GAWBS phase noise characteristics in multi-core fibers for digital coherent transmission

Naoya Takefushi; Masato Yoshida; Keisuke Kasai; Toshihiko Hirooka; Masataka Nakazawa

doi:10.1364/OE.399202

1. Introduction

Digital coherent quadrature amplitude modulation (QAM) transmission has enabled a large increase in spectral efficiency through the use of the high coherence of optical waves by encoding data on both the amplitude and phase with multiple levels. While higher multiplicity is advantageous for increasing the transmission capacity, it also increases the vulnerability to amplitude and phase noise. Therefore, highly precise compensation techniques for such signal impairments as chromatic dispersion and Kerr effects are very important for the long-haul transmission of QAM signals with high multiplicity. Recently, guided acoustic wave Brillouin scattering (GAWBS) [1,2] has been found to be a new source of signal impairment in digital coherent transmission [3–11]. GAWBS is a spontaneous forward scattering phenomenon in optical fibers, in which the propagating light interacts with the acoustic resonant vibrations of thermally excited transverse waves induced in a quartz glass under a condition of thermal equilibrium. The scattered light acts as phase and polarization noise and is responsible for a large error floor in QAM transmission with ultrahigh multiplicity such as 1024 QAM [6,8]. We previously reported the detailed measurement of GAWBS phase and polarization noise in various types of single-core fibers (SCF) and evaluated their characteristics including the dependence on the effective area and the impact on digital coherent transmission [9–11].

Recently, multi-core fibers (MCF) have received a lot of attention as a new transmission medium with a view to achieving ultrahigh-capacity space division multiplexing (SDM). In MCF, the cores are located both in center and off-center positions in the cross section. Inter-core cross-talk caused by stimulated GAWBS in a seven-core fiber has been studied theoretically and experimentally. Here, acoustic waves were stimulated in the fiber cross section by pumping a coherent light wave modulated at the GAWBS resonance frequencies [12]. However, with an SDM transmission, the signal launched into the MCF is modulated with a random data sequence, and the signal bandwidth is much wider than that of GAWBS. Therefore, the influence of the stimulated GAWBS on the SDM transmission is negligible. In contrast, spontaneous GAWBS occurs when light interacts with a transverse acoustic vibration induced in a quartz glass fiber under thermal equilibrium conditions, which may affect the SDM transmission performance.

With respect to the spontaneous GAWBS induced in MCF, the depolarization noise in a seven-core fiber has been measured with a direct detection method, where inter-core cross-talk induced by GAWBS was not observed [13]. On the other hand, we previously reported preliminary work on the measurement of phase noise induced by spontaneous GAWBS in a four-core fiber with heterodyne detection [14]. However, the influence of GAWBS phase noise on MCF transmission, especially the dependence of the GAWBS noise on the core location, has yet to be clarified.

In this paper, we present a detailed experimental and analytical evaluation of GAWBS phase noise in MCF, with a special focus on their dependence on the core geometry. We first describe GAWBS phase noise measurements in an uncoupled four-core fiber (4CF) with a 125 µm cladding and show that the phase noise caused by the R_0,m mode is greatly reduced while the noise due to higher-order modes such as TR_1,_m, TR_3,_m, and TR_4,_m is observed for the first time. We then report the GAWBS phase noise in an uncoupled 19-core fiber (19CF) with a larger cladding diameter, highlighting the dependence of the phase noise on the core locations.

2. GAWBS phase noise characteristics in uncoupled four-core fibers

We first measured the GAWBS phase noise spectrum in a 36 km-long uncoupled 4CF using heterodyne detection. The crosstalk between cores is less than −40 dB. The heterodyne detection setup is shown in Fig. 1. The 4CF had a cladding diameter of 125 µm and a core pitch of 44.2 µm [15]. Each core had a trench-assisted pure-silica core profile with an effective core area A_eff of 88-89 µm², whose cross-sectional view is shown in the inset of Fig. 1. As a transmitter and local oscillator (LO) source, we used CW fiber lasers with a linewidth of 4 kHz, which is sufficiently narrower than that of GAWBS noise (∼ MHz). At the receiver, the polarizations of the transmitted light and LO were aligned with a polarization controller, and they were then mixed with a photo diode (PD) to obtain a GAWBS phase noise spectrum as a heterodyne detected RF signal. The intermediate frequency (IF) was 1 GHz. For comparison, we also measured the GAWBS phase noise in a 35 km-long standard single-mode fiber (SSMF) whose length, cladding diameter (125 µm) and A_eff (80 µm²) are almost identical to that of 4CF.

Fig. 1. Heterodyne detection setup for observing GAWBS phase noise spectra generated in 36 km 4CF and 35 km SSMF.

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Figures 2(a) and 2(b) show the GAWBS phase noise spectrum measured in a 36 km 4CF and a 35 km SSMF, respectively. In these figures, the phase noise components resulting from the R_0,1 ∼ R_0,10 modes are highlighted by red arrows. As the SSMF and 4CF have the same cladding diameter, the resonant frequencies are the same for the two fibers. In Fig. 2(a), where the phase noise spectra at four cores are superimposed, it can be seen that the four cores exhibit the same phase noise. We also note that the power levels of the R_0,m components indicated by the arrows are greatly reduced in 4CF compared with SSMF, but a larger number of small components, which have the appearance of a continuous spectrum, are newly observed in 4CF. This makes it difficult to distinguish the peaks of the R_0,m modes. The ratio of the phase noise magnitude in both fibers, obtained by integrating the power spectrum, was 1 (SSMF): 0.917 (4CF), which indicates that the influence of GAWBS in 4CF is comparable to that in SSMF.

Fig. 2. Experimentally measured GAWBS phase noise spectra of an optical carrier transmitted in (a) a 36 km four-core fiber with a 125 µm cladding (Core #1∼#4) and (b) a 35 km SSMF.

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Next, we analyzed the GAWBS phase noise spectrum in 4CF numerically. The GAWBS-induced refractive index change Δn originates from thermally excited acoustic vibrations in silica glass under equilibrium conditions. These acoustic waves satisfy a boundary condition where the stress on the fiber surface is zero. The boundary condition that a general TR mode satisfies is given by the following equation [16]

(1)$$\left|{\begin{array}{cc} {\left( {{n^2} - 1 - \frac{{{y^2}}}{2}} \right){J_n}(\alpha y)}&{\left[ {n({n^2} - 1) - \frac{{{y^2}}}{2}} \right]{J_n}(y) - ({n^2} - 1)y{J_{n + 1}}(y)}\\ {(n - 1){J_n}(\alpha y) - \alpha y{J_{n + 1}}(\alpha y)}&{\left[ {n(n - 1) - \frac{{{y^2}}}{2}} \right]{J_n}(y) + y{J_{n + 1}}(y)} \end{array}} \right|= 0,$$

where α is the ratio of the transverse sonic velocity V_s ( = 3740 m/s) to the longitudinal sonic velocity V_d ( = 5996 m/s) in quartz glass, and J_n(y) is an n-th order Bessel function. The solution of Eq. (1), y_n,m, gives the normalized frequency of an acoustic wave, where n and m give the oscillation order in the radial and azimuthal directions, respectively. The resonant frequency f_n,m is then given by

(2)$${f_{n,m}} = \frac{{{V_s}}}{{2\pi a}}{y_{n,m}}\textrm{ },$$

where a ( = 62.5 µm) is the fiber radius. The displacement of the acoustic wave on the fiber surface is given by [16]

(3.1)$${U_r}(r,\theta ) = {C_{n,m}}{u_r}\left( {\frac{r}{a}} \right)\cos n\theta $$

(3.2)$${U_\theta }(r,\theta ) = {C_{n,m}}{u_\theta }\left( {\frac{r}{a}} \right)\sin n\theta $$

(4.1)$${u_r}\left( {\frac{r}{a}} \right) = {A_1}\frac{n}{r}{J_n}\left( {\frac{{{y_{n,m}}r}}{a}} \right) - {A_2}\frac{\partial }{{\partial r}}{J_n}\left( {\frac{{\alpha {y_{n,m}}r}}{a}} \right) $$

(4.2)$${u_\theta }\left( {\frac{r}{a}} \right) ={-} {A_1}\frac{\partial }{{\partial r}}{J_n}\left( {\frac{{{y_{n,m}}r}}{a}} \right) + {A_2}\frac{n}{r}{J_n}\left( {\frac{{\alpha {y_{n,m}}r}}{a}} \right)\textrm{ }, $$

where C_n,m is the acoustic wave amplitude caused by thermal vibration [1,2], and A₁ and A₂ are defined as

(5.1)$${A_1} = n\left( {{n^2} - 1 - \frac{{{y^2}}}{2}} \right){J_n}({\alpha y} ) $$

(5.2)$$\textrm{ }{A_2} = \left[ {n({n^2} - 1) - \frac{{{y^2}}}{2}} \right]{J_n}(y )- ({n^2} - 1)y{J_{n + 1}}(y )\textrm{ }.$$

In accordance with Hooke’s law, the displacements U_r(r,θ) and U_θ(r,θ) introduce the strains S_rr(r,θ), S_θθ(r,θ) and S_rθ(r,θ), which are given by

(6.1)$${S_{rr}}(r,\theta ) = \frac{{\partial {U_r}}}{{\partial r}} $$

(6.2)$${S_{\theta \theta }}(r,\theta ) = \frac{{{U_r}}}{r} + \frac{1}{r}\frac{{\partial {U_\theta }}}{{\partial \theta }} $$

(6.3)$${S_{r\theta }}(r,\theta ) = \frac{1}{r}\frac{{\partial {U_r}}}{{\partial \theta }} - \frac{{{U_\theta }}}{r} + \frac{{\partial {U_\theta }}}{{\partial r}}\textrm{ }. $$

The strains S_rr(r,θ), S_θθ(r,θ) and S_rθ(r,θ) induce the refractive index change Δn_n_,_m through the photoelastic effect on the fiber surface, which is given by

(7)$$\begin{aligned} \Delta n_{n,m}^{}(r,\theta ) = &\frac{{n_0^3}}{2}\{{({{P_{11}}{{\cos }^2}\theta + {P_{12}}{{\sin }^2}\theta } ){S_{rr}}(r,\theta ) + ({{P_{11}}{{\sin }^2}\theta + {P_{12}}{{\cos }^2}\theta } ){S_{\theta \theta }}(r,\theta )} \\ &\textrm{ }\textrm{ } - {({{P_{11}} - {P_{12}}} )\sin \theta \cos \theta {S_{r\theta }}(r,\theta )} \}\textrm{ }, \end{aligned}$$

where P₁₁( = 0.121) and P₁₂( = 0.270) are the strain-optic coefficients and n₀ is the core index. The profile of the refractive index changes caused by the R_0,7, TR_1,7, TR_2,7, TR_3,7, and TR_4,7 modes are shown in Figs. 3(a)–3(e), respectively.

Fig. 3. Refractive index change profiles induced by thermally excited acoustic modes. (a) R_0,7 mode, (b) TR_1,7 mode, (c) TR_2,7 mode, (d) TR_3,7 mode, and (e) TR_4,7 mode.

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The phase shift δϕ_n,m of the propagating light caused by acoustic scattering is then obtained by employing the overlap integral between the refractive index change Δn_n_,m(r,θ) induced by the TR_n_,m mode and the optical mode field profile E(r,θ) [2], which is written as

(8)$$\delta {\phi _{n,m}} = kl\int_0^{2\pi } {\int_0^a {\{{\Delta n_{n,m}^{}(r,\theta ) \times E(r,\theta )} \}rdrd\theta } }.$$

Here, k is a propagation constant and l is the fiber length. Since the refractive index difference between the core and cladding is sufficiently small, the acoustic mode profiles generated in the fiber cross section and the refractive index change of Δn_n_,m in Eq. (8) are independent of the number of core in the MCF. Then, the GAWBS phase noise depends strongly on the electrical mode field of E, which is determined by the core position in the fiber cross section. Figures 4(a)–4(e) show the mode profiles of the optical fields in SSMF and 4CF and the refractive index changes caused by the R_0,7, TR_1,7, TR_2,7, TR_3,7 and TR_4,7 modes, respectively. The vertical axis is normalized by the maximum value.

Fig. 4. Comparison of overlaps between the optical electric field E_SSMF, E_4CF and the refractive index change Δn profiles induced by thermally excited acoustic modes. (a) R_0,7 mode, (b) TR_1,7 mode, (c) TR_2,7 mode, (d) TR_3,7 mode, and (e) TR_4,7 mode.

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In SSMF, since the mode field is located in the center of a fiber and has a symmetric profile as shown by the black dashed curves, the overlap integral with Δn_0,7(r) or Δn_2,7(r), which is symmetric with a peak at the center, has a large value. On the other hand, the overlap integral between E(r) and Δn_1,7(r) or Δn_3,7(r) becomes zero, as the TR modes with odd orders have anti-symmetric profiles. Furthermore, even-ordered TR modes with m ≥ 4 also have a refractive index change profile that becomes zero at r = 0. Thus, the overlap integral with these higher-order even modes becomes very small compared with the R_0,m and TR_2,m modes. For these reasons, GAWBS phase noise due to the TR_1,m, TR_3,m and TR_4,m or higher modes is not observed in SCF with a core in the center.

In contrast, in 4CF, the mode field is located off-center as shown by the blue curve. This asymmetry causes the overlap integral with Δn_0,7(r) and Δn_2,7(r) to decrease, and this results in a reduction of the phase noise caused by the R_0,m and TR_2,m modes. Instead, the overlap integral between E(r) and the odd TR modes, such as Δn_1,7(r) or Δn_3,7(r), is no longer zero, and the phase shift caused by the odd modes, which was not observed in SCF, becomes dominant in MCF. Moreover, TR modes higher than 4th order also induce a phase shift in MCF. This explains the larger number of phase noise components measured in 4CF.

We calculated the overlap integral and analyzed the magnitude of the GAWBS phase noise in 4CF. Figure 5 shows the GAWBS phase noise calculated for the R_0,m to TR_8,m modes. Here the overlap integral is averaged with respect to θ by integrating from θ = 0 to 2π as the polarization state of the optical signal rotates randomly during propagation through a fiber. For comparison, the black curve shows the measured RF spectrum of the GAWBS noise, where the background noise (- 90 dBm) is removed as an offset from the actual spectrum shown in Fig. 2(a). The vertical axis of the numerical results is normalized by the result for the TR_2,7 mode. It can be seen that the calculated profile is in good agreement with the experimental data. The experimental results are somewhat lower than the calculated values at lower frequencies. This is due to the leakage of the acoustic wave into the outer polymer jacket in the lower modes [17,18], which is not taken into account in the present theoretical model.

Fig. 5. Comparison of the measured and calculated results for GAWBS phase noise in a 4CF.

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3. GAWBS phase noise in uncoupled 19-core fiber with large cladding

Next, we measured the GAWBS phase noise in an uncoupled MCF with a larger number of cores, i.e., 19CF [19]. The measurement setup is the same as that in Fig. 1 except that the fan-in and fan-out devices are replaced with those for 19CF. The 19CF was 31 km long, and had a cladding diameter of 240 µm. The average effective core area was 96 µm², and the core pitch was 40.4 µm. The 19 cores were aligned in a hexagonal closely packed structure, and they could be divided into four groups depending on the distance from the fiber center. Figures 6(a)–6(d) show the GAWBS phase noise spectra measured at each core for the four groups as depicted to the right of each figure. In each figure, we have superimposed the phase noise spectra at different cores belonging to the same group. As shown in Fig. 6(a), at the center core #1, we measured distinct side modes caused by the R_0m and TR_2m modes that resemble those in SSMF. On the other hand, the GAWBS spectra for the surrounding cores are quite different from that of core #1, where the cores belonging to the same group exhibit a similar GAWBS noise structure.

Fig. 6. Experimentally measured GAWBS phase noise spectra in 31 km 19CF. (a) Core #1 (center core), (b) core #2∼7 (inner off-center core group), (c) core #9, #11, #13, #15, #17, and #19 (outer off-center cores group 1), and (d) core #8, #10, #12, #14, #16, and #18 (outer off-center core group 2).

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Figure 7 shows the superposition of the GAWBS spectra at core #2, #8, and #9, which all belong to different groups. It can be seen that the cores located at different distances from the center have different GAWBS noise profiles. The calculated ratio of the phase noise magnitude was 1 (core #1) : 0.925 (core #2) : 0.929 (core #8) : 0.937 (core #9), which indicates that the influence of GAWBS in 19CF is almost independent of core position.

Fig. 7. Comparison of GAWBS phase noise spectra in 31 km 19CF at core #2, #8, and #9.

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Here we compare the GAWBS phase noise spectra at the center core #1 of a 19CF [Fig. 6(a)] and an SSMF [Fig. 2(b)]. We first note that the resonant frequencies in the 19CF are lower than in the SSMF. This is evident from Eq. (2) because the resonant frequency f_m_,n is inversely proportional to the cladding diameter, 2a, and the 19CF has a cladding almost twice that of the SSMF. Furthermore, core #1 of the 19CF has a larger number of sidebands, and the phase modulation due to higher R_0,m modes is relatively larger than that of the SSMF. To explain the reason for this result, we introduce a “relative mode field diameter”, which is defined as the RMFD normalized by the cladding diameter. 19CF has a smaller RMFD than SSMF. From our previous result [9], a fiber with a smaller MFD is easier to overlap with higher GAWBS modes, and therefore 19 CF with a smaller RMFD has a large overlap integral with higher GAWBS modes.

Finally, we used a vector signal analysis to evaluate the phase fluctuation of the optical carrier signal caused by GAWBS during a 19CF transmission over 31 km [9], where the optical signal was A/D-converted and analyzed offline with a DSP. In the DSP, first a carrier synchronized to the IF signal was generated, and the phase difference between the IF and the carrier was detected with a double balanced mixer (DBM) and by removing irrelevant components with a low pass filter. The phase difference corresponds to the phase fluctuation induced by GAWBS. Figures 8(a)–8(d) show a statistical histogram of phase fluctuation of IF signals measured at core #1, #2, #8, and #9. It can be seen that they follow a Gaussian distribution, and the variance is almost the same among the cores regardless of their location, where the ratio is 1 (core #1) : 0.964 (core #2) : 0.973 (core #8) : 0.976 (core #9). These values are in good agreement with the ratio of the integrated value of the spectrum described above. The result with a 30 km SSMF, which we newly measured for comparison, is also shown in Fig. 8(e), where the variance is comparable to that of 19CF.

Fig. 8. Histograms of optical carrier phase noise after transmission over a 31 km 19CF. (a) Core #1, (b) core #2, (c) core #8, (d) core #9, and (e) 30 km SSMF.

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19CF and SSMF suffer from a comparable amount of GAWBS phase noise despite their large difference in the cladding diameter for the following reason. The RMFD is larger in SSMF than in 19CF, which results in less efficient excitation of the higher order modes in SSMF. On the other hand, GAWBS induces a larger index change in fibers with a smaller cladding diameter because the amplitude of the acoustic wave C_n_,m is in inverse proportion to the cladding diameter [1,2]:

(9)$${C_{n,m}} = \sqrt {\frac{{{k_B}T}}{{\pi l\rho {\Omega _{n,m}}{a^2}{B_{n,m}}}}} \textrm{ },$$

where k_B is the Boltzmann constant, T is the absolute temperature, ρ is the density, Ω_n,m is the angular frequency of the TR_n_,m mode, and B_n,m is the integral defined as

(10)$${B_{n,m}} = \int_0^1 {\{{{{[{{u_r}(x)} ]}^2} + {{[{{u_\theta }(x)} ]}^2}} \}\textrm{ }} xdx\textrm{ }\textrm{.}$$

Taking these two factors into account, i.e., the RMFD and the dependence of the acoustic wave amplitude on the cladding diameter, the GAWBS phase noise is found to be insensitive to the cladding diameter, that is 1/[RMFD] × C_n_,m is independent of a. These results indicate that MCF transmission is also impaired by the GAWBS phase noise whose magnitude is comparable to that of SSMF regardless of the core geometry and cladding diameter, and thus GAWBS noise compensation is also very important with respect to MCF transmission.

4. Conclusion

We described our theoretical and experimental analyses of the GAWBS phase noise in MCF. We found that off-center cores in an MCF are dominantly affected by higher-order TR_n_,m modes rather than the R_0,m mode unlike in a center core. The spectral profile of the GAWBS noise strongly depends on the distance of the core from the center. Regardless of these core-dependent GAWBS characteristics, we found that the transmission impairments caused by GAWBS are almost independent of the core geometry. These results indicate that GAWBS phase noise is also a source of impairment in an MCF transmission regardless of the structure, and therefore, GAWBS noise compensation techniques will play an important role in SDM transmission.

Funding

Ministry of Internal Affairs and Communications (MIC), Research and Development of Innovative Optical Network Technology for a Novel Social Infrastructure (JPMI00316) (Technological Theme II : OCEANS).

Acknowledgments

The 36 km four-core fiber was supplied by Sumitomo Electric Industries, Ltd. The 31 km 19-core fiber was supplied by Furukawa Electric Co., Ltd.

Disclosures

The authors declare that they have no conflicts of interest.

References

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GAWBS phase noise characteristics in multi-core fibers for digital coherent transmission

Abstract

1. Introduction

2. GAWBS phase noise characteristics in uncoupled four-core fibers

3. GAWBS phase noise in uncoupled 19-core fiber with large cladding

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (8)

Equations (15)

Optics Express