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Abstract
Two frequency combs emitting from a single cavity are of great potential in the field of dual-comb spectroscopy because they are mutually coherent and therefore the common mode noise can be suppressed naturally. However, it is difficult to fully and flexibly control the repetition frequency difference in most of the all-optical schemes. In this paper, a birefringence-compensation Kerr resonator is proposed for the mutual dual-comb generation. It is shown that by offset aligning the fast and slow axis with appropriate fiber length, the total birefringence of the cavity can be equalized while the local one keeps at a high level. Theoretical investigations reveal that the polarization decoupled mutual dual-comb can be generated with nearly the same power level and arbitrary repetition frequency difference. Additionally, a numerical model of polarization-maintaining fiber (PMF) with near-zero dispersion is proposed for the proof of the concept. Based on this fiber, the coherent polarization-decoupled dual-comb with 10-dB bandwidth of 33 nm can be obtained. And the repetition frequency difference can be flexibly tuned compared to the cavity without offset alignment.
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1. Introduction
Kerr nonlinear optical resonator, one of the simplest systems supporting complex nonlinear optical phenomena, can generate cavity solitons (CSs) with ultrashort pulse width [1–3], which can be also used as optical frequency combs [4–6]. A promising application of optical frequency combs is the so-called dual-comb spectroscopy, which can be carried out without diffractive elements compared to the traditional Fourier transform method [7–9]. At present, the technical routes of mutual coherent dual-comb mainly include the electro-optic method and the all-optical method [10]. A single continuous light was modulated by a mutually locked signal to produce dual-combs with a spectrum width of 40 nm [11]. However, the bandwidth of the electro-optic dual-comb is limited by the response time of the modulator. Further extending the bandwidth relies on the nonlinear broadening. The mutual coherence dual-comb can also be obtained from mode-locked lasers by direction multiplexing [12,13], wavelength multiplexing [14], and polarization multiplexing [15]. In these approaches, gain fiber and the saturable absorber are required to ensure ultrashort pulse generation. Because the optical fiber supports two orthogonal polarization states, in recent decades, increasing interest has been devoted to the investigations on the polarization properties of the CSs. It is reported that the weakly birefringent fiber resonator injected by linearly polarized light and the results show that the resonator can sustain the two distinct polarization-locked vector solitons with unbalanced power [16]. It is shown that in the condition of strong nonlinearity, different states can coexist in the two orthogonal polarization axis, such as modulation instability and soliton, soliton and soliton state [17]. In a near-zero birefringence fiber ring cavity, the dynamic process of the two polarization states may experience the so-called spontaneous symmetry breaking when the driving parameters of the system i.e. the pumping power or the cavity detuning cross the pitchfork bifurcation point [18]. Many studies focus on the dynamics of soliton-bound states in birefringent resonators [19,20]. The polarization dynamics of vector CSs in birefringent fiber resonators and the effect of group velocity mismatch on the polarization of CS have also been reported [21,22]. The formation of dark vector dissipative solitons in the presence of nonlinear polarization mode coupling in normal dispersion resonators has also been studied [23]. The generation of CSs in the birefringent cavity is also related to the emergence of vectorial modulational instability patterns. It is shown that new kinds of instabilities can occur for co-propagating fields that interact through nonlinear cross-phase modulation [24]. Polarization multiplexing in the passive ring resonator using the birefringence effect is a potential solution to realize the mutual coherent dual-comb. At this moment, the research on ultrafast dynamics in fiber ring resonators is mainly focused on weakly birefringent fibers. The light fields of two polarization states are coupled and bound with each other with the same repetition frequency, which is not favorable for dual-comb applications. Additionally, the bandwidth is also limited by the large dispersion in conventional fibers. Therefore, how to generate broadband CSs in both two polarizations axis and decouple them is key for dual-comb generation in passive fiber ring cavities.
In this paper, we show that even though the polarization-decoupled CSs can be generated in a strong birefringence fiber cavity composed of polarization-maintaining fiber (PMF), the repetition frequency difference is still difficult to maintain at a low level which is essential for expanding the effective bandwidth of the dual-comb spectroscopy. To overcome the limitation, offset alignment of the two sections of PMF is considered to compensate for the large group-velocity mismatch. Based on this method, the two polarization states carrying CSs can be decoupled due to the large local birefringence, while their power is equalized since the total cavity birefringence is nearly the same. To verify the concept, a step-index-photonic-crystal hybrid PMF (SIPCPMF) is designed to achieve both strong birefringence and low dispersion. Based on this fiber, the vector dynamics of the CSs are studied in detail. It is shown that the proposed scheme can be a highly potential approach for generating broadband polarization-multiplexed mutual coherent dual-comb with controllable repetition frequency difference.
2. Theoretical model and method
To interpret the evolution of the light field in the cavity for each polarization mode, a Kerr resonator is considered with two orthogonal polarization mode families. Similar to the theoretical simulation model in [25], coupled Lugiato-Lefever equations (LLEs) are used to describe the evolution of the intracavity optical field corresponding to both polarization modes. In dimensionless form, these equations are written as follows,
Fig. 1. Schematic diagrams of optical fiber resonators. (a) Traditional fiber ring cavity, (b) offset fiber ring cavity.
In the following discussions, the excitation of CSs is achieved by adding intensity perturbation to the continuous-wave pump. The steady-state output characteristics in the resonant ring of four fiber rings are depicted in Fig. 2, consisting of isotropic optical fiber, weak birefringence fiber, strong birefringence fiber, and offset fiber ring cavity, respectively. For the low birefringence fiber cavity, according to the results in [17–19], the four-wave mixing coherence term between polarization components is ignored. The relationship between the nonlinear resonance peaks and the detuning for two polarization states is shown in Fig. 2(a)-(d), and Fig. 2(e)-(h) represent the time-domain waveforms corresponding to the detuning at the black dashed line. For isotropic optical fiber rings, the two polarization states exhibit good symmetry, but the dominant effect of the nonlinearity still causes the two optical fields to form bound states with the same group velocity. The resonance peaks of the two polarization states in the weakly birefringence fiber cavity are tilted under the effect of Kerr nonlinearity, which can excite CSs simultaneously, but the relative detuning is different, resulting in unbalanced pulse power of the two polarization states. In strong birefringence fiber cavities, due to strong resonance offset, it is difficult to excite solitons on both polarization states simultaneously. To overcome the above problems, an offset fiber ring cavity is proposed, in which two sections of strong birefringence fibers with close lengths are fused by fast-slow axis rotation. Since the optical paths of the two polarization states are almost identical, the overall wave vector mismatch is near zero and the resonances almost overlap with each other. While in each localized segment of the cavity, the two orthogonal polarization components are decoupled due to strong birefringence. Therefore, the repetition frequency difference Δfrep is adjustable by judiciously controlling the length of the two fiber segments.
Fig. 2. Typical steady-state output characteristics for different fiber resonator structures. (a) Traditional zero birefringence fiber ring (X= 5, Δk = 0, B = 1.6 [25]), (b) weak birefringence fiber ring (X = 40, Δk = 10, B = 1.3 [17]), (c) strong birefringence fiber ring cavity (X = 40, Δk = 22.5, B = 2/3 [26]) (d) offset fiber ring cavity (X= 5, Δk = 0, B = 2/3). (e)-(h) Profiles of the CSs in four different cases.
In the traditional birefringence fiber ring cavity, when Δβ1 is relatively small, intracavity temporal evolution for CSs is illustrated in Fig. 3. When Δβ1 is 0.5, the amplitude modulation of two polarization states will not only excite the pulse on the respective polarization axis but also excite a weaker amplitude one in the orthogonal polarization direction, as shown in Fig. 3(a)-(c). As the propagation distance increases, the weaker pulse gradually increases to the same amplitude as the generated pulses. In this case, the pulses of two polarization states appear as a vector state locked in group velocity. When the pulse is circulated to 15 and 50 roundtrips, the Stokes parameters of some CSs are plotted in Fig. 3. The Stokes parameters of CSs are shown in Fig. 3(f)-(i) at the 15th roundtrip. The Stokes parameters of CSs are shown in Fig. 3(l)-(o) at the 50th roundtrip. ${\textrm{S}_0}\textrm{ = |}\overline v {\textrm{|}^\textrm{2}}\textrm{ + |}\overline u {\textrm{|}^\textrm{2}}$ represents the total polarization of the beam, ${\textrm{S}_1}\textrm{ = |}\overline v {\textrm{|}^\textrm{2}}\textrm{ - |}\overline u {\textrm{|}^\textrm{2}}$ represents the difference between the horizontally polarized flux components and the vertically polarized flux components, ${\textrm{S}_2}\textrm{ = }\overline v \textrm{*} \times \overline u + \overline v \times \overline u *$ represents the flux components along with the π/4 and 3π/4 directions, and ${\textrm{S}_3} = {i}\left( {\overline v \textrm{*} \times \overline u - \overline v \times \overline u *} \right)$ measures the difference between right and left circularly polarized flux components. For the Stokes parameters at the 15th roundtrip, one CS is dominated by the linear flux component with horizontal polarization (S1>0), but the other one is dominated by the linear flux component with vertical polarization (S1<0). Both CSs are dominated by linear flux components along π/4 (S2>0). From the viewpoint of circular polarization, the two CSs are right and left circularly polarized, respectively. For the Stokes parameters at the 50th roundtrip, the leading edge of the two CSs has vertical polarization (S1<0) and the trailing edge has horizontal polarization (S1>0). Both CSs are dominated by linear polarization along π/4 (S2>0). The leading edges of the two CSs have right circular polarization (S3>0) and the trailing edges have left circular polarization (S3<0).
Fig. 3. Intracavity temporal evolution for CSs in a traditional fiber ring cavity, under Δβ1 = 0.5, X= 4.5, Δ=2.8, Δk = 1. (a)-(c) Evolutions of the intracavity electric fields of polarization u, ν, and total light field, respectively. (d)-(e) Profiles of the CSs, (f)-(i) Stokes parameters at the 15th roundtrip. (j)-(k) Profiles of the CSs, (l)-(o) Stokes parameters at the 50th roundtrip.
In the offset fiber ring cavity, the evolution characteristics of solitons under four different length ratios corresponding to I, II, III, and IV in Fig. 5 are presented in Fig. 4. With the increase of the length difference, the speed difference of the pulse drift of two polarization states also increases, indicating that the Δfrep is also gradually increasing. The Stokes parameter of polarization-decoupled CSs is shown in Fig. 4(i)-(l). One CS is dominated by horizontal polarization (S1>0), and the other is dominated by vertical polarization (S1<0). Therefore, they are polarization-decoupled. In the numerical simulation, the dependence between Δfrep of the polarization multiplexing soliton and the group velocity parameter difference Δβ1 in traditional strong birefringence fiber ring cavity structure is shown in Fig. 5(a). When a light pulse propagates in a traditional no-offset fiber ring, the two polarization states are easily coupled to each other and the Δfrep is zero, under the condition of relatively small Δβ1. Next, when Δβ1 reaches a critical value, Δfrep becomes to increase with Δβ1. On the contrary, if the offset alignment of two optical fibers is considered, the repetition frequency difference can also be controlled by the length difference, as shown in Fig. 5(b). Therefore, in the conventional fiber cavity, the repetition frequency difference is dominated by the birefringence, which is determined by the fiber itself. While by using offset alignment, a new dimension of fiber length difference is introduced. Based on a strong birefringent fiber, one can achieve polarization-multiplexed frequency comb with various Δfrep flexibly.
Fig. 4. (a)-(d) Profiles of the CSs in offset fiber ring cavity corresponding to I, II, III, and IV in Fig. 5, respectively. (e)-(h) Evolution of the intracavity electric fields. (i)-(l) Stokes parameters in Fig. 4(c).
Fig. 5. (a) Function relationship between Δfrep and Δβ1 in the traditional fiber ring cavity, under X= 4.5, Δ=2.8, Δk = 1. (b) In offset fiber ring cavity (m = L/200, L is the cavity length under normalization), under X= 4.5, Δ=3, Δk = 0.
The comparison between the numerical results of the coupled LLEs and the classical vectorial Ikeda map model is shown in Fig. 6(a)-(b). It is found that the polarization-decoupled CSs do not change much. The dynamic behavior of the light field in the cavity around 10th roundtrips near the evolution intersection of two orthogonal polarization modes is shown in Fig. 6(c). It is found that although there are multiple collisions, they don’t disturb much to the evolution trends of each polarization. And the evolution of the field along a single cavity roundtrip is plotted in Fig. 6(d). Due to the exchange group velocity in a single cavity, the transmission velocity in two fiber section is different, and there is a dynamic process of acceleration and deceleration.
Fig. 6. (a) Profiles of the CSs, (b) the spectra obtained by LLEs model and Ikeda map model. (c) Evolution and collision process of the light field in the cavity. (d) The evolution of the field along a single cavity roundtrip.
3. Fiber design
3.1 Fiber structure
In order to verify the concept in section 2, we propose a fiber design with both high refringence and low dispersion, as shown in Fig. 7. The fiber is a step-index and photonic crystal hybrid structure with two stress rods to induce birefringence. The fiber dispersion profile can be influenced by the combined contribution of the core and the rods made of high refractive index material with triangular lattice. The two stress rods and ellipse core induce birefringence to achieve polarization maintenance. The fiber is composed of pure, Ge-doped, and B-doped fused silica glass. The fiber core is Ge-doped with short and long half axes a/2 and b/2, respectively, and surrounded by four layers of individual high refractive index rods. The pitch of any two adjacent rings inclusions is denoted by P. The first layer of holes with Ge concentration of 9.8% has a diameter of d1, and the second to four layers of high refractive index rings with Ge concentration of 10.39% have a diameter of d2. The centers of the two stress rods are about 42 µm away from the fiber center, and their diameters are about 30 µm, respectively. Material refractive indices are calculated based on the Sellmeier equation [27]. The material of the core is pure SiO2 with Ge concentration of 15% and its refractive index can be calculated as 1.426 according to the mixed Sellmeier equation [28]. In addition, the process parameters of the optical fiber in the preparation process are summarized in Table 1. After the stress rod material is doped with B2O3, the thermal expansion coefficient α increases by nearly an order of magnitude, which leads to stress generation across the fiber.
Table 1. Process Parameters
3.2 Fiber property analysis
The birefringence of the fiber is caused by the combined effects of asymmetric geometry, internal thermal stress, and elastic optical effects. When the fiber was cooled down from 1100°C (reference temperature) to 20°C (working temperature), the Von Mises stress distribution on the cross-section and the difference between fast-axis and slow-axis directions are illustrated in Fig. 8. It can be seen that the Von Mises Stress around the stress rods is the strongest, while the one between the adjacent stress rods extends to each other and pass through the core, resulting in the formation of stress birefringence in the core. The difference between the stress tensor in the X direction and that in the Y direction is the stress birefringence, as shown in Fig. 8(b).
Fig. 8. The Von Mises Stress and stress birefringence. (a) The Von Mises Stress distribution of the all-fiber cross-section, (b) thermal stress difference (σx-σy) distribution, where σx and σy are the stress tensor in the X direction and Y direction, respectively.
By optimizing the fiber parameters, flattened near-zero dispersion fiber was designed and its fiber properties were analyzed as shown in Fig. 8, with a = 2.5 µm, b = 2.8 µm, P = 4.9 µm, d1 = 1.9 µm, and d2 = 2.9 µm. Figure 9(a) and (b) reveal the electric field distribution of two orthogonal polarization modes denoted with LP01,x and LP01,y at the wavelength of 1550 nm, respectively. The red arrow indicates the direction of the electric field. As is shown in Fig. 9(c), the effective refractive index (ERI) of X-polarized and Y-polarized modes are 1.44894 and 1.44837, respectively. It is found that the ERI of the X-polarized mode is slightly larger than that of the Y-polarized mode, which corresponds to the slow axis and the fast axis, respectively. The birefringence of the fiber is characterized as 5.58×10−4 ∼ 6.02×10−4 in the wavelength range of 1200 nm ∼ 2000 nm. The ERI of two orthogonal polarization modes decreases monotonically in the whole scanning wavelength range, while the birefringence increases very slightly. The fast axis and the slow axis have different ERI, which leads to the slightly different dispersion curve distribution of the two polarization states. The dispersion profiles of the two polarization states are shown in Fig. 9(d). Generally, the group velocity is expressed by β1. And group-velocity mismatch is defined as the difference of group velocity derivatives, as shown in Eq. (3).
Fig. 9. (a)-(b) Electric field distribution of X-polarized mode and Y-polarized mode at 1550 nm. ERI of the two modes is 1.4489 and 1.4484, respectively. (c) ERI variations of LP01,x (black dotted line) and LP01,y (red dotted line) modes, and birefringence in such a wide wavelength range (blue dashed line). (d) Slow axis (black line) and fast axis (red line) dispersion curve. (e) The group velocity of the slow axis (black dotted line), fast axis (red dotted line), and GVM of two orthogonal modes (blue line).
4. CSs in the polarization-maintaining cavity
4.1 CSs in single-polarization
First, a single polarization state is considered based on the slow axis parameters of the designed fiber, and its characteristics are analyzed in Fig. 10. Since the second-order dispersion (SOD) is relatively small, the effect of higher-order dispersion on Kerr CSs must be taken into account. Above all, the intensity profile and the spectrum of CSs with only SOD are shown in Fig. 10(a)-(b). The blue solid line is the spectrum obtained by adding third-order dispersion (TOD) based on SOD and the black dotted line is the spectra obtained by considering SOD, TOD, and the fourth-order dispersion. By comparing the spectral characteristic of different order dispersion, the introduction of higher-order dispersion leads to the reduction of spectral width. The 10 dB-bandwidth of the optical comb structure with only SOD is 130 nm. The curves with higher-order dispersion are consistent to a large extent, which denotes that the performance is dominated by TOD. Furthermore, it can be found that the strong dispersive wave (DW) is appeared at the short wavelength side considering higher-order dispersion. The peak power and FWHM of the CSs observed are 30.4 W and 132 fs, respectively, as shown in Fig. 10(c). In addition to the appearance of a relatively strong short-wavelength DW peak at about 1509 nm, the spectrum demonstrates a slight but conspicuous long-wavelength sideband around 1593 nm as well. The latter corresponds to the symmetrical detuning (relative to the pump frequency) of the strong DW at 1509 nm. According to the phase-matching condition [29] of the DW as Eq. (4)
Fig. 10. (a) Intensity profile of CSs with SOD. (b) The spectrum with different order dispersion. (c) Simulated intensity profile of CSs with SOD and TOD. (d) Spectrum, the red dotted line indicates a pair of dispersive waves of 1509 nm and 1593 nm (Δf represents frequency shift from the pump frequency fp).
4.2 Polarization-multiplexed CSs generation
The polarization-multiplexed fiber ring has more plentiful dynamics and is suitable for the dual-comb generation. The nonlinear dynamic model of the laser field in the offset fiber ring cavity can be described by the following coupled LLEs:
Table 2. Relative physical parameters
According to the results of the discussion on the generation of CSs in a single polarization ring, only SOD and TOD are considered. First, the dynamics of solitons in conventional strong birefringence fiber rings are studied. With the parameters of the designed fiber, the resonance characteristics differ significantly and CSs are difficult to intersect, as shown in Fig. 11(a). Meanwhile, the soliton states of the two polarization states at different detuning amounts are depicted in Fig. 11(b)-(c). According to Eq 4, it can be obtained that the phase-matching condition of the DW is related to the detuning. In strong birefringence fiber, the misalignment of the resonance peaks results in different detuning of the two polarization states, which in turn changes the position of DWs. With larger detuning, the dispersive wave is farther away from the pump frequency and the amplitude of the DW is weaker.
Fig. 11. (a) Steady-state output characteristics in a traditional strong birefringence fiber ring. (b)-(c) Only one polarization state soliton exists for different detuning conditions.
In the case of length symmetry, to simultaneously excite CSs, we set δ1 = 0.032 m−1 located at the effective red-detuned domain corresponding to the black dashed line in Fig. 12(a). The time-domain evolution and spectrum of polarization-multiplexed soliton are illustrated in Fig. 12(b) and (d). It can be clearly seen that the relative loop drift of the two soliton pulses is consistent in Fig. 12(c), and the soliton movement in the time window is due to the existence of higher-order dispersion. Although the peak powers of the two solitons are almost equal, they have the same repetition frequency, which makes it impossible to obtain the required dual-comb. The introduction of higher-order dispersion will lead to DW. According to the phase-matching condition of DW, the resonate wavelength marked by the black dotted line in Fig. 12(d) is the DW. The appearance of Kelly sidebands is due to the interaction of two polarization states with different dispersions in the cavity. According to the phase-matching condition of Kelly sidebands [30]:
Fig. 12. Characterizations of pulses in offset fiber ring cavity with L1 = L2. (a) The intracavity power varies with cavity detuning, black dashed line locates at an effective red-detuned domain. (b) The time-domain waveform at the 300th roundtrip. (c) Time-domain evolution diagram of solitons on orthogonal polarization states. (d) The spectrum at the 300th roundtrip.
In order to obtain two soliton pulses with unequal repetition frequencies, the length difference of the two fibers is controlled within a relatively small range. The dynamic process of polarization-multiplexed soliton output from two length asymmetric fibers is shown in Fig. 13. The intracavity power varies with cavity detuning as is illustrated in Fig. 13(a). The time-domain evolution of two polarization solitons with the 300 roundtrips is revealed in Fig. 13(c), which indicates that the drift velocity is inconsistent, resulting in the intersection of soliton evolution trajectories. Compared with the polarization state ν, the pulse of polarization state u shifts to the right with a fixed slope, which proves that the two polarized pulses have different repetition frequencies. The time-domain pulse and the spectrum of the CSs evolution to the 300th roundtrip in the asymmetric fiber ring are shown in Fig. 13(b) and (d), respectively. Coherent polarization-decoupled dual-comb with 10-dB bandwidth of 33 nm can be obtained.
Fig. 13. Characterizations of pulses in offset fiber ring cavity with different lengths. (a) The intracavity power varies with cavity detuning. (b) Time-domain waveform corresponding to the 300th roundtrip. (c) Time-domain evolution diagram of solitons on orthogonal polarization states. (d) The spectrum at the 300th roundtrip.
Comparing the simulation results of LLEs model and Ikeda map model, although the pulse positions are different, the time-domain CSs, spectra shape, and repetition rate are basically the same as shown in Fig. 14. Finally, the characteristics of polarization-multiplexed solitons with different length differences are compared and discussed, as shown in Fig. 15. With the increase of the length difference, the repetition frequency difference also raises slightly. Further adjusting the length difference at 0 m, 0.11 m, 0.22 m, and 0.33 m, the repetition frequency difference of the two pulse solitons is 0 Hz, 6.86 Hz, 13.72 Hz, and 20.57 Hz, respectively. At the same time, the difference between the peak powers of polarization-decoupled dual-comb is no more than 1.5 W. This verifies that the Δfrep can be flexibly controlled in this offset-aligned cavity. A limitation of the method is that the repetition rate difference between the two combs cannot be adjusted at will in experiments. To overcome this limitation, inserting a fiber stretcher whose length can be controlled by external electrical voltage in the cavity can be a potential approach [31].
Fig. 14. (a) Profiles of the CSs, (b) the spectrums with LLEs model and Ikeda map model. (c) Time-domain evolution diagram of solitons on orthogonal polarization states with LLEs. (d) Time-domain evolution diagram with Ikeda map model.
5. Conclusion
In summary, we theoretically analyze that strong birefringence offset fiber ring cavity can be used to produce polarization-decoupled solitons. To extend the effective bandwidth of the dual-comb source, we design flattened near-zero dispersion PMF which is composed of both step-index and photonic crystal structure. The results show that the fiber has a flattened near-zero dispersion profile by selecting appropriate Ge doping concentration and carefully arranging the cladding pore distribution. Furthermore, the order of birefringence of the SIPCPMF can reach 10−4, which verifies that the proposed fiber has superior PMF performance. The application of this fiber in the generation of CSs in passive rings is investigated further. Finally, by controlling the length of two optical fibers in the offset fiber ring cavity, it is helpful to flexibly adjust the repetition frequency difference, which has a great prospect in the generation and application of dual-comb.
Funding
Wuhan National Laboratory for Optoelectronics (2019WNLOKF005); National Natural Science Foundation of China (61605179); Natural Science Foundation of Hubei Province (2019CFB598, 2021CFA001).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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