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We propose a novel mode multiplexer based on phase plates followed by a Mach-Zehnder interferometer (MZI) with image inversion. After the higher-order modes are selectively converted from fundamental linear-polarized (LP) modes by the phase plates, the converted modes are coupled without fundamental loss using MZI with image inversion, in which the original spatial pattern and inverted pattern of the optical signal are interfered. Our scheme is also applicable to the coupling of degenerated LP modes such as LP11a and LP11b. First, we numerically and experimentally evaluate the performance of the mode converter based on phase plates. The mode converter is suitable as long as the five LP modes such as LP01, LP11ab and LP21ab are sustained in a few-mode fiber (FMF), although the crosstalk due to excitation of undesirable modes is unavoidable when the higher-order modes over LP02 are sustained in FMF. Next, we develop and characterize the proposed mode multiplexers based on phase plates and MZIs with image inversion. The insertion loss is suppressed to around 3 dB for mode multiplexing of LP11a and LP11b. Using a fabricated mode multiplexer for LP31a and LP31b, we measure the bit-error rate performance of single-polarization mode-multiplexed quadrature-phase shift keying optical signals.
© 2015 Optical Society of America
1. Introduction
The transmission capacity in single-mode fibers (SMFs) has increased rapidly by ultra-dense wavelength-division multiplexing (WDM) techniques [1, 2], the advanced modulation format with digital coherent reception [3, 4], and forward error correction (FEC) [5]. Transmission experiments with over 100-Tbit/s capacity have been reported [6, 7], while the total input power of all WDM channels before the fiber is required to be over 1 Watt in order to maintain the optical signal-to-noise ratio (OSNR). This is now approaching a fundamental limitation due to light-induced catastrophic damage in fiber [8] and fiber nonlinearity [9].
The space division multiplexing technique in few-mode fibers (FMFs) [10–12] is one technique used to overcome such limitation in the single-core SMF, and the transmission experiments have already been reported [13–15]. In the mode-multiplexed systems, the mode multiplexer is required to convert multiple optical signals on many SMFs into multiple modes in a single FMF. As shown in Fig. 1, such function could be achieved by mode conversion from fundamental linear-polarized (LP) modes to higher-order LP modes followed by mode coupling of the converted modes.
Various schemes for mode multiplexing have been reported and applied to the transmission experiments. They can be categorized by the operation principle and the device used for mode conversion, as summarized in Table 1. The most well-known technique is based on the use of a hologram such as a spatial light modulator (SLM), which manipulates spatial intensity and phase distribution of a light beam to create those of the desirable mode [16–19]. The conversion error due to the spatial resolution limit of the SLM pixel is significant, although the mode coupling is also possible without fundamental loss in principle. A much simpler scheme is the use of a phase plate manipulating just the phase distribution of the beam [20–22]. Since a photolithographic process can be used to create the phase mask, the very high resolution of the phase pattern is achieved. It is widely used for experiments because of its simplicity, even although there is not only relatively large error of the mode conversion [19, 22] but also large loss for the external mode coupling such as a beam splitter (BS). By using multiple phase plates with the optical Fourier transform, manipulation of not only the phase pattern but also the intensity pattern was recently reported [23]. Another approach is based on coupling between waveguide modes that satisfy the phase matching condition. Using a coupler composed of two approximate waveguides, two modes having the same effective refractive indexes can be coupled, achieving in mode multiplexing as well as two mode coupling [24–27]. Using couplers based on planer lightwave circuits (PLCs) [25] or optical fibers [26, 27], the low-loss and low-crosstalk characteristics have been demonstrated, while coupling of the degenerated modes with the same effective indexes is not possible. There have been reports about mode conversion based on long-period grating (LPG) in FMF so as to maintain the phase matching between the coupled modes, although the external mode coupling is required [10, 28]. Recently, adiabatic transition in a photonic lantern that converts multiple beams into the super-mode in FMF has attracted attention [13, 15, 29–31]. It has been introduced to coupled-mode transmission experiments, while the multiple-input multiple-output (MIIMO) processing is required to compensate the signal crosstalk due to the mode coupling. The selective mode multiplexing using the photonic lantern has been reported [31].
Table 1. Classification of mode multiplexers proposed so far.
We propose a novel mode multiplexer based on phase plates and Mach-Zehnder interferometer (MZI) with image inversion, in which the original spatial image and the inverted image of the incoming beam are interfered [32]. Instead of BSs for mode coupling, the MZI with image inversion is used without any fundamental loss. This scheme is applicable to multiplex five modes such as LP01, LP11a, LP11b, LP21a and LP21b. In this paper, we numerically and experimentally evaluate the performance of the mode converter based on phase plates. The results show that the crosstalk from undesirable modes cannot be suppressed to less than −20 dB when the higher-order modes over LP02 are sustained in FMF. Next, we experimentally evaluate the performance of the proposed mode multiplexer based on phase plates followed by MZI with image inversion. We measure the bit-error rate (BER) performance of single-polarization quadrature-phase shift keying (QPSK) signals over LP31a and LP31b modes using our proposed multiplexer.
2. Configuration of our proposed mode multiplexer
Figure 2(a) shows the schematic configuration of the proposed mode multiplexer, which is composed of a mode conversion part and a mode coupling part. In the mode conversion part, Gaussian-like beams of optical signals output from SMFs are converted into the higher-order modes using the phase plate, whose phase pattern is created so that the π phase jump matches that of the desirable mode, as shown in Fig. 2(b). In the mode coupling part, the converted modes are combined by MZI with image inversion function, whose schematic diagram is shown in Fig. 3(a). The image inversion function is inserted in only one path of MZI, and it is easily given by the difference in the number of reflections between two optical paths of MZI, as shown in Fig. 3(b). With the image inversion, the spatial pattern is inverted with respect to the x axis, as in the case shown in Fig. 3(a). Here, we consider that odd and even beam patterns with respect to the x axis are launched into the different input ports of MZI. By the image inversion, the odd pattern is inverted with respect to the x axis, whereas the even pattern is not changed. After the beam combiner, the even and odd patterns are interfered and coupled into only one output port, resulting in the ideal mode coupling without any fundamental loss. This operation does not have polarization dependence as long as the polarization in the optical paths is maintained.
Fig. 3 (a) Schematic diagram and (b) free-space configuration of Mach-Zehnder interferometer with image inversion. BS: beam splitter, M: mirror.
Next, we consider the demultiplexing operation when the mode-multiplexed signals are launched into our proposed scheme from the output port of MZI. Fist, the odd pattern and even pattern are separated by the MZI with image inversion. After that, the desirable mode is converted into the fundamental LP01 mode by the phase plate. Our scheme is also applicable to the mode demultiplexing.
Figure 4 indicates application of the proposed scheme to multiplexing of five types of LP modes. This is the tree configuration of mode multiplexers based on the phase plates and MZIs with image inversion function. First, LP01 and LP21a converted by the phase plates are coupled by MZI with image inversion with respect to the vertical axis. Second, their modes and LP11a are combined by image inversion with respect to the y axis. Using another multiplexer with respect to the y axis, LP11b and LP21b are combined. The patterns of LP01, LP21a and LP11a are the even functions with respect to the x axis, while those of LP11b and LP21b are the odd function. Their coupling can be achieved by the image inversion with respect to the x axis. Note that each layer of the tree configuration has the different direction of the axis of the image inversion. In the case of Fig. 4, the image inversion with respect to the horizontal axis is used for the first layer. For the second and third layers, the y axis and the x axis are used, respectively. Consequently, the multiplexing of five types of LP modes is achieved using four types of the proposal multiplexers.
Fig. 4 Configuration of the mode multiplexer for five modes such as LP01, LP11a, LP11b, LP21a and LP21b.
3. Performance evaluation of higher-order mode converter based on phase plates
First, we numerically evaluate the crosstalk characteristics of the higher-order mode conversion based on phase plates [22]. The configuration of the mode conversion is shown in Fig. 5(a). The beam output from SMF is collimated, and then it passes through a phase plate. After that, the converted beam is launched into FMF using a lens. Here, we consider the input edge of FMF placed in the Fourier plane for its simplicity, although it is possible to locate it at the image plane of the phase plate. The spatial distribution of the complex amplitude A(x,y) of the converted beam at the focal point is described by the orthogonal combination of propagation modes and radiation modes in FMF, as shown in Fig. 5(b). It is given by
where the complex amplitude distribution of a propagation mode i and a radiation mode j are Mi(x,y) and Nj(x, y), respectively, and their coupling complex coefficients are ηi and ζj. The number of the propagation modes in FMF is m – 1. Here, we neglect the radiation modes assuming that they would not be coupled to the propagation modes. The coupling efficiency for the excitation of mode i in FMF is described by |ηi|2 [19, 21, 22]. Using A(x,y) and Mi(x,y), which are numerically obtained based on the finite element method (FEM) and the beam propagation method (BPM) [33], the coupling efficiency |ηi|2 is calculated and written aswhere * denotes the complex conjugation.Fig. 5 (a) Configuration of mode converter based on the phase plate. (b) Model for excitation of modes in FMF.
Here, we consider that the converted beams by five types of phase plates whose phase patterns are shown in Fig. 2(b) are launched into FMF, in which ten modes (LP01, LP11ab, LP21ab, LP02, LP31ab, and LP12ab) are sustained. The spatial intensity patterns of ideal modes of FMF and the converted beams by the phase plates are shown in Figs. 6(a) and 6(b). These were obtained by the calculation based on FEM and BPM [33]. There was a slight difference between the ideal LP modes and the converted beams. The difference would have originated from the incomplete conversion of the phase plates, which cannot manipulate the intensity patterns. In order to investigate the crosstalk originated from such conversion errors, we calculated the efficiencies coupled to the LP modes when the converted beams after phase plates are launched into FMF. Although the calculation results were already reported in our previous paper [22], these are introduced here in order to make us understand well the characteristics of the phase plate for the higher-order mode conversion. Figure 7 shows the dependence of the calculated efficiency on the beam waist. The vertical and the horizontal axis are normalized by the optimized efficiency and the beam waist for the minimal crosstalk from the undesirable modes, respectively. For the normalized beam waist of less than 1.0, the calculation error would be relatively large due to the limitation of the spatial resolution, which is determined by the wavelength of the beam in BPM. The calculation results of LP11b, LP21a, LP31a, and LP12b, which are not shown in Fig. 7, denote the same tendency of their degenerated modes of LP11a, LP21b, LP31b, and LP12a, respectively.
Fig. 7 The normalized efficiency coupled to each LP mode when launching the ideal (a) LP01, (b) LP11a, (c) LP21b, (d) LP02, (e) LP31b, or (f) LP12a mode of FMF. : LP01, : LP11a, : LP11b, :LP21a, : LP21b, : LP02, : LP31a, : LP31b, : LP12a, and : LP12b.
Figure 7(a) indicates the calculated results when the Gaussian beam was launched without any phase plate. The coupling to the undesirable LP02 mode is more enhanced when the waist of the input beam is far from the optimum. Figure 7(d) shows the calculated efficiencies for launching the LP02-like beam converted by the phase plate with the circular pattern. The crosstalk due to the coupling of the LP01 mode is significant when the input beam waist is far from the optimum point. The adjustment of the input beam waist within ± 5% is required to suppress the crosstalk from other LP modes to be less than −20 dB. Such high-accuracy adjustment of the beam waist seems to be distant.
The calculation results for launching the LP11a-like beam after the phase plate are shown in Fig. 7(b). The LP12a mode would be coupled if the waist of the input beam is far from the optimum. Note that LP31b coupling is significant in this case. It is not possible to suppress the crosstalk from the coupling of LP31b to less than −20 dB even though the input beam waist is carefully adjusted [20, 22]. This is because the spatial patterns of the converted beams after the phase plates have longer tail components, which are different from ideal LP11a, as shown in Figs. 6(a) and 6(b). In the same manner, the undesirable LP31b mode would be coupled when the LP12a-like beam is launched, as shown in Fig. 7(f).
Figures 7(c) and 7(e) show the calculated coupling efficiencies for input of the LP21b-like and LP31b-like beams after the phase plates, respectively. The coupling of other LP modes can be suppressed to be less than −20 dB even though the input beam waist is not carefully adjusted. Although the spatial patterns after the phase plates seem to be different from those of the ideal LP21b and LP31b, the conversion error would be coupled to radiation modes rather than propagation modes.
These results suggest that the mode converter based on the phase plate is suitable for FMF in which the propagated modes are limited to the lower five LP modes (LP01, LP11a, LP11b, LP21a and LP21b), while the crosstalk from undesirable modes is unavoidable when the higher-order modes over LP02 are sustained in FMF.
Next, we experimentally evaluated the crosstalk characteristics of the mode conversion based on phase plates using the S2 measurement [34]. The experimental setup is shown in Fig. 8(a). A CW light was generated from a tunable laser, and it was input to a mode converter. The mode converter was composed of a phase plate, collimator lenses, SMF and FMF, as shown in Fig. 5(a). In this experiment, we used FMF in which ten modes (LP01, LP11ab, LP21ab, LP02, LP31ab, and LP12ab) are sustained. The near-field patterns of the converted beams after the mode conversion were measured by a 2-dimension InGaAs detector. Performing the Fourier transformation of the wavelength dependence of the pattern, we obtained the intensity impulse response after the mode convertor with FMF. The impulse response of each mode can be separated, because each mode would propagate with the different group delay in FMF. Extracting the component corresponding to each mode, the crosstalk power and the spatial intensity pattern can be estimated [34].
Fig. 8 (a) The experimental setup for S2 measurement, and the spatial intensity patterns of the observed LP modes when the phase plate for (b) LP11 or (c) LP31 is used.
Figure 8(b) shows the intensity patterns of the observed modes when we used the mode converter for LP11. We found the mainly-excited LP11 mode with the undesirable modes. The coupling to LP31 and LP12 modes was clearly observed, as was expected from the calculated results shown in Fig. 7(b). Although there was coupling to LP21, it would be because of misalignment of the free-space optics.
Figure 8(c) shows the measured results for the mode conversion to LP31. In this case, there were no undesirable modes, although the mainly-excited LP31 mode was clearly found. These experimental results are as expected from the calculated results as shown in Fig. 7(e).
4. Performance evaluation of our proposed mode multiplexer
We fabricated two types of proposed mode multiplexers for LP11ab and LP31ab. These multiplexers consisted of phase plates for the mode conversion of LP11 or LP31 and MZI with the image inversion function for coupling degenerated modes. Figure 9(a) shows the configuration of the fabricated multiplexer. A lightwave output from SMF was collimated, and then it was converted into the desirable mode by a phase plate for LP11 or LP31. The lightwave launched into one SMF port was converted into LP11a or LP31a, and another SMF port was converted into LP11b or LP31b. After that, two degenerated modes were combined by MZI with the image inversion. Both paths of MZI were several-cm long, and they were maintained by the Peltier-based temperature controller to ensure their lengths were matched. Figure 9(b) shows a photograph of the developed module, whose size is 160 mm × 120 mm × 55 mm.
Figures 10(a) and 10(b) show the near-field patterns of LP11ab and LP31ab modes measured at the FMF side, when a CW light at the wavelength of 1550 nm was launched into each SMF port. The multiplexing operation was successfully performed. Compared with LP31ab, the pattern of LP11ab was slightly distorted because the undesirable modes would be excited as mentioned in Section III. The wavelength dependence of the insertion loss for LP11a and LP11b modes is plotted by dots and closed triangles in Fig. 11, respectively. In this case, we tuned the optical paths of MZI by temperature controller so that the minimal loss was obtained. For comparison, open circles and open triangles indicate the results using a conventional mode converter for LP11a and LP11b. The conventional scheme is based on phase plates with BSs [12]. In the case of the conventional scheme, the insertion loss was about 5 dB including the fundamental loss of BS. In contrast, the loss of the proposed module was less than 3 dB. The mode conversion error due to imperfection of the phase plate and the coupling loss to FMF would be included in both cases. The insertion loss of the proposed module for LP31a and LP31b modes was measured to around 6.5 dB at the wavelength of 1550 nm. This mode-dependent loss would be originated from the fiber coupling and the conversion error in the phase plate rather than MZI. The mode dependent loss of the MZI would be close to the same as that of BS in principle.
Fig. 10 The near-field patterns of the output beams from our proposed multiplexer. (a) LP11a and LP11b, (b) LP31a and LP31b.
Fig. 11 Wavelength dependence of the insertion loss of the LP11 mode multiplexer. Dots and closed triangles: LP11a and LP11b in the case of the proposed scheme using MZI with image inversion, open circles and open triangles: LP11a and LP11b in the case of the conventional scheme using the beam splitter (BS).
Finally, we evaluated the BER performance of single-polarization QPSK signals over LP31a and LP31b modes using the proposed mode multiplexer in the back-to-back configuration. Figure 12 shows the experimental setup. Two 32-Gbaud single-polarization QPSK signals were launched into two input SMF ports of the proposed LP31 mode multiplexer. After that, the LP31a and LP31b modes were separated by a conventional multiplexer based on the phase plates with BS. These demultiplexed signals were received by the digital coherent receiver. The 2 × 2 complex MIMO equalizer was performed to discriminate between LP31a and LP31b modes in the offline digital signal processing. The measured BERs of the QPSK signals over the LP31a and LP31b modes are plotted by the dots in Fig. 13. For comparison, the open circles indicate the measured results of the QPSK signal without mode multiplexing. In this experiment, the received OSNR was changed by loading amplified spontaneous emission (ASE) noise in the transmitter side. No significant BER penalty was observed due to the mode multiplexing of LP31a and LP31b.
Fig. 12 The experimental setup for BER measurement of single-polarization QPSK signals over LP31a and LP31b modes.
Fig. 13 Measured bit-error rates of single-polarization QPSK signals with and without mode multiplexing of LP31a and LP31b.
5. Conclusion
We proposed a mode multiplexer based on phase plates and MZI with image inversion function. The desirable higher-order modes are converted from LP01 using the phase plates, and then they are coupled without the fundamental loss using MZI with image inversion. First, we evaluated the performance of the mode converter based on phase plates. The calculation and experimental results show that the mode converter is suitable as long as the propagated modes are limited to five LP modes (LP01, LP11ab and LP21ab) in FMF. The crosstalk due to the coupling of undesirable modes is unavoidable when the higher-order modes over LP02 are sustained in FMF. Next, we evaluated the performance of the proposed mode multiplexer for degenerated LP modes. For mode multiplexing of LP11a and LP11b, the insertion loss was suppressed to around 3 dB. In addition, we measured BER of the single-polarization QPSK signals over LP31a and LP31b modes using the proposed mode multiplexer. There was no remarkable BER penalty due to the mode multiplexing.
Acknowledgments
We would like to thank Mr. R. Otani from Sigmakoki Co. Ltd. for fabricating MZI with image inversion. A part of the research results have been achieved by the Commissioned Research of the National Institute of Information and Communications Technology (NICT), Japan.
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