1. Introduction

To deal with further expanding traffic, the optical communication system has continued to progress through various technologies, such as polarization division multiplexing, wavelength division multiplexing, and spatial division multiplexing (SDM). Among them, the SDM technology, including multicore fiber (MCF) and few-mode fiber (FMF) approaches, is a key technology for keeping the evolution of transmission capacity [1,2]. The FMF supports a limited number of linearly-polarized (LP) modes, e.g. 2LP (LP₀₁ and LP_11a/b modes) or 4LP (LP₀₁, LP₁₁, LP₂₁, and LP₀₂ modes) [3]. Although each mode can individually transmit the optical signal, there arises a difference in the transmission quality because the higher-order modes have generally larger propagation losses compared with the fundamental LP₀₁ mode [4,5]. It is called mode-dependent loss (MDL) that arises from various factors, such as propagation losses, bending losses, mode mismatch at the connection, and mode-dependent gain in the optical amplifier. A reduction of the MDL is important in the long-haul SDM transmission since it degrades the performance of the optical MIMO (multiple-input multiple-output) processing [6]. So far, an approach to equalize the MDL has been recently proposed [7–12]. In [8–10,12], by using a mode-selective Mach-Zehnder interferometer (MZI) using the silica planar lightwave circuit (silica-PLC) platform, the transmission of the LP₀₁ mode is controlled while keeping a higher transmission of the LP_11a/b modes. Although the MDL control by a micro heater has been successfully demonstrated, the wavelength dependency of MDL inevitably causes because the amount of phase shift by the thermo-optical (TO) effect depends on the wavelength. In addition, a directional coupler (DC) in the mode-selective MZ has also wavelength dependency, which may be a problem in broadband operation.

In this paper, we propose a new configuration of the silica-PLC mode equalizer as shown in Fig. 1. As described later in detail, almost all the components rely on the principle of adiabatic mode conversion in the proposed mode equalizer, which leads to the broadband operation as well as the high fabrication tolerance. It is a unique point that, instead of separating the LP₀₁ mode, we utilize the selective mode conversion from the LP₀₁ mode to the LP₂₁ mode, because the LP₂₁ mode will be eventually eliminated due to the cutoff condition in the FMF.

 

Fig. 1. Schematic of a proposed silica-PLC MDL equalizer and the corresponding mode evolutions in the device.

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This paper is organized as follows. In Section 2, the concept to achieve broadband operation as well as overviewed device operation are explained in detail. In Section 3, the concrete device designs of the LP₁₁ mode rotator, Y-branch waveguide, 3-mode exchanger, and the waveguide in the MZI arm are numerically investigated. In Section 4, the characteristics of the overall device as shown in Fig. 1 are evaluated, where some configurations of the MZI arm are examined. Finally, it is numerically demonstrated that the proposed concept of broadband design of the MDL equalizer is validated.

2. Device operation principle

2.1 Overviewed device operation

The proposed device as shown in Fig. 1 is composed of two 3-mode exchangers and the MZI using two Y-branch waveguides. The device is connected to 2LP-mode fibers at the input and output ports, where the fiber modes and rectangular waveguide modes are mutually converted e.g. from LP₀₁, LP_11a, and LP_11b modes in the FMF to E₁₁, E₂₁, and E₁₂ modes in the silica-PLC waveguide, respectively. The subscripts m (n) of the E_mn mode represents the mode order for the horizontal (vertical) direction. The ideal mode evolution in each component is depicted below the schematic. As indicating the mode evolutions by arrows, the left mode exchanger converts from E₂₁, E₁₂, and E₂₂ modes to E₁₂, E₂₂, and E₂₁ modes, respectively. Noting that, the right mode exchanger has an inverse operation in contrast with the left mode exchanger, which will be described in Subsection 2.2. As we can see from the flow of mode evolutions, an important point is that the input LP₀₁ mode does not interfere with the output LP_11a/b modes, and vice versa. The detailed behavior is explained as follows.

At first, it is considered when the input LP₀₁ mode is input from the left FMF. The LP₀₁ mode reaches the left mode exchanger as the E₁₁ mode. It passes through the left mode exchanger without the mode conversion because this mode exchanger affects the E₁₂, E₂₁, and E₂₂ modes. For the input E₁₁ mode to the left Y-branch waveguide in the MZI, the in-phase E₁₁ mode is excited as shown in Fig. 2 (a), which is the well-known adiabatic mode conversion as in [13]. The equally divided E₁₁ modes are given the phase difference of Δφ by the delay line or by heating up. If these lightwaves are combined at the right Y-branch waveguide, as we can understand from the facts of Figs. 2(a) and (b) and the optical linearity, the E₁₁ and E₂₁ modes can be excited with the power ratio of cos²(Δφ/2) : sin²(Δφ/2) for E₁₁ and E₂₁ modes (see Appendix A). These modes pass through the right mode exchanger, resulting in only a change from E₂₁ mode to E₂₂ mode. Although the finally excited E₁₁ and E₂₂ modes can be converted to the LP₀₁ and LP₂₁ modes in the right FMF, the LP₂₁ mode will be eliminated in the 2LP-mode fiber due to the cut-off condition. It means that, by using the proposed silica-PLC mode equalizer, an arbitrary loss can be caused to the input LP₀₁ mode by controlling Δφ in MZI.

 

Fig. 2. Mode evolutions in the Y-branch waveguide. (a) E₁₁ mode to in-phase E₁₁ mode, (b) E₂₁ mode to out-phase E₁₁ mode, (c) E₁₂ mode to in-phase E₁₂ mode, and (d) E₂₂ mode to out-phase E₁₂ mode.

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On the other hand, as the mode evolution can be seen from Fig. 1, the LP_11a/b modes input from the left FMF remain as LP_11a/b modes with a mode mixing corresponding to Δφ (see Appendix A). In these mode evolutions, similarly to Figs. 2(a) and (b) for the interference between E₁₁ and E₂₁ modes in the MZI, the interferences as shown in Figs. 2(c) and (d) occur. In this way, ideally, the proposed device can control the MDL in the 2LP transmission system by controlling the transmission of the LP₀₁ mode without affecting the loss of LP_11a/b modes. Although the LP_11a/b modes rotate corresponding to the attenuation of the LP₀₁ mode, there is no concern since the LP_11a/b modes rotate also in the FMF.

As seen from the operation principle explained above, the objective of the left 3-mode exchanger in Fig. 1 is to ensure that the input LP₀₁ mode does not interfere with the input LP_11a/b modes in the MZI. In other words, as long as the left 3-mode exchanger converts the input LP_11a/b modes to the E₁₂ and E₂₂ modes, the excitation ratio of the E₁₂ and E₂₂ modes does not matter. One may wonder why the E₂₁/E₂₂ mode exchanger is not used and the 3-mode exchanger is used instead. One may also consider that even if the 3-mode exchanger is replaced with the E₁₁/E₁₂ mode exchanger, the proposed device will work well. According to the operation principle, these suggestions are correct. However, the E₁₁/E₁₂ and E₂₁/E₂₂ mode exchangers, which convert the vertical order of the mode, are generally realized by the top grating as in [14], leading to large wavelength dependence and scattering losses. To obtain broadband and low-loss characteristics, the 3-mode exchanger seems to be preferable as explained in the following subsection. Furthermore, we also note that the Y-branch waveguides can be replaced by such a tapered waveguide as in [15], which can work the same as the Y-branch waveguides.

2.2 Operation principle of 3-mode exchanger

As explained in the previous subsection, to ensure the designated mode interference in the proposed device, the 3-mode exchanger is a key component. As shown in Fig. 1, this mode exchanger has the cyclic mode conversion, which can separate into two different pairs of two-mode switching such as LP₁₁ mode rotators or the long-period grating mode converters. To achieve broadband operation, the adiabatic mode converter is preferable, e.g. the LP₁₁ mode rotators using tapered trenches [16], and so on. As promising one of the possible configurations, we newly propose a 3-mode exchanger as shown in Fig. 3(a), in which the E₁₂/E₂₂ mode exchanger and the tapered LP₁₁ mode rotator [16] (E₁₂/E₂₁ mode exchanger) are concatenated. In the schematics, the blue- and orange-filled areas denote the core and trench region, respectively. Especially, to realize the adiabatic mode exchange, the E₁₂/E₂₂ mode exchanger proposed here is composed of the MZI using two Y-branch waveguides and the cascaded LP₁₁ mode rotators in each MZI arm. It may be strange to concatenate two LP₁₁ mode rotators because the LP_11b mode is returned to the LP_11b mode via the LP_11a mode, which means that the mode field change does not occur at all. However, just as illustrated, these two upper and lower cascaded rotators have a different symmetry, which leads to the relative phase shift of π between the upper and lower arms for only the LP_11a/b modes (see Appendix B), and thus, the mode evolutions in Fig. 3(b) are obtained. These behaviors of the 3-mode exchanger can be expressed as

 

Fig. 3. Schematics of proposed mode exchangers and the corresponding mode evolutions in the device. (a) 3-mode exchanger and (b) E₁₂/E₂₂ mode exchanger.

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3. Numerical design

In this section, to investigate the characteristics of the proposed device as shown in Fig. 1 and its components as shown in Fig. 3(a), the LP₁₁ mode rotator and the Y-branch waveguide are numerically designed by using the scalar finite-element (SFE) guided mode solver [17] and the SFE beam propagation method (SFE-BPM) solver [18]. After that, the characteristics of the 3-mode exchanger are investigated, and then the phase shift in the MZI is also investigated.

Throughout the paper, the structural parameters for the silica-PLC platform are fixed as follows: the waveguide height of h = 10 µm, the trench depth of h_t = 2 µm, and the relative refractive index difference of Δ = 1%. The cladding refractive index, n_cl, is set by the Sellmeier equation [19], and then the core refractive index, n_co, is determined by the definition of Δ ≡ (n_co² – n_cl²)/(2n_co²). As following, the characteristics of 3-mode exchanger is investigated, and then the phase shift in the MZI is also investigated.

3.1 LP₁₁ mode rotator based on tapered trenches

The LP₁₁ mode rotators used in the 3-mode exchanger are designed here. As seen from Fig. 3(a), the objective of LP₁₁ mode rotators is slightly different between the E₁₂/E₂₂ mode exchanger and the E₁₂/E₂₁ mode exchanger. Focusing on the E₁₂/E₂₁ mode exchanger in the left 3-mode exchanger in Fig. 1 (here we call it Rot-1), three modes (the E₁₁, E₂₁, and E₂₂ modes) are assumed to be input because the LP₂₁ mode is not launched from the left 2LP-mode fiber. It is preferable to place the symmetric tapered trench first, and then the asymmetric taper trench is connected behind it because the E₁₂ mode does not exist, as discussed in [16]. In addition, unlike in [16], we should care to the guide of the E₂₂ mode with low loss, which is easy to couple to some higher order modes. Therefore, we designed the Rot-1 by the straight tapered trenches. Whereas, the MZI in the E₁₂/E₂₂ mode exchanger has also LP₁₁ mode rotators (here we call it Rot-2). When concatenating two Rot-2s, the symmetric tapered trench is not needed. In the Rot-2, only the E₁₁ and E₁₂ modes are assumed to be input, and thus the fast quasi-adiabatic (FAQUAD) tapered structure [20], which is one design scheme of the shortcut to adiabaticity, can be easily used for the broadband operation with a small footprint.

Figures 4(a) and (b) show the effective index n_eff in the silica-PLC waveguide with asymmetric and symmetric trench as a function of the trench width w_t, where the wavelength is λ = 1550 nm and the waveguide width is set to w = 9 µm satisfying w : h ≈ h – h_t : w. The cross-sectional waveguide geometries are depicted in the middle left and right, and the electric field distributions are shown below the graph. If the w_t is gradually changed, the guided mode is modulated along the lines. The 1st mode corresponds to the E₁₁ mode, which hardly couples to other modes. The 2nd and 3rd modes correspond to the E₁₂ and E₂₁ modes. As we can understand by following the green or blue lines, the E₁₂ and E₂₁ modes are exchanged by the adiabatic mode conversion. The 4th mode corresponds to E₂₂ mode and other higher-order modes (e.g. E₃₁ or E₁₃ modes). It is difficult to specify which pair of modes couple, and thus here we do not use the FAQUAD tapered structure in designing the Rot-1. For the Rot-2, the FAQUAD approach can be easily used, and the obtained trench shape is shown in Fig. 5. The horizontal axis is the normalized propagation position, which is z divided by the tapered length L_tp. It is obtained so that the 2nd and 3rd modes do not couple and the E₁₂ (E₂₁) mode is successfully converted to the E₂₁ (E₁₂) mode. By using such a curved trench shape, a high mode transition can be obtained in the Rot-2.

 

Fig. 4. Effective index n_eff in the silica-PLC waveguide with (a) asymmetric and (b) symmetric trench as a function of the trench width w_t, where waveguide width is w = 9 µm and the wavelength is λ = 1550 nm.

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Fig. 5. Optimized trench width w_opt for Rot-2.

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Figures 6(a)-(c) show the transmission in the LP₁₁ mode rotator obtained by the SFE-BPM solver as a function of the taper length L_tp, where the wavelength is λ = 1550 nm. For the symmetric straight tapered trench (w_t = wz/L_tp) as shown in Fig. 6(a), the transmissions of all modes saturate larger than −0.01 dB for L_tp > 1 mm. Whereas, a much larger L_tp is required to obtain high mode conversion efficiencies for the asymmetric tapered trench because the strong mode coupling arises due to the structural asymmetry. For the asymmetric straight tapered trench (w_t = wz/L_tp) as shown in Fig. 6(b), to obtain a transmission larger than −0.1 dB (−0.01 dB) for all modes, L_tp > 4 mm (8 mm) is required. The characteristics of the asymmetric tapered trench applied to the FAQUAD are shown in Fig. 6(c), where the trench shape in Fig. 5 is used. At L_tp = 1.7 mm, a transmission larger than −0.01 dB is obtained except for the E₂₂ mode. On the contrary, the E₂₂ mode is difficult to reach a high transmission in the visible range in Fig. 6(c). After this, for the parameter of the Rot-1, L_tp = 1 mm and 5 mm are used in the symmetric trench and asymmetric trench, respectively, and for the parameter of the Rot-2, L_tp = 1.7 mm is used for the tapered trench based on the FAQUAD design. Figures 7(a)-(c) show the transmission spectra of the partially divided components used in the 3-mode exchanger. Thanks to the adiabatic mode conversion, the almost ideal wavelength insensitivity is confirmed.

 

Fig. 6. Transmission in the LP₁₁ mode rotator as a function of the taper length L_tp, where the wavelength is λ = 1550 nm.

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Fig. 7. Transmission spectra of the partially divided components used in the 3-mode exchanger. The corresponding structure is depicted below each graph.

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3.2 Tapered Y-branch waveguide

By considering the configuration of Figs. 1 and 3, it is preferable to match the input and output width in the Y-branch waveguide as shown in Fig. 8(a). The length of the Y-branch waveguide is set to L_Y = 1 mm, and the parameter determining the separation between upper and lower waveguides is set to s_Y = 10 µm. This geometry is obtained by placing two sine-curved waveguides with the x-axis symmetry. For the lower half part in Fig. 8(a), the position of the side of core, x₀ and x₁ (x₁ > x₀), are defined as

 

Fig. 8. (a) Schematic of the Y-branch waveguide and (b) its transmission spectra.

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3.3 3-mode exchanger

By combining the Y-branch waveguide described in Subsection 3.2 and LP₁₁ mode rotator described in Subsection 3.1, the device structure in Fig. 3(a) is determined. The device length of the 3-mode exchanger becomes 11.4 mm. Figures 9(a) and (b) show the transmission spectra of the designed 3-mode exchanger. We can confirm the broadband operation. Especially in the range from 1460 to 1625 nm (S + C + L band), the transmission of all modes larger than −0.08 dB. The transmission of the E₂₂ mode tends to drastically degrade at the longer wavelength because the E₂₂ mode get close to the cutoff. If selecting much longer L_Y and L_tp and the higher-Δ (higher-height) core parameter, such a degradation in the band edge will be improved.

 

Fig. 9. Transmission spectra of the designed 3-mode exchanger as shown in Fig. 3(a). (a) Plot in the broadband range and (b) enlarged graph from 1450 nm to 1650 nm

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3.4 Wavelength dependence in MZI

In terms of wavelength dependence, one concern remains. That is, in the center MZI in Fig. 1, there is no idea to make the arbitrary phase shift Δφ by particularly the adiabatic mode conversion. Generally, the phase shift Δφ in the MZI arm is given by the TO effect or the delay lines. For example, when Δφ is given by the TO effect, it can be expressed as

Whereas, when passively controlling Δφ by the difference of waveguide width Δw, it is given as

 

Fig. 10. (a) Group index as a function of the width w, where the wavelength is λ = 1550 nm. (b) Pair of w and w + Δw corresponding to N_w = N_w+Δw and the required L_Δw for Δφ_Δw = π.

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Fig. 11. Configuration example of MZI arms satisfying Eq. (8) for Δw ≠ 0.

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Although further reduction of wavelength dependence is possible such as introducing the cascaded MZI so-called wavelength-insensitive coupler (WINC) as in [22], even the short and simple method above described has sufficient wavelength-insensitive operation as described in the following section.

4. Device characteristics and discussion

In this section, we confirm the characteristics of the proposed MDL equalizer as shown in Fig. 1. To investigate the device characteristics, the following transfer matrix, T_{MDL‐Equalizer}, is calculated as

At first, we investigate the passive design, in which the relative phase shift Δφ in the MZI arm is given by Eq. (10), corresponding to the waveguide width change as shown in Fig. 11. Since the length of the 3-mode exchanger is 11.4 mm and the maximum length of the MZI is about 2.8 mm, the overall length of MDL equalizer is estimated as 25.6 mm. Figures 12(a)-(c) show the transmission spectra of the device for T_obj = 50% when launching the E₁₁, E₁₂, and E₂₂ modes. Noting that, T_obj corresponds to the only E₁₁ mode at λ = 1550 nm, and it is adjusted by L_Δw (L_Δw = 390 µm for T_obj = 50%). As shown in Fig. 12(a), the transmissions of the E₁₁ and E₂₂ modes when launching the E₁₁ mode is almost 50% throughout the whole bandwidth. On the other hand, as shown in Figs. 12(b) and (c), the transmissions of the E₁₂ and E₂₁ modes are almost 50% because the phase shift differences Δφ for between the E₁₁ and E₁₂ modes in the MZI arms are not so large. Note that, the mode conversion between the E₁₂ and E₂₁ modes corresponds to simply the rotation of the LP_11a/b mode in the FMF, and it does not affect the MDL control. To evaluate the differential mode attenuation (DMA), the comparison of the transmission of the E₁₁ mode when launching the E₁₁ mode and the summation of the E₁₂ and E₂₁ modes when launching the E₁₂ or E₂₁ mode are shown in Fig. 13(a). The insertion losses of E₁₂ and E₂₁ modes are sufficiently small, which is about −0.15 ∼ −0.3 dB. The transmission of E₁₁ mode is about −2.9 ∼ −3.2 dB, agreeing well with the objective transmission. Figure 13(b) shows the crosstalk spectra. The worst crosstalk is seen for the E₂₁ mode input, which is −16 dB at 1675 nm.

 

Fig. 12. Transmission spectra of the MDL equalizer for T_obj = 50% (passive design) when launching E₁₁, E₁₂, and E₂₂ modes.

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Fig. 13. (a) Transmission and (b) crosstalk spectra of the MDL equalizer for T_obj = 50% (passive design) when launching E₁₁, E₁₂, and E₂₂ modes, where transmission of E₁₂ and E₂₂ modes are summed up.

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Figure 14(a) shows the transmission spectra of E₁₁ mode when launching E₁₁ mode for T_obj = 100% (0 dB), 70% (−1.5 dB), 50% (−3 dB), 25% (−6 dB), and 13% (−9 dB). As decrease the T_obj, the wavelength dependence slightly increases. Nevertheless, the broadband wavelength insensitive operation can be seen. Figure 14(b) shows the crosstalk spectra when launching E₂₁ for various T_obj. The large change is not seen even if the T_obj is changed.

 

Fig. 14. (a) Transmission spectra of E₁₁ mode when launching E₁₁ mode and (b) crosstalk spectra when launching E₂₁ for various T_obj (passive design).

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Next, we consider the relative phase shift Δφ in the MZI arm is given by heating up the waveguide, in which Δφ is actively controlled although the wavelength dependence arises corresponding to Eq. (9). In this case, T_obj is only adjusted by ΔT, and thus Δw = 0, and thus the required ΔT for a fixed the heated length L_TO is given as

 

Fig. 15. Transmission spectra of E₁₁ mode when launching E₁₁ mode for various T_obj (active operation).

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Fig. 16. Transmission spectra of E₁₁ mode when launching E₁₁ mode for various T_obj (passive design and active operation), where transmission of (a) 50% and (b) 13% are set for the passive design, and the active operation is assumed to reach various T_obj.

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5. Conclusion

In this paper, we newly proposed a silica-PLC MDL equalizer for the 2LP-mode transmission system, which has two 3-mode exchangers and a simple MZI for adjusting the transmission of the LP₀₁ mode. Because of the adiabatic mode conversion of the 3-mode exchangers, as expected from the proposed concept, the MDL equalizer enables controlling the MDL with a considerably broadband operation. We also would like to emphasize that the phase shifter by the cascaded LP₁₁ mode rotators in the E₁₂/E₂₂ mode exchanger is novel and effective, which is a key component for broadband operation of the proposed MDL equalizer. The numerical results proved that the proposed device enables the MDL equalization for the wavelength from 1200 nm to 1650 nm keeping the transmission of LP₁₁ modes larger than −0.4 dB at least for arbitral objective transmissions.

Appendix A

By defining x_s and y_s (s ∈ {E₁₁, E₁₂, E₂₁, E₂₂}) as the input and output complex amplitudes corresponding to the mode s, the transfer matrix in the MZI at the center in Fig. 1 can be expressed as

(A1)$${\left( {\begin{array}{cccc} {{y_{\textrm{E11}}}}&{{y_{\textrm{E21}}}}&{{y_{\textrm{E12}}}}&{{y_{\textrm{E22}}}} \end{array}} \right)^\textrm{T}} = {\mathbf T}_\textrm{Y}^\textrm{T}\left( {\begin{array}{cc} {{{\mathbf T}_{\textrm{Arm-U}}}}&{}\\ {}&{{{\mathbf T}_{\textrm{Arm-L}}}} \end{array}} \right){{\mathbf T}_\textrm{Y}}{\left( {\begin{array}{cccc} {{x_{\textrm{E11}}}}&{{x_{\textrm{E21}}}}&{{x_{\textrm{E12}}}}&{{x_{\textrm{E22}}}} \end{array}} \right)^\textrm{T}},$$

(A2)$${\mathbf T}_\textrm{Y}^\textrm{T} = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{cccccccc} {\exp ({j{\phi_{11}}} )}&0&0&0&\exp ({j{\phi_{11}}} )&0&0&0\\ {\exp ({j{\phi_{21}}} )}&0&0&0&{ - \exp ({j{\phi_{21}}} )}&0&0&0\\ 0&0&{\exp ({j{\phi_{12}}} )}&0&0&0&{\exp ({j{\phi_{12}}} )}&0\\ 0&0&{\exp ({j{\phi_{22}}} )}&0&0&0&{ - \exp ({j{\phi_{22}}} )}&0 \end{array}} \right)$$

(A3)$${{\mathbf T}_{\textrm{Arm-U}}} = \left( {\begin{array}{cccc} {\exp ({j\Delta {\varphi_{11}}} )}&{}&{}&{}\\ {}&{\exp ({j\Delta {\varphi_{21}}} )}&{}&{}\\ {}&{}&{\exp ({j\Delta {\varphi_{12}}} )}&{}\\ {}&{}&{}&{\exp ({j\Delta {\varphi_{22}}} )} \end{array}} \right){{\mathbf T}_{\textrm{Arm-L}}}$$

(A4)$${{\mathbf T}_{\textrm{Arm-L}}} = \left( {\begin{array}{cccc} {\exp ({j{\varphi_{11}}} )}&{}&{}&{}\\ {}&{\exp ({j{\varphi_{21}}} )}&{}&{}\\ {}&{}&{\exp ({j{\varphi_{12}}} )}&{}\\ {}&{}&{}&{\exp ({j{\varphi_{22}}} )} \end{array}} \right)$$

where ϕ_i and φ_i (i ∈ {11, 21, 12, 22}) are the amounts of phase shift of the E_i mode in the Y-branch waveguide and the arm of the MZI, respectively, and Δφ_i (i ∈ {11, 21, 12, 22}) is the phase shift arising in the upper arm in the MZI. The superscript of T denotes the transpose. When the E₁₁ mode is launched (x_E11 = 1 and x_E12 = x_E21 = x_E22 = 0), we can obtain |y_E11|² = cos²(Δφ₁₁/2) and |y_E21|² = sin²(Δφ₁₁/2). Similarly, when the E_1n mode is launched, we can obtain |y_E1n|² = cos²(Δφ_1n/2) and |y_E2n|² = sin²(Δφ_1n/2). Inversely, when the E_2n mode is launched, we can obtain |y_E1n|² = sin²(Δφ_1n/2) and |y_E2n|² = cos²(Δφ_1n/2). Since the E₁₂ and E₂₂ modes correspond to the LP_11a/b mode in the left 2LP-mode fiber in Fig. 1, it acts like the LP_11a/b mode rotator with a rotation angle of Δφ₁₂/2. Although ϕ_i and φ_i have wavelength dependencies, even if considering these wavelength dependencies, the above mode rotation operation is not affected. However, it is indispensable to take care of the wavelength dependence of Δφ_i in the proposed MDL equalizer.

Appendix B

Here, we consider the transfer matrix expression for several configurations including the LP₁₁ mode rotators as shown in Figs. 17(a)-(g), and it will be explained why the relative phase shift of π arises between the two types of cascaded LP₁₁ mode rotators as shown in Fig. 3(b) (namely, Figs. 17(d) and (g)). Although all of Figs. 17(a)-(c) represents a single LP₁₁ mode rotator, Figs. 17(b) and (c) are z-symmetric and x-symmetric structures against Fig. 17(a), respectively. In the ideal case, the transfer matrix for Fig. 17(a), T_a, can be expressed as

(B1)$${\left( {\begin{array}{cccc} {{y_{\textrm{E11}}}}&{{y_{\textrm{E21}}}}&{{y_{\textrm{E12}}}}&{{y_{\textrm{E22}}}} \end{array}} \right)^\textrm{T}} = {{\mathbf T}_\textrm{a}}{\left( {\begin{array}{cccc} {{x_{\textrm{E11}}}}&{{x_{\textrm{E21}}}}&{{x_{\textrm{E12}}}}&{{x_{\textrm{E22}}}} \end{array}} \right)^\textrm{T}},$$

(B2)$${{\mathbf T}_\textrm{a}} = \left( {\begin{array}{cccc} {\exp ({j{\theta_1}} )}&{}&{}&{}\\ {}&{}&{\exp ({j{\theta_3}} )}&{}\\ {}&{\exp ({j{\theta_2}} )}&{}&{}\\ {}&{}&{}&{\exp ({j{\theta_4}} )} \end{array}} \right),$$

where x_s and y_s are the same for Eq. (A1), and θ_i (i ∈ {1, 2, 3, 4}) is the amount of phase shift for each mode. According to the Lorentz reciprocity theorem, the transfer matrix for Fig. 17(b), T_b, is given by the transpose of T_a, namely,

(B3)$${{\mathbf T}_\textrm{b}} = {\mathbf T}_\textrm{a}^\textrm{T} = \left( {\begin{array}{cccc} {\exp ({j{\theta_1}} )}&{}&{}&{}\\ {}&{}&{\exp ({j{\theta_2}} )}&{}\\ {}&{\exp ({j{\theta_3}} )}&{}&{}\\ {}&{}&{}&{\exp ({j{\theta_4}} )} \end{array}} \right).$$

Whereas, the transfer matrix for Fig. 17(c), T_c, can be expressed by

(B4)$${{\mathbf T}_\textrm{c}} ={{\mathbf C}_{\textrm{sym}}}{{\mathbf T}_\textrm{a}}{{\mathbf C}_{\textrm{sym}}} =\left( {\begin{array}{cccc} {\exp ({j{\theta_1}} )}&{}&{}&{}\\ {}&{}&{ - \exp ({j{\theta_3}} )}&{}\\ {}&{ - \exp ({j{\theta_2}} )}&{}&{}\\ {}&{}&{}&{\exp ({j{\theta_4}} )} \end{array}} \right),$$

(B5)$${{\mathbf C}_{\textrm{sym}}} =\left( {\begin{array}{cccc} 1&{}&{}&{}\\ {}&{ - 1}&{}&{}\\ {}&{}&1&{}\\ {}&{}&{}&{ - 1} \end{array}} \right),$$

which is derived from the relationship of the x-symmetry of the structure and the electro-magnetic field in the isotropic material. As illustrated in Fig. 17(d), since it is the cascaded structure of Figs. 17(a) and (b), the transfer matrix for Fig. 17(d), T_d, is given by

(B6)$${{\mathbf T}_\textrm{d}} = {{\mathbf T}_\textrm{b}}{{\mathbf T}_\textrm{a}} = \left( {\begin{array}{cccc} {\exp ({2j{\theta_1}} )}&{}&{}&{}\\ {}&{\exp ({2j{\theta_2}} )}&{}&{}\\ {}&{}&{\exp ({2j{\theta_3}} )}&{}\\ {}&{}&{}&{\exp ({2j{\theta_4}} )} \end{array}} \right).$$

Whereas, the transfer matrix for Fig. 17(e), T_e, is given by

(B7)$${{\mathbf T}_\textrm{e}} = {{\mathbf T}_\textrm{b}}{{\mathbf T}_\textrm{c}} = \left( {\begin{array}{cccc} {\exp ({2j{\theta_1}} )}&{}&{}&{}\\ {}&{ - \exp ({2j{\theta_2}} )}&{}&{}\\ {}&{}&{ - \exp ({2j{\theta_3}} )}&{}\\ {}&{}&{}&{\exp ({2j{\theta_4}} )} \end{array}} \right).$$

From Eqs. (B6) and (B7), the signs for the E₁₂ and E₂₁ modes are different, indicating the phase difference of π. On the other hand, there are no phase differences for E₁₁ and E₂₂ modes, clearly indicating the phase difference of 0. We would note that θ_i has wavelength dependence. Nonetheless, the above phase differences (0 or π) are maintained since their wavelength dependencies are canceled. Also, the transfer matrix for Figs. 17(f) and (g) are given as

(B8)$${{\mathbf T}_\textrm{f}} = {{\mathbf C}_{\textrm{sym}}}{{\mathbf T}_\textrm{d}}{{\mathbf C}_{\textrm{sym}}} = {{\mathbf T}_\textrm{d}},$$

(B9)$${{\mathbf T}_\textrm{g}} = {{\mathbf C}_{\textrm{sym}}}{{\mathbf T}_\textrm{e}}{{\mathbf C}_{\textrm{sym}}} = {{\mathbf T}_\textrm{e}}.$$

This is the reason why the relative phase difference of π arises for the E₁₂ and E₂₁ modes in the structures corresponding to Figs. 17(d) and (g). Noting that, such a behavior can be also confirmed by using the LP₁₁ mode rotator with a straight trench as in [23].

 

Fig. 17. Schematics of several configurations including LP₁₁ mode rotators.

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Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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