Design of hollow core step-index antiresonant fiber with stepped refractive indices cladding

Abstract

With the benefits of low latency, wide transmission bandwidth, and large mode field area, hollow-core antiresonant fiber (HC-ARF) has been a research hotspot in the past decade. In this paper, a hollow core step-index antiresonant fiber (HC-SARF), with stepped refractive indices cladding, is proposed and numerically demonstrated with the benefits of loss reduction and bending improvement. Glass-based capillaries with both high (n = 1.45) and low (as low as n = 1.36) refractive indices layers are introduced and formatted in the cladding air holes. Using the finite element method to perform numerical analysis of the designed fiber, results show that at the laser wavelengths of 980 and 1064 nm, the confinement loss is favorably reduced by about 6 dB/km compared with the conventional uniform cladding HC-ARF. The bending loss, around 15 cm bending radius of this fiber, is also reduced by 2 dB/km. The cladding air hole radius in this fiber is further investigated to optimize the confinement loss and the mode field diameter with single-mode transmission behavior. This proposed HC-SARF has great potential in optical fiber transmission and high energy delivery.

Key words： antiresonant fiber (ARF); stepped refractive indices; confinement loss; bending loss; laser pumping

1 Introduction

Since the transmission of light in solid-core fibers is limited by the loss and nonlinearity of the core material [1], hollow-core fibers have received extensive attention. According to different wave-guiding principles, hollow core microstructure fibers can basically be divided into several categories: hollow core Bragg fiber, hollow core photonic band gap fiber, and hollow core antiresonant fiber (HC-ARF) [2–5]. The HC-ARF has relatively simple geometry. It also currently has two acknowledged light guiding mechanisms found in research: the principle of antiresonant reflection [6] and the mode coupling theory between the cladding and core [7], for example. HC-ARF has some transmission properties that are not inferior to those of the photonic bandgap fiber. Also, its geometry indicates great simplification of the fabrication process. In addition, it has great prospects in practical applications. Researchers are mainly working on the low confinement loss HC-ARF, with birefringence for sensing, or large mode field area for high-power delivery [8–19].

The primary HC-ARF is known as the holey photonic crystal fiber with Kagome cladding reported by Benabid et al. [8]. Because of its advantages, such as wide bandwidth [9], it has gained strong worldwide interest and has gradually developed into two types: the hypocycloid and negative curvature. The hypocycloid Kagome HC-ARF has been continuously modified and improved to reduce transmission loss and to obtain a loss below 10 dB/km [10,11]. As for the negative curvature HC-ARF, it is mainly developed to achieve mid-to-far infrared light propagation and to reach a loss of tens dB/km levels [12,13]. Meanwhile, these structures came across some challenges to further reduce the overall loss due to Fano resonance [14]. To overcome the loss limitation, a nodeless HC-ARF has been proposed and demonstrated. The transmission loss of the nodeless HC-ARF, in some wavelength bands, could reach several decibels per km [15]. It is also proved that the nodeless HC-ARF, with 6–8 cladding tubes, performs superior for low-loss transmission [7].

Recently, based on the nodeless HC-ARF, the multilayer and multiring antiresonance structure has been proposed to further reduce the transmission confinement loss. Hasan et al. reported a negative curvature HC-ARF with nested elliptical elements in the cladding tubes, showing a confinement loss below 0.1 dB/km between 1000 and 1500 nm [16]. Gao et al. reported a conjoined-tube HC-ARF with about 2 dB/km confinement loss at 1512 nm [17]. Sakr et al. presented a nested nodeless HC-ARF with 700 nm wavelength bandwidth, and 6.6 dB/km confinement loss at 1550 nm [18]. To be noted, the above-mentioned HC-ARF literature focused on the geometry modification and innovation, with merely single uniform silica glass material (n≈ 1.45).

In this paper, a hollow core step-index antiresonant fiber (HC-SARF) is proposed and numerically demonstrated. Varied from the conventional uniform index HC-ARF, high (n₁ = 1.45) and low (as low as n₂ = 1.36) refractive indices layers are combined into the tube wall of the cladding air hole. Using the finite element method (FEM) to perform numerical analysis of the designed fiber, the confinement loss is favorably reduced by about 6 dB/km at the pumping laser and ultrafast laser wavelengths (980 and 1064 nm), when compared with conventional structures. The bending loss, around 15 cm bending radius, is also reduced by 2 dB/km. The cladding air hole radius in this HC-SARF is further investigated to optimize the confinement loss and the mode field diameter with single-mode transmission characteristics of the designed fiber. By changing the lower refractive index n₂ of the air tube wall and the cladding air hole radius r, the light propagation performance is further investigated and optimized. This work theoretically investigates the influence of the refractive index, the thickness of the second material, as well as the cladding tube size on fiber properties and the comparison with the conventional single-layer fiber for the loss and bending improvement. This designed holey fiber has great potential in low-loss transmission and high-power delivery for ultrafast lasers and pumping lasers.

2 Principles and methods

Litchinitser et al. proposed an antiresonant reflecting optical waveguide (ARROW) model for low-index core optical waveguides [6]. Based on the low-loss seven tubes nodeless HC-ARF [19], the nodeless tubular fiber (comprising a single ring of seven tubes) is primarily employed in the design. Innovatively, we introduced the stepped layers with high and low refractive indices in the cladding tube wall of the conventional structure.

As for the light guiding principle of the conventional single-layer cladding tube, Litchinitser et al. have given a theoretical explanation as follows [6]:

(1)\mathrm{2}nt\sqrt{n^{\mathrm{2}}-\mathrm{1}}={\displaystyle \frac{m}{\mathrm{2}}}{{\displaystyle \lambda }}_m,\hspace{0.17em}m=\mathrm{1},\mathrm{2},\mathrm{3},...

(2){{\displaystyle \lambda }}_m={\displaystyle \frac{m}{\mathrm{4}}}nt\sqrt{n^{\mathrm{2}}-\mathrm{1}},\hspace{0.17em}m=\mathrm{1},\mathrm{2},\mathrm{3},...

where n and t are the refractive index and thickness of the cladding tube, respectively. If m is odd number, the light with corresponding l_m will be reflected at the cladding interface into the core. This will result in a high transmission coefficient in the core, as the guided light. If m is even number, the light with corresponding l_m will leak into the cladding with a low transmission coefficient in the core.

For the step indices layer tube, Belardi et al. also proposed a theoretical analysis method by equivalently considering a two-layer step refractive index tube, as a single-layer tube with refractive index n_eq, and thickness t_eq as follows [20]:

(3){{\displaystyle n}}_{\mathrm{e}\mathrm{q}}={{\displaystyle n}}_{\mathrm{1}}{\displaystyle \frac{{{\displaystyle S}}_{\mathrm{1}}}{{{\displaystyle S}}_{\mathrm{1}}+{{\displaystyle S}}_{\mathrm{2}}}}+{{\displaystyle n}}_{\mathrm{2}}{\displaystyle \frac{{{\displaystyle S}}_{\mathrm{2}}}{{{\displaystyle S}}_{\mathrm{1}}+{{\displaystyle S}}_{\mathrm{2}}}},

(4){{\displaystyle t}}_{\mathrm{e}\mathrm{q}}={{\displaystyle t}}_{\mathrm{1}}+{{\displaystyle t}}_{\mathrm{2}},

where n₁ and n₂ are the refractive index of the two layers, t₁ and t₂ are the thickness of the two layers, and S₁ and S₂ are the surfaces occupied by the two layers.

Fig.1 Longitudinal structure schematic of the stepped refractive index HC-ARF based on the ARROW model

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Figure 1 shows the principle of longitudinal structure schematic of the designed HC-SARF, based on the ARROW model. The high (n₁) and low (n₂) refractive index layers of the cladding, illustrated as gray areas in Fig. 1, can be considered an Fabry–Pérot (FP) resonator. If the light with wavelength l₁ in Fig. 1 could satisfy the resonant condition, the light will leak into the cladding with a low transmission coefficient in the core. If the light with wavelength l₂ varies from the resonant condition, the light will be reflected at the cladding interface into the core. This will result in a high transmission coefficient in the core as the guided light.

Figure 2 shows the cross-sectional geometry of the proposed HC-SARF. The core is filled with air with a central air hole radius of R = 20.1 µm. The cladding consists of seven identical non-contact air tubes with an outer radius of r = 12 µm. The air tube wall is composed of silica and similar glass material, and it is divided into two layers, both with a thickness of t = 0.18 µm. The outer layer has a higher refractive index layer with n₁ = 1.45, while the inner layer has a lower refractive index of n₂ = 1.36–1.44. The available refractive index value of glass materials can be found in some related research and literatures [21]. With these parameters, the FEM is utilized to perform numerical analysis of the designed fiber. The confinement loss, the bending loss, as well as the single-mode transmission characteristics, are analyzed and discussed.

Fig.2 Cross section of HC-SARF with stepped high and low refractive indices cladding. (a) Overall transverse geometry. (b) Enlarged stepped indices area

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3 Simulation and discussion

The fundamental mode of the designed HC-SARF is initially simulated. First, the low refractive index region, n₂ of the cladding air hole tube wall, is changed from 1.45, 1.43, and 1.41 to 1.36, while the n₁ is set to 1.45. It becomes a conventional HC-ARF when n₂ = 1.45. Secondly, the cladding air hole radius r is analyzed to further discuss and optimize the fiber characteristics. Figure 3 shows the circular-symmetric Gaussian shape fundamental mode field distribution of the designed HC-ARF, which indicates a promising light confinement and guidance in the air core.

3.1 Confinement loss with refractive index contrast

Fig.3 Fundamental mode field distribution of the designed HC-SARF with stepped refractive indices cladding. (a) Two-dimensional image. (b) Three-dimensional profile

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Fig.4 Confinement loss with respect to wavelength for the designed HC-SARF with different indices contrast

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To obtain improved transmission characteristics beyond the conventional HC-ARF, we simulated and analyzed the confinement loss and the bending loss of the HC-SARF. This process was carried out by varying the refractive index n₂ of the inner layer of the cladding air tube wall. Figure 4 shows the fundamental mode confinement loss of the designed HC-SARF from 800 to 2000 nm, while changing inner layer refractive index n₂. It is apparent that the confinement loss decreases with lower n₂ between 800 and 1100 nm. Especially for the pumping lasers and the ultrafast laser wavelengths of 980 and 1064 nm, the confinement loss is significantly reduced. Therefore, it is concluded that the designed HC-ARF, with stepped refractive indices cladding, can achieve minor confinement loss.

3.2 Bending loss with refractive index contrast

Due to coupling between cladding modes and core modes, the conventional HC-ARF usually comes across with serious bending issues [22]. It is acknowledged that there is trade-off between bending loss and confinement loss of the HC-ARF. To reduce the confinement loss, the core diameter of the HC-ARF is generally ten times the wavelength value. This leads to the sharply increased bending loss, once the bending radius is below 10 cm. This effect will seriously restrict the practical applications of the HC-ARF. The bending loss (BL) in holey fiber could be derived as [23]

(5){{\displaystyle n}}_{\mathrm{e}\mathrm{q}}\left(x,y\right)=n\left(x,y\right){\mathrm{e}}^{\frac{x}{{{\displaystyle R}}_{\mathrm{b}}}},

(6)\mathrm{B}\mathrm{L}={\displaystyle \frac{\mathrm{20}}{\ln \mathrm{10}}}\cdot {\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{\lambda }}\mathrm{Im}\left({{\displaystyle n}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\right)=\mathrm{8.686}\times {\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{\lambda }}\mathrm{Im}\left({{\displaystyle n}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\right),

where n_eq(x,y) is the equivalent cross-sectional refractive index distribution of the bending HC-SARF, n(x,y) is the equivalent cross-sectional refractive index of straight HC-SARF, x is coordinate of the bending direction, R_b is bending radius, and n_eff is the effective refractive index of the fundamental mode.

Based on Eqs. (5) and (6), the bending loss of the proposed HC-SARF is simulated and analyzed with respect to the bending radius. Figure 5 shows the shifted fundamental mode field distribution of the bending HC-SARF, and the bending orientation is along the positive direction of x-axis. Figure 6 presents the bending loss of the designed HC-SARF with different n₂ at the operating wavelength of 1064 nm, where it exhibits the minimum confinement loss in Fig. 4. As shown in Fig. 6(a), the designed HC-SARF has lower bending loss with the decreased n₂. When n₂ = 1.36, the bending loss is reduced by about 2 dB for the bending radius between 14 and 20 cm, as shown in Fig. 6(b). Therefore, it could be concluded that the designed high and low stepped refractive indices cladding has a strong contribution toward reducing the bending loss of the HC-ARF.

Fig.5 Fundamental mode field distribution of the bending HC-SARF in the x-axis orientation

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Fig.6 Bending loss with respect to bending radius for the designed HC-SARF. (a) Between 8 and 20 cm. (b) Enlarged section between 14 and 20 cm

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3.3 Confinement loss with cladding air hole radius

The confinement loss of this HC-SARF could also be modified and optimized with the cladding air hole radius, if in the setting of the lower refractive index, n₂ = 1.43, and in the central air core radius, R = 20.1 µm. Figure 7 illustrates the relationship between the fundamental mode confinement loss and the cladding air hole radius. The confinement loss first decreases and then rises with the increased r and reaches the minimum value at r = 12 µm. Hence, it is recommended to have the air hole radius equivalent to around 12 µm in this design.

Fig.7 Fundamental mode confinement loss with respect to cladding air hole radius r for the designed HC-SARF

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3.4 Single-mode transmission performance related to cladding air hole radius

While maintaining the lower refractive index n₂ = 1.43 and the central air core radius R = 20.1 µm, by changing the radius r of the cladding air hole, the mode coupling theory is employed [7] to investigate the influence of the cladding air hole radius on the single mode maintaining and fundamental mode field diameter of the designed HC-SARF. Figure 8(a) depicts the effective refractive index n_eff of LP₀₁ and LP₁₁ mode in the core, as well as the LP₀₁ mode in the cladding from 800 to 2000 nm, and the cladding air hole radius r = 11, 12, and 13 cm. Considering the data in Fig. 8(a), the n_eff of LP_11-core (in red color) and LP_01-cladding (in blue color) is adjacent, which indicates that the higher-order mode LP_11-core in the core is more likely to be coupled into the cladding mode LP_01-cladding. Therefore, the fundamental core mode LP_01-core could be well maintained. Figure 8(b) shows the detailed n_eff difference between LP_11-core and LP_01-cladding, with respect to the cladding air hole radius. As the cladding air hole radius r increases from 10 to 13 cm, the n_eff contrast decreases, allowing LP_11-core to be easily coupled with the LP_01-cladding. This results in maintaining transmission of the single mode.

Considering the 1064-nm light in laser pumping applications, the high-power density delivery capability is essential for this HC-SARF implementation. Here, the mode field diameter (MFD) is also analyzed and discussed for the designed HC-SARF. Figure 9 shows the relationship between the LP_01-core MFD and the cladding air hole radius. The MFD remains around a level of 30 µm, and it slightly decreases with the increased r.

Fig.8 Mode effective refractive index n_eff of the core and cladding mode. (a) Relationship with wavelengths for r = 11, 12, and 13 cm. (b) Detailed n_eff contrast between LP_11-core and LP_01-cladding

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Fig.9 Fundamental mode field diameter with respect to cladding air hole radius for the designed HC-SARF

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Originated from the conventional stacking and drawing process for photonic crystal fibers (PCFs), current fabrication techniques for the HC-ARF can achieve the single-layer tube thickness of 200 to 300 nm and a low confinement loss [17–20]. Moreover, the fluoride-doped glass material has shown the decreased refractive index to be below 1.36 [21]. Hence, it is possible to develop fiber fabrication methods applied to the proposed design, by introducing material deposition techniques and stacking and drawing the PCF fabrication process.

4 Conclusions

A unique HC-SARF design, with stepped refractive indices cladding, has been proposed and numerically analyzed. This work theoretically investigates the influence of the refractive index and the thickness of the step-index material, as well as the cladding tube size on fiber properties and the comparison with the conventional single-layer HC-ARF. With a dedicated and optimized design of the fiber geometry, it shows the advanced light transmission characteristics, such as reduced confinement loss and bending loss. The confinement loss is reduced by about 6 dB/km compared with the conventional HC-ARF at the 980 and 1064 nm wavelength for pumping lasers and ultrafast lasers. The bending loss (around 14 cm bending radius of this fiber) is also reduced by 2 dB/km. Since the design concept is universal for the antiresonant structure, it can be directly applied to ultra-low loss HC-ARF and for high-power delivery applications, showing great potential in the fields of optical communication and laser delivery.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 62075074), and the National Key R&D Program of China (Nos. 2018YFF01011800 and 2018YFB2201901).

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