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We numerically and experimentally evaluate the performance of higher-order mode conversion based on phase plates for 10-mode fibers (10MFs). The phase plates have the phase jump of π between multiple planes, which match the phase patterns of linearly polarized (LP) modes of 10MF. First, we numerically investigate the effects of the fabrication errors such as the phase-difference error and the slope in the phase jump of the phase plate. The simulation results for the mode conversion to LP11 indicate that such errors make the spatial pattern of the converted beam asymmetric. In order to maintain the symmetric pattern, the phase-difference error is required to be less than ± 2%, and the ratio of the slope width to the input beam waist should be suppressed to be less than 0.05. Next, we calculate the coupling power efficiencies of the excitation of LP modes in 10MF when the converted beams after the phase plate are launched into 10MF using a lens. As the calculation results, highly accurate adjustment of the input beam waist is required to suppress the crosstalk due to coupling of undesirable LP modes by less than −20 dB. For mode excitation of LP11 or LP12, crosstalk of more than −20 dB is not avoidable even if the input beam waist is carefully adjusted. In contrast, the crosstalk for the mode excitation of LP21 or LP31 is easily suppressed to be less than −20 dB without careful adjustment of the input beam waist. These results suggest that phase plates are not applicable to mode conversion to LP11 and LP12 in 10MF while they are suitable for conversion to LP02, LP21 and LP31. Finally, we experimentally demonstrate conversion from LP01 to LP21 and LP31 modes in 10MF using phase plates. We obtain nearly ideal LP21 and LP31 modes with the small crosstalk due to the coupling of the other undesirable LP modes.
© 2014 Optical Society of America
1. Introduction
The transmission capacity over a single-mode fiber (SMF) is rapidly approaching a fundamental limitation due to light-induced catastrophic damage in a fiber (fiber fuse) [1] and fiber nonlinearity [2], whereas the demand for the traffic is still exponentially increasing. Space-division multiplexing (SDM) in few-mode fibers (FMFs) is one technique for overcoming such limits in SMF, and SDM transmission experiments in FMFs have been recently demonstrated [3–10].
For such mode-multiplexed systems, a spatial mode multiplexer is needed in order to multiplex multiple signals on many SMFs into the multiple modes of a single FMF. This requires the function of the mode conversion from the fundamental linearly polarized (LP) mode LP01 of SMF to the higher-order LP modes in FMF. Although it is not always necessary to convert individual modes of FMF for the mode-multiplexed systems based on the multiple-input multiple-output (MIMO) processing [11], mode-selective conversion [12–15] is useful for equalizing mode dependent effects such as differential group delay and mode dependent loss or gain [14].
One simple scheme for selective mode conversion is based on the use of phase plates whose phase profiles match the phase patterns of the higher-order modes [15]. Based on free-space optics, a collimated beam from the output of SMF passes through the phase plate. By launching the converted beam into FMF, the individual mode can be selectively excited in FMF. The mode conversion to LP21 has been demonstrated by using a phase plate with a phase jump of π between four quad planes [6]. For conversion to the further higher-order modes such as LP02, LP31 and LP12, more complicated phase patterns are required, and their performance has not been investigated.
In this paper, we numerically and experimentally evaluate the performance of mode conversion based on phase plates from LP01 in SMF to the ten modes, which are LP01, LP11a, LP11b, LP21a, LP21b, LP02, LP31a, LP31b, LP12a, LP12b in a 10-mode fiber (10MF). First, we numerically investigate the mode conversion to LP11 based on the phase plate with the phase jump of π between two half planes. Here, we consider fabrication errors such as a phase-difference error and the linear slope in the phase jump. The simulation results show that such errors make the spatial patterns of the converted modes asymmetric. Next, we calculate the crosstalk characteristics for excitation of ten modes using five types of phase plates in 10MF. Highly accurate adjustment of the input beam diameter is required for crosstalk suppression of less than −20 dB in mode excitation of LP01 or LP02. For excitation of LP11 or LP12, we cannot suppress crosstalk by less than −20 dB due to excitation of LP31 because of imperfection of the mode conversion based on phase plates. In contrast, the crosstalk for excitation of LP21 or LP31 can be easily suppressed without careful adjustment of the input beam diameter. Finally, we experimentally demonstrate the mode conversion to LP21 and LP31 in 10MF. Using the phase plate with a steep slope in the phase jump, we obtain nearly ideal spatial patterns of LP21 and LP31 in 10MF.
This paper is organized as follows: Section 2 presents a simulation model of mode conversion by using phase plates. Section 3 discusses the effects of the fabrication errors of phase plates on the mode conversion to LP11. Section 4 discusses numerically crosstalk characteristics for the excitation of ten modes when the converted beams after phase plates are launched into 10MF. Using the fabricated phase plates, the experiment results of mode conversion to LP21 and LP31 in 10MF are described in Section 5. Finally, Section 6 concludes this paper.
2. Simulation model of mode excitation in 10-mode fibers
Here, we discuss selective excitation of an individual mode in a step-index-type 10MF. The ten modes correspond to six types of LP modes (LP01, LP11, LP21, LP02, LP31, and LP12) and the degenerate spatial modes such as LP11a and LP11b. The spatial intensity patterns are shown in Fig. 1. The configuration of higher-order mode conversion is shown in Fig. 2. An output beam from SMF is collimated to a Gaussian beam, and then it passes through a phase plate. Finally, it is launched into 10MF using a lens. The input edge of 10MF is located at the focal point of the lens. Although it is possible to locate it at the image plane of the phase plate [8], we consider the input edge placing in the Fourier plane for its simplicity. Based on the beam propagation method (BPM) [16], we calculated the complex amplitude distributions of converted beams after phase plates at the focal point and the far-field patterns after the converted beams propagated for the free space of 5 cm. Here, we neglect the free-space propagation between the lens and the phase plate assuming the propagation of the ideal collimated beam, while the distance between the lens and the phase plate should be set to be the focal point of the lens in the real application.
We consider five types of phase plates for mode conversion to LP11, LP21, LP02, LP31 and LP12. These plates have multiple planes with different thickness corresponding to a phase jump of π. The phase mask patterns are shown in Fig. 3(a). For instance, the phase plate for conversion to LP11 has the phase jump π between two half planes. Here, we consider a phase-difference error and a linear slope in the phase jump, as shown in Fig. 3(b). We define δ and S as a normalized phase-difference error and the width of a linear slope in the phase jump, respectively.
Fig. 3 (a) Phase patterns and (b) thickness profile of phase plates for higher-order mode conversion.
Next, we consider the coupling efficiency of mode excitation in 10MF when the converted beam after the phase plate is launched into 10MF using the lens. The spatial distribution of the complex amplitude A(x,y) of the converted beam at the focal point can be described by the orthogonal combination of propagation modes and radiation modes in 10MF, as shown in Fig. 4. It is given by
where Mi(x,y) is the complex amplitude distribution of a propagation mode i, and Nj(x, y) is that of a radiation mode j. The coupling complex coefficients for the propagation mode i and the radiation mode j are ηi and ζj, respectively. 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 10MF can be described by |ηi|2 [8, 13]. Using A(x,y) and Mi(x,y), which are numerically obtained based on BPM and the finite element method (FEM) [16], the coupling efficiency |ηi|2 can be calculated and written as3. Effects of fabrication errors of phase plates
First, we numerically investigate the impact of the phase-difference error δ and the slope width S in the phase jump of the phase plate. Here, we consider that the Gaussian beam from LP01 of SMF is converted into the LP11-like beam using the phase plate with two half planes followed by the lens.
Figure 5(a) shows the calculated spatial intensity patterns of the converted beams at the focal spot when δ = 0, 1.1, 2.2, 3.3, 4.4, and 5.6%. We can see clearly two-peak patterns in the converted spatial mode for small δ. The patterns are similar to the well-known intensity pattern of LP11. The profiles of the intensity patterns along the x-axis are shown in Fig. 6(a). It is notable that the intensity patterns become more asymmetric when δ is increased. Figure 7 indicates the δ dependence of the ratio between the two peak powers in the intensity pattern. In order to maintain the ratio over 0.9, δ of less than 2% is required. The acceptable error agrees with the trend of thickness-difference errors reported by [8]. Note that the acceptable phase-difference error corresponds to the operation bandwidth of the phase plates. Assuming that the optimized wavelength is 1550 nm, the operation bandwidth of the phase plate can be expected to be over 60 nm. Indeed, the full-C-band operation of mode conversion from LP01 to LP11 based on phase plates has been reported [7].The far-field patterns after the phase plates are shown in Fig. 5(b), when δ = 0, 1.1, 2.2, 3.3, 4.4, and 5.6%. The intensity profiles along the x-axis are shown in Fig. 6(b). We can see the small dependence of the patterns on δ, in contrast to dependency of the intensity patterns at the focal point.
Fig. 5 (a) Calculated spatial intensity patterns of the converted mode at the focal spot, and (b) far-field patterns after the phase plate. δ is the phase-difference error of the phase jump of the phase plate.
Fig. 6 (a) Calculated intensity profiles of spatial patterns of converted beams at the focal spot, and (b) profiles of far-field patterns after the phase plate. δ is the phase-difference error of the phase plate.
Fig. 7 Calculated ratio between two peaks of the intensity spatial patterns at the focal point as a function of phase-difference error δ.
Next, we calculated the dependence on the slope width normalized by the input beam waist S/ω0 of 0.0, 0.1, 0.2, 0.4, 0.6 or 0.8. The calculated intensity patterns at the focal point and the far-field patterns after the phase plate are shown in Figs. 8(a) and 8(b), respectively. We found the similar patterns to the ideal intensity pattern of LP11 for small S/ω0, although the patterns were more distorted with larger S/ω0. Note that the increase of the slope width gives rise to the distortion of both the spot patterns and the far-field patterns. Their intensity profiles are shown in Figs. 9(a) and 9(b). The ratios between the two peaks of the intensity profiles at the focal point are plotted in Fig. 10. The normalized slope width is required to be less than 0.05 to maintain the symmetric pattern with the peak ratio of more than 0.9.
Fig. 8 (a) Calculated spatial intensity patterns of the converted mode at the focal spot, and (b) far-field patterns after the phase plate. The parameter is the slope width S normalized by beam waist ω0.
Fig. 9 (a) Intensity profiles of converted patterns at the focal point, and (b) profiles of far-field patterns after phase plates. S is the slope width and ω0 is the beam waist.
4. Simulation of selective mode excitation in 10-mode fiber
We simulated the coupling efficiency of the excited mode when the converted beam after the phase plate was launched into 10MF in order to evaluate the crosstalk due to the coupling of undesirable modes. Here, we calculated the dependency on the beam waist at the focal spot, at which the input edge of 10MF is located, when ideal LP modes and the converted beams after phase plates were launched.
4.1 Input of ideal LP modes
Assuming that it is possible to convert to the ideal LP modes by the phase plates, we first calculated the efficiency coupled to each LP mode when the ideal LP mode was launched into 10MF. The dependency on the beam waist of the launched LP mode is shown in Fig. 11. The calculated results in the vertical axis are normalized by the ideal coupling efficiency when each ideal LP mode is launched. In the horizontal axis, the input beam waist normalized by the beam waist of each ideal LP mode is indicated.
Fig. 11 The normalized efficiency coupled to each LP mode when launching the ideal (a) LP01, (b) LP11a, (c) LP21a, (d) LP31a, (e) LP02, or (f) LP12a mode of 10MF. : LP01, : LP11a, : LP11b, :LP21a, : LP21b, : LP02, : LP31a, : LP31b, : LP12a, and : LP12b.
Figure 11(a) shows the calculated results when the ideal LP01 mode was launched. The LP01 mode can be excited just without any coupling to other modes, provided that the beam waist is ideal. When the beam waist is not optimized, the crosstalk by the coupling of LP02 is not negligible. For input of the LP02 mode, the results are shown in Fig. 11(d). The LP01 coupling is enhanced when the beam waist is not optimized. These results suggest that the coupling between LP01 and LP02 is very sensitive. This is expected from the fact that the center parts of LP01 and LP02 are overlapped, as shown in Fig. 1. The results for launching LP11a or LP12a are shown in Figs. 11(b) and 11(f), respectively. We found that the LP11a and LP12a are easily coupled to each other if the beam diameter of the input mode is not optimized. This is because their special patterns are strongly overlapped, similarly to the coupling between LP01 and LP02 as mentioned above.
The calculated efficiencies for launching the ideal LP21a and LP31a are shown in Figs. 11(c) and 11(e), respectively. It is apparent that the same mode as the launched mode is efficiently excited without any crosstalk due to the excitation of other LP modes. Note that this is almost independent on the beam waist of the input mode.
The calculation results of LP11b, LP21b, LP31b, and LP12b, which are not shown in Fig. 11, denote the same tendency of their degenerated modes of LP11a, LP21a, LP31a, and LP12a, respectively.
4.2. Input of converted beams by phase plates
Next, we consider that the converted beams by five types of phase plates whose phase patterns are shown in Fig. 3(a) are launched into 10MF. Figure 12 illustrates the spatial intensity patterns of the converted beams by phase plates at the focal spot. We found a slight difference between the converted beams and the ideal LP modes as shown in Fig. 1. In order to investigate numerically the crosstalk originated from such conversion errors, we calculated the efficiencies coupled to the LP modes for launching the converted beams after phase plates. Figure 13 shows the dependence of the calculated efficiency on the beam waist normalized by the ideal waist of the desirable LP mode. Similarly to Fig. 11, the calculated results are normalized by the optimized efficiency for the minimal crosstalk from the undesirable modes.
Fig. 13 The normalized efficiency coupled to each LP mode when launching the converted beam for (a) LP01, (b) LP11a, (c) LP21b, (d) LP02, (e) LP31b, or (f) LP12b mode of 10MF. : LP01, : LP11a, : LP11b, :LP21a, : LP21b, : LP02, : LP31a, : LP31b, : LP12a, and : LP12b.
Figure 13(a) indicates the calculated results when the Gaussian beam was launched without any phase plate. We can excite the LP01 mode only without coupling to other modes as long as the waist of the input beam is optimized. The coupling to the LP02 mode is more enhanced when the waist of the input beam is far from the optimum. Figure 13(d) shows the calculated efficiencies for launching the LP02-like beam converted by the phase plate with the circular pattern. The LP02 excitation without small coupling to other LP modes is possible at the optimal beam waist input, while the LP01 mode coupling is not negligible when the input beam waist is far from the optimum point. This coupling between LP01 and LP02 is similar to the results when the ideal LP01 and LP02 mode were launched, as shown in Figs. 11(a) and 11(d), respectively. 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.
The calculation results for launching the LP11a-like beam after the phase plate are shown in Fig. 13(b). The LP12a mode would be coupled if the waist of the input beam is far from the optimum. Note that LP31b coupling is not negligible in this case. It is not possible to suppress the crosstalk from the coupling of LP31b to be less than −20 dB even though the input beam waist is carefully adjusted [13]. 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 Fig. 1 and Fig. 12. The difference leads to the crosstalk by the coupling of LP31b. In the same manner, the undesirable LP31b mode would be coupled when the LP12a-like beam is launched, as shown in Fig. 13(f).
Figures 13(c) and 13(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 without carefully adjusting the input beam waist. Although the spatial patterns 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.
5. Experiments of higher-mode conversion to LP21 and LP31 in 10MF
As mentioned in Section 4, the LP21 and LP31 modes can be converted with the small crosstalk from other LP modes in 10MF. Finally, we experimentally demonstrated the mode conversion to LP21 and LP31 in 10MF by launching the converted beam after the phase plates having four quad planes and six-divided planes, which are illustrated by the second left and the second right in Fig. 3(a), respectively. In addition, we evaluated the effects of the slope width in the phase jump of the phase plates experimentally.
The experimental setup is shown in Fig. 14. A CW light output from SMF was collimated with a beam waist of around 1 mm, and then the collimated beam passed through a phase plate. The converted beam after the phase plate was launched into 10MF using a lens. The input edge of 10MF was located at the focal point of the lens. Using a CCD camera, we measured the far-field pattern after the phase plate and the output spatial pattern from 10MF.
The phase plates were fabricated from SiO2-glass plates with phase patterns matching the phases of ideal LP21 and LP31. Here, we used two types of processes to create the phase patterns for evaluation of the effects of the slope in the π phase jump, as mentioned in Section 3. One was the use of the physical vapor deposition. This created phase pattern had a relatively large slope width around 200 µm. The other was based on the dry etching process. In the phase pattern fabricated by this process, the slope width was maintained to be less than 0.1 µm. For both processes, the phase jump error could be suppressed to be less than ± 2%.
Figure 15(a) shows the far-field pattern of the LP21-like beams converted by the phase plates with the slope width of 200 µm. We can see the asymmetry in the spatial pattern, which is similar trend to the simulation results for the LP11 conversion, as shown in Fig. 8(b). Using the phase plates with a slope width of less than 0.1 µm, the measured far-field pattern is shown in Fig. 15(b), and the calculated pattern is indicated in Fig. 15(c). The symmetric pattern was observed, and it agreed with the simulation result. These results suggest that the asymmetry of the pattern of the converted beam originates from the slope in the phase jump of the used phase plate. In addition, we measured the far-field pattern of the LP31-like beam converted by the phase plate with six-divided planes. The slope in the phase jump was maintained to be less than 0.1 µm. The measured result is shown in Fig. 16(a), revealing a very nice symmetric pattern. The calculated pattern after the phase plate for LP31 conversion is shown in Fig. 16(b). It is apparent that there is good agreement between the experimental and calculated results.
Fig. 15 Measured far-field pattern after phase plates for excitation of LP21. The use of the phase plates with the slope width of (a) more than 200 µm, and (b) less than 0.1 µm. (c) Calculated far-field pattern.
The measured output spatial patterns from 10MF with a length of 1 km using the phase plates for LP21 and LP31 conversion are shown in Figs. 17(a) and 17(b), respectively. There are clearly four and six peaks of patterns, which are very similar to the well-known intensity patterns of the LP21 and LP31 modes, respectively. These suggest that coupling of other undesirable LP modes is sufficiently suppressed without careful adjustment of the input beam waist. This is consistent with the simulation results as mentioned in Section 4.
6. Conclusion
We numerically and experimentally investigated the conversion of the higher-order modes in 10-mode fiber (10MF) using the phase plates. First, we numerically investigated the effects of fabrication errors of phase plates such as the phase-difference error and the slope in the phase jump for the LP11 conversion. Such errors make the special patterns of converted beams after phase plates asymmetry. Next, we simulated the excitation of LP modes when the converted beams by the phase plates were launched into 10MF. We indicated the difference between the patterns of converted beams by using phase plates and those of ideal LP modes. The difference gives rise to the coupling of undesired LP modes, and the crosstalk due to the coupling of undesirable LP modes was numerically evaluated. Adjustment of the input beam waist with an accuracy of less than ± 5% is required for the coupling of LP01 or LP02 in order to suppress the crosstalk. In contrast, the crosstalk cannot be suppressed to be less than −20 dB for the excitation of the LP11 or LP12 mode, even though the input beam waist is optimized. On the other hand, it is easily possible for excitation of the LP21 and LP31 modes to suppress the crosstalk without careful adjustment of the input beam waist. Finally, we experimentally demonstrated the conversion to LP21 and LP31 in 10MF using the phase plates. We observed output intensity patterns from 10MF that were very similar to those of the ideal well-known LP21 and LP31 modes.
Acknowledgments
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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