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Abstract
Surface modes (SM) are highly spatially localized modes existing at the core-cladding interface of photonic-bandgap hollow-core fiber (PBG-HCF). When coupling with SM, the air modes (AM) in the core would suffer a higher confinement loss despite being spectrally within the cladding photonic bandgap, and would be highly dispersive around the avoided crossing (anti-crossing) wavelength. In this paper, we numerically explored how such avoided crossings can play an important role in the tuning of the temperature dependence of group delay of AM of PBG-HCF. At higher temperatures, both the thermo-optic effect and thermal expansion contribute to the redshift of avoided crossing wavelength, giving rise to a temperature dependence of the AM dispersion. Numerical simulations show that the redshift of avoided crossing can significantly tune the thermal coefficient of delay (TCD) of PBG-HCF from −400 ps/km/K to 400 ps/km/K, approximately −120 ppm/K to 120 ppm/K. In comparison with the known tuning mechanism by thermal-induced redshift of photonic bandgap [Fokoua et al., Optica 4, 659, 2017], the tuning of TCD by SM coupling presents a much broader tuning range and higher efficiency. Our finding may provide a new route to design PBG-HCF for propagation time sensitive applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In a solid-core silica optical fiber, the thermo-optic effect, thermal expansion and elasto-optic effect (due to the thermal stress) together contribute to a strong dependence of modal effective refractive index on the environmental temperature [1,2]. As a result, the fluctuation of ambient temperature inevitably results in thermal-related phase noise on light transmitted through the fiber, which is hazardous for propagation time sensitive applications, e.g. long-distance/distributive time and frequency transfer [3,4], highly accurate time/data synchronization [5] and broadband optical communication networks [6,7].
The thermal coefficient of delay (TCD) is used to characterize the temperature-dependence of group delay of fiber, which is about 37 ps/km/K for a bare SMF28 without fiber coating [8]. The thermo-optic effect of fiber material contributes approximate 95% of TCD in conventional optical fiber, while the other 5% is due to the thermal fiber elongation [6]. By carefully tuning the thermal expansion coefficient of the coating material, the thermal contraction of the coating reduces the TCD of SMF28 to a few ps/km/K, in a limited temperature range from 0 °C to 40 °C [2]. Meanwhile, more efforts have been devoted to the development of time-phase synchronization techniques instead to dissolve the challenges of optical fiber thermal noise [9–11]. It is noted that most of those active compensation methods are applied only for slowly-time-varying thermal phase noises, owing to the limited bandwidth of electronic devices and circuits [12].
The thermal dependence of group delay in standard fibers arises from the intrinsic material thermal response and is impossible to eliminate directly. In contrast, low-loss hollow-core fibers (HCFs) can confine over 99% of light field in the air/vacuum core, and so the impact of fiber material becomes significantly reduced or even negligible [13]. Low-loss HCFs, e.g. PBG-HCF and antiresonant hollow-core fiber (AR-HCF), exhibit low dispersion, low nonlinearity and low material absorption at the same time, when compared to standard fibers.
The low thermal sensitivity of PBG-HCF was firstly experimentally reported by R. Slavik et al. in 2015 and its TCD measured as 2 ps/km/K, which is about 1/18 of SMF-28 [8]. Later, G. A. Cranch et al. reported the use of commercial PBG-HCF for thermal phase noise suppression for the first time [14]. In 2017, E. N. Fokoua et al. studied and demonstrated that at the long-wavelength band edge of the transmission window, the thermally induced red shift of the bandgap could balance the thermal expansion of PBG-HCF length and result in a zero or even negative TCD at specific wavelengths. However, this method comes with inevitable leakage loss, which limits the use of a long fiber length in practical applications [6]. In 2019, R. Slavik et al. demonstrated ±50 mrad/m/°C phase temperature sensitivity coefficient of PBG-HCF in the temperature range of −73 °C ∼ −69 °C, which is three orders of magnitude lower than that of SMF-28 [15]. At the same time, they found PBG-HCF with the open ends in the air would have the fiber thermal sensitivity further decreased. By controlling the density of the air inside PBG-HCF through pressure, close-to-zero sensitivity (within 0.01 ppm/°C) was observed over 100 nm bandwidth at 113 °C [16]. In 2020, the influence of coating material on the thermal properties of PBG-HCF was studied at temperature in the range of −180 °C- 25 °C. The coating significantly influences a fibers overall thermal phase sensitivity, especially at low temperatures, because of the transition from a rubbery state at room temperature to stiff glassy state at low temperatures [17].
The low thermal sensitivity and the tunability of TCD by thermal-induced photonic-bandgap shift makes PBG-HCF of great potential for propagation time sensitive applications. In this paper, we report a new route of tuning the temperature-dependent group delay of PBG-HCF by using the surface mode (SM) coupling. The numerical simulation shows that at different wavelength offset from the avoided crossing where SM coupling appears, the TCD of PBG-HCF can be extensively tuned from −400 ps/km/K to 400 ps/km/K. More importantly, the local loss introduced by SM coupling is predicted to be moderately lower than near the bandedge of transmission window, which may indicate a pathway to practical applications [18,19]. The birefringence of PBG-HCF introduced by SM coupling is also discussed in the paper.
2. Principle
Figure 1(a) shows the dispersion of supermode at the avoided crossing in a typical PBG-HCF when SM couples with air modes (AM) at 0°C and 40°C. At a higher temperature, the redshift of the avoided crossing results in the increase of modal effective refractive index.
Fig. 1. (a) Simulated supermode dispersions at the avoided crossing of typical PBG-HCF at 0°C and 40°C. The dashed lines show dispersions of independent air (dash) and surface (dot) modes without coupling. (b) The normalized effective refractive indices of TE0m modes in an air-cladding silica glass slab waveguide at different temperatures when thermo-optic effect and thermal expansion are considered.
The redshift of avoided crossing at higher temperature is attributed to the redshift of SM dispersion due to the thermo-optic effect and thermal expansion. We apply the model of a slab waveguide to approximate the highly localized SM supported by the thin core wall in PBG-HCF. Without losing generality, TE modes are investigated where TM modes are expected to follow a similar trend.
Figure 1(b) plots the dispersion curves of TE modes at different temperature for a silica slab waveguide clad by the air. At a higher temperature, dispersion curves of all TE modes exhibit a similar tendency of redshift. Here we include the thermo-optic effect and thermal expansion in the calculation. Details are found in the Appendix.
The redshift of SM dispersion can be characterized by the shift of cutoff wavelength. The cutoff wavelength of TE0m mode is defined by [20],
Here, d is the thickness of the core wall, nco and ncl are the refractive index. For the core of silica glass, its thermo-optic coefficient ε = 11×10−6 K−1 is bigger than thermal expansion coefficient α = 0.55×10−6 K−1 by over one order of magnitude [21]. Therefore the thermo-optic effect is expected to dominate the redshift phenomenon. Neglecting the thermal induced core wall thickness expansion, the shift of cutoff wavelength is derived from Eq. (1) as,
We assume the redshift of avoided crossing equals Δλm and for ΔT <<T0 where T0 is the ambient temperature, the shifted supermode dispersion curve maintains its profile. Then we have,
Naturally we link the dependence of supermode effective refractive index on temperature with the supermode dispersion as follows,
The TCD is defined by [6],
Eq. (4) and (5) show that the longitudinal change of fiber length by thermal expansion, the redshift factor of SM/avoided crossing and the dispersion of supermode decide the TCD altogether. Near the avoided crossing, the dispersion of supermode usually experience a strong variation, which offers the possibility of tailoring TCD of PBG-HCF in an unprecedented range.
3. Simulation and discussion
3.1 Simulation models and method
An accurate depiction of tuning of TCD by SM in PBG-HCF is numerically demonstrated by using the finite-element method (FEM). The multi-physics FEM software COMSOL is used where the silica glass is used as the host material of fiber and both the thermal expansion and the thermo-optic effect are included in the simulation. The effective index of the fundamental-like supermode at the avoided crossing as function of wavelengths are obtained. In the simulation, the ambient temperature is increased from 0°C to 40°C in steps of 10°C. By fitting the group refractive index (GRI) as function of temperature, the derivative of GRI for temperature is obtained. TCD is finally calculated by Eq. (5).
Two kinds of 7-cell and 19-cell PBG-HCFs are modelled as shown in Fig. 2. The geometric parameters are summarized in Table 1. The refractive indices of silica glass and air are assumed to be 1.45 and 1 at 20°C, respectively. All relevant mechanical and thermal properties are presented in Table 2. To introduce SM effectively, the core wall of PBG-HCF is assumed thicker than the strut of cladding [19]. Λ = 4 and D/Λ = 0.97 are selected for pitch and core-diameter-to-pitch ration which decides the transmission window centered around 1.55 µm wavelength.
Fig. 2. Models of (a)7-cell and (b)19-cell PBG-HCFs in the simulation. Two models share the same structural parameters except the core diameter Rc, Λ is the pitch, d1, d2, and d3 are the chamfer diameters of the air hole in the cladding, the inner side of the air hole in the innermost circle, and the core, respectively. t1, t3 and t2 are the wall thickness of the core in the horizontal and vertical directions, and the cladding struct, respectively.
Table 1. Structure parameters of the 7-cell and 19-cell PBG-HCF
Table 2. Material parameters of the silica and air [6]
3.2 Simulations and discussion of TCD
In our models, the core wall thicknesses are different in x and y directions, resulting in modal birefringence similar to [18]. Figure 3 and 4 summarize all the simulation results which are categorized by mode polarization.
Fig. 3. Simulation results of TCD of 7-cell PBG-HCF. Left: x-polarization. Right: y-polarization. (a) The effective refractive index and confinement loss of supermodes near the band gap at 20°C. Rainbow dotted lines: The effective refractive indices; Black solid line: The confinement loss of the core-like supermode. (b) The GRI near the avoided crossing regions at temperature from 0°C to 40°C (c) Calculated TCD of 7-cell PBG-HCF at the avoided crossing.
Fig. 4. Simulation results of TCD of the 19-cell PBG-HCF. Left: x-polarization. Right: y-polarization. (a) The effective refractive index and loss of supermodes near the band gap at 20°C. Rainbow dotted lines: The effective refractive indices; Black solid line: The confinement loss of the core-like supermode. (b) Calculated TCD of 19-cell PBG-HCF at the avoided crossing.
3.2.1 7-cell PBG-HCF
For 7-cell PBG-HCF, one single avoided crossing is identified in the transmission window from 1.5 µm to 1.8 µm, where the coupling of SM with x-polarized AM occurs at a shorter wavelength than the y polarized. The modal effective refractive index and confinement loss at 20 °C are shown in Fig. 3(a-I) and 3(a-II).
When the temperature rises from 0 °C to 40 °C, the avoided crossing shifts towards the longer wavelength as analyzed in Section 2. The calculated GRI at the avoided crossing regions for different temperatures are shown in Fig. 3(b-I) and(b-II). The redshift of GRI becomes more significant at wavelengths approaching the center of the avoided crossing resulting in a larger absolute value of TCD. The calculated TCD ranged from −400 ps/km/K to 400 ps/km/K which is a much broader range than the tuning by the redshift of photonic bandgap at the band edge as in [6].
Away from the avoided crossing, TCD quickly drops and passes zero. At wavelengths further apart, the modulation of AM dispersion by the coupling of SM is expected to be much weaker and eventually disappears. In this region, the longitudinal thermal expansion starts to dominate TCD of PBG-HCF.
3.2.2 19-cell PBG-HCF
TCD evolution near the avoided crossing in the 19-cell PBG-HCF is shown in Fig. 4. The birefringence of 19-cell PBG-HCF becomes significant in the vicinity of avoided crossing. For y polarization, avoided crossing I and II are so close that TCD experiences a dramatic change.
As [22] points out, the formation of a bandgap is the consequence of strong coupling of massive cladding modes of PBG-HCF. We argue that the mechanism of SM redshift can be also applied to the redshift of photonic bandgap of PBG-HCF. Near the photonic bandgap edge, the interaction between the AM and specific cladding modes becomes stronger so that the dispersion of AM is modulated and thus contributes to the tuning of TCD. At a higher temperature, the increase of refractive index of cladding material is expected to shift the dispersion curves of cladding modes toward the longer wavelength side, and tune the AM near the band edge in a way as similar to around the avoided crossing. In fact, our TCD curve between two neighboring avoided crossings tuned by SM coupling appears almost the same characteristic as TCD tuned by the redshift of photonic bandgap without SM in [6].
3.2.3 Discussion of surface-mode loss
As shown in Fig. 3 and Fig. 4, the simulated confinement losses around the avoided crossing are of the order of 10−3 dB/m. Such low level of extra loss introduced by surface mode was also reported in [23,24]. It is also note that tens or hundreds of dB/km losses due to the surface mode coupling were experimentally observed [19,25], which ultimately limits the applicable fiber length in use. We attribute such difference to the local core wall thickness and the surface-mode coupling strength with core modes [25]. A thicker core wall and a smaller core size tend to bring up the surface-mode induced loss. More detailed consideration is beyond the scope of this paper.
Figure 5 shows the simulated confinement loss of x-polarized fundamental modes in 7-cell and 19-cell PBG-HCFs with different core wall thicknesses. When the core wall thickness is tuned from half of cladding struct thickness to 1.5 times while the other cladding unchanged, the surface-mode introduced loss rise in both fibers. For the same core wall thickness, 7-cell PBG-HCF presents a higher surface mode loss attributed to a greater coupling strength due to a small core diameter [25]. The suppression of surface-mode loss level is feasible by using a large core design and carefully selecting a proper core wall thickness. More effort will be necessary to optimize the design of PBG-HCF for the balance of TCD tuning range and loss cost of surface mode.
Fig. 5. The simulated confinement loss of x-polarized fundamental mode in (a) 7-cell and (b) 19-cell PBG-HCF at 20°C. The ratio of core wall thickness over the struct thickness of cladding (T) is 0.5, 1 and 1.5.
3.3 On temperature dependence of birefringence
Without SM coupling, the birefringence of PBG-HCF is mainly the geometric birefringence determined by the core shape. The thermal induced core expansion in the simulation is so small that the birefringence is regarded as constant. With SM coupling, the birefringence of PBG-HCF is found much enhanced near the avoided crossing because of its strong polarization dependence [18]. Consequently the redshift of avoided crossing is also expected to result in a notable temperature dependence of birefringence.
Similarly we define the thermal coefficient of (phase) birefringence (TCB) to describe temperature dependence of birefringence as,
As in Eq. (3), we have,
Again we link the dependence of birefringence on temperature with the birefringence dispersion as follows,
The polarization mode dispersion(PMD) defined by the group delay difference per unit length between two orthogonal polarization modes, can be linked with birefringence B as,
Then it is not difficult to demonstrate that derivative of PMD with respect to temperature is exactly the difference between TCDs of two orthogonal polarizations.
Figure 6 shows the birefringence, TCB and the derivative of birefringence with wavelength of 7-cell and 19-cell PBG-HCFs. The maximum of birefringence of PBG-HCF reaches 10−3 at the avoided crossing wavelengths. It is noted that TCB exactly follow the trend of the derivative of birefringence with wavelength but with a negative sign. As the birefringence becomes smooth and flat, the TCB approaches zero as shown.
Fig. 6. Simulation results of the birefringence, TCB and the derivative of birefringence with wavelength. (a) 7-cell PBG-HCF. (b) 19-cell PBG-HCF. Red solid line: The birefringence of the core-like supermode near the band gap at 20°C. Black solid line: The TCB of the fundamental mode-like supermode. Blue solid line: -dB/dλ of the core-like supermode near the band gap at 20°C. Rectangular shaded are centers of the avoided crossing for x and y polarizations.
4. Conclusion
In this paper, we numerically demonstrate that by utilizing the redshift phenomenon of avoided crossing, TCD of PBG-HCF can be flexibly tuned by SM coupling. The tuning range of TCD is greatly extended from −400 ps/km/K to 400 ps/km/K comparing with the reported redshift of photonic bandgap in [6]. The temperature dependence of birefringence of PBG-HCF in the vicinity of SM coupling is also discussed. We point it out that the application of our method for PBG-HCF design for the practical TCD control over a long fiber length will depend on the suppression of increase of local loss by surface mode coupling, which will take more efforts to explore in both theory and experiment in future.
Appendix
Figure 1(b) plots the general solution of the eigenvalue equations of TE0m modes in a slab waveguide as approximation of SM of PBG-HCF. The eigenvalue equations of TE0m modes above and below cut-off are written as [22],
Here, ${U^2} = {(\frac{{2\pi }}{\lambda}\frac{d}{2})^2}(n_{co}^2 - n_{eff}^2)$, ${W^2} = {(\frac{{2\pi }}{\lambda }\frac{d}{2})^2}(n_{eff}^2 - n_{cl}^2)$, ${V^2} = {U^2} + {W^2}$, nco and ncl are the refractive indices of core and cladding and neff is the effective refractive index of TE0m mode. For a silica glass slab waveguide with the air cladding, nco and d changes because of thermo-optic effect and thermal expansion at different temperatures,
Here, n0 and d0 are the refractive index and the thickness of the silica slab waveguide at the reference temperature T0. ε and α are the thermo-optic effect coefficient and thermal expansion coefficient of silica, respectively. It is noted that thermal effects of air are neglected.
Funding
Natural Science Foundation of Hebei Province (F2021203002); National Key Research and Development Program of China (2020YFB1312802); Chinese Academy of Sciences (Pioneer Hundred Talents Program, ZDBS-LY- JSC020); National Natural Science Foundation of China (61935002); International Science and Technology Cooperation Programme (2018YFE0115600).
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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