begins to increase rapidly, the field evolves into the third stage, in which another bright soliton is formed. Its background power is high, but the pulse relative intensity is weak, which is attributed to the dominant 3PA and FC effect. When the microresonator is in a positive detuning stage, a new balance among 3PA, FC effect, nonlinearity, and detuning is established, which causes the dark soliton appearance. This condition can exist for a long time, which means that it is a relatively stable state with 3PA.
In addition, we study the tuning process with 4PA. The object of tuning is still an air-clad, etchless silicon microresonator as discussed in Fig. 4, and the range of frequency detuning is −4 to 4. The power variation curve is shown in Fig. 7. It can be roughly divided into four stages according to the different field distributions. At the initial stage, a powerful pulse is generated in the microresonator, and the power increases with the scanning time. When the power increases to a certain value, the pulse splits due to the high power density. After a period of evolution and power decay, the field divides into two pulses in the first descent of the second stage. In the subsequent interval, the peak power increases because of the continuous compression of the pulse width of the double pulse. The 4PA and FC effect is enhanced, the sharps of the field generates, which leads to the rapid power decay. The weak and irregular field distribution is formed in the third stage. Although the power increases slightly in the region near zero detuning due to the small detuning loss, the field is still irregular. Finally, under the action of 4PA, FC effect and positive detuning, the field form in the fourth stage is presented. On the basis of the preceding analysis, the soliton can generate in the microresonator during the process of frequency detuning with multiphoton absorption. Therefore, solitons can be effectively excited in the microresonator by means of detuning regulation.
5. Conclusion
In conclusion, based on the LLE, the influence of frequency detuning on the field over several wavelength ranges in the silicon microresonator with multiphoton absorption and FC effect is investigated. Frequency detuning exhibits different degrees of impacts on soliton generation with diverse multiphoton absorption (2PA, 3PA, and 4PA). In the operating wavelength range of approximatel 1.5 µm, 2PA is dominant, and the field with different frequency detuning demonstrates bright soliton, dual pulses, and dark soliton, respectively. It compensates for the inability to form a regular field in the microresonator without 2PA. The large frequency detuning favors a broadband spectrum. For a SiO2-cladding, etchless silicon microresonator with pump wavelength 2.4 µm, in the weak detuning range, 3PA and the incidental FC is conducive to bright and dark solitons. In an air-clad, etchless silicon microresonator, which is pumped at 4 µm, 4PA is particularly strong. In the interval of detuning near zero, bright and dark solitons are excited separately. It is resulted that with the aid of different frequency detuning, multiphoton absorption and FC effect are propitious to soliton generation.
To facilitate the actual soliton excitation, detuning regulation of solitons with multiphoton absorption and FC effects is also studied. In the regulation of frequency detuning process with 2PA, a progressively enhanced soliton can be formed in the region near zero detuning. 3PA can generate bright soliton and dark solitons in different detuning intervals. Finally, bright solitons and double pulses can be observed in the microresonator with 4PA. Thus, the detuning regulation is beneficial to excite the bright soliton with multiphoton absorption and FC effect. These research results promote the significant development of the silicon microresonator, the multiphoton absorption and FC effect of which cannot be neglected.
Funding
National Natural Science Foundation of China (52175503, 52005147); National Key Research and Development Program of China (2019YFE0107400).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper arenot publicly available at this time but may be obtained from the authors upon reasonable request.
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