1 Introduction

A polarization beam splitter (PBS) splits light into two beams of a single polarization state and is a fundamental optical device in optical communication systems. Coupling length, bandwidth and higher order mode extinction ratio (HOMER) are important indicators of the performance of PBSs. Since L. Zhang proposed a polarization beam splitter based on a dual-core microstructure fiber in 2003, it has become an increasingly popular research topic today [1, 2].

In recent years, to break through the bottleneck of traditional solid optical fibers due to material defects, a hollow-core fiber (HCF) using air as the transmission medium is expected. HCFs are characterized by a unique transverse cladding structure that effectively confines the light beam to the air-core, thus allowing the light beam to propagate only along the longitudinal direction of the core for the purpose of optical communication using air as a medium. Air as a special transmission medium for air-core optical fibers is widely used in optical fiber gyroscopes [3], high power pulse transmission [4] and infrared transmission [5] due to its low dispersion and low delay, high damage threshold and low temperature sensitivity.

HCFs can be divided into two categories according to the principle of light conduction [6]. One is the hollow-core photonic bandgap fiber (HC-PBGF), which is based on the photonic bandgap effect, and the other is the hollow-core anti-resonant fiber (HC-ARF) or negative curvature fiber. Compared to bandgap hollow-core micro-structured fibers, hollow-core anti-resonant micro-structured fibers do not have more stringent requirements on the structural cladding of the fiber, and the structural design is flexible and versatile, thus promising a wider low-loss transmission bandwidth and more flexible extension of its coupling characteristics to realize hollow-core PBS as well as couplers, which have become the subject of much research interest today.

As an important passive optoelectronic device, fiber optic polarization beam splitter can split a beam of light into two orthogonal beams of linearly polarized light, and has more extensive applications in optical communication systems and other fields [7,8,9,10,11,12]. The dual-core fiber (DCF) uses the coupling of the optical extinction field between the two cores to achieve multiple choices of optical power, wavelength, polarization state, etc., and is widely used in the development of polarization beam splitters [13]. Polarization beam splitters and couplers based on DCFs are essential integrated components for fiber optic communications [14], fiber optic gyroscopes [15], fiber optic sensors [16] and fiber lasers. They are increasingly becoming a research hotspot for today’s work. In 2002, a hollow-core microstructure with a Kagome cladding structure was proposed [17], which is called HC-ARF because its light conduction mechanism is based on the restricted coupling theory. In 2016, Liu et al. proposed two kinds of no-mode DHC-ARFs, which broke the limitation of the cladding wall on the power transmission between the two cores and strongly indicated the possibility of using this fiber structure for realizing fiber beam splitters [18], making the HC-ARF rapidly become a new research hotspot for hollow-core micro-structured fibers. Huang et al. first fabricated a DHC-ARF and tested the coupling characteristics of the fiber in 2017, but the component length was not conducive to integration into fiber systems, and the more pronounced wavelength dependence of the constructed fiber also posed a hindrance to the realization of ultra-bandwidth performance [19]. Two years later, Zhao et al. proposed a DHC-ARF PBS with two elliptical tubes, which was able to achieve a bandwidth of 310 nm [20], but this also caused some difficulties in practical fabrication due to the introduction of the elliptical structure. At 2021, Jia et al. proposed a DHC-ARF PBS with a length of 4.42 cm and a bandwidth of 460 nm [21]. The expansion of coupling length and bandwidth performance has been a hot topic of research in previous work, and the single-mode performance has been a key issue in the development of ultra-wideband DHC-ARF based PBSs. However, the wavelength dependence exhibited by most fiber structures makes it difficult to have wider bandwidths, and the introduction of elliptical structures has also created some difficulties for the practical fabrication of fiber PBSs. Therefore, the introduction of a DHC-ARF PBS with a wider bandwidth while ensuring excellent single-mode characteristics is a great challenge.

In this paper, we propose an ultra-broadband DHC-ARF PBS by introducing two longer U-shaped quartz tubes to divide the core into two mutually symmetrical cores A and B, leaving a narrow gap between them as a mode coupling channel. This structure was introduced to have a significant impact on broadening the bandwidth of the constructed DHC-ARF PBS, effectively reducing the wavelength dependence and allowing a greater independence of the wavelength from other parameters. It also enables the structures we designed to have an ultra-wide bandwidth of 510 nm, which was not available in all previous studies. At the same time, the introduction of the circular nested tube and U-shaped nested tube structures effectively reduces the loss of the base film, thus enhancing its single-mode performance very well. The HOMER between the lowest loss of HOMs and the highest loss of FMs is consistently above 100 or even 1000 in the wavelength bands considered, and reaches 1376 at 1.55 μm, which indicates that our constructed DHC-ARF PBS also has excellent single-mode properties.

2 Structure and principle

Figure 1 shows the structure of the proposed DHC-ARF PBS cross-section. The core region consists of two quartz tubes with radius and thickness set to \(R\) and \(t_{0}\). The cladding region of the DHC-ARF consists of six circularly nested quartz tubes and two U-shaped nested quartz tubes, which are arranged around the inner wall of the outer quartz tube and are connected to each other. Each U-shaped nested tube can be viewed as consisting of a portion that resembles a rectangle and a semicircle near the core of the fiber. The radius of the outer sector of the U-shaped nesting tube near the core of the fiber is \(r_{2}\), the radius of the outer quartz tube in the vertical direction of the fiber core is of size \(r_{0}\), and the radius of the outer quartz tube at the remaining four corners is \(r_{1}\). Affected by the manufacturing process, there can be a relatively narrow gap between each round tube and the U-shaped tube. The purpose of introducing an air gap of width g between the two middle U-shaped quartz tubes is to form a power transfer channel between core A and core B. The ratio of the radius of the inner nested quartz tube to the radius of its outer quartz tube is set to \(k\), the radius \(r_{ni}\) of the inner nested quartz tube and the radius \(r_{i}\) of its outer quartz tube should satisfy \(r_{ni} = k{*}r_{i}\), where \(k\) is defined as a constant. The thickness of the quartz tube wall is indicated by the letter \(t\). When the air gap width g and r2 are determined, the distance between the center of the semicircle and the fiber core is g/2 + r2, so that the position of the whole U-tube can be determined. We need to control the values of r2 and g not to be too large, to ensure that the quartz nested tube with radius r1 can be tangent to the portion that resembles a rectangle instead of the semicircle of the U-shaped nested tube, which is the first tangent condition it needs to satisfy. At the same time, if the air gap width g is too large, it will make the coupling channel between the two cores wider, which is not conducive to the polarization beam splitting of the light. Based on the above considerations, we set the values of r2 and g in the ranges of 6 μm < r2 < 10 μm and 1 μm < g < 5 μm, respectively. When the size of the quartz nested tubes in the vertical direction of the fiber cores with r0 as the radius is determined, the quartz nested tubes with r1 as the radius need to be ensured that it is tangent to it as well, and this is the second tangent condition that it needs to be satisfied. Since the quartz nested tube with radius r1 must satisfy both of these tangent conditions, r1 can be uniquely determined through the system of equations when the parameters r2, r0, and g are determined. On the other hand, the thickness of the quartz tube wall cannot be too thick or too thin to prevent not only the energy in the core leaking into the cladding tube, but also the modes in the core to resonate with the modes in the cladding. Therefore, we set the initial value of t to about 0.5 μm. In addition, since larger device sizes result in longer coupling lengths, we set the device radius R to 30 μm which is suitable. Subsequently, we will also carry out a detailed optimization of each parameter to determine the best values for each parameter, ensuring that the best performance is obtained from the designed structure.

Fig. 1
figure 1

Schematic cross-section of the designed DHC-ARF PBS

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The propagation characteristics of a DHC-ARF PBS can be calculated using a finite-element algorithm. A perfectly matched layer is added to the outer side of the PBS quartz tube of the DHC-ARF, with its thickness and refractive index set to \(C = 15\) μm and \(n\_{\text{silica}} + 0.03\) [22]. Because of the large refractive index of the casing wall with silica, the minimum mesh element divided in the finite-element simulation should be small. The function of PML is to construct a domain with open boundaries for the designed structure, which is convenient for the subsequent finite-element simulation. In addition, PML has a relatively homogeneous structure, so the minimum grid unit divided can be larger. This helps increase the speed and efficiency of research calculations. Based on the above analysis, the grid size of the silica is set to \(\lambda /6\), the grid size of the air is set to \(\lambda /4\), and the grid size of the perfect match layer is set to \(\lambda /2\). The refractive index of the silica material can be obtained from the following Sell Meier equation [23]:

$$n_{{{\text{Silica}}}} (\lambda ) = \sqrt {1 + \frac{{A_{1} \lambda^{2} }}{{\lambda^{2} - B_{1}^{2} }} + \frac{{A_{2} \lambda^{2} }}{{\lambda^{2} - B_{2}^{2} }} + \frac{{A_{3} \lambda^{2} }}{{\lambda^{2} - B_{3}^{2} }}} ,$$
(1)

where A1 = 0.6961663, A2 = 0.4079426, A3 = 0.897479, B1 = 0.00684043, B2 = 0.1162414, B3 = 9.896161, and \(\lambda\) is the wavelength of the incident light wave in microns.

The above-proposed DHC-ARF PBS follows the same waveguide mechanism as the HC-ARF, with the resonant wavelength \(\lambda_{{{\text{res}}}}\) determined by the thickness t of the cladding tube, both satisfying [24]

$$\lambda _{{{\text{res}}}} = 2t\sqrt {n^{2} - 1} /m,m = 1,2, \ldots$$
(2)

where m denotes an arbitrary positive integer and n denotes the refractive index of the glass.

According to the mode coupling theory [25], four fundamental supermodels exist in the proposed DHC-ARF PBS, which are x-pol odd mode, x-even mode, y-pol odd mode and y-even mode, respectively, as shown in Fig. 2 for the mode field distribution of the four fundamental supermodels when the incident light wavelength is 1.55 μm. The mode field (\({\text{LP}}_{11}\) mode) distribution of the minimum loss HOMs at this time is also shown in Fig. 2.

Fig. 2
figure 2

Simulation results for four fundamental modes (x-pol odd mode, x-pol even mode, y-pol odd mode, y-pol even mode) and low-loss high order mode fields (\({\text{LP}}_{11} - 1\) and \({\text{LP}}_{11} - 2\)) at an incident light wavelength of 1.55 μm

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Coupling length is defined as the transmission length at which optical power in one core can be fully transmitted to the other core. The coupling length of x and y modes is calculated by the following formula [26]:

$$L_{c}^{x} = \frac{\lambda }{{|2(n_{{{\text{even}}}}^{x} - n_{{{\text{odd}}}}^{x} )|}}$$
(3)
$$L_{c}^{y} = \frac{\lambda }{{|2(n_{{{\text{even}}}}^{y} - n_{{{\text{odd}}}}^{y} )|}},$$
(4)

where \(\lambda\) is the wavelength of the incident light, \(n_{{{\text{odd}}}}^{x} ,n_{{{\text{even}}}}^{x}\) and \(n_{{{\text{odd}}}}^{y} ,n_{{{\text{even}}}}^{y}\) are the effective refractive index of the x and y odd–even modes, respectively.

The coupling length ratio can be expressed as [27]

$${\text{CLR}} = \frac{{L_{c}^{x} }}{{L_{c}^{y} }}.$$
(5)

To achieve the performance of a polarization beam splitter based on a double hollow-core anti-resonant fiber, the CLR should be as close as possible to 0.5 or 2.

HOMER can be used to characterize the single-mode properties of an optical fiber with a value equal to the ratio of the higher order mode (e.g., \({\text{LP}}_{11}\)) limiting loss to the base film limiting loss. That is [28]

$${\text{HOMER = }}\frac{{{\text{Confinement}}\;{\text{loss}}\;{\text{of}}\;{\text{LP}}_{11} }}{{{\text{Confinement}}\;{\text{loss}}\;{\text{of}}\;{\text{LP}}_{01} }}.$$
(6)

When light is incident on core A, the normalized output power expressions for X parity mode and Y parity mode at core A and core B are [29]

$$P_{{{\text{out}},A}}^{x,y} = P_{{{\text{in}}}} \cos^{2} \left( {\frac{\pi }{2}\frac{L}{{L_{c}^{x,y} }}} \right)$$
(7)
$$P_{{{\text{out}},B}}^{x,y} = P_{{{\text{in}}}} \sin^{2} \left( {\frac{\pi }{2}\frac{L}{{L_{c}^{x,y} }}} \right),$$
(8)

where \(L\) is the propagation distance.

ER is a key metric for evaluating PBS performance and is defined as the ratio of the power of one polarization to the other in the same core, which can be expressed, as an example, for core A as [30]

$${\text{ER}} = 10\log_{10} \frac{{P_{{{\text{out}},B}}^{y} }}{{P_{{{\text{out}},A}}^{x} }}.$$
(9)

When the ER is less than − 20 dB, the separation of the Y-mode from core A and the PBS can be achieved. Therefore, the wavelength range satisfying − 20 dB or less is considered the bandwidth of the PBS. Since core A and core B are symmetrically distributed, we only need to consider the variation of ER with the light wavelength when light is incident on one of the cores.

3 Simulation results and discussion

To enhance the performance of the DHC-ARF PBS to a large extent, a wide bandwidth needs to be achieved, and the coupling length should be wavelength-insensitive and as short as possible. At the same time, the fiber will exhibit single-mode characteristics when the first higher order mode limiting loss is significantly higher than the fundamental modes limiting loss, which means that we consider it to have single-mode performance when the HOMER is large enough. Based on this consideration, a nested structure is introduced in the outer quartz tube to reduce the base film limiting loss and thus improve HOMER.

To make the coupling ratio more stable around 0.5 or 2, and to obtain a wider bandwidth and better single-mode performance, the structural parameters of the DHC-ARF PBS need to be optimized. We use the finite-element method combined with the perfect matching layer to set the boundary conditions to simulate the coupling length, coupling ratio, single-mode characteristics and other characteristic parameters under different parameter conditions. In the following simulation, the incident wavelength is set to 1.55 μm, and the initial parameters are set to R = 30 µm, r0 = 7.5 µm, r2 = 7.6 µm, g = 3.8 µm, t = 0.52 µm, k = 0.56 and C = 15 µm.

Figure 3a and b show the variation and trend of the coupling length and coupling ratio of X and Y modes, as well as the limiting loss and HOMER of the base film and the first higher order mode when g increases from 1.8 to 4.6 μm. It can be seen from the figure that as the gap g between the two U-shaped quartz tubes becomes larger, both \({L}_{c}^{x}\) and \({L}_{c}^{y}\) decrease significantly, but the coupling ratio tends to increase and then decrease. When g increases to about 3.4 μm, the CLR can reach the ideal coupling ratio of about 2. Meanwhile, as g increases, the corresponding limiting losses of the base film and the first higher order mode slowly decrease, and HOMER also shows a certain fluctuation trend. This is due to the fact that the increase in g leads to a wider channel for coupling light from one core to the other, making it easier to achieve polarized beam splitting of the light, and hence a correspondingly shorter coupling length. At the same time, the widening of the channel in the core region also results in less core energy leaking into the cladding, and hence less loss. On the other hand, the increase of g moves the outer cladding of the fiber away from the core, and the increased core region makes it easy for phase mismatch to occur between the core and cladding modes, which suppresses the coupling strength between them and helps to further reduce the loss. Considering the value of the coupling ratio and the trend of HOMER, it is necessary to select a suitable value of g.

Fig. 3
figure 3

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when g increases from 1.8 to 4.6 μm

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Figure 4a shows the variation of the corresponding coupling length and coupling ratio when \(k\) increases from 0.44 to 0.68, where \(k\) is defined as the ratio of the radius of the inner nested quartz tube to the radius of the outer quartz tube. In a previous study [31], the nested tube structure was introduced in the outer silica cladding tube to reduce the losses during transmission. Therefore, to reduce the sensitivity of the coupling length to wavelength and to further extend the bandwidth of the DHC-ARF PBS, we also introduced the nested tube structure. We reflect the variation of the nested tube size by the variation of the defined constant \(k\). It can be seen from Fig. 4a that the coupling length of X and Y modes decreases approximately linearly with increasing \(k\), while the CLR increases with increasing \(k\), but the growth rate becomes gradually slower. Figure 4b shows the relationship between the limiting loss, HOMER and \(k\) for the \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) modes. From Fig. 4b, it can be seen that when \(k\) increases from 0.44 to 0.68, the limiting loss of \({\mathrm{LP}}_{01}\) mode decreases and then increases. This is because the increase in k causes the radius of the nested tubes in the cladding to increase, which reduces the space of the cladding tubes and the light in the core is less likely to leak into the cladding. This can reduce the coupling length and the loss of the fundamental mode to some extent. However, when k increases to a certain extent, the cladding tube is closer to the core, increasing the degree of bending of the cladding tube near the core. This is more likely to cause resonance between the modes in the core and the modes in the cladding tube, which in turn causes the light to leak into the cladding, thus increasing the loss of the higher order mode and the fundamental mode. The limiting loss of \({\mathrm{LP}}_{11}\) mode increases and then decreases, and when the value of \(k\) is around 0.58, the limiting loss of \({\mathrm{LP}}_{11}\) mode reaches a relatively high level, and the corresponding HOMER value is also larger, showing an obvious single-mode performance. The limiting loss of \({\mathrm{LP}}_{01}\) mode has been maintained at a lower level for any value of k. At the same time, for some values of \(k\), the corresponding HOMER is smaller and has no obvious single-mode characteristics. From the above analysis, although the value of \(k\) has no obvious effect on CLR, but there is a significant effect on \({\mathrm{LP}}_{11}\) mode and HOMER, so the determination of \(k\) is very critical.

Fig. 4
figure 4

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when \(k\) increases from 0.44 to 0.68

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Figure 5a shows the variation of coupling length and coupling ratio with \(R\). From the figure, we can see that the coupling length of both X and Y modes gradually increases with the increase of \(R\) from 27 to 33 μm, and the growth rate is getting faster, while the coupling ratio CLR keeps decreasing with the increase of both. From Fig. 5b, we can see that the limiting loss of \({\mathrm{LP}}_{11}\) mode has an overall increasing trend with the change of \(R\), and the limiting loss of \({\mathrm{LP}}_{01}\) mode has an overall decreasing trend with the change of \(R\). This may be due to the fact that as R becomes larger and the device size increases, the microscopic distribution of the electromagnetic field energy of the guided mode changes in the fiber. This not only leads to a change in the percentage of energy in the silica and air materials, but also to a change in the effective refractive index coefficients of the corresponding modes. Therefore, the coupling ratio and mode loss of the material are changed to some extent. Meanwhile, we can also learn that the value of HOMER is around 100 when \(R\) increases to about 28.5 μm, which indicates that the DHC-ARF PBS can present single-mode characteristics at this time. Therefore, the value of \(R\) also plays a crucial role in the single-mode performance of the fiber polarization beam splitter.

Fig. 5
figure 5

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when \(R\) increases from 27 to 33 μm

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Figure 6a and b show the variation of each parameter with \({r}_{0}\). From Fig. 6a, it can be seen that \({L}_{c}^{x}\) and \({L}_{c}^{y}\) keep decreasing as \({r}_{0}\) grows from 6.5 to 8.5 μm. For the coupling ratio CLR, it reaches exactly near our ideal expected value of 2.0 when \({r}_{0}\) grows to 8.5 μm. This may be due to the fact that the larger r0 increases the width of the coupling channel of the fiber core, which makes it easier to achieve polarized beam splitting of the light. At the same time, the increase in the vertical tube size also enhances the birefringence characteristics of the designed DHC-ARF to some extent, resulting in a shorter coupling length. Therefore, from the coupling length and coupling ratio analysis alone, we expect the larger value of \({r}_{0}\) the better. However, if the value of \({r}_{0}\) is too large, it may cause HOMER to be too low and thus the single-mode characteristics of the PBS will be lost. Considering the above analysis, we need to select a suitable \({r}_{0}\) when considering each performance of the designed device.

Fig. 6
figure 6

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when \({r}_{0}\) increases from 6.5 to 8.5 μm

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Figure 7a shows the variation of \({L}_{c}^{x}\), \({L}_{c}^{y}\) and the coupling ratio with \({r}_{2}\). When \({r}_{2}\) increases from 6.6 to 8.6 μm, the value of \({L}_{c}^{x}\) shows a slow increase, but the overall change is not large. The value of \({L}_{c}^{y}\) keeps increasing with the change of \({r}_{2}\), and the value of CLR also increases with the increase of \({r}_{2}\). When the value of \({r}_{2}\) reaches about 8.5 μm, the CLR reaches the ideal value of about 2.0. This is probably due to the fact that the larger r2 increases the size of the U-tube in the horizontal direction. This narrows the coupling channel in the x-pol direction, leading to a more difficult coupling of the x-pol mode from one core to another and a consequent increase in the coupling length. From Fig. 7b, the overall trend of the limiting loss of the base film and higher order modes is not significant with the change of \({r}_{2}\), and the value of HOMER has been maintained above \({10}^{2}\) orders of magnitude, which indicates that the corresponding PBS has its single-mode performance when the \({r}_{2}\) value increases from 6.6 to 8.6 μm.

Fig. 7
figure 7

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when \({r}_{2}\) increases from 6.6 to 8.6 μm

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t denotes the thickness of the quartz tube wall. According to the resonance conditions listed above in Eq. (2), the increase of t leads to the red shift of the resonance wavelength. In addition, the confinement loss of the fundamental mode at the resonance wavelength is higher, so the operating bandwidth of the DHC-ARF should be avoided to be selected near the resonance wavelength. Based on the above considerations, t around 0.5 μm is a relatively ideal value. Figure 8a and b show the subsequent changes of each parameter as t changes from 0.44 to 0.64 μm. The coupling lengths of both X and Y modes decrease when the t increases. This may be attributed to the increase in t which makes the wall of the quartz tube thicker and, therefore, makes the light in the core less likely to leak into the cladding. This attenuates the energy loss and also reduces the coupling length. The coupling ratio reaches around 2.0 when the value of t is around 0.58 μm. The value of HOMER remains above 100 as t changes, which indicates that it exhibits single-mode performance for any value of t in the interval from 0.44 to 0.64 μm.

Fig. 8
figure 8

a Coupling length of the x-pol and y-pol core modes and CLR, and b the confinement losses of the modes \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER when \(t\) increases from 0.44 to 0.64 μm

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4 PBS based on a DHC-ARF

Based on the above discussion and analysis results, the parameters of the structured polarization beam splitter are finally determined as follows: R = 30 µm, r0 = 7.5 µm, r2 = 7.6 µm, g = 3.8 µm, t = 0.52 µm, \(k = 0.56\), and the thickness of the perfect matching layer C = 15 µm. Under the above parameters, the variation of \({L}_{c}^{x}\),\({L}_{c}^{y}\), and the coupling ratio with the incident wavelength from 1.3 to 2.0 μm is shown in Fig. 9. From the figure, it can be seen that \({L}_{c}^{x}\) increases gradually from 1.3 to 1.7 μm and then stabilizes. \({L}_{c}^{y}\) increases gradually from 1.3 to 1.9 μm, but there is a significant change in the higher wavelength band. The coupling ratio, on the other hand, keeps decreasing with the increase of the optical wavelength, but also climbs significantly around 1.95 μm. From the data results, it is possible that the performance of the constructed anti-resonant fiber polarization beam splitter is lost in the higher wavelength band above 1.9 μm. Our designed DHC-ARF PBS has the following outstanding performance.

Fig. 9
figure 9

Coupling length of the x-pol and y-pol core modes and CLR in the wavelength range from 1.3 to 2.0 μm

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4.1 Ultra-wide bandwidth

Figure 10 shows the variation of the normalized output power of core A and core B with respect to the propagation distance when the incident light with a wavelength of 1.55 μm is incident to core A. In Fig. 10, the normalized output power of core A x-pol light is close to the maximum value when the propagation distance reaches 8.1 cm, while the normalized output power of core A y-pol light decreases to 0. In Fig. 10, when the propagation distance reaches 8.1 cm, the normalized output power of x-pol light in core A is close to the maximum, while the normalized output power of y-pol light in core A decreases to 0. In core B, when the propagation distance is 8.1 cm, the normalized output power of x-pol light becomes 0, and the normalized output power of y-pol light reaches the maximum currently. At this time, x-pol light exists only in A-core and y-pol light exists only in B-core, indicating that x-pol light and y-pol light are well separated in A-core and B-core.

Fig. 10
figure 10

Normalized output powers of cores A and B as functions of the propagation distance

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ER is an important measure of the operating performance of the PBS. When the power of one polarization mode is 100 times that of the other one, i.e., the ER is less than − 20 dB, the separation of the two mode beams is considered to be achieved. At this point, the polarization beam splitting of the light is achieved and the corresponding wavelength range is the operating bandwidth of the designed PBS. The variation of ER with the light wavelength is shown in Fig. 11. The values of the individual fiber structure parameters set have been described at the beginning of Chapter 4. In Fig. 11, polarization ERs of − 51.51 dB and − 48.30 dB are achieved at wavelengths of 1.5 μm and 1.86 μm, respectively. The polarization ER is less than − 20 dB in the wavelength range of 1.43–1.94 μm. As a result, the proposed DHC-ARF PBS covers four communication bands and has a bandwidth of up to 510 nm, which is far beyond the reach of previous studies.

Fig. 11
figure 11

Polarization ER as a function of wavelength

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In the following sections, we discuss the possible reasons for achieving this ultra-wide bandwidth. In reference [21], JIA et al. designed a kind of anti-resonant fiber, which is named sakura type anti-resonant fiber because the six nested quartz tubes resemble cherry blossoms. The polarization beam splitter can reach the bandwidth of 460 nm, and its performance structure has achieved a great breakthrough. They use two identical circular quartz-embedded tubes to leave a narrow air gap in the center of the fiber core as a channel to achieve fiber polarization fraction. Unlike their proposed cross-section structure, our proposed structure uses two U-shaped tubes that extend directly from the fiber wall to create this narrow air gap directly in the fiber core. We guess that the change of this structure has a great impact on the realization of the ultra-wide bandwidth. Therefore, we analyze the change of the parameter air gap width g, which is greatly affected by the change of this structure (Fig. 12).

Fig. 12
figure 12

Effect of air gap width g on coupling length and CLR of x and y polarization splitters in two different fiber structures

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As can be seen from the comparison of the two figures above, both structures become monotonically decreasing with increasing g on the coupling length of the x and y polarization states, but there is a significant difference in the rate of change. In the structure proposed in the reference [21], when g increases from 2.6 to 5.0 μm, it decreases monotonically from 4.53 cm and 7.69 cm to 1.91 cm and 1.19 cm, respectively, and the CLR varies between 0.52 and 0.62. In our proposed structure, when g increases from 1.8 to 4.6 μm, it decreases monotonically from 21.0 cm and 34.8 cm to 2.2 cm and 3.5 cm correspondingly, the CLR shows a trend of increasing and then decreasing. Through the above analysis, the motivation of citing two longer U-tubes is that they greatly increase the impact of the air gap width g on the performance of the designed beam splitter. On the other hand, the influence of wavelength changes on the coupling length and bandwidth of the designed beam splitter is weakened, and the dependence on wavelength is reduced. It is of great significance to optimize each performance index of the beam splitter.

The following two figures show the mode field coupling at g = 2.6 μm and g = 3.8 μm at a wavelength of 1.55 μm, respectively. As shown in the left figure, when g = 2.6 μm, part of the mode field energy in the y-direction is coupled into the cladding structure of the U-tube, which is not conducive to the performance of the designed anti-resonant fiber polarization beam splitter. When g increases to our chosen value of 3.8 μm, as shown in the right figure, most of the energy is concentrated in the middle of the core within the width of the air gap and is less likely to be coupled into the U-tube cladding structure, which is beneficial to the performance improvement of parameters such as the bandwidth (Fig. 13).

Fig. 13
figure 13

Schematic diagram of the mode field energy at g = 2.6 μm and g = 3.8 μm for λ = 1.55 μm

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Finally, from the perspective of the mode coupling theory, we artificially reduce the size of the vertical tube of the designed fiber cross-section and increase the size of the tubes on the horizontal side of the core area by introducing two U-shaped quartz tubes extending directly from the core wall, thus enhancing the birefringence characteristics of the designed anti-resonant fiber polarization beam splitter and further ensuring that the two orthogonal polarized beams can be better separated within a short coupling length. This further ensures a better separation of the two orthogonal polarized beams within a shorter coupling length and achieves the beam splitting effect of the polarized light. The above analysis further argues that the introduction of two U-shaped quartz nesting structures has a significant effect on widening the bandwidth of the polarization beam splitter.

4.2 Excellent single-mode characteristics

HOMER is an important reference when designing single-mode devices based on DHC-ARF, and it can avoid the influence of the remaining unwanted modes when the device is connected to the fiber system. Figure 14 shows the constraint loss as a function of wavelength for \({\mathrm{LP}}_{01}\), \({\mathrm{LP}}_{11}\) and HOMER modes. In Fig. 14, the constraint loss of \({\mathrm{LP}}_{01}\) mode is always around 1 dB/m in the wavelength range of 1.35–1.62 μm, and the constraint loss of \({\mathrm{LP}}_{11}\) mode is always higher than 1000 dB/m in the wavelength range of 1.20–1.70 μm. HOMER value is greater than 100 in the wavelength range of 1.3–1.82 μm, and reaches 1376 near 1.55 μm, indicating that the designed DHC-ARF PBS has excellent single-mode characteristics.

Fig. 14
figure 14

Confinement losses of the mode \({\mathrm{LP}}_{01}\) and \({\mathrm{LP}}_{11}\) and HOMER in the wavelength range from 1.3 to 2.0 μm

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We have compared the two parameters, bandwidth and HOMER, in our study with other studies in recent years. The results are shown in Table 1.

Table 1 Comparison of our study with recent studies
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As can be seen from Table 1, our DHC-ARF PBS can achieve a bandwidth of up to 510 nm, which is absolutely leading the way in recent years’ research. At the same time, HOMER is well above 100 in the operating band and above 1000 near the 1.55 μm center band, indicating that our DHC-ARF PBS also has excellent single-mode characteristics.

In the following, we provide some brief discussions on the fabrication of the designed anti-resonant fiber polarization beam splitter. The main difficulty in the preparation of our proposed DHC-ARF is that it contains three different sizes of quartz nested tubes, which are difficult to ensure the tangibility between two due to the fabrication process, and there may be nodes between adjacent cladding tubes. In the literature [33], Huang et al. proposed a split-clad HC-ARF, this structure allows for multiple thicker nodes between adjacent pipes in the actual manufacturing process. However, the final experimental data showed that the core shape was maintained even though the wall boundary shape was deformed at the nodes. The measured values of the losses of the prepared HC-ARF were in general agreement with the values calculated in the experimental simulations of the ideal structure of the HC-ARF. Also in the literature [21], JIA et al. explored the effect of these nodes and the possibility of round tubes becoming elliptical during preparation, and upon comparing the loss and HOMER values of the two PBSs. The results show that there was only a relatively slight difference, suggesting that the deformation of the nested tubes during the actual fabrication process has very little effect on the ultra-wideband and single-mode characteristics.

Second, considering that a parallel glass tube with a U-shaped structure on both sides of the center may be deformed due to its difficulty in maintaining the ideal shape, in the literature [34], Zheng et al. discuss the effect of the rotation angle θ of a parallel glass tube with a U-shaped structure on the limiting loss and single-mode characteristics, and the final simulation results show that the tolerance of this U-shaped nested tube for θ is approximately 15°, which is perfectly achievable in today’s fabrication processes. Most of today’s DHC-ARF beam splitter-based devices have been successfully fabricated using stacking and drawing methods. Using existing stacking and drawing [35] and 3D-printing techniques [36], DHC-ARFs with a variety of complex structures such as conjoined tubes [35], nested tubes [37] and elliptical arc clad tubes [38] can be successfully fabricated. The post-processing of our proposed structure can also be performed after fabrication. In summary, we believe that our proposed structure can be fabricated within tolerance using existing fabrication processes to achieve DHC-ARF-based fiber polarization beam splitting.

5 Conclusion

Based on the previous discussion, an ultra-wideband hollow-core coupler for dual hollow-core micro-structured fibers is proposed in this paper. This paper carefully analyzes the effects of each performance parameter of the polarization beam splitter on x-pol and y-pol polarization coupling lengths, coupling ratios and single-mode performance. The results indicate that the ER value of our designed device remains less than − 20 dB in the wavelength range of 1.43–1.94 μm, which indicates that the designed DHC-ARF PBS has an ultra-wideband of up to 510 nm width. At the same time, HOMER reaches 2008 at 1.42 μm, and it is greater than 100 in the wavelength range of 1.3–1.82 μm. It is worth mentioning that the HOMER is greater than 1000 in the wavelength range of 1.3–1.61 μm, which indicates that the involved anti-resonant fiber polarization beam splitter also has excellent single-mode performance. In summary, the proposed ultra-wideband hollow-core coupler with dual hollow-core micro-structured fiber has an ultra-wideband width of 510 nm and excellent single-mode performance, and this is unprecedented in previous studies. The proposed DHC-ARF PBS is believed to have good prospects for application in optical communication systems.