## Abstract

In this paper, we present a mode-selective coupler based on a dual-core all-solid photonic bandgap fiber (AS-PBGF). Because they are all-solid, AS-PBGF-based mode converters are easier to splice to other fibers than those based on air-hole photonic crystal fibers. Mode conversions between the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes, ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{21}$ modes, and ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{02}$ modes are obtained at the wavelength $\lambda =1550\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. The 3 dB wavelength bandwidth of these mode converters are 47.8, 20.3, and 20.3 nm, respectively.

© 2015 Chinese Laser Press

## 1. INTRODUCTION

Mode conversion is a crucial part of mode division multiplexing (MDM) optical fiber communication systems [1]. In MDM systems, we should first convert the fundamental fiber modes to higher-order modes, and then couple these higher-order modes into few-mode fibers (FMFs) as independent transmission channels. Two kinds of methods have been proposed to realize the issue of mode conversion. One method is to use the free-space optical system [2,3]; the other is based on waveguide structure [4 –8]. Compared to the method of free-space optics, the solutions based on waveguide structure have lower loss and higher level of integration, and are easier to conform to the all-fiber communication system. In fact, fiber-based couplers were demonstrated as an efficient mode converter many years ago [6]. Recently, in the context of MDM, they have attracted attention again [7,8].

Photonic crystal fiber (PCF) is a good choice of waveguide structure to solve the issue of mode conversion. In 2007, Lai *et al.* demonstrated two all-fiber mode converters based on PCF transitions [9]. Both converters rely on adiabatic propagation, and mode conversions can be achieved across a wide wavelength range. Chen and Zhou proposed another mode converter realized by mode coupling between a large core and a small core in an asymmetric dual-core air-hole PCF with a bandwidth of 14 nm [10]. They solved the issue of mode conversion very well, but the structure of an air-hole PCF will inevitably lead to air-hole collapse when welding with few-mode fibers.

In this paper, a new type of fiber coupler based on an all-solid photonic bandgap fiber (AS-PBGF) is proposed. AS-PBGF is another kind of PCF, in which an array of isolated high-index rods are inset in a low-index background and the core region is a defect introduced by removing several high-index rods [11 –14]. Compared with air-hole PCF and hollow-core PBGF, this fiber is easier to fabricate and splice to other fibers [12]. Furthermore, the bend loss of AS-PBGF would be greatly improved by introducing an index-depressed layer around the high-index rods in the fiber cladding [13].

In this work, we realize efficient mode conversion in an AS-PBGF by selective mode coupling between two fiber cores. Mode conversions between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes, ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{21}$ modes, and ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{02}$ modes are realized at the wavelength $\lambda =1550\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ by satisfying the phase-matching condition. The 3 dB wavelength bandwidths of these mode converters are 47.8 and 20.3 nm, respectively.

## 2. PRINCIPLE OF THE MODE CONVERTER

The AS-PBGF-based mode coupler is illustrated in Fig. 1. In the structure of Fig. 1, there are two large cores of the same diameter and different refractive index. The refractive index of the left core is equal to the refractive index of the background (${n}_{0}=1.45$). The refractive index of the right core is denoted by ${n}_{r}$. The left core supports four modes (${\mathrm{LP}}_{01}$, ${\mathrm{LP}}_{11}$, ${\mathrm{LP}}_{21}$, and ${\mathrm{LP}}_{02}$) at the particular wavelength of 1550 nm, which is the window used in optical fiber communication. The high-index rods around the cores are designed to be distributed circularly. This design is possible in reality and makes the mode field closer to that of conventional FMF. In this design, the background index (pure silica) of the PBGF displayed in gray and the high-index rods displayed in black are assumed to be 1.45 and 1.7, respectively. Recently, many methods of fabricating AS-PBGF have been reported [11,12]. We can realize the structure by a special stack and draw technique. All of these refractive indices can be achieved by doping different materials in silica. The period ($\mathrm{\Lambda}$) of the PCF structure is 4 μm, and the high-index rods’ filling fraction in the transverse section is $\text{fill}=\frac{\pi {d}^{2}}{2\sqrt{3}{\mathrm{\Lambda}}^{2}}$, which is set as 0.15. The diameter of high-index rods $d$ can be calculated by this equation. The diameter of the core is $\text{width}=2\sqrt{3}\mathrm{\Lambda}$.

For the purpose of converting the ${\mathrm{LP}}_{01}$ mode in the right core to ${\mathrm{LP}}_{11}$ mode (${\mathrm{LP}}_{21}$ and ${\mathrm{LP}}_{02}$ modes in the left core), we need to find the phase-matching condition between these two modes (${\beta}_{01}={\beta}_{mn}$) [15]. As we know that $\beta ={n}_{\text{eff}}k$, where ${n}_{\text{eff}}$ is the effective index of fiber mode which can be calculated by the finite-difference beam propagation method (BPM), $k=2\pi /\lambda $ is the wave vector in the vacuum.

Next, we will find the phase-matching condition between the fundamental mode and the higher-order modes. In Fig. 2, we present the relationship between the ${n}_{\text{eff}}$ of the ${\mathrm{LP}}_{01}$ mode in the right core and the refractive index of the right core (${n}_{r}$), at the wavelength of 1550 nm; the ${n}_{\text{eff}}$ of the higher-order modes in the left core is presented too. They are constant values and correspond to a straight line, respectively. They intersect with the curve of the ${\mathrm{LP}}_{01}$ mode at point A (${n}_{r}={n}_{1}$), point B (${n}_{r}={n}_{2}$), and point C (${n}_{r}={n}_{3}$). Refractive index ${n}_{1}$ is 0.22% lower than ${n}_{0}$, refractive index ${n}_{2}$ is 0.51% lower than ${n}_{0}$, and refractive index ${n}_{3}$ is 0.60% lower than ${n}_{0}$. Finally, we find the phase-matching points (A, B, and C) between the fundamental and higher-order modes.

As we know, PBGF characterizes photonic band gap. In Fig. 3, we present the gap structure of this design and illustrate modal dispersion curves for the fundamental mode in the right core and the higher-order mode in the left core. These modal dispersion curves exist in the fourth gap of this design. As we can see from Fig. 3, phase-matching points between the ${\mathrm{LP}}_{01}$ mode and higher-order modes occur around the 1550 nm wavelength.

## 3. SIMULATION RESULTS

Figure 4 is the schematic drawing of three cascade mode-selective couplers. Coupler 1 realizes mode conversion between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$. Coupler 2 realizes mode conversion between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{21}$. Coupler 3 realizes mode conversion between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{02}$. Refractive indices ${n}_{1}$, ${n}_{2}$, and ${n}_{3}$ correspond to points A, B, and C in Fig. 2. Figure 5 shows the curves of coupling efficiency between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$, ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{21}$, and ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{02}$ along with the propagation direction. According to Fig. 5, the beat length of Couplers 1–3 is ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$, respectively. The beat length is not only related to the structure of the coupler and mode fields, but also to the wavelength and the distance between cores [16]. The mode-coupling losses in Couplers 1, 2, and 3 are 0.013, 0.044, and 0.106 dB, respectively, at the beat length.

To test the mode-conversion performance of those couplers, we perform numerical simulations using BPM. Figure 6 presents mode field at different positions along the propagation direction, where Fig. 6(a) refers to Coupler 1, Fig. 6(b) to Coupler 2, and Fig. 6(c) to Coupler 3. Fundamental modes initially launched in the right core are gradually converted to higher-order modes in the left core. ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$ correspond to the beat length given in Fig. 5. As we can see from Fig. 6, Couplers 1–3 can realize efficient mode conversion. In addition, the converted mode fields of AS-PBGF are similar to the corresponding mode fields of FMF [9].

Figure 7 shows the wavelength bandwidth properties of Couplers 1–3. As we can see, the bandwidths of these mode couplers are ${B}_{1}=47.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, ${B}_{2}=20.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ around 1550 nm. Obviously, the bandwidth of Coupler 1 is wider than that of Coupler 2 and the bandwidth of Coupler 2 is similar to that of Coupler 3; the reason is that coupling efficiency is always related to the ${n}_{\text{eff}}$ mismatch between the mode in the right core and the mode in the left core. As we can see from Fig. 3, the dispersion curve of the higher-order modes is steeper than that of the lower-order mode, so the ${n}_{\text{eff}}$ mismatch between the ${\mathrm{LP}}_{02}$ and ${\mathrm{LP}}_{01}$ modes is faster than that between the ${\mathrm{LP}}_{11}$ and ${\mathrm{LP}}_{01}$ modes when the wavelength deviates from 1550 nm.

## 4. MODE (DE)MULTIPLEXER

In this section, a parallel mode multiplexer based on three-core AS-PBGF is introduced. The schematic drawing and cross-section of this coupler is presented in Fig. 8. As shown in Fig. 8, unlike cascading-style mode multiplexers, this multiplexer simultaneously couples two fiber modes into the few-mode core. The parameters (including background index, fill proportion, core width, and period) of this structure are equal to those of the dual-core coupler which is introduced in Section 3.

As we can see from Fig. 9, the power is not totally converted to higher-order modes. Actually, in this simulation, the power coupled to the ${\mathrm{LP}}_{11}$ and ${\mathrm{LP}}_{02}$ modes is a little more than 90%; the reason is that the length of the device ${L}_{c}=\frac{{L}_{1}+{L}_{3}}{2}$ is unequal to the beat lengths ${L}_{1}$ or ${L}_{3}$. The wavelength bandwidth of this multiplexer is 44.8 nm for converting ${\mathrm{LP}}_{01}$ mode to ${\mathrm{LP}}_{11}$ mode and 15.1 nm for converting ${\mathrm{LP}}_{01}$ mode to ${\mathrm{LP}}_{02}$ mode, and this bandwidth is smaller than that of the cascading-style multiplexer.

## 5. CONCLUSION

In this paper, a novel mode converter based on mode coupling in a dual-core AS-PBGF is proposed. Through selectively changing the core refractive index, the phase-matching conditions are guaranteed between two modes. Coupling loss is lower than 0.306 dB. The wavelength bandwidth of this mode converter is higher than 20.3 nm, better than that based on conventional air-hole PCF. An extension to three-core mode coupler in a parallel structure has been proposed and discussed. This three-core coupler can simultaneously combine the processes of mode conversion and mode multiplexing. Generally, this kind of mode converter has potential applications in future MDM optical fiber communication systems.

## ACKNOWLEDGMENTS

This work is supported in part by the National Basic Research Program of China (973 Program) under Grant 2014CB340102, in part by the National Natural Science Foundation under Grants 61271191 and 61271193, and in part by the Fund of State Key Laboratory of Information Photonics and Optical Communications.

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