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Abstract
A broadband single-polarization single-mode (SPSM) hollow core anti-resonant photonic crystal fiber (HC-ARPCF) is proposed and analyzed by the finite element method in this paper. The HC-ARPCF design consisted of outer semicircular cladding tubes and inner circular cladding tubes. The SPSM behavior is achieved through controlling the effective material absorption loss (EML) by loading epsilon negative (ENG) material in the selected semicircular cladding tubes. Optimization of the configuration parameters is conducted to yield a large loss difference (LD) between one of the two orthogonally polarized fundamental modes and all the other unwanted modes. Therefore, only one desired mode will exist after a proper propagation distance, i.e., SPSM guidance. Specially, the optimal design provides a 288 nm (from 1408 nm to 1676 nm and from 1680 nm to 1700 nm) bandwidth in terms of 40 dB/m minimum LD (MLD) and 168 nm (from 1452 nm to 1620 nm) bandwidth in terms of 100 dB/m MLD. Furthermore, this fiber also exhibits a large effective mode area and near-zero dispersion properties over the entire operation bandwidth. The proposed HC-ARPCF may find its applications in polarization maintaining and high-power laser systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a significant characteristic of the electromagnetic (EM) wave, polarization effect exhibits an indispensable role in numerous applications, such as optical communication system [1], optical sensor [2], narrow linewidth fiber laser [3], and fiber optic gyroscope (ROG) [4]. However, the performance of ordinary single mode fibers used in most optical fiber systems is unsatisfactory considering random polarization fluctuations introduced by the fiber fabrication inaccuracy and external interferences, like stress, bend, twist, and elongation. Moreover, polarization mode dispersion (PMD) is another factor associated with polarization that can influence the normal operation of optical systems. In order to solve the above issues, highly birefringent polarization maintaining (PM) fibers like PANDA and elliptical cladding fibers [5,6], which can propagate two polarization fundamental modes in different phase velocities. Nevertheless, the achieved birefringence values are insufficient for special demands [7,8].
In recent years, photonic crystal fibers (PCFs), also called micro-structured fibers (MSFs), have been widely concerned due to their design flexibilities and remarkable performances in various characteristics, e.g., endless single mode [9], ultra-high nonlinearity [10], large effective mode area [11], and tunable dispersion [12]. Furthermore, PCFs can yield much higher birefringence values than traditional PM fibers so that enhance PM performance. However, the PMD and crosstalk in highly birefringent PCFs can still exist with different propagation constants [13]. To solve these problems, the other type of PM PCFs named single-polarization single-mode (SPSM) PCFs is proposed. In SPSM PCFs, only one selected polarization mode can be effectively propagated due to large mode loss difference (LD) or mode cutoff. To date, two main types of SPSM PCFs are reported, one is solid core PCFs (SC-PCFs), the second is hollow core PCFs (HC-PCFs).
SPSM SC-PCFs, generally, called total-internal-reflection guided PCFs, can operate SPSM behavior through two methods. One is based on index-matching coupling technique, which utilizes index-matched cladding defects to differentiate the attenuation rates of the two orthogonally polarized fundamental modes (FMs) [14]. The other method is to introduce different cut-off frequencies by arranging asymmetric size or number of air holes [15]. Although SC-PCFs can achieve SPSM guidance, controlling the dispersion of substrate materials in SC-PCFs is still a serious challenge. Consequently, this kind of SPSM PCFs is limited to apply in nonlinear optical systems and coherent communication systems [16].
On the other hand, HC-PCFs guiding EM waves through the air core, can realize near-zero dispersion, ultra-low nonlinearity, large effective mode area, low loss, and insensitivity to stress, which reduce the impact of issues of SC-PCFs [17]. Due to such advantages, achieving SPSM guidance and highly sensitivity biosensor [18] in hollow core fibers (HCFs) have attracted enormous attentions in recent years. The simplest hollow core fiber is consisted of air core and solid cladding, in which the total internal reflection condition cannot be satisfied. Consequently, the light is leaking during propagation. Hollow core photonic bandgap fiber (HC-PBGF) is another type of HCFs guiding EM waves in the air core by bandgap effect [19]. To achieve bandgap SPSM guiding, the PCFs cladding must be designed with a restricted and complex arrangement [20]. Moreover, HC-PBGFs can only work around special bandgap wavelength ranges, which are usually very narrow. Compared with HC-PBGFs, hollow core anti-resonant PCFs (HC-ARPCFs), proposed in the past few years, have been followed with continuously increasing interest. In HC-ARPCFs, both core modes and cladding modes can exist. At modes coupling points (resonant wavelengths), the FMs are coupled with the cladding modes resulting in the largest confinement loss (CL). By suppressing the modes coupling between the FMs and cladding modes, the CL of the FMs can be minimized as a low value, thus, FMs are confined to propagate in the air core [21]. In contrast to HC-PBGFs, the structures of SPSM HC-ARPCFs are simple and the operation bandwidth is wide. Although recent works of SPSM HC-ARPCFs provided high values of polarization extinction ratio (PER) and high-order mode extinction ratio (HOMER), their real LDs are not enough to satisfy short distance applications [22–24].
As mentioned, minimum loss difference (MLD), i.e., the lowest loss value between the wanted mode and unwanted modes, plays a key role in maintaining and evaluating SPSM performance in PCFs. The total loss (TL) of PCFs is generally consisted of two components, CL and effective material absorption loss (EML). The CL is the normally main factor of the loss characteristic because the EML of optical fiber is usually quite low and can be ignored [7]. In our previous work, we achieved SPSM SC-PCFs in terahertz regime by controlling EML [7,25,26], however, the characteristics of SC-PCFs can introduce high loss and large dispersion due to the limitation of materials in the terahertz regime. Compared to the terahertz regime, the optical communication regime has more practical and broad application prospects, which can facilitate the fabrication and experiment. Moreover, HC-ARPCF is a great candidate for breaking through the tradition optical fiber limitations due to the special structure and guiding mechanism. Nonetheless, to the best of our knowledge, there is no study of controlling EML to realize SPSM property in HC-ARPCF in the optical communication regime.
In this paper, we propose a novel symmetric double-ring cladding HC-ARPCF. The outer cladding is comprised with semicircular tubes to minimize the overall scale of our PCF, while the inner cladding consists of circular tubes that can suppress high-order (HO) modes. Then, SPSM guidance is achieved by filling an epsilon negative (ENG) material to achieve a large MLD by controlling the EML of our fiber. With the optimized parameters, the fiber can maintain 288 nm and 168 nm SPSM bandwidth in conditions of MLD $\ge$ 40 dB/m and MLD $\ge$ 100 dB/m, respectively, ranging from 1.3 to 1.7 µm. Moreover, we investigate the effective mode area and dispersion characteristics. Finally, a comparison between the proposed HC-ARPCF with other reported works is presented.
2. Configuration and optimization
Figure 1 shows the cross-section view of our initial design. Double-ring silica substrate structure is combined with eight outer semicircular cladding tubes and seven inner circular tubes. Semicircular tubes can effectively reduce the size of the diameter for integration [27]. Moreover, compared to the claddings with circular tubes or semi-elliptical tubes, the cladding composed of semicircular tubes not only can provide better design flexibility, but also introduce less manufacturing difficulty [28]. Two large and five small circular tubes are applied in the inner cladding to suppress HO modes and enhance reflection at the boundary between the air core and cladding. The relative permittivity of silica can be obtained by the Sellmeier equation, which can be regarded as a constant value (2.1) in our interested wavelength range [29]. EML is finely controlled by coating ENG ring in strategically selected semicircular tube, which can couple the energy out of the FM whose polarization direction is perpendicular to the ENG-loaded ring. Generally, ENG characteristic can be obtained by metamaterials [30,31], or natural materials [32–34]. In the optical regime, conductive metal oxides (CMOs), which have low loss factors and tuneable carrier densities in contrast to the plasmonic noble metals materials, e.g., gold and silver, are the most attractive ENG materials for various applications. In this work, a typical CMO named aluminum-doped zinc oxide (AZO) which behaves low loss characteristic near the wavelength of 1.55 µm is adopted. The relative permittivity of AZO can be obtained by Drude-Lorentz model [33],
For the configuration of this work, the outer semicircular silica tubes have the same diameter $d$ and thickness $t$. The inner silica tubes have two sizes. Five small tubes have the same diameter $d_{1}$ and thickness $t_{1}$, while two middle tubes have a larger diameter $d_{2}$ and thickness $t_{2}$. The diameter $D$ is fixed as 44 µm in this work to avoid the overlap between the cladding tubes, while suppressing the high-order mode. The thickness of ENG ring coated in the top semicircular tubes is $t_{ENG}$. Considering the convenience for parameter optimization, the configuration parameters $d$, $d_{1}$ and $d_{2}$ are chosen as 25 µm, 10 µm and 12.5 µm, respectively. The thicknesses $t$, $t_{1}$, $t_{2}$, $t_{ENG}$ are all set as 0.38 µm. Moreover, the fiber cladding is covered by the protective layer and PML layer. The thicknesses of the protective layer and PML are fixed at 6.5 µm. The study of this work is conducted by a full-wave vectorial finite element method (FEM) based commercial software, COMSOL Multiphysics. Besides, a perfectly matched layer (PML) boundary is employed to absorb any radiations leaking out of the proposed fiber.
Figure 2 shows the E-field distributions of the XP, YP, and two HO modes at 1.49 µm. It is clearly observed from the figure that the E-field of the XP and YP modes are confined in the core region, but, with different distributions. To be specific, a portion of the E-field of the YP mode is attracted to the ENG ring due to the guidance theory of HC-ARPCF and boundary condition. Since the ENG material is lossy, the TL of the YP mode should be high. In contrast to the YP mode, the XP has its E-field mainly confined in the air core. It is also noted that the E-field of the two HO modes exhibiting lowest loss among all the HO modes is either coupled to the nearby circular tubes or attracted to the ENG-loaded semicircular tube. Therefore, the TL of the XP mode is much lower than that of the YP and HO modes. The TLs of the XP, YP and HO modes as functions of wavelength are displayed in Fig. 3(a). The loss curve of the HO mode presented in the figure is actually a composite of the lower loss value between the two HO modes. Because these two HO modes alternately affect SPSM performance in our configuration within wavelength range from 1.3 to 1.7 µm.
Fig. 3. (a) Total loss of XP, YP and HO modes. (b) Meaning of Minimum Loss Difference (MLD). (c) Minimum loss difference in the initial structure.
In this work, the performance of the SPSM HC-ARPCF will be evaluated by the MLD value at the most widely used wavelength range in optical communication. The LD between the XP and YP modes is represented by LD (YP-XP), while the LD between the XP and HO modes is represented by LD (HO-XP). As shown in Fig. 3(b) and Fig. 3(c), the meaning of MLD is the lower value between LD (YP-XP) and LD (HO-XP). To evaluate the SPSM performance of the proposed fiber, the operation bandwidth is defined as the MLD $\ge$ 40dB/m and MLD $\ge$ 100dB/m for different scenarios. This means the LD between the wanted mode and unwanted mode would be as high as 40 dB or 100 dB after 1-meter length of propagating through the proposed PCF.
2.1 Optimization for thicknesses of the tubes
The most significant parameter is the thickness of semicircular tubes in the outer cladding [35]. Because it is the main factor that can influence the TL of all the modes supported by the PCF. HC-ARPCFs take advantage of the coherent reflection of light between the cladding tubes in the fiber to confine the light in the air core and propagate along the axis. The role of the cladding tubes in HC-ARPCFs is like the F-P cavity resonator, which leads to multi-peaks in the propagation spectrum. Since the existence of these peaks, the propagation spectrum can be divided into several high reflection regions, also known as the anti-resonant windows. In these windows, grazing incidence from the hollow core will lead to high reflection, which greatly reduces the CL of the fiber. The resonant wavelength can be calculated by
where $t$ is the tube thickness, $m$ is the positive integer of resonant order, and $n$ is the refractive index of silica. In our simulations, although, the resonance wavelength calculated by Eq. (2) is slightly shifted from the stopband central wavelength due to the existence of ENG material, Eq. (2) still has theoretical significance for our proposed fiber. Besides, the interested wavelength range for our work is the first anti-resonant pass band which is the widest anti-resonant low loss window [36].In order to obtain the basic loss characteristic, $t$ is fine tuned to adjust the resonance wavelength. Parameters $d$, $d_{1}$ and $d_{2}$ are set to 25 µm, 10 µm and 12.5 µm, respectively, and thicknesses $t_{1}$, $t_{2}$, $t_{ENG}$ are all equal to $t$ in the optimization proceeding of $t$. The value of $t_{0}$ is set to be 0.38 µm as the initial value of $t$. As shown in Fig. 4, the shift of the loss peaks of the modes are nearly linear for different values of thickness $t$. The peak point is affected not only by the tube thickness, but also by the ENG materials [37]. Since 1.55 µm is widely used as the optical communication wavelength, finer adjustments of $t$ are investigated, i.e., $t$ = $t_{0}$, 1.05$t_{0}$ and 1.1$t_{0}$, to make the center of the SPSM operation range around this wavelength. The MLD values for these three values of $t$ are shown in Fig. 4(d). Moreover, considering the two MLD conditions, the SPSM bandwidths for different values of $t$ are listed in Table 1. It can be observed from Fig. 4(d) and Table 1 that the best thickness is $t$ = 1.1$t_{0}$ since the center wavelength of the SPSM bandwidth interval is 1.55 µm along with the highest MLD. Besides, the bandwidth of MLD $\ge$ 100 dB/m is quite wide. Therefore, we set $t$ = 1.1$t_{0}$ = 0.418 µm as the optimal thickness of the semicircular tubes.
Fig. 4. Effect of thickness $t$ on (a) the total loss of XP, (b) the total loss of YP, (c) the total loss of HO modes, (d) MLD.
Table 1. SPSM bandwidth for different MLD conditions and thickness $t$.
Apart from the thickness of the semicircular infrastructure, other cladding tubes can also affect resonant wavelengths according to Eq. (2). The loss characteristic of this fiber is superimposed by the resonant influences of all tubes, therefore, the MLD of the whole fiber can also be ameliorated by optimizing the thicknesses of the inner cladding tubes. Subsequently, parameter sweeps of $t_{1}$ are conducted by changing the value from 0.9$t$ to 1.1$t$ with the other dimensions fixed, i.e., $d$ = 25 µm, $d_{1}$ = 10 µm, $d_{2}$ = 12.5 µm and $t$ = $t_{2}$ = $t_{ENG}$ = 0.418 µm. It is noted from Fig. 5 that the peak of the MLD is slightly red shifted as the value of $t_{1}$ increases. Moreover, the variations of $t_{1}$ barely affect the overall MLD, thus, the SPSM bandwidth is stable. In order to reduce manufacturing difficulty, $t_{1}$ is set as the same as the value of $t$, which is 0.418 µm.
Additionally, the parameter sweeps of $t_{2}$ are conducted by tuning its value from 0.9$t$ to 1.1$t$ with a step length of 0.1$t$. As shown in Fig. 6, the peak around 1.55 µm will be slightly red shift with the increase of $t_{1}$. Moreover, the amplitudes of MLD peaks are fluctuated due to the resonance wavelengths are different for varying thicknesses. To evaluate the fiber performance, the SPSM bandwidths for different $t_{2}$ are showed in Table 2. Although bandwidths of the case $t_{2}$=1.1$t$ are slightly better than other cases, $t_{2}$ is still set as t to ease the complexity of the fiber.
Table 2. SPSM bandwidth with different $t_{2}$.
Finally, considering fully filling material in the air hole might be easier than coating a film inside the tube, we investigate the case where the ENG material is fully filled in the semicircular air hole and compare with the ENG ring case. The cross-section view of ENG fully filled configuration is shown in Fig. 7(a). To straightly illustrate the performance and necessity of the ENG, the MLD values of different cases, i.e., ENG absent, ENG ring, and ENG fully filled, are shown in Fig. 7(b). It can be obtained from the figure that the existence of ENG is essential to realize wideband SPSM propagation. Besides, whatever the ENG exists in the fiber as a ring or being fully filled, the MLD values are quite stable. This is probably because the existence of the ENG affects the resonant condition at the interface between the air and substrate. From the perspective of practice, ENG material is set to be fully filled in the following work.
Fig. 7. (a) Cross-section view of the ENG fully filled configuration. (b) MLD values as function of wavelength for ENG absent, ENG ring, and ENG fully filled cases.
2.2 Optimization of core size
The effects of other factors on the fiber’s loss characteristic are optimized as well, including the diameters $d_{1}$ and $d_{2}$ of the inner cladding tubes, which are related to the fiber core size. In the process of majorization, the diameter $d_{1}$ is chosen to be 9.5, 10, and 10.5 µm, respectively, while $d_{2}$ and other parameters are fixed as 12.5 µm. As the diameter increases, the MLD value of the fiber increases as shown in Fig. 8(a). Compared with the case $d_{1}$ = 10 µm, the value of MLD for $d_{1}$=10.5 µm is much higher, thus, the fiber has better SPSM property. Then, $d_{1}$ is fixed at 10.5 µm in the majorization of $d_{2}$. As shown in Fig. 8(b), the fiber has the best performance as $d_{2}$=12.5 µm. This is due to the fact that the HO modes are usually suppressed in the narrow core. However, it is difficult to further increase $d_{2}$ for avoiding the tubes overlap. To ease the difficulty of manufacturing, diameters $d_{1}$ and $d_{2}$ of two circular tubes are selected as 10.5 µm and 12.5 µm, respectively. With these parameters, the achieved SPSM bandwidths for MLD $\ge$ 40 dB/m and MLD $\ge$ 100 dB/m are 252 and 152 nm, respectively.
Fig. 8. MLD between the wanted and unwanted modes for different (a) $d_{1}$ and (b) $d_{2}$ over the wavelengths of interest.
2.3 Optimization for the symmetric structure
In order to further simplify the structure, we remove the bottom inner cladding tube and fully fill the ENG material in the semicircular tube at the corresponding position without changing the structural parameters. The consequent cross-section view of the symmetric configuration is shown in Fig. 9. The loss property and MLD values of the final PCF design is displayed in Fig. 10. The TL of the wanted mode can be as low as 4.19 dB/m. According to the SPSM window (MLD $\ge$ 100 dB/m) shown in Fig. 10(b), the SPSM bandwidth is wider than the asymmetric configuration, i.e., 168 nm VS 152 nm. Because loading ENG material in the extra semicircular tube along the y-axis can further enhance the EML of the YP mode. According to the above optimization process, the optimal structure and parameters are obtained. Specifically, SPSM regions as MLD $\ge$ 40 dB/m are in the wavelength range from 1408 nm to 1676 nm and from 1680 nm to 1700 nm. It should be noted that even we raise the SPSM condition as MLD $\ge$ 100 dB/m, the operation band, i.e., from 1452 to 1620 nm, can still cover S-, C-, and L-Bands, which are the most common bands in optical communication systems.
Fig. 10. (a) Total loss of XP, YP and HO modes of the symmetric ENG filled configuration. (b) Minimum loss difference and SPSM window for MLD $\ge$ 100dB/m of the symmetric ENG filled structure.
2.4 Other features of optimized PCF
With the finalized configuration, we also investigate other significant properties of the PCF in this work. The effective mode area $A_{eff}$ can be used to evaluate the amount of cross-sectional area occupied by the mode, which is calculated by Eq. (3). The $A_{eff}$ of the proposed PCF is plotted in Fig. 11. As shown in the figure, the value of $A_{eff}$ is quite large across the entire wavelength range of interest. Notably, the results even reach the level of some PCF works aiming to achieve large effective mode area [11,38]. The large effective mode area performance of proposed SPSM PCF could be contributed to applications, like high power lasers.
Furthermore, the dispersion property of the proposed PCF is also investigated, and the result can be found in Fig. 12. With the optimal parameters, the proposed PCF yields a near-zero and flat dispersion across a wide wavelength band ranging from 1.3 to 1.7 µm. Within this band, the average dispersion is as low as 0.0078 ps$\cdot$km$^{-1}\cdot$nm$^{-1}$ due to the propagation of light in the air core.
3. Fabrication discussion and performance comparison
In 2018, a HC-ARPCF with similar structure as ours has been fabricated with advanced fiber-fabrication techniques [39]. This successful case indicates that the proposed SPSM HC-ARPCF with fully filled ENG material in this paper can be potentially fabricated. To build a PCF, techniques like stacking [12], drilling [40] and die-extrusion [41] are implemented to produce a preform, and then, the preform is drawn into a fiber through a hot drawing process at a suitably high temperature. Moreover, 3D printing is a potential method to achieve fibers fabrication with lower cost and more convenience [27]. To the best of our knowledge, various methods such as high-pressure microfluidic chemical deposition [42], the capillary effect method [43], high-temperature pressure-assisted technique [44] and pressure-assisted splicing technique [44,45] could be applied for filling ENG CMOs. Especially, pressure-assisted splicing technique is highly suitable for our fiber fabricating and investigated comprehensively in the aforementioned references. Therefore, implementation of the proposed PCF design would be very likely completed with the current existing manufacturing technologies.
In addition, the fabrication robustness of our PCF is also considered, which will contribute to our experiments in the following work. During the fabrication process, the thickness fluctuation of tubes may occur unintentionally, which is the most common inaccuracy. To estimate the effect of thickness fluctuation on the fabrication robustness, one can analyze the parameter sweeps in the previous section. According to the results of parameter optimization, the SPSM bandwidth is hardly affected but the window will be drifted about 65 nm caused by the $\pm$10$\%$ manufacturing error associated with the thickness value. Even the $\pm$10$\%$ manufacturing error occurred, the wavelength of 1550 nm can still be covered by the broadband SPSM bandwidth. Thus, the design can tolerate fabrication error well.
Finally, a performance comparison between the result of our work and other state-of-the-art works is shown in Table 3. Fiber types and SPSM bandwidths under different conditions are listed. Compared with other SPSM HC-ARPCF works with high values of PER and HOMER, our fiber can obtain a better performance in real LD. Besides, the achieved near-zero dispersion and large effective mode area, which are hardly obtained in SC-PCFs, can help the fiber find applications in more operational scenarios. Most importantly, the SPSM bandwidth of the proposed design is the widest among all the works.
Table 3. Comparison between our SPSM HC-ARPCF with other state-of-the-art SPSM works.
4. Conclusion
In this paper, we proposed a novel broadband single-polarization single-mode dual-rings hollow core anti-resonant photonic crystal fiber, which presents large values of minimum loss difference between the wanted mode (XP) and unwanted modes (YP and HO modes). After optimization, eight semicircular substrate tubes and six circular substrate tubes are introduced as outer and inner claddings, respectively. Two central inner circular tubes are chosen to have a larger diameter to suppress HO modes. To obtain significantly enhanced MLD values by manipulating EML, ENG material is introduced to be fully filled in two outer semicircular tubes along in the E-field direction of the YP mode. It was demonstrated that the optimal design achieves a wide SPSM bandwidth of 288 nm (from 1408 nm to 1676 nm and from 1680 nm to 1700 nm) for MLD $\ge$ 40 dB/m and 168 nm (from 1452 nm to 1620 nm) for MLD $\ge$ 100 dB/m. Moreover, the bandwidth of the proposed PCF in condition of MLD $\ge$ 100 dB/m covers S-, C-, and L-Bands, which are the most common bands in optical communication and network systems. Furthermore, results of the simulation also indicated that the PCF has a near-zero and flat dispersion characteristic, along with a large effective mode area. In addition, we discussed the fabrication possibility and compared the proposed work with other SPSM PCFs. To sum up, the proposed PCF with these attractive performance characteristics is a competitive candidate for a variety of optical guiding applications requiring polarization purity.
Funding
Shenzhen Research Foundation (JCYJ20170413152328 742).
Acknowledgments
We thank the editors and the anonymous reviewers for their careful reading and valuable comments that were very helpful for revision and improving this paper.
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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