1 Introduction

In the past two decades, internet traffic demand was mainly met by wavelength-division multiplexing (WDM) technology. Similar to the multiple-input-multiple-output (MIMO) architecture for wireless communication, space is the degree of freedom that is being considered for optical fiber communications [1, 2]. Space-division multiplexing (SDM), based on mode-division multiplexing (MDM) or multicore fibers (MCFs) has attracted much attention recently. For the MDM system, each mode in the few-mode optical fibers can be applied as a signal transmission channel, which greatly expands the capacity of optical fibers.

Mode conversion and multiplexing/demultiplexing are important for the mode division multiplexing systems. Various techniques have been proposed for the combination or separation of the modes in few-mode optical fibers, which includes the phase plates or spatial light modulators with free-space optics [3,4,5,6,7,8], fiber or waveguide coupling devices [6, 9,10,11,12,13,14], volume holographic approaches [15], optical filtering of spatial samples taken from a multi-mode fiber aperture [16], silicon photonics [17,18,19], and photonic lantern made by adiabatically merging several single-mode cores into one multimode core [20]. Tapered velocity mode-selective couplers based on the adiabatic condition have been also proposed for ultra-wideband mode-division multiplexing of few-mode waveguides, the mode-selective functionality is demonstrated in slab geometry, with the principles generalizing to planar or fiber-based devices [21]. As a basic device for mode division multiplexing application, mode converters with variant configurations have been proposed [10, 22,23,24,25,26,27,28]. In particular, a mode-selective spatial multiplexer/demultiplexer based on a multi-plane light converter has been proposed [29], which is fully compatible with a wavelength and space-division multiplexed optical transmission line. Generally, mode converters are applied for conversion between the fundamental mode and a higher-order mode.

Few-mode waveguides based on asymmetric Y Junctions are also promising for mode multiplexing/demultiplexing applications [30,31,32,33,34,35]. The mode conversion characteristics are based on the principle of modal effective index matching, therefore, generally wide bandwidth can be achieved. Especially, cascaded asymmetric Y-junctions has been proposed for 1 × 4 mode multiplexing/demultiplexing [33]. According to the principle of modal effective index matching, the fundamental mode of a narrow arm can adiabatically excite the higher-order mode of its stem in the multiplexing case, and a high-order mode in the stem can be adiabatically separated from other lower-order modes and exit as the fundamental mode of the narrow arm in the demultiplexing case [33]. It is well known that the classification of the higher-order modes of the optical fibers and the rectangular waveguides are different. It would be more directly significant if mode multiplexing/demultiplexing is based on few-mode optical fibers. In this paper, we proposed the design of a few-mode optical device based on optical fibers, mode conversion can be achieved between the fundamental mode and the higher-order mode with wide bandwidth. In addition, the proposed device can be cascaded as a wide-bandwidth mode multiplexer/demultiplexer. It is found that the modes in the main core can be converted to the other modes owing to the existence of the assistant core, and the bandwidth of the configuration is expanded by applying a specially designed configuration.

2 Fiber configuration and fundamental mode to high-order mode conversion

Figure 1 shows a basic configuration of the proposed optical device. An assistant core is applied to tailor the mode characteristics of the main core. One end of the assistant core is attached to the side of the main core. The main core can support the guidance of high-order modes, whereas the assistant core is generally single-mode guided. The center axes of the two cores are located in a plane and form a small angle θ. Light can, therefore, be launched from the assistant core or the two ends of the main core. The refractive indexes of the main core n m and the cladding n c are setting as n m = 1.455, n c = 1.45, respectively. The diameter of the main core is d m = 20 μm, which ensures the guidance of the LP01, LP11, LP21 and LP02 modes for the wavelength range we investigated. As we know, the effective indexes n 01, n 11, n 21, and n 02 of the LP01, LP11, LP21 and LP02 modes in the main core should meet the condition of n 01 > n 11 > n 21 > n 02. The diameter of the assistant core is fixed at d s = 6.5 μm. The refractive index of the assistant core is defined as n s, which is determined by Δs, where Δs = n s − n c is the index contrast between the assistant core and the cladding. The default operating wavelength is set as 1.55 μm. The length of the main core is setting to be L = 80 mm. The center-to-center distance between the two cores at the top end is fixed at D m = 35 μm.

Fig. 1
figure 1

Profile of the basic configuration

Full size image

A full-vectorial finite-difference beam propagation method (BPM) [36] with transparent boundary conditions [37] is applied to investigate the transmission characteristics of the device. Owing to the low-index contrast characteristics of the configuration, we will only need consider the x-polarized state of the mode, similar characteristics can be achieved for the y-polarized state. The simulation results of the x-polarized state are presented in this article. The mode conversion characteristic of the device is demonstrated first by setting the index contrast of the assistant core to be Δs = 0.0083. Figure 2 shows the mode field profiles of the device by launching light from Port B. The fundamental mode of the assistant core is gradually transferred into the LP11 mode of the main core. Such phenomenon is similar to the digital optical switch [38], where the light is guided by adiabatically evolving the input mode to the mode of the destination arm with the increased refractive index. Therefore, the LP01 mode in the assistant core will be converted to a mode in the main core with lower effective index. It should be noted that LP11 mode is a four-fold degeneracy mode, which can be separated into LP11a and LP11b modes. The converted mode shown in Fig. 2 is denoted as LP11a mode. The conversion characteristics of LP11b mode will be discussed in Sect. 4.

Fig. 2
figure 2

Mode field profiles of the device by launching light from the assistant core for the fiber with a Z = 0, b Z = (2/3)L, c Z = (3/4)L, and d Z = L

Full size image

As we have stated, the LP01 mode of the assistant core can be converted to a specified mode in the main core. This can be achieved by simply adjust the parameters of the assistant core. For simplicity, we will only change the refractive index of the assistant core n a. Figure 3 shows the mode output powers of the main core as functions of Δs. The LP01 mode of the assistant core will be converted to a specific mode of the main core as long as the index contrast is fallen in a specific region. The boundary of the regions is denoted with a, b, c, and d, which shows the positions where the effective index of the LP01 mode of the assistant core equals with those of the LP02, LP21, LP11, and LP01 modes of the main core, respectively. We can see high-efficiency mode conversion can be achieved as long as the effective index of the LP01 mode in the assistant core is higher than that of the mode in the main core. High conversion efficiency can be achieved even if the refractive index of the assistant core is varied in a wide range, therefore, the device allows relatively large fabrication tolerance. Such characteristics are similar to the characteristics digital optical switch, where the refractive index of the branch waveguide can be changed in a wide range with little influence on the transmission characteristics of the waveguide. For the region where the LP01 mode in the assistant core is lower than all the modes in the main core, the input light will be dispelled. Another interesting phenomenon is that the input light will excite two modes with equal powers if the index contrast is chosen to be the positions of b, c, or d.

Fig. 3
figure 3

Mode output powers of the main core as a function of Δs, where a, b, c, and d denotes the positions where the effective index of the LP01 mode in the assistant core equals with those of the LP02, LP21, LP11 and LP01 modes in the main core, respectively

Full size image

As we expected, mode conversion can happen with wide wavelength range. For the mode conversion of the LP01, LP11, LP21 and LP02 modes, a simple selection for the refractive index of the assistant cores is setting Δs to be the middle value of the corresponding region shown in Fig. 3. The assistant cores (AC) are denoted as AC1, AC2, AC3, AC4, and the index contrast of which are 0.0047, 0.0063, 0.0083, and 0.0095, respectively. Figure 4 shows the mode output powers of the main core as functions of wavelength. As expected, the light launched into AC1, AC2, AC3, and AC4 can be converted into the LP02, LP21, LP11, and LP01 modes, respectively. Crosstalk lower than −18.2 dB and insertion loss lower than −0.041 dB can be achieved in a wide wavelength range of 1460–1625 nm, which covers the S, C, and L bands. Figure 5 shows the mode effective indexes of the main core (MC) and the assistant cores. The relationship between the effective indexes of the modes in the main core and the assistant cores agrees well with the mode conversion characteristics of the modes. For example, the effective index of the LP01 mode in AC1 meets the condition of n 21 > n s > n 02 in the wavelength range of 1.44–1.68 μm. At that wavelength range, the LP01 mode launched from AC1 is converted to the LP02 mode of the main core. At short wavelength, it will be converted to the LP31 mode which is in accord with the condition of n 02 > n s > n 31, and at long wavelength, it is converted to the LP21 mode accompanied by the condition of n 11 > n s > n 21. Similarly, conditions hold for the other assistant cores. Apparently, the operating bandwidth can be expanded if the parameters of the configuration are adjusted so that the effective index of the LP01 mode in the assistant core is localized in the region formed by the two adjacent modes of the main core in a wide wavelength range.

Fig. 4
figure 4

Mode output powers of the main core as a function of wavelength for the assistant core with a Δs = 0.0047, b Δs = 0.0063, c Δs = 0.0083, and d Δs = 0.0095

Full size image
Fig. 5
figure 5

Mode effective indexes of the main core and the assistant cores as functions of wavelength

Full size image

3 High-order mode to low-order mode conversion

We can also launch the modes from Port A. As an example, the configuration with AC2 is applied. The mode transmission characteristics of the modes in the main core are investigated by launching individual modes into the main core. As shown in Fig. 6, the LP01 and LP11 mode will be directly outputted from the top end of the main core. The LP21 mode will be converted into the LP01 mode of the assistant core and outputted from the assistant core. The mode conversion characteristic can be easily understood based on the principle of reversibility of light. As we have demonstrated, the LP01 mode of the assistant core will be converted to the LP21 mode of the main core if the light is launched from the assistant core. The launched LP02 mode will be converted to the LP21 mode of the main core. In general, the modes with effective indexes higher than that of the assistant core will be outputted with the same mode forms. The mode with a lower effective index will be converted to the LP01 mode of the assistant core. The other high-order modes will be converted to a higher-order mode. Figure 7 shows the mode output powers in the main core and the assistant core. As expected, low insertion loss and low cross-talk can be achieved in the operating wavelength range shown in the figure.

Fig. 6
figure 6

Mode field transition of the device by launching of a the LP01 mode, b the LP11 mode; c the LP21 mode, and d the LP02 mode into the main core from Port A. Left: the mode field transition along the propagation distance Z, Top-right: mode field at the output end, Bottom-right: mode field at the input end

Full size image
Fig. 7
figure 7

Mode output powers from Port C of the main core by launching a the LP01 mode, b the LP11 mode, c the LP21 mode, d the LP02 mode into the main core from Port A

Full size image

4 High-order mode to low-order mode conversion mode multiplexing design

Mode multiplexer/demultiplexer can be formed based on the proposed configuration. Although it seems a multiplexer/demultiplexer can be achieved simply by adding assistant cores on the configuration. However, just as we have shown in the previous sections, a mode transmitted through the interface area of the assistant core and the main core may be converted to the other modes or filtered out. Therefore, it is important to arrange appropriately the assistant cores to achieve effectively mode multiplexing/demultiplexing.

A configuration proposed for mode multiplexing/demultiplexing is shown in Fig. 8. The mode demultiplexing characteristics will be investigated first. Six assistant cores namely, Branch 1–6 are applied to separate the LP01, LP02 modes, and the degeneracy LP11, LP21 modes. As has been discussed previously, a mode propagated in the main core will not be converted to another mode if the effective index of the mode is higher than that of the LP01 of the assistant core. Therefore, the LP02 mode should be separated first, while the LP01 mode should be separated at last. That is, Branch 1 ~ 4 should be AC1, AC2, AC3, and AC4, respectively. And the LP02, LP21a, LP11a, and LP01 modes can be separated.

Fig. 8
figure 8

Configuration of the proposed multiplexer/demultiplexer

Full size image

We can simplify the configuration by applying a group of assistant cores with the same refractive index to separate/combine the modes. Just as we have shown in Sect. 2, a mode transmission in the main core will be converted into a lower-order mode if it passes through the cross area of the main core and the assistant core, as long as the effective index of the assistant core is higher than that of the lower-order mode. Therefore, if we apply AC4 to all of Branch 1–4, then the LP01, LP11a, LP21a, and LP02 modes will be guided to Branch 1–4, respectively.

On the other hand, the degeneracy LP11 and LP21 modes should be taken into special consideration. To convert the LP11b and LP21b modes, the assistant cores should be rotated by 90° and 45°, respectively. However, mode conversion can occur between the LP11a and LP11b modes if the rotate angle θ is setting as 45°. Therefore, the LP21b mode cannot be separated first. Owing to the fact that the LP21 mode has order 4 of rotation of symmetry, the assistant cores for the separation of the LP11a and LP11b modes will not influence the propagation of the LP21 mode. Therefore, the LP11b mode should be separated first. That is, AC3 and AC2 are applied to Branch 5 and 6, respectively. The arrangement of the assistant cores is shown in Fig. 8. It can be looked as a cascade of six configurations as shown in Fig. 1, only the refractive indexes of each assistant cores are arranged as stated and the length of the assistant cores are extended to ensure the output of mode powers from the same Z position.

The mode field transition of the demultiplexer by launching different modes into the main core is shown in Fig. 9. It shows clearly the transition of the modes of the main core into the specific assistant cores. As shown in Fig. 9a, the LP01 mode launched into the main core will be converted to the LP01 mode of the Branch 1. If the LP11a mode is launched into the main core, it will be converted to the LP01 mode of the main core first. The LP01 mode will then be guided to the fourth assistant core and outputted. Similarly, condition holds for the LP21a mode and LP02 mode. That is the LP21a mode will be converted to the LP11a, LP01 mode of the main core, first, before it converted into the LP01 mode of the assistant core. And the LP02 mode will be converted to the LP21a, LP11a, and LP01 mode of the main core first. The LP11b and LP21b modes will not be influenced by the first four assistant cores, therefore, the modes will pass through the interface regions of Branch 1–4 directly, and finally, they will be outputted from the fifth and sixth assistant cores, respectively.

Fig. 9
figure 9

Mode field transition of the device by launching of a the LP01 mode, b the LP11amode, c the LP21a mode, d the LP02 mode, e LP11b mode, and f the LP21b mode into the main core from the bottom. Left: the mode field transition along the propagation distance Z, Top-right: mode field at the output end, Bottom-right: mode field at the input end. The device is rotated by 90° and 45° along the center axis of the main core, respectively, to clearly show the mode field transition of the LP11b and LP21b modes

Full size image

The operating bandwidth of the demultiplexer is determined such that the mode output powers is higher than −0.05 dB and the output powers of the unwanted modes are lower than −20 dB. As shown in Fig. 10, the operating wavelength range spanned from 1340 to 1660 nm, leading to a wide bandwidth of 320 nm. The operating wavelength range of Branch 1–4 is limited by the fact that AC4 can only operate with wavelengths shorter than 1660 nm, where the condition of n s > n 02 can be met. On the other hand, the operating wavelength range of Branch 5 is limited by the fact that the required mode index condition can only be met for wavelength longer than 1340 nm. As a comparison, we have also calculated the bandwidth of the configuration applying AC1–4 to separate the LP02, LP21a, LP11a, and LP01 modes. The operating wavelength range is limited to 1440–1660 nm. The bandwidth is mainly limited by Branch 1(AC1). As shown in Fig. 5, the relationship of n 21 > n s > n 02 can only be met at the wavelength range of 1440–1680 nm. By reciprocity, the LP01 mode in assistant cores can selectively excite modes in the main core. Therefore, mode multiplexing can be achieved by applying the same configuration.

Fig. 10
figure 10

Output powers of each mode in the main core by launching a the LP01 mode, b the LP11amode, c the LP21amode, d the LP02 mode, e LP11b mode, and f the LP21b mode into the main core from the bottom

Full size image

Previously, direct writing of three-dimensional waveguides and long-period grating based on silica optical fibers by the femtosecond Laser have been demonstrated experimentally [39,40,41,42,43]. Ultrafast-laser inscription of a three-dimensional fan-out device for multicore fiber coupling applications has been proposed as early as 2007 [44]. In particular, femtosecond laser inscription of asymmetric directional couplers in an optical fiber has been demonstrated. Precise alignment of femtosecond laser tracks in optical fiber is shown to enable controllable optical tapping of the fiber core waveguide light with fiber cladding photonic circuits [45]. Such technology enables the interconnection of light propagating in pre-existing waveguides with laser-formed devices. Our proposed structure is composed of a main core and several assistant cores. Therefore, a few-mode optical fiber should be fabricated first using the conventional optical fiber fabrication process, then the assistant cores can be produced by applying the femtosecond Laser. For the configuration shown in Fig. 8, Branch 1–4 should be fabricated firstly. To write Branch 6 and 5, the fiber should be rotated along the central axis of the main core by 90° and 135°, respectively. We are also working the design of the device based on waveguide configuration, where more versatile fabrication techniques can be applied.

5 Conclusion

In conclusion, we proposed the design of a few-mode optical waveguide device in this article, which is able to convert the modes in the few-mode optical fibers. The multiple assistant core configuration is able to multiplexing/demultiplexing modes including the degeneracy modes. It is also interesting that the modes propagating in the main core will be converted to the other modes of the main core, which means the mode multiplexer/demultiplexer cannot be achieved by simply cascading the single assistant core configurations. A specially designed mode multiplexer/demultiplexer with four same assistant cores are, therefore, proposed for wide bandwidth mode multiplexing/demultiplexing. One of the important advantages is the wide bandwidth operating characteristics. In fact, the proposed structure shows crosstalk lower than − 20 dB and insertion loss lower than − 0.05 dB can be achieved in a wide wavelength range of 1340–1660 nm. In addition, the transmission characteristics of the device show large tolerance on the structural parameters of the cores, which will be advantageous for the fabrication of the device. Therefore, such technique can find important applications in the field of mode multiplexing systems.