Temporal evolution and spectral characteristics of dual field in a double-pumped microcavity

In recent years, precision measurement schemes based on dual-optical frequency combs have attracted extensive interest because of their excellent measurement speed and efficiency [1–4]. Especially for the absolute distance measurement, dead zones can be effectively avoided by taking advantage of dual-frequency combs [5–9]. The conventional dual-frequency combs employ mode-locked lasers, the repetition frequency difference between which can be adjusted within a certain range [10, 11]. However, because of the inherent characteristics of the structure, their repetition frequency is generally limited to a few GHz at most [12, 13]. Microcavity-based dual frequency combs are employed to improve the speed and effectiveness of measurement results [14–17]. Owing to the ultra-high repetition frequencies, the dual combs allow for optical ranging of the in-flight dynamic projectiles moving at 150 m s⁻¹ [18]. Dual-comb spectroscopy also has been a potent method in the study of spectroscopy for obtaining practically immediate Raman and optical spectra with unheard-of resolution [19]. The dual-frequency comb system built on microcavities has also been used to capture images of stationary and moving subjects [20]. A unique hyperspectral digital holography is made possible by the combination of broad spectral bandwidth and strong temporal coherence of the dual-comb system [21].

Currently, generating dual microcavity-based frequency combs primarily relies on two independent microcavities, which are pumped by two lasers [18, 22, 23]. This structure determines that the repetition frequency of the dual combs, which depend on the structural parameters of microcavities, cannot be altered. In addition, the frequency locking system of the dual microcavities would be more complex. For simplicity and low cost, a microcavity is used to generate dual combs. In such a scheme, clockwise and counterclockwise modes in the microcavity can be simultaneously excited by a single pump [24, 25]. Owing to the cross-phase modulation (XPM), only a small frequency difference between the dual combs can be achieved [26]. In addition, another pumping technique for a single microcavity is suggested. Two separate lasers with completely orthogonal polarizations pump the microcavity [27–29]. Two modes can be excited inside a microcavity using this configuration. Only soliton formation with dual pumps was covered in earlier research. The structure and dispersion of the microcavity, which determines the resonant frequencies of the two modes, are similar [30]. However, the detuning of the two modes is different because the characteristics of each of the two pumps may be adjusted separately. Thus, the dual propagating modes, which have various detuning parameters, can function in diverse states. Two combs are formed as a consequence, and the range of their repetition frequency disparities can exceed several free spectral ranges (FSRs). Thus, a large repetition frequency difference between the dual combs can be achieved through the separate control of the two pumps.

To generate dual-frequency combs by using a single microcavity, this study investigates the effect of double-pumped microcavity parameters on temporal evolution and spectral characteristics of dual fields. The dual combs with orthogonal polarization, which originates from dual polarization vertical pumps, can be distinguished by a polarizing beam-splitter (PBS). Results show that dual fields can be obtained by use of two pump lasers with different polarizations, and the two fields include not only the dual soliton fields with the same FSRs, the coexistence of solitons and Turing patterns, which is equivalent to dual-frequency combs with a large repetition frequency difference. Compared with a soliton field, in the spectral domain, Turing patterns exhibit wider frequency comb intervals. Owing to their robustness against perturbations and optimal spectral purity [31], Turing patterns provide a creative platform for high-capacity communication [32], on-chip optical squeezing [33], and other applications. And in dual dual-frequency combs applications, the beat frequency between soliton and Turing patterns can obtain a richer frequency signal [34]. Furthermore, another dual combs with two solitons also can be generated with two similar positive detuning parameters. This method allows two identical combs to exist in a single microcavity. Moreover, dual Turing patterns, breathers, and multi-pulse can be activated by some special detuning conditions. Thus, the dual fields can be controlled by regulating two detunings.

In addition, the effects of other microcavity parameters on the dual fields is studied. We found that solitons of narrow pulses and Turing patterns appear when approaching zero dispersion. Excessive negative dispersion leads to DC fields. On the other hand, breathers and varied multi-pulses are experienced in succession as pumping power increases, and the other field can maintain Turing patterns. Extremely strong pumping leads to chaos. These results have important implications for the selection of microcavity parameters with dual pumps and can promote the remarkable development of dual combs in a double-pumped microcavity.

The mechanism of a double-pumped microcavity is illustrated in figure 1. Dual pumps with TE and TM modes are denoted as pumps A and B. The two orthogonally polarized pumps are fed into a microcavity by using an optical coupler. Then, the TE and TM fields are excited, which are distinguished as fields A and B in this paper. The dual propagating fields inside the microcavity interact through XPM effects [35]. The fields with different modes can be detected separately with the aid of a PBS. To describe the evolution of the dual fields A and B, we introduce the coupled LLEs as follows [30]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial A\left(\phi ,\tau \right)}{\partial \tau }=-\left(1+i{\alpha }_{A}\right)A+i{\beta }_{A}\displaystyle \frac{{\partial }^{2}A}{\partial {\phi }^{2}}+i\left({\left|A\right|}^{2}+\sigma {\left|B\right|}^{2}\right)A+\gamma \displaystyle \frac{\partial A}{\partial \phi }+{F}_{A}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial B\left(\phi ,\tau \right)}{\partial \tau }=-\left(1+i{\alpha }_{B}\right)B+i{\beta }_{B}\displaystyle \frac{{\partial }^{2}B}{\partial {\phi }^{2}}+i\left({\left|B\right|}^{2}+\sigma {\left|A\right|}^{2}\right)B-\gamma \displaystyle \frac{\partial B}{\partial \phi }+{F}_{B}\end{eqnarray} \tag{ 2 }$$

where ϕ is the azimuth angle of the microcavities, τ is the slow time, β* relates to the dispersion parameter, σ represents the XPM coefficient, γ = (D_1b−D_1a)/κ, κ is the microcavity decay, and D_1* is Taylor's expansion coefficient of the resonant frequencies ω_μ for the μ-th mode:

$$\begin{eqnarray}&&{\omega }_{\mu }={\omega }_{0}+{D}_{1}\mu +\displaystyle \frac{1}{2}{D}_{2}{\mu }^{2}+\mathrm{..}.+\displaystyle \frac{1}{n!}{D}_{n}{\mu }^{n}+\cdots \end{eqnarray} \tag{ 3 }$$

D₁ indicates the FSR of the microcavities. F* is the intensity of the pump and α* is the detuning parameter:

$$\begin{eqnarray}&&{\alpha }_{* }=-\displaystyle \frac{2\left({\omega }_{* }-{\omega }_{0}\right)}{{\rm{\Delta }}{\omega }_{tot}}\end{eqnarray} \tag{ 4 }$$

where ω* and ω₀ represent the pumping frequency and resonant frequency of the microcavity, respectively, and Δω_tot is the total linewidth. The symbol (*) represents A or B. By solving equations (1) and (2), we can obtain the evolution of the dual fields A and B in the microcavities separately.

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As the microcavity in this study is pumped by two independent lasers, the two pump wavelengths can be individually regulated to achieve different detuning parameters. This method means that α_A and α_B can be set to distinct values in the simulation. Therefore, the detuning parameters are as follows: α_A = 2.8 and α_B = 0.8. In addition, the other parameters of the microcavity are β_A = β_B = −0.05, F_A = F_B = 1.5, γ = 0.01, and σ = 2/3 because of dual pump modes with orthogonal polarizations [36]. In addition, since we are concerned about the time evolution and spectral properties of the fields in a double-pumped microcavity, the noise problem is ignored. By solving equations (1) and (2), we can obtain the temporal evolutions of fields A and B as shown in figures 2(a) and (b). Field A develops bright solitons with a stronger detuning effect, whereas field B evolves into Turing patterns because of the weaker detuning effect. Thus, the steady state of solitons and Turing patterns exist in the microcavity. The final spatial distribution of the dual fields and the corresponding spectra are illustrated in figures 2(c) and 2(d). Owing to the XPM effect between the dual orthogonally polarized fields, a single powerful pulse of field A is surrounded by some faint pulses, the positions of which correspond to the pulse positions in the Turing patterns. Field A exhibits a broadband comb spectrum and field B shows a typical Turing pattern spectrum with adjacent peaks spaced nine modes apart, corresponding to the number of pulses. Consequently, the dual combs with a large repetition frequency difference are generated in a single microcavity. Visible burrs are seen on the spectrum because of the difference in the intensity of the Turing pattern pulses in field B. To emphasize the role of dual pumps, the temporal evolution of single field by using a single laser as pump is demonstrated in figure3, where only one pump (TE mode) is employed and other parameters are the same as in figure 2. The final field distribution is shown in figure 3(b), the fluctuations of filed intensity is on the order of 10⁻¹⁴ W, which can be viewed as a DC. And the spectrum also contains only one optical mode. Thus, there is no stable frequency combs excited in a microcavity without dual pumps. Double pumping can not only excite the dual combs with a large repetition frequency difference, but also effectively compensate for the disadvantage that the combs in the micro-cavity cannot be expected to be excited.

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Another form of dual combs can also be created by modifying the detuning conditions. We set the detuning parameters to α_A = 3 and α_B = 2.4, and the other parameters are identical to those in figure 2. The dual field evolutions are shown in figures 4(a) and (b). Field B evidently shows bright solitons and the state of field A is the same. However, the intensity of the solitons is relatively weak; thus, the pulse shape is not obvious. The spatial distribution in figure 4(c) shows that the pulses in field B are much weaker than those in field A. The spectra corresponding to the two pulses are displayed in figure 4(d), and dual combs also show the same repetition frequency in a microcavity. However, the spectrum of field A is much broader than that of B because the latter has a weak pulse. Based on the above analysis, dual combs can be generated in a single microcavity when the detuning parameters are available. In general, a strong detuning and a weak detuning facilitate the coexistence of solitons and Turing patterns. Stronger detuning is beneficial for dual solitons. In other words, different detuning conditions can result in various states of the dual combs.

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In addition, other distinct forms of fields are generated with different detuning conditions. Based on the two detuning parameters of the dual-pumped microcavity, a 2D diagram is presented in figure 5(a), which indicates the effect of detuning parameters on the field distributions. As equations (1) and (2) are almost identical, only the positive and negative signs of the first-order derivative terms on the right side of the equal signs are distinct from each other. Thus, the graph in figure 5(a) is approximately axially equivalent. Most of the area, which has two weaker positive detunings, exhibits dual Turing patterns, as shown in figure 5(b). A visible difference exists in the intensity of the dual Turing patterns, with field B having only half the power of A. Therefore, the spectrum of field B is weaker than that of field A. On the basis of two Turing patterns, one detuning is enhanced, and the coexistence of the soliton and Turing patterns can be obtained, the details of which are already demonstrated in figure 2. This mechanism is located in two narrow green areas in figure 5(a). When one of the detunings is further enhanced, only the DC fields are shown in figure 5(e) because of the strong detuning effect. Only one mode exists in the spectra. In addition, this state can be obtained when both detuning parameters are negative. That is, two negative detunings lead to DC fields.

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Some particular fields are also shown in figure 5(a). In the upper right corner of the purple area of figure 5(a), both fields exhibit periodically oscillating pulses, which are described as soliton breathers, which is shown in figure5(c). Their powers are equal and the spectra are both broadband combs. According to previous research, the breathers can be constructed by special pump power and frequency detunings [37]. The instability of the breathers is due to the dispersion and soliton self-frequency shifting, which is caused by the inertia of optical nonlinearity. The breathers are also valuable for the application of nonlinear microscopy [38], frequency comb metrology [39], and others. One of the detunings is strengthened on the basis of the dual breathers. At this point, the dispersion, detuning, nonlinearity, and XPM effects in the microcavity are balanced and the instability of the breathers disappears. The dual solitons with a significant power difference are generated, as shown in figure 4. As shown in figure 5(a), the region where the double solitons exist is very small and both positive detunings are large. Therefore, in practice, two large positive detunings are required to excite dual solitons simultaneously.

Moreover, some scattered black areas, which are dual multi-pulse, are distributed in figure 5(a). Figure 5(d) presents the evolution of these fields and the final field distributions and spectra, where field B contains three apparent pulses. However, field A is also multi-pulse and has weak intensity. The distribution of pulses is irregular, the middle two pulses are comparatively strong, and then gradually decay to the two sides. Thus, the spectrum performs without regularity. According to the results in figure 5, the effect of the two detunings on the field evolutions can be analyzed. The results of dual combs accounted for only a small fraction of the detuning map, where one of the fields maintains a large positive detuning. Two weaker positive detunings favor the generation of dual Turing patterns, which dominate most of the region in figure 5(a). Both negative detuning and very large positive detuning can lead to dual DC fields. Furthermore, breathers or multi-pulse can appear under some special conditions. Thus, two different field distributions can be excited in the microcavity by regulating the detuning parameters.

The detuning effect plays a crucial role in the evolution of the dual fields. The dispersion of the microcavity also determines the form of the fields. Therefore, the effect of dispersion on the optical field is also studied. In equations (1) and (2), two dispersion parameters, β_A and β_B , are for the dual fields. However, the dispersion of microcavities contains material and structural dispersion. The dual fields transmit in the identical microcavity with the same material and structural parameters. Thus, the two dispersions are equal, i.e., β_A = β_B = β. In general, the anomalous dispersion regime facilitates soliton combs. As a result, the range of the dispersion parameter is confined to negative. β is assumed to be equal to −0.11 and the other parameters are the same as those in figure 2. The evolutions of the dual fields are shown in figures 6(a-1) and (a-2). In the initial stage, solitons and Turing patterns still coexist. However, because of the strong dispersion effect, both solitons and Turing patterns convert DC fields with a slight random noise simultaneously, as shown in figures 6(a-3). The spectra are pumping mode accompanying noise, as shown in figures 6(a-4). When the dispersion is equal to −0.03 (figure 6(b)), field B can create Turing patterns, but no regular distribution in field A is excited. Merely because of the coupling effect between the two fields, the power of field B couples into field A, the distribution of which is inconspicuous Turing patterns. When the dispersion approaches 0 (figure 6(c)), the soliton of the narrower pulse is excited in field A. Field B presents a different kind of Turing pattern, the pulse number of which increases markedly. According to the preceding analysis, dispersion plays an essential role in the excitation of solitons.

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The influence of pumping intensity on dual combs is also investigated. This study assumes that the dual pump parameters FA and FB are the same, both represented by F. Still based on the significant coexistence of solitons and Turing patterns in figure 2, the intensity of the pump F is altered. The temporal evolutions of dual fields with various pump parameters are demonstrated in figure 7. The pump parameter F is reduced to 1.3 (figure 7(a)), the Turing patterns in field B are still preserved because of the appropriate detunings. The original solitons of field A cannot be excited because of the weak pump. However, field A develops faint Turing patterns as a result of the coupling effect between the dual fields. In figures 7(a-4), the interval between the two adjacent peaks is nine modes for fields A and B. The weak power of field B leads to lower spectral intensity.

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The increase of pump power results in instability inside the microcavity. In figure 7(b), the pump parameter F is equivalent to 1.6, which is slightly higher than that in figure 2. Field A evolves into breathers. Despite the instability of the breathers, the spectrum still exhibits broadband combs. Field B continues to have Turing patterns, the spectrum of which is more pronounced because of increased pump power. In figure 7(c), pumping power is further increased and multi-pulse appears in field A. The intensity of the pulses varies with round trip number, which indicates that the instability in the microcavity is enhanced. Field B basically maintains the form of the Turing patterns, but the intensity also changes because of the very strong nonlinear effects in the microcavity. When the pumping parameter F is increased to 2 (figure 7(d)), the severe nonlinearity leads to dual chaotic fields. The distributions of the dual fields at any moment are irregular and the spectra are also random, while the power of each comb tooth is constantly changing. The chaos originates from the unstable Turing patterns, which is a consequence of bifurcations starting with unstable primary combs and the following generation of secondary, even higher-order combs. This process terminates until a totally chaotic state is reached. In practical applications, the pump power of a double-pump microcavity should be restrained to prevent chaos.

Based on the coupled LLE, the temporal evolution and spectral characteristics of dual fields in a double-pumped microcavity were investigated. The results showed that dual orthogonally polarized pumping facilitated the dual combs in a microcavity. When the two detuning parameters were suitable, the coexistence of solitons and Turing patterns could be excited. That is, dual-frequency combs with a large repetition frequency difference were generated simultaneously by using a microcavity. Two positive detuning parameters were similar and two soliton pulses with different intensities were formed. Thus, dual-frequency combs with the identical FSR were obtained, which can be distinguished by a PBS because of the orthogonal polarizations.

To facilitate the generation of dual combs, this study exhibited the influence map of microcavity parameters on dual fields. In addition to the results obtained, other special fields were observed where one of the detuning parameters was the larger value. Dual Turing patterns occupied most of the map, which meant two weaker positive detunings were beneficial for the dual Turing patterns. In addition, breathers and multi-pulses could be excited under some special conditions. According to the preceding analysis, the dual fields could be controlled by regulating the two detunings.

Furthermore, the temporal evolutions and spectra of the dual fields with various dispersions were also demonstrated. Dual combs were excited in an anomalous dispersion regime. Extremely strong negative dispersion could only cause DC fields. The dispersion effect was diminished, and detuning and nonlinearity dominated in the dual field evolution. As a result, solitons of narrow pulses and Turing patterns appeared when approaching zero dispersion.

Moreover, the effect of pump intensity on the dual fields was also given. The pump power was so low that only the Turing patterns of field B were excited. With the increase of pumping power, the nonlinear effect in the microcavity was enhanced. Field A experienced breathers and variable multi-pulses in sequence while field B maintained the Turing patterns. Extremely strong pumping led to chaos. These results had important implications for the selection of microcavity parameters with dual pumps, and promoted the remarkable development of a double-pumped microcavity.

All data that support the findings of this study are included within the article (and any supplementary files).

This work was supported by National Natural Science Foundation of China (52175503).