1. Introduction

Dispersion compensator near the cutoff of higher order mode by converting the fundamental mode LP₀₁ to the second order mode LP₁₁ [1, 2], optical add-drop multiplexer (OADM) [3], optical switch [4], optical modulator or filter [5, 6], are all implemented based on the mode coupling or mode conversion by using dual-mode fibers, in which only LP₀₁ and LP₁₁ modes are confined, and the mode selective coupler or mode converter [7] are employed to perform the mode conversion or mode selection between two confined modes. Both LP₀₁ and LP₁₁ even modes are confined in the elliptical-core fiber, which is usually used in dual-mode fiber sensing systems [8–12].

A triple-core waveguide is simulated and employed to separate LP₀₁ mode and LP₁₁ mode from a two-mode fiber. When both modes are put into the central port, they will come out on different output- side ports with the mode extinction ratio over 30dB. A device based on such a triple-core waveguide can be used for mode selection or mode split.

2. Modes in the triple-core waveguide

A triple-core waveguide, which is illustrated on the left panel of Fig. 1, has been employed in many kinds of optical devices, such as optical fiber interferometers [13], optical fiber resonator [14] and optical buffer [15], in which the waveguide is usually considered as a collinear 3x3 fiber coupler (shown on the right panel of Fig. 1), and the mode coupling properties are simulated by using the weakly-coupling theory [16]. The triple-core waveguide has three same cores collinearly arranged on the x axis, and it is characterized by the structure parameters of core radius a, core-distance d and core/cladding relative refractive index difference Δ, which are set as a=4.1μm, d=10μm, Δ=0.36% in our simulations referring to some practical devices. The mode properties are simulated by the full-vector compact supercell method [17] and the FEM method with PML boundary conditions.

 

Fig. 1. A triple-core waveguide (left panel) and a collinear 3x3 fiber coupler (right panel).

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2.1 LP₀₁ mode

The mode indexes of 20 lower-order modes at the wavelength 0.55μm, 1.0μm and 1.55μm are calculated and illustrated in Fig. 2. It is obvious that there are only 6 confined modes at 1.55μm of which the mode indexes are greater than the cladding index (1.45 here). The transverse electric field vectors of the three x-polarized modes of the 6 confined modes are shown in Fig. 3, which are the mode number 2, 3 and 5 in Fig. 2. It can be seen that the mode field in every core is similar to the mode LP₀₁ in a standard single mode fiber (SMF). Hence then they are analogically named as L${\mathrm{P}}_{01}^{\mathrm{x}1}$, L${\mathrm{P}}_{01}^{\mathrm{x}2}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$, and the three y-polarized modes can be similarly named as L${\mathrm{P}}_{01}^{\mathrm{y}1}$, L${\mathrm{P}}_{01}^{\mathrm{y}2}$ and L${\mathrm{P}}_{01}^{\mathrm{y}3}$.

All the modes are non-degenerated because the triple-core waveguide has C_2v symmetry [19], but the three mode-pairs, (L${\mathrm{P}}_{01}^{\mathrm{x}1}$ and L${\mathrm{P}}_{01}^{\mathrm{y}1}$), (L${\mathrm{P}}_{01}^{\mathrm{x}2}$ and L${\mathrm{P}}_{01}^{\mathrm{y}2}$), (L${\mathrm{P}}_{01}^{\mathrm{x}3}$ and L${\mathrm{P}}_{01}^{\mathrm{y}3}$), are still approximately degenerated because of the almost same mode-index shown in Fig. 2. At 1.55μm, the beat length of every mode pairs are 1.28m, 2.71m and 5.57m, which are much greater than the waveguide length (usually in order of millimeter or centimeter), the polarization evolution can be hence neglected. For the case of simplicity, only x-polarized modes are discussed in the following.

 

Fig. 2. Mode indexes of 20 lower-order modes at 0.55μm, 1.0μm and 1.55μm

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Fig. 3. x-polarized fundamental modes L${\mathrm{P}}_{01}^{\mathrm{x}1}$, L${\mathrm{P}}_{01}^{\mathrm{x}2}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ at 0.55μm.

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2.2 LP₁₁ mode

All the 20 lower-order modes in Fig. 2 are confined at 0.55μm, and the first 6 modes are also LP₀₁ modes. The next 12 modes are the second order modes, and the modes of number 8, 9, 11, 13, 15, 18 are x-polarized modes, of which the transverse electric field are shown in Fig. 4. The six modes are named as L${\mathrm{P}}_{11}^{\text{xe}1}$, L${\mathrm{P}}_{11}^{\text{xo}1}$, L${\mathrm{P}}_{11}^{\text{xo}2}$, L${\mathrm{P}}_{11}^{\text{xo}3}$, L${\mathrm{P}}_{11}^{\text{xe}2}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$, where ‘e’ or ‘o’ means even mode or odd mode analogically to the conventional elliptical core fiber [7–9].

In fact, a step index optical fiber with the parameters same as above can support only two modes (LP₀₁ and LP₁₁) in the wavelength range of between about 0.827μm and 1.318μm, so the triple-core waveguide support single mode at 1.55μm, and multi-mode at 0.55μm. As shown in Fig. 2, the triple-core waveguide is a two-mode device at 1.0μm. In order to clearly show the mode coupling, especially for the LP₁₁ mode, the case of 0.55μm rather than 1.0μm is considered in our manuscript because the mode is confined better at shorter wavelength.

 

Fig. 4. The second-order x polarized modes at 0.55μm.

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3. Mode coupling

3.1 LP₀₁ mode

When the fundamental mode L${\mathrm{P}}_{01}^{\mathrm{x}}$, the transverse electric field shown in Fig. 5(a), inputs on the central port 2, it can be regarded as that both L${\mathrm{P}}_{01}^{\mathrm{x}1}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ modes are stimulated simultaneously with a phase difference π, and the amplitudes of both modes are exactly same. If the waveguide length, i.e., the coupler length, is equal to the beat length $L_B^{01}$ between both modes, they will output keeping the phase difference π, and the incident L${\mathrm{P}}_{01}^{\mathrm{x}}$ mode will output on port 5. If the waveguide length is $L_B^{01}$/2, they will output in phase, and the total light will output on both port 4 and port 6 equally, which is shown in Fig. 5(b).

 

Fig. 5. The input (a) and output (b) of L${\mathrm{P}}_{01}^{\mathrm{x}}$ mode when incident on the central port 2.

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When the fundamental mode L${\mathrm{P}}_{01}^{\mathrm{x}}$ inputs on port 1 or port 3, the transverse electric field shown in Fig. 6(a), it can be regarded as that L${\mathrm{P}}_{01}^{\mathrm{x}1}$, L${\mathrm{P}}_{01}^{\mathrm{x}2}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ modes are stimulated simultaneously, they are in phase, and the amplitudes of both L${\mathrm{P}}_{01}^{\mathrm{x}1}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ modes in both side-core areas are exactly half of the L${\mathrm{P}}_{01}^{\mathrm{x}2}$ mode. If the waveguide length is $L_B^{01}$/2, a phase difference of π will occur between L${\mathrm{P}}_{01}^{\text{xl}}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ modes, the output on port 5 will be exactly determined by the sum of L${\mathrm{P}}_{01}^{\mathrm{x}1}$ and L${\mathrm{P}}_{01}^{\mathrm{x}3}$ modes, while the mode L${\mathrm{P}}_{01}^{\mathrm{x}2}$ itself dominates the output on port 4 and port 6, as shown in Fig. 6(b).

In general, if the waveguide length is $L_B^{01}$/2, the fundamental mode outputs on port 4 and 6 equally when incident on port 2, and outputs on ports 4, 5 and 6 with a particular ratio when incident on port 1 or 3.

 

Fig. 6. The input (a) and output (b) of L${\mathrm{P}}_{01}^{\mathrm{x}}$ mode when incident on port 1.

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3.2 LP₁₁ mode

When the second order even mode L${\mathrm{P}}_{11}^{\text{ex}}$ input on port 2, the transverse electric field shown in Fig. 7(a), it can be regarded as that both L${\mathrm{P}}_{11}^{\text{xe}1}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$ modes are stimulated simultaneously with a phase difference π, and the amplitudes of both modes are exactly same. If the waveguide length is equal to the beat length $L_B^{11}$ between both modes, they will output keeping the phase difference π, and the incident L${\mathrm{P}}_{11}^{\text{ex}}$ mode will output on port 5. If the waveguide length is $L_B^{11}$/2, they will output in phase, and the total light will output on both port 4 and 6 equally, which is plotted in Fig. 7(b) to show the split of L${\mathrm{P}}_{11}^{\text{ex}}$ mode.

 

Fig.7. The input (a) and output (b) of L${\mathrm{P}}_{11}^{\text{ex}}$ mode when incident on the central port 2.

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When the L${\mathrm{P}}_{11}^{\text{ex}}$ mode incident on port 1 or port 3, the transverse electric field shown in Fig. 8(a), it can be regarded as that L${\mathrm{P}}_{11}^{\text{xe}1}$, L${\mathrm{P}}_{11}^{\text{xe}2}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$ modes are stimulated simultaneously, they are in phase, and the amplitudes of both L${\mathrm{P}}_{11}^{\text{xe}1}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$ modes in both side-core areas are exactly half of the L${\mathrm{P}}_{11}^{\text{xe}2}$ mode. If the waveguide length is $L_B^{11}$/2, a phase difference of π will occur between L${\mathrm{P}}_{11}^{\text{xe}1}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$ modes, the output on port 5 will be exactly determined by the sum of L${\mathrm{P}}_{11}^{\text{xe}1}$ and L${\mathrm{P}}_{11}^{\text{xe}3}$ modes, while the mode L${\mathrm{P}}_{11}^{\text{xe}2}$ itself dominates the output electric field on port 4 and 6, as shown in Fig. 8(b).

In general, if the waveguide length is $L_B^{11}$/2, the second order even mode outputs on port 4 and 6 equally when incident on port 2, and outputs on ports 4, 5 and 6 with a particular ratio when incident on port 1 or 3.

 

Fig. 8. The input (a) and output (b) of L${\mathrm{P}}_{11}^{\text{ex}}$ mode when incident on port 1.

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4. Mode splitter

When the fundamental mode L${\mathrm{P}}_{01}^{\mathrm{x}}$ and the second order even mode L${\mathrm{P}}_{11}^{\text{ex}}$ simultaneously input on port 2, mode pairs of (L${\mathrm{P}}_{01}^{\mathrm{x}1}$, L${\mathrm{P}}_{01}^{\mathrm{x}3}$) and (L${\mathrm{P}}_{11}^{\text{ex}1}$, L${\mathrm{P}}_{11}^{\text{ex}3}$) will be stimulated. L${\mathrm{P}}_{01}^{\mathrm{x}}$ will output on port 5, and L${\mathrm{P}}_{11}^{\text{ex}}$ will equally output on port 4 and port 6, when the phase differences of both mode pairs due to the waveguide (its length is z) are

From Eq. (1), it can be obtained that

It means that modes L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$ can be separated when the beat length ratio can be expressed as a ratio of an odd number to an even number. Taking the discussed triple-core waveguide as an example, $L_B^{01}$=48.746mm, $L_B^{11}$=8.857mm at wavelength 0.55μm, the beat length ratio is about 11/2, which means k ₀₁=1 and k ₁₁=5. If the waveguide length is $L_B^{01}$, L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$ modes will output on port 5 and ports (4, 6), respectively.

In order to evaluate the performance of the mode splitter, the mode extinction ratio (ER _mn), defined as Eq. (3), is used to scale the suppression of the mode LP_mn accompany with the output mode LP_m’n’.

where P_mn and P_m’n’ , defined as P_mn ≈∫∫|E_mn |²/2dxdy, denote the power of modes LP_mn and LP_m’n’.

It is supposed that L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$, coming from an elliptical-core two-mode fiber, are in phase when they input on the port 2 of a triple-core waveguide, the incident electric field and the intensity pattern are shown in Fig. 9. The incident ER ₀₁ and ER ₁₁ in the central core are 2.92dB and 3.10dB, respectively. Both extinction ratios must be exactly 3dB when the energies of both modes are exactly same. The deviation of the incident ER from 3dB is due to the energy difference between both modes, which also leads to such an intensity pattern as shown in Fig. 9.

When the waveguide length is $L_B^{01}$, L${\mathrm{P}}_{01}^{\mathrm{x}}$ will output on port 5 with ER ₁₁ about 31.40dB, and L${\mathrm{P}}_{11}^{\text{ex}}$ mode will output on port 4 and port 6 equally with ER ₀₁ about 30.68dB. Fig. 10 shows the output electric field vector and the intensity pattern. It is obvious that the fundamental mode and the second order even mode are well split by the triple-core waveguide with the ER over 30dB. In fact, the properties of the mode splitter do not depend on the phase difference between both input modes.

 

Fig. 9. the electrical field and the intensity pattern on the input side when both L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$ modes incident on port 2.

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Fig. 10. the electrical field and the intensity pattern on the output side when both L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$ modes incident on port 2.

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5. Conclusion

The mode index and mode field of the triple-core waveguide are obtained by the compact supercell method and the FEM method. The coupling properties of LP₁₁ mode is analyzed, for the first time to our knowledge, which is similar to that of LP₀₁ mode. Both LP₀₁ and LP₁₁ modes can be separated based on their different coupling lengths in a triple-core waveguide. When both modes input on the central port 2, they will output on different ports by optimizing the waveguide length. For example, when the waveguide length is $L_B^{01}$ (about 48.74mm in this case), L${\mathrm{P}}_{01}^{\mathrm{x}}$ and L${\mathrm{P}}_{11}^{\text{ex}}$ modes will output individually on port 5 and (4, 6), the mode extinction ratio can be over 30dB. A device based on the triple-core waveguide could be widely employed in kinds of two-mode fiber devices for mode selection or mode split.