1. Introduction

Of late, the application of few mode fibers has aroused great interest among the researchers. In mode division multiplexing (MDM) system, each mode in few-mode optical fibers acts as a single information transmission channel, as a result of which the communication capacities of few-mode optical fibers improve [1–4]. Therefore, this approach can overcome the limitation of traditional single mode fibers in communication capacity. In addition, few-mode optical fibers can be applied for optical fiber sensing [5,6], dispersion compensation [7], high-power laser transmission [8] and ultra-low bending loss transmission [9,10], etc. Tailoring the modes in few-mode optical fibers, therefore, is important for such applications [11–13].

Mode filters, which can selectively filter modes in few mode fibers, can find important applications in mode multiplexing systems, in the same way the optical filters do in wavelength division multiplexing systems. For example, the launched mode and converted mode are propagated along the same fiber core for mode converters, based on long period fiber gratings [14,15], and filtering the launched mode can effectively improve the cross-talks. Mode filters can also be applied in suppressing cross-talks in add-drop mode multiplexers [16,17].

Previously, mode-selective characteristics of an optical fiber, consisting of a high-index core and a photonic bandgap cladding, surrounded by a high-index outer ring, were investigated [18]. It was found that when the effective index of the core mode becomes large enough to cross the cladding-mode band, strong coupling occurs between the core mode and the cladding mode, consequent to which the core mode suffers from a large confinement loss. One of the key features of this technique is the introduction of loss mechanism of the cladding modes. Previously, a high-index outer-cladding that could destroy the total internal reflection mechanism of the cladding modes is proposed, which causes high leakage loss to the cladding modes. However, the modal loss of this fiber is not very large; therefore, a long segment of optical fiber should be applied to cause a sufficiently high-loss for the specific mode.

The characteristics of surface plasmon (SPP) have aroused great interest in recent years. Plasmonic optical fibers were investigated as sensors [19–22], and polarization and birefringent devices [23–25]. Owing to the high-loss and strong polarization-dependent coupling characteristics of the SPP mode, it is now possible to achieve compact low cross-talk polarization spliters/filters [26].

For this study, gold wires were introduced to enhance the loss of unwanted mode in optical fibers. Besides, a specific core mode was selectively filtered out by coupling with the cladding modes, and the high-loss characteristic is caused by the coupling of the cladding modes with the SPP mode.

2. Numerical investigation

2.1 Effect of SPP on the mode losses of a single high-index rod

By way of preliminary analysis, the effect of SPP on the mode losses of a single high-index rod is studied firstly. The configuration used in this study is shown in Fig. 1. Gold was chosen as the metal, because it causes high loss and can be fabricated by stack-and-draw technique [27] or splicing-based pressure-assisted melt-filling technique [28]. Array of parallel gold nanowires can also be formed by pumping molten gold and silver into the narrow hollow channels of “holey” photonic crystal fibers. The diameter of the gold wire was d_Au, and that of the high-index rod d_rod, the refractive index of the rod was n_rod, the center-to-center distance between the gold wire and the high-index rod was d_cc. The basic structure parameters are as follows: d_rod = 5.425 μm, n_rod = 1.479, n_back = 1.45, d_cc = 6.4μm. The dielectric constants of gold were obtained from a public website [29], whose data is based on a published article [30]. The default operating wavelength was taken as λ = 1550nm, which is the most commonly explored operating wavelength for mode-division multiplexing [31,32], at that wave length, the high-index rod supports the guidance of the LP₀₁ and LP₁₁ modes.

 

Fig. 1 Cross-section of an optical fiber composed of a high-index rod and a gold wire.

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We solve the modes in this article by a full vectorial finite-element method with perfect matched layer boundary conditions [33,34], which provides the mode field distribution and the complex effective indexes of the modes. High-density meshes with quadratic element numbers in the range of 30000~70000 are used. The effective indexes of the modes are the real part of the effective index, whereas the modal losses are analyzed through the calculation of the complex effective indices of the fibers. The relationship of loss $L$and the imaginary part of effective index,$\Im (n_{eff})$, is given by the equation [35] $L=\left(20\right)/\left(\ln (10)\right)\left(2\pi \right)/\left(\lambda \right)\Im (n_{eff})$.The mode field distribution of x-polarized LP₀₁ and LP₁₁ modes coupled with the first-order SPP mode of high-index rod with d_Au = 0.8μm are shown in Fig. 2. The effective indexes of the LP₀₁ mode of the high-index rod was at a large distance from that of the first-order SPP mode, and that explains why they are only weakly coupled, as shown in Fig. 3(a).

 

Fig. 2 Normalized Electric field distributions of the configuration shown in Fig. 1 (d_Au = 0.8μm), with (a) the fundamental core mode, (b) the coupling mode of the LP_11a mode and the SPP mode, (c) the coupling mode of the LP_11b mode and the SPP mode, (d) the fundamental SPP mode, (e) the first- order SPP mode.

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Fig. 3 Dependence of (a) effective index and (b) confinement losses on the diameter of metal wires d_Au.

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Figure 3(b) shows that the loss of LP₀₁ mode increases with increase in the diameter of gold wire, whereas the loss of the LP₁₁ mode increases initially, but decreases with increase in the diameter of gold wire, and this can be explained by Fig. 3(a). The effective index of the fundamental SPP mode is generally very large. Therefore, the index difference between the fundamental SPP mode and the modes of high-index rod is also huge and its effect on the mode losses of the high-index rod is weak. On the other hand, the effective index of the first-order SPP mode is closer to the indexes of LP₀₁ and LP₁₁ modes. Thus, the first-order SPP mode exerts a strong influence on the LP₀₁ and LP₁₁ modes. With increase in the diameter of the gold wire, the index difference between the first-order SPP mode and the LP₀₁ mode decreases, and the mode coupling between them increases leading to a higher loss of the LP₀₁ mode. The index difference between the first-order SPP mode and the LP₁₁ mode continues to drop until the diameter of the gold wire reaches 0.76 μm, when the index matched point is reached. For larger gold wire, the index difference increases with increase in the diameter of the gold wire. Therefore, the coupling of the two modes increases and then declines, resulting in transmission losses showing similar variations, under similar conditions demonstrated recently [27]. The losses of LP₁₁ mode are always large, because the field of LP₁₁ mode expands more strongly to the cladding, and thus the mode coupling between the LP₁₁ mode and the SPP mode becomes easier. There are some differences between the mode losses of the two polarizations, although their major trends are similar, as shown in Fig. 3(b). The loss difference is mainly contributed to the fact that the losses and field distribution of the SPP mode are polarization dependent, which leads to the different overlap between the two coupling modes. Polarization-dependent coupling is a well-known characteristic of SPP modes [19,27]. Peoples have applied such characteristic to design polarization-dependent devices [24,36].

2.2 Mode filtering characteristics of photonic bandgap optical fiber, with gold-wire ring

The proposed mode filter is composed of a high-index core, and periodic arranged high-index rods, surrounded by gold wires. The use of periodic arranged high-index rod is well-known for forming supermodes, whose effective indexes expand along the effective mode index of the single high-index rod. Selective mode filtering is based on the coupling of a core mode with the supermodes of the high-index rods. Gold wire can cause high loss to the guided modes, as explained in Sec. 2.1. For this reason, the gold wire array was introduced along the periphery of the high index rods, which cause high losses to the supermodes.

Figure 4 shows the cross section of the proposed fiber. The following assumptions were made for the analysis: background index of n_clad = 1.45; three rings of high-index rods with a rod index of n_rod = 1.479; period of the high-index rods of Λ = 7.75μm; diameter of the rods of d_rod = 0.7Λ, core diameter of d_core = 12μm, and a high index core with index of n_core = 1.462. In the absence of the high-index rods and gold wires, the step-index core supported the transmission of four modes- the LP₀₁, LP₁₁, LP₂₁, and LP₀₂ modes, at 1550 nm wavelength. The gold wires formed a hexagonal array; the center-to-center distance between the gold wire at the corner and the fiber core was set as Λ₁ = 36μm, and that between the two adjacent gold wires as Λ₂ = 0.2Λ₁.

 

Fig. 4 The cross-section of the proposed mode-filtering optical fiber, which is composed of a high-index fiber core and periodic arranged high-index rods, surrounded by gold wires.

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Figure 5 shows the effective index curves of the core modes and the cladding supermodes. The cladding mode bands 1 and 2 were formed by the mode coupling of LP₀₁ and LP₁₁ modes, respectively, of the high-index rods. For the wavelength range shown in Fig. 5, there are over 40 groups of cladding modes with different effective indexes in cladding mode band 2, which can be explained as due to the splitting of LP₁₁ mode of a single rod into a large number of supermodes by bringing a large number of identical rods together. From Fig. 5 it can be seen that, over the wavelength range of 1510~1600 nm, the LP₁₁ core mode lies in the cladding-mode band, LP₀₁ core mode in the bandgap region 1, and LP₂₁ and LP₀₂ core modes in the bandgap region 2.

 

Fig. 5 Dispersion characteristics of the cladding modes and the core modes for the optical fiber shown in Fig. 4.

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Figure 6 shows the mode field distributions of the fiber after setting the diameter of gold wires as d_Au = 1μm. As LP₁₁ mode fell into the cladding-mode band, it coupled strongly with the cladding mode, as shown in Fig. 6(b). On the other hand, LP₀₁, LP₂₁, and LP₀₂ modes were confined mainly to the core, as shown in Fig. 6. The situation is similar to that of a confined mode in a conventional all-solid bandgap fiber.

 

Fig. 6 Normalized Electric field distributions of (a) LP₀₁ mode, (b) LP₁₁ mode, (c) LP₂₁ mode and (b) LP₀₂ mode of the proposed fiber at wavelength 1550 nm.

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Figure 7 shows the relationship between the confinement losses of the core modes and the diameter of gold wire d_Au. LP₁₁ and LP₂₁ core modes are four degenerate modes, which means that two modes of the same group exist for each polarization state. The degenerate modes should have similar confinement losses, depending on the symmetry of the structure. For this reason, the characteristics of only one of the degenerate modes had to be analyzed. The results of the x-polarized state are provided first, for simplicity. The loss of LP₁₁ core mode reaches the maximum at d_Au = 0.85μm, which is almost equal to the index matched point of the SPP mode and the cladding mode. The effective index of LP₀₁ core mode is higher than that of supermodes’ modes in cladding-mode band 2; therefore, the coupling of LP₀₁ core mode with the supermodes is difficult, as a result of which the confinement loss is small. Because LP₂₁ and LP₀₂ modes of the core have lower effective indexes, and are only confined by the photonic bandgap effect, their mode field distributions expanded more deeply into the cladding, as shown in Fig. 6. Therefore, they couple with the cladding supermodes more easily, leading to higher mode losses. For the gold wire with diameter in the range of 0.8-1.2 μm, the loss of LP₁₁ mode is larger than 5dB/cm and that of the other modes less than 0.01dB/cm (see Fig. 7). This implies that the fiber has high tolerance for the diameter of gold wires. In other words, high loss can be achieved for the supermodes even when the difference between the effective indexes of the supermodes and the SPP mode is large. Therefore, by propagating along a segment of 4 cm-long optical fiber, the power of LP₁₁ mode can be reduced by more than 20 dB, and that of the other modes by less than 0.04 dB. Therefore, this configuration can effectively filter out the LP₁₁ mode, while simultaneously keeping the transmission losses low for other modes.

 

Fig. 7 Confinement losses of the core modes as a function of the diameter of metal wires d_Au.

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Although the loss of LP₁₁ mode is maximum for d_Au = 0.85μm, the losses of the LP₂₁ and LP₀₂ modes of the fiber are also high at that position. Therefore, the diameter of the gold wire was set to d_Au = 1μm, where the loss of LP₁₁ mode was still high, but the losses of other modes were lower. Then, the influence of center-to-center distance between the gold wires and the fiber core Λ₁ on mode losses was investigated and the results are shown in Fig. 8. From this figure it can be seen that the losses of core modes increase with reduction in Λ₁. The confinement loss ofLP₁₁ mode increases rapidly with reduction of Λ₁ and finally becomes saturated, as Λ₁ is small enough. The main reason for this is the twin effect of shrinking distance on the loss of LP₁₁ mode. On the one hand, the decrease of Λ₁ value leads to enhanced interaction between the cladding mode and the SPP mode, so as to strengthen the mode coupling; on the other hand, it also affects the effective index of the SPP mode, because, as the refractive index of the surrounding gold wires increases, the effective index of the SPP mode also will increase. Therefore, the index difference between LP₁₁ and SPP modes will increase as well, thereby weakening the mode coupling. A large Λ₁ value can be chosen to ensure a large enough loss ratio of LP₁₁ mode with other core modes, because the losses of other core modes increase with decrease of Λ₁.

 

Fig. 8 Dependence of confinement loss of the core modes on the center-to-center distance between the gold wires and the fiber core Λ₁.

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Based on the above investigation, the optimized configuration parameters were set as d_Au = 1 μm, and Λ₁ = 35.6 μm. Figure 9 shows that the mode losses depend on the wavelength. The loss of LP₁₁ mode at 1550 nm wavelength is 8.6 dB/cm and that of other modes is lower than 0.005 dB/cm. The loss of LP₂₁ and LP₀₂ modes increases rapidly with increase in the wavelength. As a result, in the long wavelength region, the effective indexes of the two modes become closer to the cladding mode band, and the mode fields extend more strongly to the cladding region, enhancing the mode coupling. For the wavelength range of 1534-1570 nm, as shown in Fig. 9, the loss of LP₁₁ core mode is larger than 4 dB/cm and the leakage loss of other modes is less than 0.01dB/cm. This means that the loss ratio of LP₁₁ and several other modes is larger than 400. Consequently, a segment of the proposed 5 cm long fiber effectively filters out the LP₁₁ core mode. To achieve wider operating bandwidth, the losses of LP₂₁ and LP₀₂ modes can be increased further by increasing the ring number of high-index rods [18].

 

Fig. 9 Dependence of the losses of the core modes on wavelength λ.

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3. Conclusion

The coupling relationship between the SPP mode formed by a metal wire and a high-index rod, as well as its influence on the loss of guided mode, was investigated. The SPP mode was found to have a strong effect on the mode transmission loss of the guided modes, even when the index difference between the SPP mode and the guided mode is large. This loss mechanism was applied to the implementation of selective filtering of optical fiber mode. The results show that the insertion of gold wire greatly enhances the loss of the cladding modes in the fiber. The energy of the core mode exchanges with that of the SPP mode of the gold wire, when the core mode is coupled to the cladding modes, which causes strong attenuation of the core mode. Selective filtering of LP₁₁ mode of the core has been demonstrated numerically in this paper. Following a similar procedure, the other fiber core modes also can be filtered. A segment of the proposed 5 cm long fiber effectively filters out the LP₁₁ mode in the wavelength range of 1534-1570 nm, because of the high-loss feature of gold wires, which covers the entire C band. The proposed fiber can find application in mode-sensitive systems to manipulate the mode selectively.